Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.6 → 0.2

Time: 7.2min

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -3.6736184683334172 \cdot 10^{-7} \lor \neg \left(x \le 1.0202326982250869 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{\frac{x \cdot \sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double VAR;
	if (((x <= -3.673618468333417e-07) || !(x <= 1.020232698225087e-16))) {
		VAR = ((((double) (x * ((double) sin(y)))) / y) / z);
	} else {
		VAR = (x / (z / (((double) sin(y)) / y)));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	2.6
Target	0.3
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if x < -3.6736184683334172e-7 or 1.0202326982250869e-16 < x

   1. Initial program 0.2

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

   3. Applied associate-*r/ 0.2

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

   if -3.6736184683334172e-7 < x < 1.0202326982250869e-16

   1. Initial program 4.8

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

   3. Applied associate-/l* 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.6736184683334172 \cdot 10^{-7} \lor \neg \left(x \le 1.0202326982250869 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{\frac{x \cdot \sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(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))