Average Error: 7.8 → 0.7
Time: 18.5s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.14904344650051927 \cdot 10^{47} \lor \neg \left(z \le 396504734138806976\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{x \cdot y}{z}, \frac{y}{x \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.14904344650051927 \cdot 10^{47} \lor \neg \left(z \le 396504734138806976\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{x \cdot y}{z}, \frac{y}{x \cdot z}\right)\\

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

\end{array}
double code(double x, double y, double z) {
	return ((double) (((double) (((double) cosh(x)) * ((double) (y / x)))) / z));
}
↓
double code(double x, double y, double z) {
	double VAR;
	if (((z <= -1.1490434465005193e+47) || !(z <= 3.96504734138807e+17))) {
		VAR = ((double) fma(0.5, ((double) (((double) (x * y)) / z)), ((double) (y / ((double) (x * z))))));
	} else {
		VAR = ((double) (((double) (((double) cosh(x)) * ((double) (y / x)))) / z));
	}
	return VAR;
}

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
Target0.4
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.1490434465005193e+47 or 3.96504734138807e+17 < z

    1. Initial program 12.9

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

    2. Taylor expanded around 0 0.8

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

    3. Simplified0.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{x \cdot y}{z}, \frac{y}{x \cdot z}\right)}\]

    if -1.1490434465005193e+47 < z < 3.96504734138807e+17

    1. Initial program 0.5

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.14904344650051927 \cdot 10^{47} \lor \neg \left(z \le 396504734138806976\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{x \cdot y}{z}, \frac{y}{x \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 +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 1.0385305359351529e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

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