Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.8 → 0.6

Time: 4.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \le -1.487036224053662 \cdot 10^{-309} \lor \neg \left(\frac{x \cdot \frac{\sin y}{y}}{z} \le 2.1227229608366263 \cdot 10^{-242}\right):\\ \;\;\;\;\frac{x \cdot \left(\sin y \cdot \frac{1}{y}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\frac{\sin y}{y}}{\sqrt[3]{z}}\\ \end{array}\]

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) (((double) (x * ((double) (((double) sin(y)) / y)))) / z)) <= -1.48703622405366e-309) || !(((double) (((double) (x * ((double) (((double) sin(y)) / y)))) / z)) <= 2.1227229608366263e-242))) {
		VAR = ((double) (((double) (x * ((double) (((double) sin(y)) * ((double) (1.0 / y)))))) / z));
	} else {
		VAR = ((double) (((double) (x / ((double) (((double) cbrt(z)) * ((double) cbrt(z)))))) * ((double) (((double) (((double) sin(y)) / y)) / ((double) cbrt(z))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	2.8
Target	0.3
Herbie	0.6

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

Derivation

1. Split input into 2 regimes

2. if (/ (* x (/ (sin y) y)) z) < -1.487036224053662e-309 or 2.1227229608366263e-242 < (/ (* x (/ (sin y) y)) z)

   1. Initial program 0.4

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

   3. Applied div-inv 0.5

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

   if -1.487036224053662e-309 < (/ (* x (/ (sin y) y)) z) < 2.1227229608366263e-242

   1. Initial program 8.6

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

   3. Applied add-cube-cbrt 8.7

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

   4. Applied times-frac 0.6

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \le -1.487036224053662 \cdot 10^{-309} \lor \neg \left(\frac{x \cdot \frac{\sin y}{y}}{z} \le 2.1227229608366263 \cdot 10^{-242}\right):\\ \;\;\;\;\frac{x \cdot \left(\sin y \cdot \frac{1}{y}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\frac{\sin y}{y}}{\sqrt[3]{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020173 
(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))