Average Error: 7.5 → 0.4
Time: 51.6s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.986686670783298043324591018464960578634 \cdot 10^{-19}:\\ \;\;\;\;\frac{y \cdot \cosh x}{x \cdot z}\\ \mathbf{elif}\;z \le 7.189025362679728768411324688191991410865 \cdot 10^{-40}:\\ \;\;\;\;\left(\frac{y}{x} \cdot \cosh x\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \cosh x}{x \cdot z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.986686670783298043324591018464960578634 \cdot 10^{-19}:\\
\;\;\;\;\frac{y \cdot \cosh x}{x \cdot z}\\

\mathbf{elif}\;z \le 7.189025362679728768411324688191991410865 \cdot 10^{-40}:\\
\;\;\;\;\left(\frac{y}{x} \cdot \cosh x\right) \cdot \frac{1}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r23091705 = x;
        double r23091706 = cosh(r23091705);
        double r23091707 = y;
        double r23091708 = r23091707 / r23091705;
        double r23091709 = r23091706 * r23091708;
        double r23091710 = z;
        double r23091711 = r23091709 / r23091710;
        return r23091711;
}
↓
double f(double x, double y, double z) {
        double r23091712 = z;
        double r23091713 = -1.986686670783298e-19;
        bool r23091714 = r23091712 <= r23091713;
        double r23091715 = y;
        double r23091716 = x;
        double r23091717 = cosh(r23091716);
        double r23091718 = r23091715 * r23091717;
        double r23091719 = r23091716 * r23091712;
        double r23091720 = r23091718 / r23091719;
        double r23091721 = 7.189025362679729e-40;
        bool r23091722 = r23091712 <= r23091721;
        double r23091723 = r23091715 / r23091716;
        double r23091724 = r23091723 * r23091717;
        double r23091725 = 1.0;
        double r23091726 = r23091725 / r23091712;
        double r23091727 = r23091724 * r23091726;
        double r23091728 = r23091722 ? r23091727 : r23091720;
        double r23091729 = r23091714 ? r23091720 : r23091728;
        return r23091729;
}

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

Original7.5
Target0.4
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.986686670783298e-19 or 7.189025362679729e-40 < z

    1. Initial program 10.8

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

    3. Applied div-inv10.8

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

    5. Applied associate-*r/10.8

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

    6. Applied frac-times0.4

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

    7. Simplified0.4

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

    if -1.986686670783298e-19 < z < 7.189025362679729e-40

    1. Initial program 0.3

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

    3. Applied div-inv0.4

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.986686670783298043324591018464960578634 \cdot 10^{-19}:\\ \;\;\;\;\frac{y \cdot \cosh x}{x \cdot z}\\ \mathbf{elif}\;z \le 7.189025362679728768411324688191991410865 \cdot 10^{-40}:\\ \;\;\;\;\left(\frac{y}{x} \cdot \cosh x\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \cosh x}{x \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +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))