Average Error: 7.4 → 0.4
Time: 20.4s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.1569808277222708 \cdot 10^{-65}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\cosh x} \cdot \sqrt[3]{\cosh x}\right) \cdot \left(y \cdot \sqrt[3]{\cosh x}\right)}{x \cdot z}\\ \mathbf{elif}\;z \le 2.102710110923131 \cdot 10^{-20}:\\ \;\;\;\;\frac{\left(\frac{\frac{1}{2}}{e^{x}} + \frac{1}{2} \cdot e^{x}\right) \cdot \frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\cosh x} \cdot \sqrt[3]{\cosh x}\right) \cdot \left(y \cdot \sqrt[3]{\cosh x}\right)}{x \cdot z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.1569808277222708 \cdot 10^{-65}:\\
\;\;\;\;\frac{\left(\sqrt[3]{\cosh x} \cdot \sqrt[3]{\cosh x}\right) \cdot \left(y \cdot \sqrt[3]{\cosh x}\right)}{x \cdot z}\\

\mathbf{elif}\;z \le 2.102710110923131 \cdot 10^{-20}:\\
\;\;\;\;\frac{\left(\frac{\frac{1}{2}}{e^{x}} + \frac{1}{2} \cdot e^{x}\right) \cdot \frac{y}{z}}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\sqrt[3]{\cosh x} \cdot \sqrt[3]{\cosh x}\right) \cdot \left(y \cdot \sqrt[3]{\cosh x}\right)}{x \cdot z}\\

\end{array}
double f(double x, double y, double z) {
        double r26979685 = x;
        double r26979686 = cosh(r26979685);
        double r26979687 = y;
        double r26979688 = r26979687 / r26979685;
        double r26979689 = r26979686 * r26979688;
        double r26979690 = z;
        double r26979691 = r26979689 / r26979690;
        return r26979691;
}
↓
double f(double x, double y, double z) {
        double r26979692 = z;
        double r26979693 = -3.1569808277222708e-65;
        bool r26979694 = r26979692 <= r26979693;
        double r26979695 = x;
        double r26979696 = cosh(r26979695);
        double r26979697 = cbrt(r26979696);
        double r26979698 = r26979697 * r26979697;
        double r26979699 = y;
        double r26979700 = r26979699 * r26979697;
        double r26979701 = r26979698 * r26979700;
        double r26979702 = r26979695 * r26979692;
        double r26979703 = r26979701 / r26979702;
        double r26979704 = 2.102710110923131e-20;
        bool r26979705 = r26979692 <= r26979704;
        double r26979706 = 0.5;
        double r26979707 = exp(r26979695);
        double r26979708 = r26979706 / r26979707;
        double r26979709 = r26979706 * r26979707;
        double r26979710 = r26979708 + r26979709;
        double r26979711 = r26979699 / r26979692;
        double r26979712 = r26979710 * r26979711;
        double r26979713 = r26979712 / r26979695;
        double r26979714 = r26979705 ? r26979713 : r26979703;
        double r26979715 = r26979694 ? r26979703 : r26979714;
        return r26979715;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.4
Target0.5
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.1569808277222708e-65 or 2.102710110923131e-20 < z

    1. Initial program 10.3

      \[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt10.3

      \[\leadsto \frac{\color{blue}{\left(\left(\sqrt[3]{\cosh x} \cdot \sqrt[3]{\cosh x}\right) \cdot \sqrt[3]{\cosh x}\right)} \cdot \frac{y}{x}}{z}\]

    4. Applied associate-*l*10.3

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\cosh x} \cdot \sqrt[3]{\cosh x}\right) \cdot \left(\sqrt[3]{\cosh x} \cdot \frac{y}{x}\right)}}{z}\]
    5. Using strategy rm

    6. Applied associate-*r/10.3

      \[\leadsto \frac{\left(\sqrt[3]{\cosh x} \cdot \sqrt[3]{\cosh x}\right) \cdot \color{blue}{\frac{\sqrt[3]{\cosh x} \cdot y}{x}}}{z}\]

    7. Applied associate-*r/10.3

      \[\leadsto \frac{\color{blue}{\frac{\left(\sqrt[3]{\cosh x} \cdot \sqrt[3]{\cosh x}\right) \cdot \left(\sqrt[3]{\cosh x} \cdot y\right)}{x}}}{z}\]

    8. Applied associate-/l/0.5

      \[\leadsto \color{blue}{\frac{\left(\sqrt[3]{\cosh x} \cdot \sqrt[3]{\cosh x}\right) \cdot \left(\sqrt[3]{\cosh x} \cdot y\right)}{z \cdot x}}\]

    if -3.1569808277222708e-65 < z < 2.102710110923131e-20

    1. Initial program 0.3

      \[\frac{\cosh x \cdot \frac{y}{x}}{z}\]

    2. Taylor expanded around inf 22.4

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{2} \cdot e^{x} + \frac{1}{2} \cdot e^{-x}\right) \cdot y}{x \cdot z}}\]

    3. Simplified0.3

      \[\leadsto \color{blue}{\frac{\left(e^{x} \cdot \frac{1}{2} + \frac{\frac{1}{2}}{e^{x}}\right) \cdot \frac{y}{z}}{x}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.1569808277222708 \cdot 10^{-65}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\cosh x} \cdot \sqrt[3]{\cosh x}\right) \cdot \left(y \cdot \sqrt[3]{\cosh x}\right)}{x \cdot z}\\ \mathbf{elif}\;z \le 2.102710110923131 \cdot 10^{-20}:\\ \;\;\;\;\frac{\left(\frac{\frac{1}{2}}{e^{x}} + \frac{1}{2} \cdot e^{x}\right) \cdot \frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\cosh x} \cdot \sqrt[3]{\cosh x}\right) \cdot \left(y \cdot \sqrt[3]{\cosh x}\right)}{x \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"

  :herbie-target
  (if (< y -4.618902267687042e-52) (* (/ (/ y z) x) (cosh x)) (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

  (/ (* (cosh x) (/ y x)) z))