Average Error: 7.6 → 0.9
Time: 4.0s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.4618237390497889 \cdot 10^{62} \lor \neg \left(z \le 6.97590778484872185 \cdot 10^{25}\right):\\ \;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{-1 \cdot x} + e^{x}}{2} \cdot \frac{y}{x}}{z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.4618237390497889 \cdot 10^{62} \lor \neg \left(z \le 6.97590778484872185 \cdot 10^{25}\right):\\
\;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{e^{-1 \cdot x} + e^{x}}{2} \cdot \frac{y}{x}}{z}\\

\end{array}
double f(double x, double y, double z) {
        double r502926 = x;
        double r502927 = cosh(r502926);
        double r502928 = y;
        double r502929 = r502928 / r502926;
        double r502930 = r502927 * r502929;
        double r502931 = z;
        double r502932 = r502930 / r502931;
        return r502932;
}
↓
double f(double x, double y, double z) {
        double r502933 = z;
        double r502934 = -1.461823739049789e+62;
        bool r502935 = r502933 <= r502934;
        double r502936 = 6.975907784848722e+25;
        bool r502937 = r502933 <= r502936;
        double r502938 = !r502937;
        bool r502939 = r502935 || r502938;
        double r502940 = 0.5;
        double r502941 = x;
        double r502942 = y;
        double r502943 = r502941 * r502942;
        double r502944 = r502943 / r502933;
        double r502945 = r502940 * r502944;
        double r502946 = r502941 * r502933;
        double r502947 = r502942 / r502946;
        double r502948 = r502945 + r502947;
        double r502949 = -1.0;
        double r502950 = r502949 * r502941;
        double r502951 = exp(r502950);
        double r502952 = exp(r502941);
        double r502953 = r502951 + r502952;
        double r502954 = 2.0;
        double r502955 = r502953 / r502954;
        double r502956 = r502942 / r502941;
        double r502957 = r502955 * r502956;
        double r502958 = r502957 / r502933;
        double r502959 = r502939 ? r502948 : r502958;
        return r502959;
}

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

Original7.6
Target0.4
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.461823739049789e+62 or 6.975907784848722e+25 < z

    1. Initial program 12.7

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

    2. Taylor expanded around 0 1.0

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

    if -1.461823739049789e+62 < z < 6.975907784848722e+25

    1. Initial program 0.9

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

    3. Applied cosh-def0.9

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

    4. Simplified0.9

      \[\leadsto \frac{\frac{\color{blue}{e^{-1 \cdot x} + e^{x}}}{2} \cdot \frac{y}{x}}{z}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.4618237390497889 \cdot 10^{62} \lor \neg \left(z \le 6.97590778484872185 \cdot 10^{25}\right):\\ \;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{-1 \cdot x} + e^{x}}{2} \cdot \frac{y}{x}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(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))