Average Error: 7.3 → 0.3
Time: 3.8s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le -1.569646754895685203707472211986928556371 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}\\ \mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le 1.43550247534837939212643563586416747364 \cdot 10^{308}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \left(y \cdot \frac{\frac{1}{z}}{x}\right)\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le -1.569646754895685203707472211986928556371 \cdot 10^{-6}:\\
\;\;\;\;\frac{\frac{y}{z} \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}\\

\mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le 1.43550247534837939212643563586416747364 \cdot 10^{308}:\\
\;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r440490 = x;
        double r440491 = cosh(r440490);
        double r440492 = y;
        double r440493 = r440492 / r440490;
        double r440494 = r440491 * r440493;
        double r440495 = z;
        double r440496 = r440494 / r440495;
        return r440496;
}
↓
double f(double x, double y, double z) {
        double r440497 = x;
        double r440498 = cosh(r440497);
        double r440499 = y;
        double r440500 = r440499 / r440497;
        double r440501 = r440498 * r440500;
        double r440502 = z;
        double r440503 = r440501 / r440502;
        double r440504 = -1.5696467548956852e-06;
        bool r440505 = r440503 <= r440504;
        double r440506 = r440499 / r440502;
        double r440507 = exp(r440497);
        double r440508 = 0.5;
        double r440509 = r440508 / r440507;
        double r440510 = fma(r440507, r440508, r440509);
        double r440511 = r440506 * r440510;
        double r440512 = r440511 / r440497;
        double r440513 = 1.4355024753483794e+308;
        bool r440514 = r440503 <= r440513;
        double r440515 = 1.0;
        double r440516 = r440515 / r440502;
        double r440517 = r440516 / r440497;
        double r440518 = r440499 * r440517;
        double r440519 = r440498 * r440518;
        double r440520 = r440514 ? r440503 : r440519;
        double r440521 = r440505 ? r440512 : r440520;
        return r440521;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original7.3
Target0.4
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935152855236908684227749499669 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (* (cosh x) (/ y x)) z) < -1.5696467548956852e-06

    1. Initial program 12.2

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

    3. Applied *-un-lft-identity12.2

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

    4. Applied times-frac12.2

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

    5. Simplified12.2

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

    6. Simplified11.5

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

    7. Taylor expanded around inf 11.5

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

    8. Simplified0.3

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

    if -1.5696467548956852e-06 < (/ (* (cosh x) (/ y x)) z) < 1.4355024753483794e+308

    1. Initial program 0.2

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

    if 1.4355024753483794e+308 < (/ (* (cosh x) (/ y x)) z)

    1. Initial program 63.9

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

    3. Applied *-un-lft-identity63.9

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

    4. Applied times-frac63.7

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

    5. Simplified63.7

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

    6. Simplified0.4

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

    8. Applied div-inv0.5

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

    10. Applied add-cube-cbrt0.5

      \[\leadsto \cosh x \cdot \left(y \cdot \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{x \cdot z}\right)\]

    11. Applied times-frac0.6

      \[\leadsto \cosh x \cdot \left(y \cdot \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{x} \cdot \frac{\sqrt[3]{1}}{z}\right)}\right)\]

    12. Simplified0.6

      \[\leadsto \cosh x \cdot \left(y \cdot \left(\color{blue}{\frac{1}{x}} \cdot \frac{\sqrt[3]{1}}{z}\right)\right)\]

    13. Simplified0.6

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

    15. Applied associate-*l/0.5

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

    16. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le -1.569646754895685203707472211986928556371 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}\\ \mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le 1.43550247534837939212643563586416747364 \cdot 10^{308}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \left(y \cdot \frac{\frac{1}{z}}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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))