Average Error: 7.6 → 0.3
Time: 4.4s
Precision: binary64
\[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -40841048426.870995:\\ \;\;\;\;\cosh x \cdot \frac{y}{z \cdot x}\\ \mathbf{elif}\;z \leq 7.323266843150271 \cdot 10^{-31}:\\ \;\;\;\;\begin{array}{l} t_0 := \sqrt{\cosh x}\\ \frac{t_0 \cdot \left(t_0 \cdot \frac{y}{x}\right)}{z} \end{array}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot \left(e^{x} + \frac{1}{e^{x}}\right)}{z \cdot x}\\ \end{array} \]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -40841048426.870995:\\
\;\;\;\;\cosh x \cdot \frac{y}{z \cdot x}\\

\mathbf{elif}\;z \leq 7.323266843150271 \cdot 10^{-31}:\\
\;\;\;\;\begin{array}{l}
t_0 := \sqrt{\cosh x}\\
\frac{t_0 \cdot \left(t_0 \cdot \frac{y}{x}\right)}{z}
\end{array}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \frac{y \cdot \left(e^{x} + \frac{1}{e^{x}}\right)}{z \cdot x}\\


\end{array}

(FPCore (x y z) :precision binary64 (/ (* (cosh x) (/ y x)) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= z -40841048426.870995)
   (* (cosh x) (/ y (* z x)))
   (if (<= z 7.323266843150271e-31)
     (let* ((t_0 (sqrt (cosh x)))) (/ (* t_0 (* t_0 (/ y x))) z))
     (* 0.5 (/ (* y (+ (exp x) (/ 1.0 (exp x)))) (* 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 (z <= -40841048426.870995) {
		tmp = cosh(x) * (y / (z * x));
	} else if (z <= 7.323266843150271e-31) {
		double t_0 = sqrt(cosh(x));
		tmp = (t_0 * (t_0 * (y / x))) / z;
	} else {
		tmp = 0.5 * ((y * (exp(x) + (1.0 / exp(x)))) / (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
Target0.4
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y < -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y < 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if z < -40841048426.8709946

    1. Initial program 12.1

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

    2. Applied *-un-lft-identity_binary6412.1

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

    3. Applied times-frac_binary6412.1

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

    4. Simplified12.1

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

    5. Simplified0.3

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

    if -40841048426.8709946 < z < 7.3232668431502707e-31

    1. Initial program 0.3

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

    2. Applied add-sqr-sqrt_binary640.3

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

    3. Applied associate-*l*_binary640.3

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

    4. Simplified0.3

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

    if 7.3232668431502707e-31 < z

    1. Initial program 10.6

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

    2. Taylor expanded in y around inf 0.4

      \[\leadsto \color{blue}{0.5 \cdot \frac{y \cdot \left(\frac{1}{e^{x}} + e^{x}\right)}{z \cdot x}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -40841048426.870995:\\ \;\;\;\;\cosh x \cdot \frac{y}{z \cdot x}\\ \mathbf{elif}\;z \leq 7.323266843150271 \cdot 10^{-31}:\\ \;\;\;\;\frac{\sqrt{\cosh x} \cdot \left(\sqrt{\cosh x} \cdot \frac{y}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot \left(e^{x} + \frac{1}{e^{x}}\right)}{z \cdot x}\\ \end{array} \]

Reproduce

herbie shell --seed 2021275 
(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))