Average Error: 7.6 → 0.8
Time: 4.9s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -8.70083882387234965 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \mathbf{elif}\;y \le 2.84159300253995619 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{\frac{x}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\cosh x \cdot y\right) \cdot \frac{\frac{1}{x}}{z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -8.70083882387234965 \cdot 10^{-14}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\

\mathbf{elif}\;y \le 2.84159300253995619 \cdot 10^{-56}:\\
\;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{\frac{x}{y}}}{z}\\

\mathbf{else}:\\
\;\;\;\;\left(\cosh x \cdot y\right) \cdot \frac{\frac{1}{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 ((y <= -8.70083882387235e-14)) {
		VAR = ((double) (((double) (0.5 * ((double) (((double) (x * y)) / z)))) + ((double) (y / ((double) (x * z))))));
	} else {
		double VAR_1;
		if ((y <= 2.841593002539956e-56)) {
			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) cosh(x)) * y)) * ((double) (((double) (1.0 / x)) / z))));
		}
		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

Original7.6
Target0.5
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 y < -8.70083882387235e-14

    1. Initial program 22.1

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

    2. Taylor expanded around 0 1.4

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

    if -8.70083882387235e-14 < y < 2.841593002539956e-56

    1. Initial program 0.3

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.4

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

    if 2.841593002539956e-56 < y

    1. Initial program 14.7

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

    3. Applied div-inv14.8

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

    4. Applied associate-*r*14.8

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

    6. Applied *-un-lft-identity14.8

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

    7. Applied times-frac1.2

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

    8. Simplified1.2

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -8.70083882387234965 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \mathbf{elif}\;y \le 2.84159300253995619 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{\frac{x}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\cosh x \cdot y\right) \cdot \frac{\frac{1}{x}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020122 
(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))