Average Error: 7.7 → 0.5
Time: 4.6s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4.41253613397111438599433893468190853575 \cdot 10^{77} \lor \neg \left(z \le 1023626066629829993958264012800\right):\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -4.41253613397111438599433893468190853575 \cdot 10^{77} \lor \neg \left(z \le 1023626066629829993958264012800\right):\\
\;\;\;\;\frac{y \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{z \cdot x}\\

\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 r757369 = x;
        double r757370 = cosh(r757369);
        double r757371 = y;
        double r757372 = r757371 / r757369;
        double r757373 = r757370 * r757372;
        double r757374 = z;
        double r757375 = r757373 / r757374;
        return r757375;
}
↓
double f(double x, double y, double z) {
        double r757376 = z;
        double r757377 = -4.4125361339711144e+77;
        bool r757378 = r757376 <= r757377;
        double r757379 = 1.02362606662983e+30;
        bool r757380 = r757376 <= r757379;
        double r757381 = !r757380;
        bool r757382 = r757378 || r757381;
        double r757383 = y;
        double r757384 = x;
        double r757385 = exp(r757384);
        double r757386 = 0.5;
        double r757387 = r757386 / r757385;
        double r757388 = fma(r757385, r757386, r757387);
        double r757389 = r757383 * r757388;
        double r757390 = r757376 * r757384;
        double r757391 = r757389 / r757390;
        double r757392 = r757383 / r757376;
        double r757393 = r757392 * r757388;
        double r757394 = r757393 / r757384;
        double r757395 = r757382 ? r757391 : r757394;
        return r757395;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original7.7
Target0.4
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935152855236908684227749499669 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -4.4125361339711144e+77 or 1.02362606662983e+30 < z

    1. Initial program 13.1

      \[\frac{\cosh x \cdot \frac{y}{x}}{z}\]

    2. Taylor expanded around inf 0.3

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

    3. Simplified11.8

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}}\]
    4. Using strategy rm

    5. Applied frac-times0.3

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

    if -4.4125361339711144e+77 < z < 1.02362606662983e+30

    1. Initial program 1.1

      \[\frac{\cosh x \cdot \frac{y}{x}}{z}\]

    2. Taylor expanded around inf 14.8

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

    3. Simplified0.8

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}}\]
    4. Using strategy rm

    5. Applied associate-*r/0.7

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.41253613397111438599433893468190853575 \cdot 10^{77} \lor \neg \left(z \le 1023626066629829993958264012800\right):\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +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))