Average Error: 7.7 → 0.4
Time: 4.4s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5093130071052390400 \lor \neg \left(z \le 7.93928067237076916 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{\cosh x \cdot y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5093130071052390400 \lor \neg \left(z \le 7.93928067237076916 \cdot 10^{-20}\right):\\
\;\;\;\;\frac{\cosh x \cdot y}{z \cdot x}\\

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

\end{array}
double f(double x, double y, double z) {
        double r457971 = x;
        double r457972 = cosh(r457971);
        double r457973 = y;
        double r457974 = r457973 / r457971;
        double r457975 = r457972 * r457974;
        double r457976 = z;
        double r457977 = r457975 / r457976;
        return r457977;
}
↓
double f(double x, double y, double z) {
        double r457978 = z;
        double r457979 = -5.09313007105239e+18;
        bool r457980 = r457978 <= r457979;
        double r457981 = 7.939280672370769e-20;
        bool r457982 = r457978 <= r457981;
        double r457983 = !r457982;
        bool r457984 = r457980 || r457983;
        double r457985 = x;
        double r457986 = cosh(r457985);
        double r457987 = y;
        double r457988 = r457986 * r457987;
        double r457989 = r457978 * r457985;
        double r457990 = r457988 / r457989;
        double r457991 = r457988 / r457985;
        double r457992 = r457991 / r457978;
        double r457993 = r457984 ? r457990 : r457992;
        return r457993;
}

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

Original7.7
Target0.5
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 2 regimes

  2. if z < -5.09313007105239e+18 or 7.939280672370769e-20 < z

    1. Initial program 11.7

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

    3. Applied associate-*r/11.8

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

    4. Applied associate-/l/0.3

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

    if -5.09313007105239e+18 < z < 7.939280672370769e-20

    1. Initial program 0.5

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

    3. Applied associate-*r/0.5

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

    4. Applied associate-/l/19.4

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

    6. Applied associate-/r*0.3

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

    8. Applied clear-num0.5

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{\cosh x \cdot y}{z}}}}\]
    9. Using strategy rm

    10. Applied associate-/r/0.5

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

    11. Applied associate-/r*0.6

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

    12. Simplified0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5093130071052390400 \lor \neg \left(z \le 7.93928067237076916 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{\cosh x \cdot y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +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))