Average Error: 4.6 → 2.4
Time: 8.6s
Precision: 64
\[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\]
↓
\[\mathsf{fma}\left(y, \left(z \cdot \left(\sqrt[3]{\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)} \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)}\right)\right) \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)}, x\right)\]
x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)
↓
\mathsf{fma}\left(y, \left(z \cdot \left(\sqrt[3]{\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)} \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)}\right)\right) \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)}, x\right)
double f(double x, double y, double z, double t) {
        double r258187 = x;
        double r258188 = y;
        double r258189 = z;
        double r258190 = r258188 * r258189;
        double r258191 = t;
        double r258192 = r258191 / r258188;
        double r258193 = tanh(r258192);
        double r258194 = r258187 / r258188;
        double r258195 = tanh(r258194);
        double r258196 = r258193 - r258195;
        double r258197 = r258190 * r258196;
        double r258198 = r258187 + r258197;
        return r258198;
}
↓
double f(double x, double y, double z, double t) {
        double r258199 = y;
        double r258200 = z;
        double r258201 = t;
        double r258202 = r258201 / r258199;
        double r258203 = tanh(r258202);
        double r258204 = x;
        double r258205 = r258204 / r258199;
        double r258206 = tanh(r258205);
        double r258207 = r258203 - r258206;
        double r258208 = cbrt(r258207);
        double r258209 = r258208 * r258208;
        double r258210 = r258200 * r258209;
        double r258211 = r258210 * r258208;
        double r258212 = fma(r258199, r258211, r258204);
        return r258212;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original4.6
Target2.1
Herbie2.4
\[x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\]

Derivation

  1. Initial program 4.6

    \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\]

  2. Simplified2.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt2.4

    \[\leadsto \mathsf{fma}\left(y, z \cdot \color{blue}{\left(\left(\sqrt[3]{\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)} \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)}\right) \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)}\right)}, x\right)\]

  5. Applied associate-*r*2.4

    \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(z \cdot \left(\sqrt[3]{\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)} \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)}\right)\right) \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)}}, x\right)\]

  6. Final simplification2.4

    \[\leadsto \mathsf{fma}\left(y, \left(z \cdot \left(\sqrt[3]{\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)} \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)}\right)\right) \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)}, x\right)\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y z t)
  :name "SynthBasics:moogVCF from YampaSynth-0.2"
  :precision binary64

  :herbie-target
  (+ x (* y (* z (- (tanh (/ t y)) (tanh (/ x y))))))

  (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))))