Average Error: 7.7 → 0.3
Time: 4.3s
Precision: binary64
\[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -5.365992924946917 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{2}}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_0 := y \cdot \cosh x\\ \mathbf{if}\;z \leq 149749414749544.78:\\ \;\;\;\;\frac{\frac{t_0}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{\frac{1}{z}}{x}\\ \end{array}\\ \end{array} \]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -5.365992924946917 \cdot 10^{-16}:\\
\;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{2}}{z \cdot x}\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_0 := y \cdot \cosh x\\
\mathbf{if}\;z \leq 149749414749544.78:\\
\;\;\;\;\frac{\frac{t_0}{z}}{x}\\

\mathbf{else}:\\
\;\;\;\;t_0 \cdot \frac{\frac{1}{z}}{x}\\


\end{array}\\


\end{array}

(FPCore (x y z) :precision binary64 (/ (* (cosh x) (/ y x)) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= z -5.365992924946917e-16)
   (/ (/ (* (+ (exp x) (exp (- x))) y) 2.0) (* z x))
   (let* ((t_0 (* y (cosh x))))
     (if (<= z 149749414749544.78) (/ (/ t_0 z) x) (* t_0 (/ (/ 1.0 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 <= -5.365992924946917e-16) {
		tmp = (((exp(x) + exp(-x)) * y) / 2.0) / (z * x);
	} else {
		double t_0 = y * cosh(x);
		double tmp_1;
		if (z <= 149749414749544.78) {
			tmp_1 = (t_0 / z) / x;
		} else {
			tmp_1 = t_0 * ((1.0 / z) / x);
		}
		tmp = tmp_1;
	}
	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
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y < -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y < 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if z < -5.36599292494691668e-16

    1. Initial program 11.3

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

    2. Applied associate-*r/_binary6411.3

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

    3. Applied associate-/l/_binary640.3

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

    4. Applied cosh-def_binary640.3

      \[\leadsto \frac{\color{blue}{\frac{e^{x} + e^{-x}}{2}} \cdot y}{z \cdot x} \]

    5. Applied associate-*l/_binary640.4

      \[\leadsto \frac{\color{blue}{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{2}}}{z \cdot x} \]

    if -5.36599292494691668e-16 < z < 149749414749544.781

    1. Initial program 0.3

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

    2. Applied associate-*r/_binary640.3

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

    3. Applied associate-/l/_binary6419.3

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

    4. Applied associate-/r*_binary640.3

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

    if 149749414749544.781 < z

    1. Initial program 12.2

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

    2. Applied associate-*r/_binary6412.2

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

    3. Applied associate-/l/_binary640.3

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

    4. Applied div-inv_binary640.4

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

    5. Applied associate-/r*_binary640.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.365992924946917 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{2}}{z \cdot x}\\ \mathbf{elif}\;z \leq 149749414749544.78:\\ \;\;\;\;\frac{\frac{y \cdot \cosh x}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \cosh x\right) \cdot \frac{\frac{1}{z}}{x}\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(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))