Average Error: 8.1 → 0.7
Time: 3.7s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.663392033250732015294860045286254482681 \cdot 10^{67} \lor \neg \left(z \le 1.854353753596801523136697556389482635506 \cdot 10^{70}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}\\ \end{array}\]
\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}:\\
\;\;\;\;\frac{\frac{y}{z} \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}\\

\end{array}
double f(double x, double y, double z) {
        double r440526 = x;
        double r440527 = cosh(r440526);
        double r440528 = y;
        double r440529 = r440528 / r440526;
        double r440530 = r440527 * r440529;
        double r440531 = z;
        double r440532 = r440530 / r440531;
        return r440532;
}
↓
double f(double x, double y, double z) {
        double r440533 = z;
        double r440534 = -5.663392033250732e+67;
        bool r440535 = r440533 <= r440534;
        double r440536 = 1.8543537535968015e+70;
        bool r440537 = r440533 <= r440536;
        double r440538 = !r440537;
        bool r440539 = r440535 || r440538;
        double r440540 = x;
        double r440541 = cosh(r440540);
        double r440542 = y;
        double r440543 = r440540 * r440533;
        double r440544 = r440542 / r440543;
        double r440545 = r440541 * r440544;
        double r440546 = r440542 / r440533;
        double r440547 = exp(r440540);
        double r440548 = 0.5;
        double r440549 = r440548 / r440547;
        double r440550 = fma(r440547, r440548, r440549);
        double r440551 = r440546 * r440550;
        double r440552 = r440551 / r440540;
        double r440553 = r440539 ? r440545 : r440552;
        return r440553;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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. Taylor expanded around inf 13.9

      \[\leadsto \color{blue}{\frac{y \cdot \left(\frac{1}{2} \cdot e^{x} + \frac{1}{2} \cdot e^{-x}\right)}{x \cdot z}}\]

    8. Simplified1.2

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.663392033250732015294860045286254482681 \cdot 10^{67} \lor \neg \left(z \le 1.854353753596801523136697556389482635506 \cdot 10^{70}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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))