Average Error: 4.6 → 2.1
Time: 34.3s
Precision: 64
\[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\]
↓
\[\mathsf{fma}\left(y, z \cdot \left(\left(\left(\sqrt[3]{\tanh \left(\frac{t}{y}\right)} \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right)}\right) \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right)} - \tanh \left(\frac{x}{y}\right)\right) + \tanh \left(\frac{x}{y}\right) \cdot \left(\left(-1\right) + 1\right)\right), x\right)\]
x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)
↓
\mathsf{fma}\left(y, z \cdot \left(\left(\left(\sqrt[3]{\tanh \left(\frac{t}{y}\right)} \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right)}\right) \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right)} - \tanh \left(\frac{x}{y}\right)\right) + \tanh \left(\frac{x}{y}\right) \cdot \left(\left(-1\right) + 1\right)\right), x\right)
double f(double x, double y, double z, double t) {
        double r425772 = x;
        double r425773 = y;
        double r425774 = z;
        double r425775 = r425773 * r425774;
        double r425776 = t;
        double r425777 = r425776 / r425773;
        double r425778 = tanh(r425777);
        double r425779 = r425772 / r425773;
        double r425780 = tanh(r425779);
        double r425781 = r425778 - r425780;
        double r425782 = r425775 * r425781;
        double r425783 = r425772 + r425782;
        return r425783;
}
↓
double f(double x, double y, double z, double t) {
        double r425784 = y;
        double r425785 = z;
        double r425786 = t;
        double r425787 = r425786 / r425784;
        double r425788 = tanh(r425787);
        double r425789 = cbrt(r425788);
        double r425790 = r425789 * r425789;
        double r425791 = r425790 * r425789;
        double r425792 = x;
        double r425793 = r425792 / r425784;
        double r425794 = tanh(r425793);
        double r425795 = r425791 - r425794;
        double r425796 = 1.0;
        double r425797 = -r425796;
        double r425798 = r425797 + r425796;
        double r425799 = r425794 * r425798;
        double r425800 = r425795 + r425799;
        double r425801 = r425785 * r425800;
        double r425802 = fma(r425784, r425801, r425792);
        return r425802;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original4.6
Target2.0
Herbie2.1
\[x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\]

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. Simplified2.0

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

  4. Applied add-cube-cbrt2.1

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

  5. Applied add-cube-cbrt2.2

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

  6. Applied prod-diff2.2

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

  7. Simplified2.1

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

  8. Simplified2.1

    \[\leadsto \mathsf{fma}\left(y, z \cdot \left(\left(\left(\sqrt[3]{\tanh \left(\frac{t}{y}\right)} \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right)}\right) \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right)} - \tanh \left(\frac{x}{y}\right)\right) + \color{blue}{\tanh \left(\frac{x}{y}\right) \cdot \left(\left(-1\right) + 1\right)}\right), x\right)\]

  9. Final simplification2.1

    \[\leadsto \mathsf{fma}\left(y, z \cdot \left(\left(\left(\sqrt[3]{\tanh \left(\frac{t}{y}\right)} \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right)}\right) \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right)} - \tanh \left(\frac{x}{y}\right)\right) + \tanh \left(\frac{x}{y}\right) \cdot \left(\left(-1\right) + 1\right)\right), x\right)\]

Reproduce

herbie shell --seed 2020100 +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))))))