Average Error: 7.9 → 0.5
Time: 11.6s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.72843423949862217 \cdot 10^{-14}:\\ \;\;\;\;\frac{\cosh x}{\frac{x \cdot z}{y}}\\ \mathbf{elif}\;y \le 6802291.148674449:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{\frac{y}{z}}{x}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.72843423949862217 \cdot 10^{-14}:\\
\;\;\;\;\frac{\cosh x}{\frac{x \cdot z}{y}}\\

\mathbf{elif}\;y \le 6802291.148674449:\\
\;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{\frac{y}{z}}{x}\\

\end{array}
double f(double x, double y, double z) {
        double r625703 = x;
        double r625704 = cosh(r625703);
        double r625705 = y;
        double r625706 = r625705 / r625703;
        double r625707 = r625704 * r625706;
        double r625708 = z;
        double r625709 = r625707 / r625708;
        return r625709;
}
↓
double f(double x, double y, double z) {
        double r625710 = y;
        double r625711 = -1.7284342394986222e-14;
        bool r625712 = r625710 <= r625711;
        double r625713 = x;
        double r625714 = cosh(r625713);
        double r625715 = z;
        double r625716 = r625713 * r625715;
        double r625717 = r625716 / r625710;
        double r625718 = r625714 / r625717;
        double r625719 = 6802291.148674449;
        bool r625720 = r625710 <= r625719;
        double r625721 = r625710 / r625713;
        double r625722 = r625714 * r625721;
        double r625723 = r625722 / r625715;
        double r625724 = 0.5;
        double r625725 = r625713 * r625710;
        double r625726 = r625725 / r625715;
        double r625727 = r625724 * r625726;
        double r625728 = r625710 / r625715;
        double r625729 = r625728 / r625713;
        double r625730 = r625727 + r625729;
        double r625731 = r625720 ? r625723 : r625730;
        double r625732 = r625712 ? r625718 : r625731;
        return r625732;
}

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

Original7.9
Target0.4
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -1.7284342394986222e-14

    1. Initial program 19.8

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

    3. Applied associate-/l*19.9

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

    4. Simplified18.2

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

    6. Applied *-un-lft-identity18.2

      \[\leadsto \frac{\cosh x}{\color{blue}{\left(1 \cdot z\right)} \cdot \frac{x}{y}}\]

    7. Applied associate-*l*18.2

      \[\leadsto \frac{\cosh x}{\color{blue}{1 \cdot \left(z \cdot \frac{x}{y}\right)}}\]

    8. Simplified0.4

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

    if -1.7284342394986222e-14 < y < 6802291.148674449

    1. Initial program 0.3

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

    if 6802291.148674449 < y

    1. Initial program 22.1

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

    2. Taylor expanded around 0 1.5

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}}\]
    3. Using strategy rm

    4. Applied clear-num1.6

      \[\leadsto \frac{1}{2} \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{1}{\frac{x \cdot z}{y}}}\]

    5. Taylor expanded around 0 1.5

      \[\leadsto \frac{1}{2} \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{y}{x \cdot z}}\]

    6. Simplified1.5

      \[\leadsto \frac{1}{2} \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{\frac{y}{z}}{x}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.72843423949862217 \cdot 10^{-14}:\\ \;\;\;\;\frac{\cosh x}{\frac{x \cdot z}{y}}\\ \mathbf{elif}\;y \le 6802291.148674449:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{\frac{y}{z}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(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))