Average Error: 8.1 → 0.7
Time: 4.0s
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 r451423 = x;
        double r451424 = cosh(r451423);
        double r451425 = y;
        double r451426 = r451425 / r451423;
        double r451427 = r451424 * r451426;
        double r451428 = z;
        double r451429 = r451427 / r451428;
        return r451429;
}
↓
double f(double x, double y, double z) {
        double r451430 = z;
        double r451431 = -5.663392033250732e+67;
        bool r451432 = r451430 <= r451431;
        double r451433 = 1.8543537535968015e+70;
        bool r451434 = r451430 <= r451433;
        double r451435 = !r451434;
        bool r451436 = r451432 || r451435;
        double r451437 = x;
        double r451438 = cosh(r451437);
        double r451439 = y;
        double r451440 = r451437 * r451430;
        double r451441 = r451439 / r451440;
        double r451442 = r451438 * r451441;
        double r451443 = r451439 / r451430;
        double r451444 = exp(r451437);
        double r451445 = 0.5;
        double r451446 = r451445 / r451444;
        double r451447 = fma(r451444, r451445, r451446);
        double r451448 = r451443 * r451447;
        double r451449 = r451448 / r451437;
        double r451450 = r451436 ? r451442 : r451449;
        return r451450;
}

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))