Average Error: 7.7 → 0.3
Time: 19.0s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -7446916366273207 \lor \neg \left(z \le 2.540645334512932305961620808823739516416 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{y \cdot \left(\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)\right)}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -7446916366273207 \lor \neg \left(z \le 2.540645334512932305961620808823739516416 \cdot 10^{-27}\right):\\
\;\;\;\;\frac{y \cdot \left(\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)\right)}{z \cdot x}\\

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

\end{array}
double f(double x, double y, double z) {
        double r301147 = x;
        double r301148 = cosh(r301147);
        double r301149 = y;
        double r301150 = r301149 / r301147;
        double r301151 = r301148 * r301150;
        double r301152 = z;
        double r301153 = r301151 / r301152;
        return r301153;
}
↓
double f(double x, double y, double z) {
        double r301154 = z;
        double r301155 = -7446916366273207.0;
        bool r301156 = r301154 <= r301155;
        double r301157 = 2.5406453345129323e-27;
        bool r301158 = r301154 <= r301157;
        double r301159 = !r301158;
        bool r301160 = r301156 || r301159;
        double r301161 = y;
        double r301162 = 0.5;
        double r301163 = x;
        double r301164 = exp(r301163);
        double r301165 = -r301163;
        double r301166 = exp(r301165);
        double r301167 = r301164 + r301166;
        double r301168 = r301162 * r301167;
        double r301169 = r301161 * r301168;
        double r301170 = r301154 * r301163;
        double r301171 = r301169 / r301170;
        double r301172 = cosh(r301163);
        double r301173 = r301161 / r301163;
        double r301174 = r301172 * r301173;
        double r301175 = r301174 / r301154;
        double r301176 = r301160 ? r301171 : r301175;
        return r301176;
}

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

Original7.7
Target0.4
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -7446916366273207.0 or 2.5406453345129323e-27 < z

    1. Initial program 11.4

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

    2. Taylor expanded around inf 0.4

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

    3. Simplified0.4

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

    if -7446916366273207.0 < z < 2.5406453345129323e-27

    1. Initial program 0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7446916366273207 \lor \neg \left(z \le 2.540645334512932305961620808823739516416 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{y \cdot \left(\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)\right)}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(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))