Average Error: 7.9 → 0.5
Time: 3.4s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -1.865052624181062 \cdot 10^{247}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, x \cdot \frac{y}{z}, \frac{y}{x \cdot z}\right)\\ \mathbf{elif}\;\cosh x \cdot \frac{y}{x} \le 2.4776903717072572 \cdot 10^{243}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, x \cdot \frac{y}{z}, \frac{1}{\frac{x \cdot z}{y}}\right)\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -1.865052624181062 \cdot 10^{247}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{2}, x \cdot \frac{y}{z}, \frac{y}{x \cdot z}\right)\\

\mathbf{elif}\;\cosh x \cdot \frac{y}{x} \le 2.4776903717072572 \cdot 10^{243}:\\
\;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{2}, x \cdot \frac{y}{z}, \frac{1}{\frac{x \cdot z}{y}}\right)\\

\end{array}
double f(double x, double y, double z) {
        double r589907 = x;
        double r589908 = cosh(r589907);
        double r589909 = y;
        double r589910 = r589909 / r589907;
        double r589911 = r589908 * r589910;
        double r589912 = z;
        double r589913 = r589911 / r589912;
        return r589913;
}
↓
double f(double x, double y, double z) {
        double r589914 = x;
        double r589915 = cosh(r589914);
        double r589916 = y;
        double r589917 = r589916 / r589914;
        double r589918 = r589915 * r589917;
        double r589919 = -1.865052624181062e+247;
        bool r589920 = r589918 <= r589919;
        double r589921 = 0.5;
        double r589922 = z;
        double r589923 = r589916 / r589922;
        double r589924 = r589914 * r589923;
        double r589925 = r589914 * r589922;
        double r589926 = r589916 / r589925;
        double r589927 = fma(r589921, r589924, r589926);
        double r589928 = 2.477690371707257e+243;
        bool r589929 = r589918 <= r589928;
        double r589930 = r589918 / r589922;
        double r589931 = 1.0;
        double r589932 = r589925 / r589916;
        double r589933 = r589931 / r589932;
        double r589934 = fma(r589921, r589924, r589933);
        double r589935 = r589929 ? r589930 : r589934;
        double r589936 = r589920 ? r589927 : r589935;
        return r589936;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original7.9
Target0.5
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* (cosh x) (/ y x)) < -1.865052624181062e+247

    1. Initial program 38.8

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

    2. Taylor expanded around 0 1.3

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}}\]

    3. Simplified1.3

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

    5. Applied *-un-lft-identity1.3

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

    6. Applied times-frac1.3

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

    7. Simplified1.3

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

    if -1.865052624181062e+247 < (* (cosh x) (/ y x)) < 2.477690371707257e+243

    1. Initial program 0.2

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

    if 2.477690371707257e+243 < (* (cosh x) (/ y x))

    1. Initial program 39.3

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

    2. Taylor expanded around 0 1.5

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}}\]

    3. Simplified1.5

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

    5. Applied *-un-lft-identity1.5

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

    6. Applied times-frac1.5

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

    7. Simplified1.5

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

    9. Applied clear-num1.6

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -1.865052624181062 \cdot 10^{247}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, x \cdot \frac{y}{z}, \frac{y}{x \cdot z}\right)\\ \mathbf{elif}\;\cosh x \cdot \frac{y}{x} \le 2.4776903717072572 \cdot 10^{243}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, x \cdot \frac{y}{z}, \frac{1}{\frac{x \cdot z}{y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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))