Average Error: 7.6 → 0.7
Time: 5.5s
Precision: binary64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -4.651198953700106 \cdot 10^{+87}:\\ \;\;\;\;\cosh x \cdot \left(y \cdot \frac{1}{x \cdot z}\right)\\ \mathbf{elif}\;y \leq 6.083274916861838 \cdot 10^{-33}:\\ \;\;\;\;\frac{0.5 \cdot \frac{y \cdot \left(e^{x} + e^{-x}\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{z}}{x}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -4.651198953700106 \cdot 10^{+87}:\\
\;\;\;\;\cosh x \cdot \left(y \cdot \frac{1}{x \cdot z}\right)\\

\mathbf{elif}\;y \leq 6.083274916861838 \cdot 10^{-33}:\\
\;\;\;\;\frac{0.5 \cdot \frac{y \cdot \left(e^{x} + e^{-x}\right)}{x}}{z}\\

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

\end{array}
(FPCore (x y z) :precision binary64 (/ (* (cosh x) (/ y x)) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= y -4.651198953700106e+87)
   (* (cosh x) (* y (/ 1.0 (* x z))))
   (if (<= y 6.083274916861838e-33)
     (/ (* 0.5 (/ (* y (+ (exp x) (exp (- x)))) x)) z)
     (/ (* (cosh x) (/ y z)) x))))
double code(double x, double y, double z) {
	return (cosh(x) * (y / x)) / z;
}
↓
double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.651198953700106e+87) {
		tmp = cosh(x) * (y * (1.0 / (x * z)));
	} else if (y <= 6.083274916861838e-33) {
		tmp = (0.5 * ((y * (exp(x) + exp(-x))) / x)) / z;
	} else {
		tmp = (cosh(x) * (y / z)) / x;
	}
	return tmp;
}

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

Derivation

  1. Split input into 3 regimes

  2. if y < -4.65119895370010597e87

    1. Initial program 28.7

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

    3. Applied *-un-lft-identity_binary6428.7

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

    4. Applied times-frac_binary6428.7

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

    5. Simplified28.7

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

    6. Simplified0.3

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

    8. Applied div-inv_binary640.4

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

    if -4.65119895370010597e87 < y < 6.0832749168618383e-33

    1. Initial program 0.8

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

    2. Taylor expanded around inf 0.9

      \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{\left(e^{x} + e^{-x}\right) \cdot y}{x}}}{z}\]

    if 6.0832749168618383e-33 < y

    1. Initial program 18.9

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

    3. Applied *-un-lft-identity_binary6418.9

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

    4. Applied times-frac_binary6418.8

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

    5. Simplified18.8

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

    6. Simplified0.5

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

    8. Applied *-un-lft-identity_binary640.5

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

    9. Applied times-frac_binary640.5

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

    10. Applied associate-*r*_binary640.5

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

    11. Simplified0.5

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

    13. Applied associate-*l/_binary640.4

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.651198953700106 \cdot 10^{+87}:\\ \;\;\;\;\cosh x \cdot \left(y \cdot \frac{1}{x \cdot z}\right)\\ \mathbf{elif}\;y \leq 6.083274916861838 \cdot 10^{-33}:\\ \;\;\;\;\frac{0.5 \cdot \frac{y \cdot \left(e^{x} + e^{-x}\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{z}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2021155 
(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))