Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.7 → 0.8

Time: 3.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} = -inf.0 \lor \neg \left(\cosh x \cdot \frac{y}{x} \le 3.7080047232710205 \cdot 10^{110}\right):\\ \;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double VAR;
	if (((((double) (((double) cosh(x)) * ((double) (y / x)))) <= -inf.0) || !(((double) (((double) cosh(x)) * ((double) (y / x)))) <= 3.7080047232710205e+110))) {
		VAR = ((double) (((double) (0.5 * ((double) (((double) (x * y)) / z)))) + ((double) (y / ((double) (x * z))))));
	} else {
		VAR = ((double) (((double) (((double) cosh(x)) * ((double) (y / x)))) / z));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.7
Target	0.4
Herbie	0.8

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

Derivation

1. Split input into 2 regimes

2. if (* (cosh x) (/ y x)) < -inf.0 or 3.7080047232710205e110 < (* (cosh x) (/ y x))

   1. Initial program 29.2

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

   2. Taylor expanded around 0 2.4

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}}\]

   if -inf.0 < (* (cosh x) (/ y x)) < 3.7080047232710205e110

   1. Initial program 0.2

      \[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} = -inf.0 \lor \neg \left(\cosh x \cdot \frac{y}{x} \le 3.7080047232710205 \cdot 10^{110}\right):\\ \;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020150 
(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))