Average Error: 7.8 → 0.4
Time: 12.4s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -23.9528592754304945:\\ \;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \mathbf{elif}\;y \le 7876022.030429827:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \left(e^{x} + e^{-x}\right)}{2 \cdot x}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -23.9528592754304945:\\
\;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\

\mathbf{elif}\;y \le 7876022.030429827:\\
\;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{y}{z} \cdot \left(e^{x} + e^{-x}\right)}{2 \cdot x}\\

\end{array}
double f(double x, double y, double z) {
        double r593066 = x;
        double r593067 = cosh(r593066);
        double r593068 = y;
        double r593069 = r593068 / r593066;
        double r593070 = r593067 * r593069;
        double r593071 = z;
        double r593072 = r593070 / r593071;
        return r593072;
}
↓
double f(double x, double y, double z) {
        double r593073 = y;
        double r593074 = -23.952859275430495;
        bool r593075 = r593073 <= r593074;
        double r593076 = 0.5;
        double r593077 = x;
        double r593078 = r593077 * r593073;
        double r593079 = z;
        double r593080 = r593078 / r593079;
        double r593081 = r593076 * r593080;
        double r593082 = r593077 * r593079;
        double r593083 = r593073 / r593082;
        double r593084 = r593081 + r593083;
        double r593085 = 7876022.030429827;
        bool r593086 = r593073 <= r593085;
        double r593087 = cosh(r593077);
        double r593088 = r593073 / r593077;
        double r593089 = r593087 * r593088;
        double r593090 = r593089 / r593079;
        double r593091 = r593073 / r593079;
        double r593092 = exp(r593077);
        double r593093 = -r593077;
        double r593094 = exp(r593093);
        double r593095 = r593092 + r593094;
        double r593096 = r593091 * r593095;
        double r593097 = 2.0;
        double r593098 = r593097 * r593077;
        double r593099 = r593096 / r593098;
        double r593100 = r593086 ? r593090 : r593099;
        double r593101 = r593075 ? r593084 : r593100;
        return r593101;
}

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

Original7.8
Target0.4
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -23.952859275430495

    1. Initial program 20.7

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

    3. Applied *-un-lft-identity20.7

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

    4. Applied times-frac20.7

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

    5. Simplified20.7

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

    6. Simplified0.2

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

    8. Applied *-un-lft-identity0.2

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

    9. Applied times-frac0.4

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

    10. Taylor expanded around 0 1.1

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

    if -23.952859275430495 < y < 7876022.030429827

    1. Initial program 0.3

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

    if 7876022.030429827 < y

    1. Initial program 22.1

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

    3. Applied *-un-lft-identity22.1

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

    4. Applied times-frac21.9

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

    5. Simplified21.9

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

    6. Simplified0.3

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

    8. Applied *-un-lft-identity0.3

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

    9. Applied times-frac0.3

      \[\leadsto \cosh x \cdot \color{blue}{\left(\frac{1}{x} \cdot \frac{y}{z}\right)}\]
    10. Using strategy rm

    11. Applied associate-*l/0.3

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

    12. Applied cosh-def0.3

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

    13. Applied frac-times0.3

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

    14. Simplified0.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -23.9528592754304945:\\ \;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \mathbf{elif}\;y \le 7876022.030429827:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \left(e^{x} + e^{-x}\right)}{2 \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(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))