Average Error: 8.1 → 0.6
Time: 17.6s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -33600391477343555682304:\\ \;\;\;\;\frac{\frac{1}{x}}{z} \cdot \left(\cosh x \cdot y\right)\\ \mathbf{elif}\;z \le 8.053275837647428032378095404484547122 \cdot 10^{-71}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(e^{-x} + e^{x}\right)}{z \cdot \left(2 \cdot x\right)}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -33600391477343555682304:\\
\;\;\;\;\frac{\frac{1}{x}}{z} \cdot \left(\cosh x \cdot y\right)\\

\mathbf{elif}\;z \le 8.053275837647428032378095404484547122 \cdot 10^{-71}:\\
\;\;\;\;\frac{\cosh x \cdot \frac{y}{z}}{x}\\

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

\end{array}
double f(double x, double y, double z) {
        double r21235096 = x;
        double r21235097 = cosh(r21235096);
        double r21235098 = y;
        double r21235099 = r21235098 / r21235096;
        double r21235100 = r21235097 * r21235099;
        double r21235101 = z;
        double r21235102 = r21235100 / r21235101;
        return r21235102;
}
↓
double f(double x, double y, double z) {
        double r21235103 = z;
        double r21235104 = -3.3600391477343556e+22;
        bool r21235105 = r21235103 <= r21235104;
        double r21235106 = 1.0;
        double r21235107 = x;
        double r21235108 = r21235106 / r21235107;
        double r21235109 = r21235108 / r21235103;
        double r21235110 = cosh(r21235107);
        double r21235111 = y;
        double r21235112 = r21235110 * r21235111;
        double r21235113 = r21235109 * r21235112;
        double r21235114 = 8.053275837647428e-71;
        bool r21235115 = r21235103 <= r21235114;
        double r21235116 = r21235111 / r21235103;
        double r21235117 = r21235110 * r21235116;
        double r21235118 = r21235117 / r21235107;
        double r21235119 = -r21235107;
        double r21235120 = exp(r21235119);
        double r21235121 = exp(r21235107);
        double r21235122 = r21235120 + r21235121;
        double r21235123 = r21235111 * r21235122;
        double r21235124 = 2.0;
        double r21235125 = r21235124 * r21235107;
        double r21235126 = r21235103 * r21235125;
        double r21235127 = r21235123 / r21235126;
        double r21235128 = r21235115 ? r21235118 : r21235127;
        double r21235129 = r21235105 ? r21235113 : r21235128;
        return r21235129;
}

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

Original8.1
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 3 regimes

  2. if z < -3.3600391477343556e+22

    1. Initial program 13.0

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

    3. Applied div-inv13.1

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

    4. Applied associate-*r*13.1

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

    6. Applied *-un-lft-identity13.1

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

    7. Applied times-frac0.5

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

    8. Simplified0.5

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

    if -3.3600391477343556e+22 < z < 8.053275837647428e-71

    1. Initial program 0.3

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

    3. Applied div-inv0.5

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

    5. Applied associate-*r/0.5

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

    6. Applied associate-*l/0.5

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

    7. Simplified0.4

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

    if 8.053275837647428e-71 < z

    1. Initial program 10.8

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

    3. Applied cosh-def10.8

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

    4. Applied frac-times10.8

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

    5. Applied associate-/l/0.8

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -33600391477343555682304:\\ \;\;\;\;\frac{\frac{1}{x}}{z} \cdot \left(\cosh x \cdot y\right)\\ \mathbf{elif}\;z \le 8.053275837647428032378095404484547122 \cdot 10^{-71}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(e^{-x} + e^{x}\right)}{z \cdot \left(2 \cdot x\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(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))