Average Error: 7.5 → 0.4
Time: 4.2s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.53999665919217589 \cdot 10^{-39}:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{elif}\;z \le 1.24619194950009797 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{z}}{x}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.53999665919217589 \cdot 10^{-39}:\\
\;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\

\mathbf{elif}\;z \le 1.24619194950009797 \cdot 10^{-4}:\\
\;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\

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

\end{array}
double code(double x, double y, double z) {
	return ((cosh(x) * (y / x)) / z);
}
↓
double code(double x, double y, double z) {
	double VAR;
	if ((z <= -1.539996659192176e-39)) {
		VAR = (cosh(x) * (y / (x * z)));
	} else {
		double VAR_1;
		if ((z <= 0.0001246191949500098)) {
			VAR_1 = (((cosh(x) * y) / z) / x);
		} else {
			VAR_1 = (y * ((cosh(x) / z) / x));
		}
		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

Original7.5
Target0.5
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.539996659192176e-39

    1. Initial program 10.2

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

    3. Applied *-un-lft-identity10.2

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

    4. Applied times-frac10.2

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

    5. Simplified10.2

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

    6. Simplified0.4

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

    if -1.539996659192176e-39 < z < 0.0001246191949500098

    1. Initial program 0.3

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

    3. Applied div-inv0.4

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

    5. Applied associate-*r/0.4

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

    6. Applied associate-*l/0.4

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

    7. Simplified0.3

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

    if 0.0001246191949500098 < z

    1. Initial program 11.5

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

    3. Applied associate-/l*11.9

      \[\leadsto \color{blue}{\frac{\cosh x}{\frac{z}{\frac{y}{x}}}}\]
    4. Using strategy rm

    5. Applied div-inv11.9

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

    6. Applied *-un-lft-identity11.9

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

    7. Applied times-frac0.8

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

    8. Applied *-un-lft-identity0.8

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

    9. Applied times-frac0.6

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

    10. Simplified0.5

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

    11. Simplified0.4

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.53999665919217589 \cdot 10^{-39}:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{elif}\;z \le 1.24619194950009797 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{z}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020092 +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))