Average Error: 8.2 → 0.3
Time: 18.3s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -6.42783361027247785235561078261638924498 \cdot 10^{-40}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{elif}\;z \le 1431949075770.42724609375:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -6.42783361027247785235561078261638924498 \cdot 10^{-40}:\\
\;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\

\mathbf{elif}\;z \le 1431949075770.42724609375:\\
\;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\

\end{array}
double f(double x, double y, double z) {
        double r21786905 = x;
        double r21786906 = cosh(r21786905);
        double r21786907 = y;
        double r21786908 = r21786907 / r21786905;
        double r21786909 = r21786906 * r21786908;
        double r21786910 = z;
        double r21786911 = r21786909 / r21786910;
        return r21786911;
}
↓
double f(double x, double y, double z) {
        double r21786912 = z;
        double r21786913 = -6.427833610272478e-40;
        bool r21786914 = r21786912 <= r21786913;
        double r21786915 = x;
        double r21786916 = cosh(r21786915);
        double r21786917 = y;
        double r21786918 = r21786916 * r21786917;
        double r21786919 = r21786915 * r21786912;
        double r21786920 = r21786918 / r21786919;
        double r21786921 = 1431949075770.4272;
        bool r21786922 = r21786912 <= r21786921;
        double r21786923 = r21786918 / r21786912;
        double r21786924 = r21786923 / r21786915;
        double r21786925 = r21786922 ? r21786924 : r21786920;
        double r21786926 = r21786914 ? r21786920 : r21786925;
        return r21786926;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original8.2
Target0.4
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -6.427833610272478e-40 or 1431949075770.4272 < z

    1. Initial program 12.1

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

    3. Applied div-inv12.1

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

    4. Applied associate-*r*12.1

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

    6. Applied associate-/l*0.4

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

    7. Simplified0.3

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

    if -6.427833610272478e-40 < z < 1431949075770.4272

    1. Initial program 0.3

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

    3. Applied div-inv0.4

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

    4. Applied associate-*r*0.4

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

    6. Applied div-inv0.5

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

    8. Applied un-div-inv0.4

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

    9. Applied associate-*l/0.4

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

    10. Simplified0.3

      \[\leadsto \frac{\color{blue}{\frac{y \cdot \cosh x}{z}}}{x}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.42783361027247785235561078261638924498 \cdot 10^{-40}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{elif}\;z \le 1431949075770.42724609375:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"

  :herbie-target
  (if (< y -4.618902267687042e-52) (* (/ (/ y z) x) (cosh x)) (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

  (/ (* (cosh x) (/ y x)) z))