Average Error: 4.7 → 2.6
Time: 1.7m
Precision: 64
\[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -7.259998072692759908723582781056792092665 \cdot 10^{217}:\\ \;\;\;\;x + \left(t - x\right) \cdot z\\ \mathbf{elif}\;y \le 9.400818729820711601228948134662996152541 \cdot 10^{181}:\\ \;\;\;\;\mathsf{fma}\left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right), z \cdot y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right) + \mathsf{fma}\left(-\sqrt[3]{x}, \sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot z\\ \end{array}\]
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}\;y \le -7.259998072692759908723582781056792092665 \cdot 10^{217}:\\
\;\;\;\;x + \left(t - x\right) \cdot z\\

\mathbf{elif}\;y \le 9.400818729820711601228948134662996152541 \cdot 10^{181}:\\
\;\;\;\;\mathsf{fma}\left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right), z \cdot y, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(z, t - x, x\right) + \mathsf{fma}\left(-\sqrt[3]{x}, \sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot z\\

\end{array}
double f(double x, double y, double z, double t) {
        double r13704630 = x;
        double r13704631 = y;
        double r13704632 = z;
        double r13704633 = r13704631 * r13704632;
        double r13704634 = t;
        double r13704635 = r13704634 / r13704631;
        double r13704636 = tanh(r13704635);
        double r13704637 = r13704630 / r13704631;
        double r13704638 = tanh(r13704637);
        double r13704639 = r13704636 - r13704638;
        double r13704640 = r13704633 * r13704639;
        double r13704641 = r13704630 + r13704640;
        return r13704641;
}
↓
double f(double x, double y, double z, double t) {
        double r13704642 = y;
        double r13704643 = -7.25999807269276e+217;
        bool r13704644 = r13704642 <= r13704643;
        double r13704645 = x;
        double r13704646 = t;
        double r13704647 = r13704646 - r13704645;
        double r13704648 = z;
        double r13704649 = r13704647 * r13704648;
        double r13704650 = r13704645 + r13704649;
        double r13704651 = 9.400818729820712e+181;
        bool r13704652 = r13704642 <= r13704651;
        double r13704653 = r13704646 / r13704642;
        double r13704654 = tanh(r13704653);
        double r13704655 = r13704645 / r13704642;
        double r13704656 = tanh(r13704655);
        double r13704657 = r13704654 - r13704656;
        double r13704658 = r13704648 * r13704642;
        double r13704659 = fma(r13704657, r13704658, r13704645);
        double r13704660 = fma(r13704648, r13704647, r13704645);
        double r13704661 = cbrt(r13704645);
        double r13704662 = -r13704661;
        double r13704663 = r13704661 * r13704661;
        double r13704664 = r13704661 * r13704663;
        double r13704665 = fma(r13704662, r13704663, r13704664);
        double r13704666 = r13704665 * r13704648;
        double r13704667 = r13704660 + r13704666;
        double r13704668 = r13704652 ? r13704659 : r13704667;
        double r13704669 = r13704644 ? r13704650 : r13704668;
        return r13704669;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original4.7
Target2.0
Herbie2.6
\[x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\]

Derivation

  1. Split input into 3 regimes

  2. if y < -7.25999807269276e+217

    1. Initial program 19.3

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

    2. Simplified19.3

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

    3. Taylor expanded around inf 4.4

      \[\leadsto \color{blue}{\left(t \cdot z + x\right) - x \cdot z}\]

    4. Simplified4.4

      \[\leadsto \color{blue}{x + z \cdot \left(t - x\right)}\]

    if -7.25999807269276e+217 < y < 9.400818729820712e+181

    1. Initial program 2.2

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

    2. Simplified2.2

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

    if 9.400818729820712e+181 < y

    1. Initial program 18.4

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

    2. Simplified18.4

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

    3. Taylor expanded around inf 5.2

      \[\leadsto \color{blue}{\left(t \cdot z + x\right) - x \cdot z}\]

    4. Simplified5.2

      \[\leadsto \color{blue}{x + z \cdot \left(t - x\right)}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt5.4

      \[\leadsto x + z \cdot \left(t - \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\right)\]

    7. Applied add-sqr-sqrt34.1

      \[\leadsto x + z \cdot \left(\color{blue}{\sqrt{t} \cdot \sqrt{t}} - \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)\]

    8. Applied prod-diff34.1

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

    9. Applied distribute-rgt-in34.1

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

    10. Applied associate-+r+34.1

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

    11. Simplified5.2

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

  4. Final simplification2.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -7.259998072692759908723582781056792092665 \cdot 10^{217}:\\ \;\;\;\;x + \left(t - x\right) \cdot z\\ \mathbf{elif}\;y \le 9.400818729820711601228948134662996152541 \cdot 10^{181}:\\ \;\;\;\;\mathsf{fma}\left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right), z \cdot y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right) + \mathsf{fma}\left(-\sqrt[3]{x}, \sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot z\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +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))))))