Average Error: 7.7 → 0.4
Time: 4.9s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.97809705527759387 \cdot 10^{-29}:\\ \;\;\;\;\frac{\cosh x \cdot y}{z \cdot x}\\ \mathbf{elif}\;z \le 5.38882957650543835 \cdot 10^{-20}:\\ \;\;\;\;\frac{\cosh x}{\frac{z}{y} \cdot x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{z}}{x}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.97809705527759387 \cdot 10^{-29}:\\
\;\;\;\;\frac{\cosh x \cdot y}{z \cdot x}\\

\mathbf{elif}\;z \le 5.38882957650543835 \cdot 10^{-20}:\\
\;\;\;\;\frac{\cosh x}{\frac{z}{y} \cdot 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 <= -3.978097055277594e-29)) {
		VAR = ((cosh(x) * y) / (z * x));
	} else {
		double VAR_1;
		if ((z <= 5.3888295765054384e-20)) {
			VAR_1 = (cosh(x) / ((z / y) * 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

Original7.7
Target0.4
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 < -3.978097055277594e-29

    1. Initial program 10.8

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

    3. Applied associate-*r/10.8

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

    4. Applied associate-/l/0.4

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

    if -3.978097055277594e-29 < z < 5.3888295765054384e-20

    1. Initial program 0.3

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

    3. Applied associate-/l*0.4

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

    5. Applied associate-/r/0.4

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

    if 5.3888295765054384e-20 < z

    1. Initial program 11.3

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

    3. Applied associate-/l*11.8

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

    5. Applied div-inv11.8

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

    6. Applied *-un-lft-identity11.8

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

    7. Applied times-frac0.9

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

    8. Applied *-un-lft-identity0.9

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

    9. Applied times-frac0.5

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.97809705527759387 \cdot 10^{-29}:\\ \;\;\;\;\frac{\cosh x \cdot y}{z \cdot x}\\ \mathbf{elif}\;z \le 5.38882957650543835 \cdot 10^{-20}:\\ \;\;\;\;\frac{\cosh x}{\frac{z}{y} \cdot x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{z}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020075 +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))