Average Error: 7.8 → 0.5
Time: 7.2s
Precision: binary64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq -6.594132343947064 \cdot 10^{+229}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\cosh x}{x}\\ \mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 1.1306874589075905 \cdot 10^{+300}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq -6.594132343947064 \cdot 10^{+229}:\\
\;\;\;\;\frac{y}{z} \cdot \frac{\cosh x}{x}\\

\mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 1.1306874589075905 \cdot 10^{+300}:\\
\;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\

\end{array}
(FPCore (x y z) :precision binary64 (/ (* (cosh x) (/ y x)) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= (/ (* (cosh x) (/ y x)) z) -6.594132343947064e+229)
   (* (/ y z) (/ (cosh x) x))
   (if (<= (/ (* (cosh x) (/ y x)) z) 1.1306874589075905e+300)
     (/ (* (cosh x) (/ y x)) z)
     (+ (* 0.5 (/ (* x y) z)) (/ y (* x z))))))
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 (((cosh(x) * (y / x)) / z) <= -6.594132343947064e+229) {
		tmp = (y / z) * (cosh(x) / x);
	} else if (((cosh(x) * (y / x)) / z) <= 1.1306874589075905e+300) {
		tmp = (cosh(x) * (y / x)) / z;
	} else {
		tmp = (0.5 * ((x * y) / z)) + (y / (x * z));
	}
	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

Original7.8
Target0.4
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;y < -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y < 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < -6.5941323439470641e229

    1. Initial program 34.7

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

    3. Applied associate-/l*_binary64_1400434.5

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

    5. Applied associate-/r/_binary64_140050.4

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

    6. Applied *-un-lft-identity_binary64_140590.4

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

    7. Applied times-frac_binary64_140650.5

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

    8. Simplified0.4

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

    if -6.5941323439470641e229 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.13068745890759053e300

    1. Initial program 0.3

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

    if 1.13068745890759053e300 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

    1. Initial program 59.5

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

    2. Taylor expanded around 0 3.1

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq -6.594132343947064 \cdot 10^{+229}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\cosh x}{x}\\ \mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 1.1306874589075905 \cdot 10^{+300}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2021043 
(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))