Average Error: 8.1 → 0.7
Time: 3.9s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.663392033250732015294860045286254482681 \cdot 10^{67} \lor \neg \left(z \le 1.854353753596801523136697556389482635506 \cdot 10^{70}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot {\left(\frac{\frac{y}{z}}{x}\right)}^{1}\\ \end{array}\]
\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(\frac{\frac{y}{z}}{x}\right)}^{1}\\

\end{array}
double f(double x, double y, double z) {
        double r514084 = x;
        double r514085 = cosh(r514084);
        double r514086 = y;
        double r514087 = r514086 / r514084;
        double r514088 = r514085 * r514087;
        double r514089 = z;
        double r514090 = r514088 / r514089;
        return r514090;
}
↓
double f(double x, double y, double z) {
        double r514091 = z;
        double r514092 = -5.663392033250732e+67;
        bool r514093 = r514091 <= r514092;
        double r514094 = 1.8543537535968015e+70;
        bool r514095 = r514091 <= r514094;
        double r514096 = !r514095;
        bool r514097 = r514093 || r514096;
        double r514098 = x;
        double r514099 = cosh(r514098);
        double r514100 = y;
        double r514101 = r514098 * r514091;
        double r514102 = r514100 / r514101;
        double r514103 = r514099 * r514102;
        double r514104 = r514100 / r514091;
        double r514105 = r514104 / r514098;
        double r514106 = 1.0;
        double r514107 = pow(r514105, r514106);
        double r514108 = r514099 * r514107;
        double r514109 = r514097 ? r514103 : r514108;
        return r514109;
}

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 pow11.3

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

    12. Applied pow11.3

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

    13. Applied pow-prod-down1.3

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

    14. Simplified1.2

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

  4. Final simplification0.7

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

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))