Average Error: 7.5 → 0.4
Time: 16.6s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3780519785934131560000 \lor \neg \left(z \le 2.47177184411059884 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{\left(\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)\right) \cdot y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3780519785934131560000 \lor \neg \left(z \le 2.47177184411059884 \cdot 10^{-52}\right):\\
\;\;\;\;\frac{\left(\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)\right) \cdot y}{z \cdot x}\\

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

\end{array}
double f(double x, double y, double z) {
        double r382938 = x;
        double r382939 = cosh(r382938);
        double r382940 = y;
        double r382941 = r382940 / r382938;
        double r382942 = r382939 * r382941;
        double r382943 = z;
        double r382944 = r382942 / r382943;
        return r382944;
}
↓
double f(double x, double y, double z) {
        double r382945 = z;
        double r382946 = -3.7805197859341316e+21;
        bool r382947 = r382945 <= r382946;
        double r382948 = 2.471771844110599e-52;
        bool r382949 = r382945 <= r382948;
        double r382950 = !r382949;
        bool r382951 = r382947 || r382950;
        double r382952 = 0.5;
        double r382953 = x;
        double r382954 = -r382953;
        double r382955 = exp(r382954);
        double r382956 = exp(r382953);
        double r382957 = r382955 + r382956;
        double r382958 = r382952 * r382957;
        double r382959 = y;
        double r382960 = r382958 * r382959;
        double r382961 = r382945 * r382953;
        double r382962 = r382960 / r382961;
        double r382963 = cosh(r382953);
        double r382964 = r382963 * r382959;
        double r382965 = r382964 / r382945;
        double r382966 = r382965 / r382953;
        double r382967 = r382951 ? r382962 : r382966;
        return r382967;
}

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
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.03853053593515302 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.7805197859341316e+21 or 2.471771844110599e-52 < z

    1. Initial program 11.0

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

    3. Applied associate-*r/11.0

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

    4. Applied associate-/l/0.4

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

    6. Applied clear-num0.8

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

    8. Applied *-un-lft-identity0.8

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

    9. Applied add-cube-cbrt0.8

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot \frac{z \cdot x}{\cosh x \cdot y}}\]

    10. Applied times-frac0.8

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{\frac{z \cdot x}{\cosh x \cdot y}}}\]

    11. Simplified0.8

      \[\leadsto \color{blue}{1} \cdot \frac{\sqrt[3]{1}}{\frac{z \cdot x}{\cosh x \cdot y}}\]

    12. Simplified10.1

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

    13. Taylor expanded around inf 0.4

      \[\leadsto 1 \cdot \color{blue}{\frac{y \cdot \left(\frac{1}{2} \cdot e^{x} + \frac{1}{2} \cdot e^{-x}\right)}{x \cdot z}}\]

    14. Simplified0.4

      \[\leadsto 1 \cdot \color{blue}{\frac{\left(\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)\right) \cdot y}{z \cdot x}}\]

    if -3.7805197859341316e+21 < z < 2.471771844110599e-52

    1. Initial program 0.4

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

    3. Applied associate-*r/0.4

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

    4. Applied associate-/l/20.7

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

    6. Applied associate-/r*0.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3780519785934131560000 \lor \neg \left(z \le 2.47177184411059884 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{\left(\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)\right) \cdot y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 +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))