Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.9 → 0.3

Time: 3.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -2.29078077532164661 \cdot 10^{-29} \lor \neg \left(z \le 7.68353018758329476 \cdot 10^{-20}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{x}}{z}\\ \end{array}\]

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 (((z <= -2.2907807753216466e-29) || !(z <= 7.683530187583295e-20))) {
		VAR = ((double) (((double) cosh(x)) * ((double) (y / ((double) (z * x))))));
	} else {
		VAR = ((double) (((double) cosh(x)) * ((double) (((double) (y / x)) / z))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.9
Target	0.4
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if z < -2.29078077532164661e-29 or 7.68353018758329476e-20 < z

   1. Initial program 11.3

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

   2. Simplified 0.4

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

   if -2.29078077532164661e-29 < z < 7.68353018758329476e-20

   1. Initial program 0.3

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

   2. Simplified 21.4

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

   4. Applied associate-/r* 0.3

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.29078077532164661 \cdot 10^{-29} \lor \neg \left(z \le 7.68353018758329476 \cdot 10^{-20}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{x}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(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))