Average Error: 8.1 → 0.7
Time: 3.6s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.663392033250732015294860045286254482681 \cdot 10^{67} \lor \neg \left(z \le 1.854353753596801523136697556389482635506 \cdot 10^{70}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \left(1 \cdot \frac{\frac{y}{z}}{x}\right)\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5.663392033250732015294860045286254482681 \cdot 10^{67} \lor \neg \left(z \le 1.854353753596801523136697556389482635506 \cdot 10^{70}\right):\\
\;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\

\mathbf{else}:\\
\;\;\;\;\cosh x \cdot \left(1 \cdot \frac{\frac{y}{z}}{x}\right)\\

\end{array}
double f(double x, double y, double z) {
        double r457981 = x;
        double r457982 = cosh(r457981);
        double r457983 = y;
        double r457984 = r457983 / r457981;
        double r457985 = r457982 * r457984;
        double r457986 = z;
        double r457987 = r457985 / r457986;
        return r457987;
}
↓
double f(double x, double y, double z) {
        double r457988 = z;
        double r457989 = -5.663392033250732e+67;
        bool r457990 = r457988 <= r457989;
        double r457991 = 1.8543537535968015e+70;
        bool r457992 = r457988 <= r457991;
        double r457993 = !r457992;
        bool r457994 = r457990 || r457993;
        double r457995 = x;
        double r457996 = cosh(r457995);
        double r457997 = y;
        double r457998 = r457995 * r457988;
        double r457999 = r457997 / r457998;
        double r458000 = r457996 * r457999;
        double r458001 = 1.0;
        double r458002 = r457997 / r457988;
        double r458003 = r458002 / r457995;
        double r458004 = r458001 * r458003;
        double r458005 = r457996 * r458004;
        double r458006 = r457994 ? r458000 : r458005;
        return r458006;
}

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
Target0.4
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935152855236908684227749499669 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -5.663392033250732e+67 or 1.8543537535968015e+70 < z

    1. Initial program 14.6

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

    3. Applied *-un-lft-identity14.6

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

    4. Applied times-frac14.6

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

    5. Simplified14.6

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

    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-frac11.7

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

    11. Applied frac-times0.3

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

    12. Simplified0.3

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

    if -5.663392033250732e+67 < z < 1.8543537535968015e+70

    1. Initial program 1.4

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

    3. Applied *-un-lft-identity1.4

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

    4. Applied times-frac1.4

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

    5. Simplified1.4

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

    6. Simplified13.9

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

    8. Applied *-un-lft-identity13.9

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

    9. Applied times-frac1.3

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

    11. Applied *-un-lft-identity1.3

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

    12. Applied *-un-lft-identity1.3

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

    13. Applied times-frac1.3

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

    14. Applied associate-*l*1.3

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

    15. Simplified1.2

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.663392033250732015294860045286254482681 \cdot 10^{67} \lor \neg \left(z \le 1.854353753596801523136697556389482635506 \cdot 10^{70}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \left(1 \cdot \frac{\frac{y}{z}}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(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))