Average Error: 8.1 → 0.3
Time: 8.9s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -2.36824288630637356 \cdot 10^{265}:\\ \;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)}{z} \cdot y}{x}\\ \mathbf{elif}\;\cosh x \cdot \frac{y}{x} \le 4.78464291171871 \cdot 10^{269}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)}{x \cdot \frac{z}{y}}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -2.36824288630637356 \cdot 10^{265}:\\
\;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)}{z} \cdot y}{x}\\

\mathbf{elif}\;\cosh x \cdot \frac{y}{x} \le 4.78464291171871 \cdot 10^{269}:\\
\;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)}{x \cdot \frac{z}{y}}\\

\end{array}
double f(double x, double y, double z) {
        double r668885 = x;
        double r668886 = cosh(r668885);
        double r668887 = y;
        double r668888 = r668887 / r668885;
        double r668889 = r668886 * r668888;
        double r668890 = z;
        double r668891 = r668889 / r668890;
        return r668891;
}
↓
double f(double x, double y, double z) {
        double r668892 = x;
        double r668893 = cosh(r668892);
        double r668894 = y;
        double r668895 = r668894 / r668892;
        double r668896 = r668893 * r668895;
        double r668897 = -2.3682428863063736e+265;
        bool r668898 = r668896 <= r668897;
        double r668899 = 0.5;
        double r668900 = exp(r668892);
        double r668901 = -r668892;
        double r668902 = exp(r668901);
        double r668903 = r668900 + r668902;
        double r668904 = r668899 * r668903;
        double r668905 = z;
        double r668906 = r668904 / r668905;
        double r668907 = r668906 * r668894;
        double r668908 = r668907 / r668892;
        double r668909 = 4.7846429117187147e+269;
        bool r668910 = r668896 <= r668909;
        double r668911 = r668896 / r668905;
        double r668912 = r668905 / r668894;
        double r668913 = r668892 * r668912;
        double r668914 = r668904 / r668913;
        double r668915 = r668910 ? r668911 : r668914;
        double r668916 = r668898 ? r668908 : r668915;
        return r668916;
}

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

Original8.1
Target0.5
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* (cosh x) (/ y x)) < -2.3682428863063736e+265

    1. Initial program 44.9

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

    2. Taylor expanded around inf 0.4

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

    3. Simplified0.4

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

    5. Applied times-frac0.4

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

    7. Applied associate-*l/0.4

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

    8. Simplified0.4

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

    if -2.3682428863063736e+265 < (* (cosh x) (/ y x)) < 4.7846429117187147e+269

    1. Initial program 0.2

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

    if 4.7846429117187147e+269 < (* (cosh x) (/ y x))

    1. Initial program 45.8

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

    2. Taylor expanded around inf 0.5

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

    3. Simplified0.5

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

    5. Applied associate-/l*0.4

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

    6. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -2.36824288630637356 \cdot 10^{265}:\\ \;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)}{z} \cdot y}{x}\\ \mathbf{elif}\;\cosh x \cdot \frac{y}{x} \le 4.78464291171871 \cdot 10^{269}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)}{x \cdot \frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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))