Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.9 → 0.3

Time: 3.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq -7.953933199296236 \cdot 10^{+270}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 5.100297545312038 \cdot 10^{+304}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \end{array}\]

C

double code(double x, double y, double z) {
	return (((double) (((double) cosh(x)) * (y / x))) / z);
}

↓

double code(double x, double y, double z) {
	double VAR;
	if (((((double) (((double) cosh(x)) * (y / x))) / z) <= -7.953933199296236e+270)) {
		VAR = ((double) ((((double) cosh(x)) / x) * (y / z)));
	} else {
		double VAR_1;
		if (((((double) (((double) cosh(x)) * (y / x))) / z) <= 5.100297545312038e+304)) {
			VAR_1 = (((double) (((double) cosh(x)) * (y / x))) / z);
		} else {
			VAR_1 = ((double) (((double) cosh(x)) * (y / ((double) (x * z)))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.9
Target	0.3
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if (/ (* (cosh x) (/ y x)) z) < -7.953933199296236e270

   1. Initial program 46.4

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

   2. Simplified 6.9

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

   4. Applied *-un-lft-identity 6.9

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

   5. Applied times-frac 0.3

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

   6. Applied associate-*r* 0.3

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

   7. Simplified 0.3

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

   if -7.953933199296236e270 < (/ (* (cosh x) (/ y x)) z) < 5.1002975453120378e304

   1. Initial program 0.2

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

   if 5.1002975453120378e304 < (/ (* (cosh x) (/ y x)) z)

   1. Initial program 62.7

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

   2. Simplified 1.0

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq -7.953933199296236 \cdot 10^{+270}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 5.100297545312038 \cdot 10^{+304}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020199 
(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))