Average Error: 7.9 → 0.5
Time: 12.2s
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 r577075 = x;
        double r577076 = cosh(r577075);
        double r577077 = y;
        double r577078 = r577077 / r577075;
        double r577079 = r577076 * r577078;
        double r577080 = z;
        double r577081 = r577079 / r577080;
        return r577081;
}
↓
double f(double x, double y, double z) {
        double r577082 = y;
        double r577083 = -1.7284342394986222e-14;
        bool r577084 = r577082 <= r577083;
        double r577085 = x;
        double r577086 = cosh(r577085);
        double r577087 = z;
        double r577088 = r577085 * r577087;
        double r577089 = r577088 / r577082;
        double r577090 = r577086 / r577089;
        double r577091 = 6802291.148674449;
        bool r577092 = r577082 <= r577091;
        double r577093 = r577082 / r577085;
        double r577094 = r577086 * r577093;
        double r577095 = r577094 / r577087;
        double r577096 = 0.5;
        double r577097 = r577085 * r577082;
        double r577098 = r577097 / r577087;
        double r577099 = r577096 * r577098;
        double r577100 = r577082 / r577087;
        double r577101 = r577100 / r577085;
        double r577102 = r577099 + r577101;
        double r577103 = r577092 ? r577095 : r577102;
        double r577104 = r577084 ? r577090 : r577103;
        return r577104;
}

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 pow118.2

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

    7. Applied pow118.2

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

    8. Applied pow-prod-down18.2

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

    9. Simplified0.4

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

    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))