Average Error: 7.7 → 0.4
Time: 4.1s
Precision: binary64
\[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
↓
\[\begin{array}{l} t_0 := \frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{if}\;t_0 \leq -4.862282847412182 \cdot 10^{-23}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;t_0 \leq 1.4877837998677327 \cdot 10^{+304}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{\frac{x \cdot z}{y}}\\ \end{array} \]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
t_0 := \frac{\cosh x \cdot \frac{y}{x}}{z}\\
\mathbf{if}\;t_0 \leq -4.862282847412182 \cdot 10^{-23}:\\
\;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\

\mathbf{elif}\;t_0 \leq 1.4877837998677327 \cdot 10^{+304}:\\
\;\;\;\;\cosh x \cdot \frac{\frac{y}{x}}{z}\\

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


\end{array}

(FPCore (x y z) :precision binary64 (/ (* (cosh x) (/ y x)) z))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* (cosh x) (/ y x)) z)))
   (if (<= t_0 -4.862282847412182e-23)
     (* (cosh x) (/ (/ y z) x))
     (if (<= t_0 1.4877837998677327e+304)
       (* (cosh x) (/ (/ y x) z))
       (/ (cosh x) (/ (* x z) y))))))
double code(double x, double y, double z) {
	return (cosh(x) * (y / x)) / z;
}
↓
double code(double x, double y, double z) {
	double t_0 = (cosh(x) * (y / x)) / z;
	double tmp;
	if (t_0 <= -4.862282847412182e-23) {
		tmp = cosh(x) * ((y / z) / x);
	} else if (t_0 <= 1.4877837998677327e+304) {
		tmp = cosh(x) * ((y / x) / z);
	} else {
		tmp = cosh(x) / ((x * z) / y);
	}
	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

Original7.7
Target0.5
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y < -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y < 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < -4.8622828474121821e-23

    1. Initial program 12.8

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

    2. Applied *-un-lft-identity_binary6412.8

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

    3. Applied times-frac_binary6412.7

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

    4. Simplified12.7

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

    5. Simplified11.4

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

    6. Applied associate-/r*_binary640.4

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

    if -4.8622828474121821e-23 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.48778379986773266e304

    1. Initial program 0.2

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

    2. Applied *-un-lft-identity_binary640.2

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

    3. Applied times-frac_binary640.3

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

    if 1.48778379986773266e304 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

    1. Initial program 61.4

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

    2. Applied *-un-lft-identity_binary6461.4

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

    3. Applied times-frac_binary6461.4

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

    4. Simplified61.4

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

    5. Simplified1.5

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

    6. Applied clear-num_binary641.6

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

    7. Applied un-div-inv_binary641.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq -4.862282847412182 \cdot 10^{-23}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 1.4877837998677327 \cdot 10^{+304}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{\frac{x \cdot z}{y}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022067 
(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 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

  (/ (* (cosh x) (/ y x)) z))