Average Error: 4.5 → 2.5
Time: 9.0s
Precision: 64
\[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) = -\infty:\\ \;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(y, z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y, z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(y, z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)}\\ \mathbf{elif}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \le 5.136736371015830348947780040880154350207 \cdot 10^{293}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t \cdot z}{y}, x\right)\\ \end{array}\]
x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)
↓
\begin{array}{l}
\mathbf{if}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) = -\infty:\\
\;\;\;\;\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)}\\

\mathbf{elif}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \le 5.136736371015830348947780040880154350207 \cdot 10^{293}:\\
\;\;\;\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{t \cdot z}{y}, x\right)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r255547 = x;
        double r255548 = y;
        double r255549 = z;
        double r255550 = r255548 * r255549;
        double r255551 = t;
        double r255552 = r255551 / r255548;
        double r255553 = tanh(r255552);
        double r255554 = r255547 / r255548;
        double r255555 = tanh(r255554);
        double r255556 = r255553 - r255555;
        double r255557 = r255550 * r255556;
        double r255558 = r255547 + r255557;
        return r255558;
}
↓
double f(double x, double y, double z, double t) {
        double r255559 = x;
        double r255560 = y;
        double r255561 = z;
        double r255562 = r255560 * r255561;
        double r255563 = t;
        double r255564 = r255563 / r255560;
        double r255565 = tanh(r255564);
        double r255566 = r255559 / r255560;
        double r255567 = tanh(r255566);
        double r255568 = r255565 - r255567;
        double r255569 = r255562 * r255568;
        double r255570 = r255559 + r255569;
        double r255571 = -inf.0;
        bool r255572 = r255570 <= r255571;
        double r255573 = r255561 * r255568;
        double r255574 = fma(r255560, r255573, r255559);
        double r255575 = cbrt(r255574);
        double r255576 = r255575 * r255575;
        double r255577 = r255576 * r255575;
        double r255578 = 5.13673637101583e+293;
        bool r255579 = r255570 <= r255578;
        double r255580 = r255563 * r255561;
        double r255581 = r255580 / r255560;
        double r255582 = fma(r255560, r255581, r255559);
        double r255583 = r255579 ? r255570 : r255582;
        double r255584 = r255572 ? r255577 : r255583;
        return r255584;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original4.5
Target1.9
Herbie2.5
\[x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\]

Derivation

  1. Split input into 3 regimes

  2. if (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))) < -inf.0

    1. Initial program 64.0

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

    2. Simplified1.7

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

    4. Applied add-cube-cbrt2.5

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

    if -inf.0 < (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))) < 5.13673637101583e+293

    1. Initial program 0.5

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

    if 5.13673637101583e+293 < (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y)))))

    1. Initial program 48.3

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

    2. Simplified16.9

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

    3. Taylor expanded around inf 36.6

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{t \cdot z}{y}}, x\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification2.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) = -\infty:\\ \;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(y, z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y, z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(y, z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)}\\ \mathbf{elif}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \le 5.136736371015830348947780040880154350207 \cdot 10^{293}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t \cdot z}{y}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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))))))