Average Error: 4.6 → 3.3
Time: 21.6s
Precision: 64
\[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\]
↓
\[\left(\sqrt[3]{\mathsf{fma}\left(y, z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y, z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(y, z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)}\]
x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)
↓
\left(\sqrt[3]{\mathsf{fma}\left(y, z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y, z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(y, z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)}
double f(double x, double y, double z, double t) {
        double r266770 = x;
        double r266771 = y;
        double r266772 = z;
        double r266773 = r266771 * r266772;
        double r266774 = t;
        double r266775 = r266774 / r266771;
        double r266776 = tanh(r266775);
        double r266777 = r266770 / r266771;
        double r266778 = tanh(r266777);
        double r266779 = r266776 - r266778;
        double r266780 = r266773 * r266779;
        double r266781 = r266770 + r266780;
        return r266781;
}
↓
double f(double x, double y, double z, double t) {
        double r266782 = y;
        double r266783 = z;
        double r266784 = t;
        double r266785 = r266784 / r266782;
        double r266786 = tanh(r266785);
        double r266787 = x;
        double r266788 = r266787 / r266782;
        double r266789 = tanh(r266788);
        double r266790 = r266786 - r266789;
        double r266791 = r266783 * r266790;
        double r266792 = fma(r266782, r266791, r266787);
        double r266793 = cbrt(r266792);
        double r266794 = r266793 * r266793;
        double r266795 = r266794 * r266793;
        return r266795;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original4.6
Target2.1
Herbie3.3
\[x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\]

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-cbrt3.3

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

  5. Final simplification3.3

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

Reproduce

herbie shell --seed 2019305 +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))))))