Average Error: 7.8 → 1.0
Time: 21.3s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -712742474614758 \lor \neg \left(z \le 3.970994457643005255735446389224709150426 \cdot 10^{112}\right):\\ \;\;\;\;\frac{e^{x} + e^{-x}}{z \cdot \left(2 \cdot x\right)} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -712742474614758 \lor \neg \left(z \le 3.970994457643005255735446389224709150426 \cdot 10^{112}\right):\\
\;\;\;\;\frac{e^{x} + e^{-x}}{z \cdot \left(2 \cdot x\right)} \cdot y\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}\\

\end{array}
double f(double x, double y, double z) {
        double r338922 = x;
        double r338923 = cosh(r338922);
        double r338924 = y;
        double r338925 = r338924 / r338922;
        double r338926 = r338923 * r338925;
        double r338927 = z;
        double r338928 = r338926 / r338927;
        return r338928;
}
↓
double f(double x, double y, double z) {
        double r338929 = z;
        double r338930 = -712742474614758.0;
        bool r338931 = r338929 <= r338930;
        double r338932 = 3.970994457643005e+112;
        bool r338933 = r338929 <= r338932;
        double r338934 = !r338933;
        bool r338935 = r338931 || r338934;
        double r338936 = x;
        double r338937 = exp(r338936);
        double r338938 = -r338936;
        double r338939 = exp(r338938);
        double r338940 = r338937 + r338939;
        double r338941 = 2.0;
        double r338942 = r338941 * r338936;
        double r338943 = r338929 * r338942;
        double r338944 = r338940 / r338943;
        double r338945 = y;
        double r338946 = r338944 * r338945;
        double r338947 = r338940 * r338945;
        double r338948 = r338947 / r338929;
        double r338949 = r338948 / r338942;
        double r338950 = r338935 ? r338946 : r338949;
        return r338950;
}

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

Original7.8
Target0.5
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -712742474614758.0 or 3.970994457643005e+112 < z

    1. Initial program 13.3

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

    3. Applied cosh-def13.3

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

    4. Applied frac-times13.3

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

    5. Applied associate-/l/0.4

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

    7. Applied associate-/l*0.8

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

    9. Applied associate-/r/0.4

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

    if -712742474614758.0 < z < 3.970994457643005e+112

    1. Initial program 1.8

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

    3. Applied cosh-def1.8

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

    4. Applied frac-times1.8

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

    5. Applied associate-/l/14.0

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

    7. Applied associate-/r*1.6

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -712742474614758 \lor \neg \left(z \le 3.970994457643005255735446389224709150426 \cdot 10^{112}\right):\\ \;\;\;\;\frac{e^{x} + e^{-x}}{z \cdot \left(2 \cdot x\right)} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +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 1.03853053593515302e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

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