Average Error: 8.0 → 0.6
Time: 15.7s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -7.86664282492854740730204323783673108712 \cdot 10^{65}:\\ \;\;\;\;\frac{\frac{e^{x} \cdot y + \frac{y}{e^{x}}}{2}}{x \cdot z}\\ \mathbf{elif}\;z \le 7.966049154723343466287174107454135040236 \cdot 10^{-53}:\\ \;\;\;\;\frac{\frac{\frac{e^{x} \cdot y + \frac{y}{e^{x}}}{z}}{2}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{x} \cdot y + \frac{y}{e^{x}}}{2}}{x \cdot z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -7.86664282492854740730204323783673108712 \cdot 10^{65}:\\
\;\;\;\;\frac{\frac{e^{x} \cdot y + \frac{y}{e^{x}}}{2}}{x \cdot z}\\

\mathbf{elif}\;z \le 7.966049154723343466287174107454135040236 \cdot 10^{-53}:\\
\;\;\;\;\frac{\frac{\frac{e^{x} \cdot y + \frac{y}{e^{x}}}{z}}{2}}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{e^{x} \cdot y + \frac{y}{e^{x}}}{2}}{x \cdot z}\\

\end{array}
double f(double x, double y, double z) {
        double r22985901 = x;
        double r22985902 = cosh(r22985901);
        double r22985903 = y;
        double r22985904 = r22985903 / r22985901;
        double r22985905 = r22985902 * r22985904;
        double r22985906 = z;
        double r22985907 = r22985905 / r22985906;
        return r22985907;
}
↓
double f(double x, double y, double z) {
        double r22985908 = z;
        double r22985909 = -7.866642824928547e+65;
        bool r22985910 = r22985908 <= r22985909;
        double r22985911 = x;
        double r22985912 = exp(r22985911);
        double r22985913 = y;
        double r22985914 = r22985912 * r22985913;
        double r22985915 = r22985913 / r22985912;
        double r22985916 = r22985914 + r22985915;
        double r22985917 = 2.0;
        double r22985918 = r22985916 / r22985917;
        double r22985919 = r22985911 * r22985908;
        double r22985920 = r22985918 / r22985919;
        double r22985921 = 7.9660491547233435e-53;
        bool r22985922 = r22985908 <= r22985921;
        double r22985923 = r22985916 / r22985908;
        double r22985924 = r22985923 / r22985917;
        double r22985925 = r22985924 / r22985911;
        double r22985926 = r22985922 ? r22985925 : r22985920;
        double r22985927 = r22985910 ? r22985920 : r22985926;
        return r22985927;
}

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

Original8.0
Target0.5
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -7.866642824928547e+65 or 7.9660491547233435e-53 < z

    1. Initial program 11.8

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

    3. Applied associate-*r/11.8

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

    4. Applied associate-/l/0.6

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

    6. Applied cosh-def0.6

      \[\leadsto \frac{\color{blue}{\frac{e^{x} + e^{-x}}{2}} \cdot y}{z \cdot x}\]

    7. Applied associate-*l/0.6

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

    8. Simplified0.6

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

    if -7.866642824928547e+65 < z < 7.9660491547233435e-53

    1. Initial program 1.0

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

    3. Applied associate-*r/1.0

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

    4. Applied associate-/l/17.5

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

    6. Applied cosh-def17.5

      \[\leadsto \frac{\color{blue}{\frac{e^{x} + e^{-x}}{2}} \cdot y}{z \cdot x}\]

    7. Applied associate-*l/17.5

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

    8. Simplified17.5

      \[\leadsto \frac{\frac{\color{blue}{e^{x} \cdot y + \frac{y \cdot 1}{e^{x}}}}{2}}{z \cdot x}\]
    9. Using strategy rm

    10. Applied associate-/r*0.8

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

    11. Simplified0.8

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.86664282492854740730204323783673108712 \cdot 10^{65}:\\ \;\;\;\;\frac{\frac{e^{x} \cdot y + \frac{y}{e^{x}}}{2}}{x \cdot z}\\ \mathbf{elif}\;z \le 7.966049154723343466287174107454135040236 \cdot 10^{-53}:\\ \;\;\;\;\frac{\frac{\frac{e^{x} \cdot y + \frac{y}{e^{x}}}{z}}{2}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{x} \cdot y + \frac{y}{e^{x}}}{2}}{x \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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))