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All 265 equations in the public catalog, filterable by domain, sortable by cost or depth. Cluster colors come from the K = 5 isomorphism fingerprint clustering.

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[H+] = sqrt(Ka*Ca)Chemistry5n1sqrt1:mul1

Activity-corrected pHChemistry5n1mul1:add1:sub1

Adam optimizer stepNeuroscience23n2sqrt1:pow2:mul1:div3:add1:sub1

Advection-dispersion GaussianGeology24n2exp1:sqrt1:pow1:mul8:div3:sub1

Affine Neural Network Layer (one output neuron)unknown0n1pow1:mul1:add1

Arbitrary Linear Combination a*x + b*y in 4nunknown0n1mul2:add1

Arrhenius equationChemistry5n1const

Arrhenius rate constantChemistry5n1exp1:mul1:div1

Average energy (2-level)Chemistry21n2exp2:mul1:div3

Batch normalization x-hatNeuroscience3n1div1:sub1

Beer-Lambert A = epsilon*c*lBiology4n0mul2

Beer-Lambert T = exp(-epsilon*c*l)Biology7n1exp1:mul2

Beverton-Holt model (folded)Biology11n3exp1:mul2:div1:add1:sub1

Bilinear Form x^T A yunknown0n1mul2

Biot-Savart law (magnitude)Electromagnetism12n2sin1:pow1:mul5:div1

Black-Scholes call option CEconomics47n1exp1:mul4:sub1

Black-Scholes d1 formulaEconomics20n2log1:sqrt1:pow1:mul2:div3:add2

Boltzmann entropy S = k_B ln OmegaChemistry3n1log1

Boltzmann factor exp(-E/kT)/ZChemistry13n1exp1:div2

Boltzmann Partition Functionunknown0n1exp1:pow1:div1

Boltzmann ratio (two states)Chemistry11n1exp1:div1

Bouguer gravity anomalyGeology4n0mul4

Butler-Volmer equationChemistry26n3exp2:mul5:div2:sub1

Cable equation V(x,t)Neuroscience12n1exp2:mul2:div2

Capacitor energy U = (1/2)*C*V^2Electromagnetism7n1pow1:mul2:div1

CAPM E(Ri) = Rf + beta*(E(Rm)-Rf)Economics7n2mul1:add1:sub1

Carbon-14 datingBiology8n1log1:div2

Clausius-ClapeyronChemistry17n1mul1:div3:sub1

Cobb-Douglas Y = A*K^alpha*L^(1-alpha)Economics12n1pow2:mul2

Collision theory rate constantChemistry10n1exp1:pow1:mul2:div2

Collision theory rate constantChemistry10n1pow1:div1

Competitive inhibitionBiology18n2mul2:div2:add1

Compound interest (discrete)Economics8n1pow1:mul1

Compounding inflation adjustmentEconomics7n1pow1:div1

Continuous compounding A = P*exp(rt)Economics5n1exp1:mul2

Continuously compounded yieldEconomics4n1log1:mul1:div2

Cosmological redshift zAstrophysics3n0div1:sub1

Coulomb Pairwise Sumunknown0n0mul2:div1

Coulomb's lawElectromagnetism8n0pow1:mul2:div1

Curie law chi = C/TElectromagnetism1n0div1

Cyclotron frequency omega = qB/mElectromagnetism3n0mul1:div1

Darcy's lawGeology8n1mul2:div2:sub1

DCF NPV (mixed cash flows)unknown0n1pow1:div1

Debye-Huckel ln(gamma_pm)Chemistry12n1sqrt1:mul2:add1:sub2

dG = -RT ln KChemistry7n1log1:mul1:sub1

dG = dG0 + RT ln QChemistry8n1log1:mul1:add1

Discounted Cash Flow (Mixed Positive/Negative)unknown0n1pow2:div1

Distribution coefficient KdGeology1n0div1

Doubling timeBiology4n1log1:div1

Eddington luminosityAstrophysics7n0mul4:div1

Electric field (point charge)Electromagnetism6n0pow1:mul1:div1

Electric potential energy (fixed charges)Electromagnetism5n1mul2:div1

Electrochemical potentialChemistry15n1log1:mul3:add2

ELO Rating System (full update)unknown0n1mul1:add1:sub1

EM wave E(x,t)Electromagnetism9n1sin1:mul3:sub1

Entropy of mixing (2-component)Chemistry13n2log2:mul3:add1:sub1

Enzyme turnover k_cat/[E]Biology4n0mul1

Escape velocity v_escAstrophysics8n1sqrt1:mul2:div1

Expected Value (mixed-sign outcomes)unknown0n1mul1

Exponential decay N0*exp(-lambda*t)Biology5n1exp1:mul2

Exponential growth N0*exp(rt)Biology5n1exp1:mul2

Eyring (transition state theory) equationChemistry13n1pow4:sub1

Eyring transition state kChemistry13n1exp1:mul2:div2

Faraday EMF (sinusoidal flux)Electromagnetism11n1sin1:mul4:div1:sub1

Fick's first lawBiology4n1mul1:div1:sub1

Fisher equation (exact)Economics10n1mul1

FitzHugh-Nagumo V equation (pos approx)Neuroscience19n0pow1:div1:add1:sub2

Flexural rigidity DGeology13n2pow2:mul2:div1

Flexural wavelength lambdaGeology8n2pow1:mul3:div2

Force between parallel wiresElectromagnetism7n1mul4:div1

Freundlich isothermGeology5n1pow1:mul1:div1

Friedmann equation RHSAstrophysics14n1pow3:mul5:div3:add1:sub1

Gauss's law (electric flux)Electromagnetism1n0div1

Gaussian Curvatureunknown0n1pow4:mul1:div1:add2:sub1

GBM step dSEconomics11n0mul4:add1

GELU Activation (exact form via tanh approximation)unknown0n1mul2

GELU approximationNeuroscience10n1mul2

Geothermal gradient T(z)Geology12n1pow1:mul2:div2:add1:sub1

Gibbs / Shannon Entropyunknown0n1log1:pow1:mul2:sub1

Gibbs Free Energy dG = dH - T*dSunknown0n0mul1:sub1

Gibbs free energy dG = dH - TdSChemistry4n0mul1:sub1

Gini coefficient (closed form)Economics6n1div1

Goldman-Hodgkin-Katz (2-ion)Chemistry27n2log1:mul1:div2:add2

Goldman-Hodgkin-Katz (3-ion)Biology31n2log1:mul1:div2:add2

Goldman-Hodgkin-Katz Vm (2-ion neuro)Neuroscience18n2log1:mul1:div2:add2

Gompertz growth modelBiology12n2exp2:mul3

Gordon growth model P = D/(r-g)Economics3n1div1:sub1

Gravitational potential energy UAstrophysics7n0mul2:div1:sub1

Gravitational wave strain prefactorAstrophysics10n1pow1:mul3:div1

Gutenberg-Richter (log form)Geology4n0mul1:sub1

Half-lifeBiology4n1log1:div1

Hall voltage VHElectromagnetism7n1mul3:div1

Heat flow Fourier's lawGeology5n1mul1:div1:sub1

Helmholtz Free Energyunknown0n2exp1:log2:mul2:div1:sub2

Helmholtz free energy A = -kT ln ZChemistry7n1log1:mul2:sub1

Henderson-HasselbalchChemistry10n1div1:add1:sub1

Henderson-Hasselbalch (general form)unknown0n1div1:add1:sub1

HH gating variable alpha_m (pos domain)Neuroscience20n3exp1:mul1:div2

Hill dose-responseBiology15n1pow3:mul1:div1:add1

Hill equation (general)Biology15n1pow3:div1:add1

Hill fractional saturation (n=1)Biology10n1div1:add1

Hill linearized (Hill plot)Biology16n1log1:mul1:sub1

Hill sphere radiusAstrophysics8n2pow1:mul2:div2

Hodgkin-Huxley total currentNeuroscience30n1pow2:mul6:add2:sub3

Hohmann transfer delta-V1Astrophysics17n3sqrt2:mul2:div2

Hopfield network energyNeuroscience0n1mul4:sub1

Hubble parameter H(z)Astrophysics16n2sqrt1:pow1:mul2:add1

Hubble's law v = H0*dAstrophysics2n0mul1

Inductance energy U = (1/2)*L*I^2Electromagnetism7n1pow1:mul2:div1

Integrated first-order [A](t)Chemistry7n1exp1:mul2

Integrated first-order rate lawChemistry7n1exp1

Integrated second-order rate lawChemistry9n0add1

Isostasy Airy modelGeology5n2mul1:div1:sub1

Itakura-Saito Divergenceunknown0n2log1:div2:sub2

Izhikevich u-equationNeuroscience6n1mul2:sub1

Izhikevich V equation (gen domain)Neuroscience26n0pow1:mul2:add3:sub1

Jensen-Shannon Divergenceunknown0n1mul2:div3:add2

Jensen-Shannon Divergence (N-term)unknown0n1mul2:add1

KL Divergence (N terms)unknown0n1log1:pow1:mul1:div1

KL Divergence (N-term)unknown0n1log1:mul1:div1

Langmuir adsorption isothermGeology8n1mul3:div1:add1

Larmor radiation power PElectromagnetism14n1pow3:mul4:div1

Lennard-Jones Potentialunknown0n1pow2:mul2:div2:sub1

Lenz's law epsilon = -L*dI/dtElectromagnetism4n0mul1:div1:sub1

Lineweaver-BurkBiology11n1mul1:div3:add1

Log return ln(Pt/Pt-1)Neuroscience2n1log1:div1

Log return ln(Pt/Pt-1)Economics2n1log1:div1

Log-Sigmoidunknown0n2exp1:log2:add1:sub2

Logarithm of difference ln(a - b)unknown0n1log1:sub1

Logarithm of sum ln(a + b)unknown0n1log1:add1

Logistic growth (sigmoid form)Biology14n3exp1:mul1:div1:sub1

Luminosity distance (per step)Astrophysics19n2mul2:div2

Magnetic dipole field (radial)Electromagnetism12n2cos1:pow1:mul4:div2

Magnetic energy density u = B^2/(2*mu0)Electromagnetism6n1pow1:mul1:div1

Magnetic flux Phi = B*A*cos(theta)Electromagnetism5n1cos1:mul2

Malthus recursion N(t+1) = lambda*N(t)Biology2n1mul1

Manning's equationGeology12n1pow2:mul2:div3

Mantle viscosity (dislocation creep)Geology13n1exp1:pow1:mul2:div1

Maxwell-Boltzmann distributionChemistry31n1exp1:pow3:mul6:div3

Mean Curvatureunknown0n2pow5:mul6:div2:add1:sub1

Michaelis-MentenBiology9n1mul1:div1:add1

Mineral dissolution rateGeology9n1exp1:mul2:div1

Mish Activation x * tanh(softplus(x))unknown0n3exp1:log1:mul1

Mixed inhibitionBiology27n3mul3:div3:add1

Moment magnitude MwGeology7n2mul1:div1:sub1

Motor Switching Probability (Hill Equation)Biology10n0pow3:add1:sub2

MWC allosteric model (general)Biology40n2pow3:mul4:div1:add1

MWC allosteric model (n=2)Biology34n2pow3:mul4:div1:add1

N-term LogSumExpunknown0n2exp1:log1:pow1

N-term Taylor Seriesunknown0n1pow2:mul1

Nash Equilibrium (2x2 Mixed Strategy, honest)unknown0n1div1:add1:sub3

Nash expected utility (2-outcome)Economics9n1mul2:add1

Nernst equation (RT/nF folded)Chemistry13n1log1:mul1:div2

Nernst equation (unfolded)Chemistry13n2log1:mul3:div2

Nernst single-ionBiology6n1log1:mul1:div2

Net reproductive rate R0Biology12n1mul1

Numerically stable LSE (LSE trick)unknown0n2exp1:log1:add1:sub1

One-compartment PK C(t)Biology7n1exp1:mul2

Orbital period (Kepler)Astrophysics9n2sqrt1:pow1:mul3:div1

Orbital velocity v = sqrt(GM/r)Astrophysics6n1sqrt1:div1

Partition function (2-level)Chemistry17n1exp1:div1:add1

Perpetuity PV = C/rEconomics1n0div1

pH = -log10[H+]Chemistry3n1add1:sub1

Phase velocity vp = 1/sqrt(mu*eps)Electromagnetism7n1sqrt1:mul1:div1

Phillips curve pi = pie - beta*(u-u*)Economics6n1mul2:sub2

Poynting vector magnitude S = E^2/(mu0*c)Electromagnetism6n1pow1:mul1:div1

Present value PV = FV/(1+r)^nEconomics7n1pow1:div1

Proton Hop Rate (Arrhenius form)Biology0n1const

Proton Motive ForceBiology6n0add1

Quadratic [H+] (Ka variable)Chemistry20n2sqrt1:pow1:mul2:div1:add1

Radioactive decay (geochronology)Geology7n1exp1:mul2

Rayleigh number RaGeology14n1pow1:mul5:div1

Reaction Free Energy dGunknown0n1log1:mul1:add1

ReLU via softplus LEAd(x,1)Neuroscience1n2exp1:log1:add1

Reluctance and flux R+PhiElectromagnetism6n1mul2:div2

Retardation factorGeology6n1mul1:div1:add1

Roche limitAstrophysics8n1pow1:mul2:div2

Saha equation (T live)Astrophysics13n1exp1:pow2:mul4:div3

Scale factor a(t) matter-dominatedAstrophysics6n1pow1:mul1:div2

Schwarzschild radiusAstrophysics8n0pow1:mul2:div1

Second-order rate r = k[A][B]Chemistry4n0mul2

Self-capacitance of sphereElectromagnetism2n0mul3

SGD parameter updateNeuroscience4n0mul1:sub2

Sharpe ratioEconomics3n1div1:sub1

Sigmoid activation sigma(x)Neuroscience8n2exp1:div1

Skin depth delta = sqrt(2*rho/(omega*mu))Electromagnetism8n2sqrt1:mul2:div1

Softmax (denominator + per-output numerator)unknown0n1exp2:pow1:div1

Softplus (smooth ReLU approximation)unknown0n2exp1:log1:add1

Solenoid field B = mu0*n*IElectromagnetism4n0mul2

Solow steady state k*Economics10n2pow1:mul1:div2

Specificity constant k_cat/KmBiology4n0div1

Stefan-Boltzmann luminosityAstrophysics12n0pow2:mul4

Stokes settling velocityGeology9n1pow1:mul4:div1:sub1

Subduction velocity v = d/tGeology1n0div1

sum_i exp(a_i * x + b_i) — General Exponential Family Sumunknown0n1exp1:pow1:mul1:add1

Swish Activation x * sigmoid(x)unknown0n2exp1:mul1:div1:add1

Synaptic alpha function I(t)Neuroscience8n1exp1:mul2:div2

Synodic periodAstrophysics8n1div3:sub1

Tafel equationChemistry13n1log1:mul1:add1

Theis well function (2-term approx)Geology18n1pow1:mul6:div2

Thermal de Broglie wavelengthAstrophysics10n1sqrt1:mul4:div1

Thermal subsidence y(t)Geology8n2exp1:mul1:div1:sub1

Transformer turns ratioElectromagnetism3n1mul1:div1

Two-compartment PKBiology17n1exp2:mul4:add1

Two-site bindingBiology24n1pow6:div2:add1

Two-term LogSumExpunknown0n2exp2:log1:add1

Uncompetitive inhibitionBiology18n2mul2:div2:add1

unnamedunknown0n1const

unnamedunknown0n1pow1:div1

unnamedunknown0n1const

unnamedunknown0n0sub1

unnamedunknown0n1const

unnamedChemistry13n1const

unnamedChemistry4n0sub1

unnamedChemistry7n0sub1

unnamedChemistry9n0sub1

unnamedunknown0n0sub1

unnamedunknown0n1mul1:div1:add1

unnamedunknown0n1sub2

unnamedunknown0n1log1:mul1:add1:sub1

unnamedunknown0n1mul1:add1

unnamedunknown0n1const

unnamedInformation Theory0n1const

unnamedProbability / Game Theory0n1const

unnamedEconomics (Finance)0n1div1

unnamedInformation Theory0n1sub1

unnamedInformation Theory0n1const

unnamedChemistry (Thermodynamics)0n1log1:mul1:add1

unnamedChemistry (Acid-Base)0n1sub2

unnamedEconomics0n1const

unnamedunknown0n0sub1

unnamedunknown0n1const

unnamedunknown0n0mul1:sub1

unnamedunknown0n1sub1

unnamedunknown0n1const

unnamedunknown0n0pow2:add1

unnamedunknown0n1exp2:add1

unnamedunknown0n1log1:add1

unnamedunknown0n2exp1:log1

unnamedunknown0n1mul1

unnamedunknown0n1const

unnamedunknown0n0mul2:add1

unnamedunknown0n1const

Van Slyke buffer capacityChemistry14n1pow1:mul3:div1:add3

Van't Hoff integratedChemistry17n1mul1:div3:sub2

Virial temperatureAstrophysics9n1mul5:div1

Von Mises Yield Criterionunknown0n1sqrt1:pow2:mul1:add1:sub1

Wien's displacement lawAstrophysics1n0div1

Yukawa Potential Sumunknown0n2exp1:pow1:mul2:div1