Average Error: 4.7 → 1.6
Time: 8.0s
Precision: 64
\[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) = -\infty \lor \neg \left(x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \le 2.04366666294127525 \cdot 10^{303}\right):\\ \;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(y, z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y, z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(y, \left(\sqrt[3]{z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)} \cdot \sqrt[3]{z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)}\right) \cdot \sqrt[3]{z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)}, x\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\\ \end{array}\]
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 \lor \neg \left(x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \le 2.04366666294127525 \cdot 10^{303}\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, \left(\sqrt[3]{z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)} \cdot \sqrt[3]{z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)}\right) \cdot \sqrt[3]{z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)}, x\right)}\\

\mathbf{else}:\\
\;\;\;\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r304553 = x;
        double r304554 = y;
        double r304555 = z;
        double r304556 = r304554 * r304555;
        double r304557 = t;
        double r304558 = r304557 / r304554;
        double r304559 = tanh(r304558);
        double r304560 = r304553 / r304554;
        double r304561 = tanh(r304560);
        double r304562 = r304559 - r304561;
        double r304563 = r304556 * r304562;
        double r304564 = r304553 + r304563;
        return r304564;
}
↓
double f(double x, double y, double z, double t) {
        double r304565 = x;
        double r304566 = y;
        double r304567 = z;
        double r304568 = r304566 * r304567;
        double r304569 = t;
        double r304570 = r304569 / r304566;
        double r304571 = tanh(r304570);
        double r304572 = r304565 / r304566;
        double r304573 = tanh(r304572);
        double r304574 = r304571 - r304573;
        double r304575 = r304568 * r304574;
        double r304576 = r304565 + r304575;
        double r304577 = -inf.0;
        bool r304578 = r304576 <= r304577;
        double r304579 = 2.0436666629412752e+303;
        bool r304580 = r304576 <= r304579;
        double r304581 = !r304580;
        bool r304582 = r304578 || r304581;
        double r304583 = r304567 * r304574;
        double r304584 = fma(r304566, r304583, r304565);
        double r304585 = cbrt(r304584);
        double r304586 = r304585 * r304585;
        double r304587 = cbrt(r304583);
        double r304588 = r304587 * r304587;
        double r304589 = r304588 * r304587;
        double r304590 = fma(r304566, r304589, r304565);
        double r304591 = cbrt(r304590);
        double r304592 = r304586 * r304591;
        double r304593 = r304582 ? r304592 : r304576;
        return r304593;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original4.7
Target2.1
Herbie1.6
\[x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))) < -inf.0 or 2.0436666629412752e+303 < (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y)))))

    1. Initial program 59.9

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

    2. Simplified15.2

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

    4. Applied add-cube-cbrt15.9

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

    6. Applied add-cube-cbrt15.9

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

    if -inf.0 < (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))) < 2.0436666629412752e+303

    1. Initial program 0.5

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) = -\infty \lor \neg \left(x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \le 2.04366666294127525 \cdot 10^{303}\right):\\ \;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(y, z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y, z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(y, \left(\sqrt[3]{z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)} \cdot \sqrt[3]{z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)}\right) \cdot \sqrt[3]{z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)}, x\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 +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))))))