Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.6 → 0.5

Time: 6.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \le -8.00989822851498711 \cdot 10^{-293} \lor \neg \left(x \cdot \frac{\sin y}{y} \le -0.0\right):\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \sin y}{z \cdot y}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double VAR;
	if (((((double) (x * ((double) (((double) sin(y)) / y)))) <= -8.009898228514987e-293) || !(((double) (x * ((double) (((double) sin(y)) / y)))) <= -0.0))) {
		VAR = ((double) (((double) (x * ((double) (((double) sin(y)) / y)))) / z));
	} else {
		VAR = ((double) (((double) (x * ((double) sin(y)))) / ((double) (z * y))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	2.6
Target	0.3
Herbie	0.5

\[\begin{array}{l} \mathbf{if}\;z \lt -4.21737202034271466 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z \lt 4.44670236911381103 \cdot 10^{64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (* x (/ (sin y) y)) < -8.009898228514987e-293 or -0.0 < (* x (/ (sin y) y))

   1. Initial program 0.3

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

   if -8.009898228514987e-293 < (* x (/ (sin y) y)) < -0.0

   1. Initial program 16.1

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

   3. Applied associate-*r/ 17.6

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

   4. Applied associate-/l/ 1.6

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \le -8.00989822851498711 \cdot 10^{-293} \lor \neg \left(x \cdot \frac{\sin y}{y} \le -0.0\right):\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \sin y}{z \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020148 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< z -4.2173720203427147e-29) (/ (* x (/ 1.0 (/ y (sin y)))) z) (if (< z 4.446702369113811e+64) (/ x (* z (/ y (sin y)))) (/ (* x (/ 1.0 (/ y (sin y)))) z)))

  (/ (* x (/ (sin y) y)) z))