Average Error: 7.9 → 0.8
Time: 4.8s
Precision: binary64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.57810435137455553 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{\frac{z \cdot x}{y}}\\ \mathbf{elif}\;z \le 3.57697453935832669 \cdot 10^{59}:\\ \;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{\frac{x}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{z \cdot x}}{\frac{1}{y}}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.57810435137455553 \cdot 10^{-57}:\\
\;\;\;\;\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{\frac{z \cdot x}{y}}\\

\mathbf{elif}\;z \le 3.57697453935832669 \cdot 10^{59}:\\
\;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{\frac{x}{y}}}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{z \cdot x}}{\frac{1}{y}}\\

\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 <= -1.5781043513745555e-57)) {
		VAR = ((double) (((double) (0.5 * ((double) (((double) exp(((double) (-1.0 * x)))) + ((double) exp(x)))))) / ((double) (((double) (z * x)) / y))));
	} else {
		double VAR_1;
		if ((z <= 3.5769745393583267e+59)) {
			VAR_1 = ((double) (((double) (((double) (0.5 * ((double) (((double) exp(((double) (-1.0 * x)))) + ((double) exp(x)))))) / ((double) (x / y)))) / z));
		} else {
			VAR_1 = ((double) (((double) (((double) (0.5 * ((double) (((double) exp(((double) (-1.0 * x)))) + ((double) exp(x)))))) / ((double) (z * x)))) / ((double) (1.0 / y))));
		}
		VAR = VAR_1;
	}
	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

Original7.9
Target0.4
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.57810435137455553e-57

    1. Initial program 10.3

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

    2. Taylor expanded around inf 0.6

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

    3. Simplified0.9

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

    if -1.57810435137455553e-57 < z < 3.57697453935832669e59

    1. Initial program 0.9

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

    2. Taylor expanded around inf 0.9

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

    3. Simplified1.0

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

    if 3.57697453935832669e59 < z

    1. Initial program 14.0

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.7

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

    5. Applied div-inv0.7

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

    6. Applied associate-/r*0.4

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{z \cdot x}}{\frac{1}{y}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.57810435137455553 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{\frac{z \cdot x}{y}}\\ \mathbf{elif}\;z \le 3.57697453935832669 \cdot 10^{59}:\\ \;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{\frac{x}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{z \cdot x}}{\frac{1}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020147 
(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))