Average Error: 7.7 → 0.7
Time: 9.1s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.93577210121831744 \cdot 10^{80} \lor \neg \left(z \le 2.60686388496247717 \cdot 10^{-42}\right):\\ \;\;\;\;\frac{y \cdot \left(\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)\right)}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{\frac{z}{y} \cdot x}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.93577210121831744 \cdot 10^{80} \lor \neg \left(z \le 2.60686388496247717 \cdot 10^{-42}\right):\\
\;\;\;\;\frac{y \cdot \left(\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)\right)}{z \cdot x}\\

\mathbf{else}:\\
\;\;\;\;\frac{\cosh x}{\frac{z}{y} \cdot x}\\

\end{array}
double f(double x, double y, double z) {
        double r491699 = x;
        double r491700 = cosh(r491699);
        double r491701 = y;
        double r491702 = r491701 / r491699;
        double r491703 = r491700 * r491702;
        double r491704 = z;
        double r491705 = r491703 / r491704;
        return r491705;
}
↓
double f(double x, double y, double z) {
        double r491706 = z;
        double r491707 = -1.9357721012183174e+80;
        bool r491708 = r491706 <= r491707;
        double r491709 = 2.6068638849624772e-42;
        bool r491710 = r491706 <= r491709;
        double r491711 = !r491710;
        bool r491712 = r491708 || r491711;
        double r491713 = y;
        double r491714 = 0.5;
        double r491715 = x;
        double r491716 = exp(r491715);
        double r491717 = -r491715;
        double r491718 = exp(r491717);
        double r491719 = r491716 + r491718;
        double r491720 = r491714 * r491719;
        double r491721 = r491713 * r491720;
        double r491722 = r491706 * r491715;
        double r491723 = r491721 / r491722;
        double r491724 = cosh(r491715);
        double r491725 = r491706 / r491713;
        double r491726 = r491725 * r491715;
        double r491727 = r491724 / r491726;
        double r491728 = r491712 ? r491723 : r491727;
        return r491728;
}

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

Original7.7
Target0.4
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.9357721012183174e+80 or 2.6068638849624772e-42 < z

    1. Initial program 12.0

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

    3. Applied associate-/l*12.4

      \[\leadsto \color{blue}{\frac{\cosh x}{\frac{z}{\frac{y}{x}}}}\]

    4. Simplified11.3

      \[\leadsto \frac{\cosh x}{\color{blue}{\frac{z}{y} \cdot x}}\]
    5. Using strategy rm

    6. Applied div-inv11.3

      \[\leadsto \color{blue}{\cosh x \cdot \frac{1}{\frac{z}{y} \cdot x}}\]

    7. Simplified10.5

      \[\leadsto \cosh x \cdot \color{blue}{\frac{\frac{y}{z}}{x}}\]

    8. Taylor expanded around inf 0.4

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

    9. Simplified0.4

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

    if -1.9357721012183174e+80 < z < 2.6068638849624772e-42

    1. Initial program 1.3

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

    3. Applied associate-/l*1.4

      \[\leadsto \color{blue}{\frac{\cosh x}{\frac{z}{\frac{y}{x}}}}\]

    4. Simplified1.2

      \[\leadsto \frac{\cosh x}{\color{blue}{\frac{z}{y} \cdot x}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.93577210121831744 \cdot 10^{80} \lor \neg \left(z \le 2.60686388496247717 \cdot 10^{-42}\right):\\ \;\;\;\;\frac{y \cdot \left(\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)\right)}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{\frac{z}{y} \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< y -4.618902267687042e-52) (* (/ (/ y z) x) (cosh x)) (if (< y 1.0385305359351529e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

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