Average Error: 7.9 → 0.4
Time: 20.0s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -6.394362921332223351644270586304115137557 \cdot 10^{-33}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{elif}\;z \le 2.035540255879025646360089979447494101805 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{\frac{x}{\cosh x}} \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -6.394362921332223351644270586304115137557 \cdot 10^{-33}:\\
\;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\

\mathbf{elif}\;z \le 2.035540255879025646360089979447494101805 \cdot 10^{-6}:\\
\;\;\;\;\frac{y}{\frac{x}{\cosh x}} \cdot \frac{1}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r24839921 = x;
        double r24839922 = cosh(r24839921);
        double r24839923 = y;
        double r24839924 = r24839923 / r24839921;
        double r24839925 = r24839922 * r24839924;
        double r24839926 = z;
        double r24839927 = r24839925 / r24839926;
        return r24839927;
}
↓
double f(double x, double y, double z) {
        double r24839928 = z;
        double r24839929 = -6.394362921332223e-33;
        bool r24839930 = r24839928 <= r24839929;
        double r24839931 = x;
        double r24839932 = cosh(r24839931);
        double r24839933 = y;
        double r24839934 = r24839932 * r24839933;
        double r24839935 = r24839931 * r24839928;
        double r24839936 = r24839934 / r24839935;
        double r24839937 = 2.0355402558790256e-06;
        bool r24839938 = r24839928 <= r24839937;
        double r24839939 = r24839931 / r24839932;
        double r24839940 = r24839933 / r24839939;
        double r24839941 = 1.0;
        double r24839942 = r24839941 / r24839928;
        double r24839943 = r24839940 * r24839942;
        double r24839944 = r24839938 ? r24839943 : r24839936;
        double r24839945 = r24839930 ? r24839936 : r24839944;
        return r24839945;
}

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
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -6.394362921332223e-33 or 2.0355402558790256e-06 < z

    1. Initial program 11.5

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

    3. Applied associate-*r/11.5

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

    4. Applied associate-/l/0.4

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

    if -6.394362921332223e-33 < z < 2.0355402558790256e-06

    1. Initial program 0.3

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

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

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

    4. Applied associate-/r*0.3

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

    5. Simplified0.3

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

    7. Applied div-inv0.4

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.394362921332223351644270586304115137557 \cdot 10^{-33}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{elif}\;z \le 2.035540255879025646360089979447494101805 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{\frac{x}{\cosh x}} \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"

  :herbie-target
  (if (< y -4.618902267687042e-52) (* (/ (/ y z) x) (cosh x)) (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

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