Average Error: 7.2 → 0.4
Time: 4.2s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le -3.58400891098028717 \cdot 10^{108}:\\ \;\;\;\;\frac{1}{\frac{z}{\cosh x \cdot y} \cdot x}\\ \mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le 1.5273947602591855 \cdot 10^{302}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le -3.58400891098028717 \cdot 10^{108}:\\
\;\;\;\;\frac{1}{\frac{z}{\cosh x \cdot y} \cdot x}\\

\mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le 1.5273947602591855 \cdot 10^{302}:\\
\;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r470040 = x;
        double r470041 = cosh(r470040);
        double r470042 = y;
        double r470043 = r470042 / r470040;
        double r470044 = r470041 * r470043;
        double r470045 = z;
        double r470046 = r470044 / r470045;
        return r470046;
}
↓
double f(double x, double y, double z) {
        double r470047 = x;
        double r470048 = cosh(r470047);
        double r470049 = y;
        double r470050 = r470049 / r470047;
        double r470051 = r470048 * r470050;
        double r470052 = z;
        double r470053 = r470051 / r470052;
        double r470054 = -3.584008910980287e+108;
        bool r470055 = r470053 <= r470054;
        double r470056 = 1.0;
        double r470057 = r470048 * r470049;
        double r470058 = r470052 / r470057;
        double r470059 = r470058 * r470047;
        double r470060 = r470056 / r470059;
        double r470061 = 1.5273947602591855e+302;
        bool r470062 = r470053 <= r470061;
        double r470063 = r470047 * r470052;
        double r470064 = r470049 / r470063;
        double r470065 = r470048 * r470064;
        double r470066 = r470062 ? r470053 : r470065;
        double r470067 = r470055 ? r470060 : r470066;
        return r470067;
}

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

Original7.2
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 (/ (* (cosh x) (/ y x)) z) < -3.584008910980287e+108

    1. Initial program 19.0

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

    3. Applied div-inv19.1

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

    4. Applied associate-*r*19.1

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

    6. Applied clear-num19.1

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

    7. Simplified0.6

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

    if -3.584008910980287e+108 < (/ (* (cosh x) (/ y x)) z) < 1.5273947602591855e+302

    1. Initial program 0.2

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

    3. Applied div-inv0.3

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

    4. Applied associate-*r*0.3

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

    6. Applied associate-*l*0.3

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

    7. Simplified0.2

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

    if 1.5273947602591855e+302 < (/ (* (cosh x) (/ y x)) z)

    1. Initial program 60.5

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

    3. Applied *-un-lft-identity60.5

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

    4. Applied times-frac60.5

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

    5. Simplified60.5

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

    6. Simplified2.2

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le -3.58400891098028717 \cdot 10^{108}:\\ \;\;\;\;\frac{1}{\frac{z}{\cosh x \cdot y} \cdot x}\\ \mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le 1.5273947602591855 \cdot 10^{302}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(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))