Average Error: 4.5 → 1.9
Time: 30.0s
Precision: 64
\[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) = -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)\\ \mathbf{elif}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \le 2.405759074108095109916766476323447027402 \cdot 10^{304}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot z\\ \end{array}\]
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:\\
\;\;\;\;\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 2.405759074108095109916766476323447027402 \cdot 10^{304}:\\
\;\;\;\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;x - x \cdot z\\

\end{array}
double f(double x, double y, double z, double t) {
        double r20666579 = x;
        double r20666580 = y;
        double r20666581 = z;
        double r20666582 = r20666580 * r20666581;
        double r20666583 = t;
        double r20666584 = r20666583 / r20666580;
        double r20666585 = tanh(r20666584);
        double r20666586 = r20666579 / r20666580;
        double r20666587 = tanh(r20666586);
        double r20666588 = r20666585 - r20666587;
        double r20666589 = r20666582 * r20666588;
        double r20666590 = r20666579 + r20666589;
        return r20666590;
}
↓
double f(double x, double y, double z, double t) {
        double r20666591 = x;
        double r20666592 = y;
        double r20666593 = z;
        double r20666594 = r20666592 * r20666593;
        double r20666595 = t;
        double r20666596 = r20666595 / r20666592;
        double r20666597 = tanh(r20666596);
        double r20666598 = r20666591 / r20666592;
        double r20666599 = tanh(r20666598);
        double r20666600 = r20666597 - r20666599;
        double r20666601 = r20666594 * r20666600;
        double r20666602 = r20666591 + r20666601;
        double r20666603 = -inf.0;
        bool r20666604 = r20666602 <= r20666603;
        double r20666605 = r20666593 * r20666600;
        double r20666606 = fma(r20666592, r20666605, r20666591);
        double r20666607 = 2.405759074108095e+304;
        bool r20666608 = r20666602 <= r20666607;
        double r20666609 = r20666591 * r20666593;
        double r20666610 = r20666591 - r20666609;
        double r20666611 = r20666608 ? r20666602 : r20666610;
        double r20666612 = r20666604 ? r20666606 : r20666611;
        return r20666612;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original4.5
Target1.9
Herbie1.9
\[x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\]

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. Simplified1.1

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

    if -inf.0 < (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))) < 2.405759074108095e+304

    1. Initial program 0.6

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

    if 2.405759074108095e+304 < (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y)))))

    1. Initial program 59.0

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

    2. Simplified20.9

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

    3. Taylor expanded around inf 32.3

      \[\leadsto \color{blue}{x - x \cdot z}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) = -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)\\ \mathbf{elif}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \le 2.405759074108095109916766476323447027402 \cdot 10^{304}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot z\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z t)
  :name "SynthBasics:moogVCF from YampaSynth-0.2"

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

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