Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 8.0 → 0.6

Time: 3.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -6.19754678860515829 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{\frac{z \cdot x}{y}}\\ \mathbf{elif}\;y \le 12219.624869563231:\\ \;\;\;\;\frac{\left(\sqrt[3]{\cosh x} \cdot \sqrt[3]{\cosh x}\right) \cdot \left(\sqrt[3]{\cosh x} \cdot \frac{y}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \end{array}\]

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 ((y <= -6.197546788605158e-22)) {
		VAR = ((double) (((double) (0.5 * ((double) (((double) exp(((double) (-1.0 * x)))) + ((double) exp(x)))))) / ((double) (((double) (z * x)) / y))));
	} else {
		double VAR_1;
		if ((y <= 12219.624869563231)) {
			VAR_1 = ((double) (((double) (((double) (((double) cbrt(((double) cosh(x)))) * ((double) cbrt(((double) cosh(x)))))) * ((double) (((double) cbrt(((double) cosh(x)))) * ((double) (y / x)))))) / z));
		} else {
			VAR_1 = ((double) (((double) (0.5 * ((double) (((double) (x * y)) / z)))) + ((double) (y / ((double) (x * z))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	8.0
Target	0.5
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if y < -6.197546788605158e-22

   1. Initial program 20.4

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

   2. Taylor expanded around inf 0.4

      \[\leadsto \color{blue}{\frac{y \cdot \left(\frac{1}{2} \cdot e^{x} + \frac{1}{2} \cdot e^{-x}\right)}{x \cdot z}}\]

   3. Simplified 0.4

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{\frac{z \cdot x}{y}}}\]

   if -6.197546788605158e-22 < y < 12219.624869563231

   1. Initial program 0.3

      \[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 0.3

      \[\leadsto \frac{\color{blue}{\left(\left(\sqrt[3]{\cosh x} \cdot \sqrt[3]{\cosh x}\right) \cdot \sqrt[3]{\cosh x}\right)} \cdot \frac{y}{x}}{z}\]

   4. Applied associate-*l* 0.3

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\cosh x} \cdot \sqrt[3]{\cosh x}\right) \cdot \left(\sqrt[3]{\cosh x} \cdot \frac{y}{x}\right)}}{z}\]

   if 12219.624869563231 < y

   1. Initial program 22.5

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

   2. Taylor expanded around 0 1.6

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -6.19754678860515829 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{\frac{z \cdot x}{y}}\\ \mathbf{elif}\;y \le 12219.624869563231:\\ \;\;\;\;\frac{\left(\sqrt[3]{\cosh x} \cdot \sqrt[3]{\cosh x}\right) \cdot \left(\sqrt[3]{\cosh x} \cdot \frac{y}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020120 
(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))