Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.7 → 0.3

Time: 7.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -3.305797093227607 \cdot 10^{-238}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{elif}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq 1.623749272138731 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array}\]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x (/ (sin y) y)) z))

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (* x (/ (sin y) y)) z) -3.305797093227607e-238)
   (/ (* x (/ (sin y) y)) z)
   (if (<= (/ (* x (/ (sin y) y)) z) 1.623749272138731e-26)
     (* (/ (sin y) y) (/ x z))
     (/ x (/ z (/ (sin y) y))))))

C

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

↓

double code(double x, double y, double z) {
	double tmp;
	if (((x * (sin(y) / y)) / z) <= -3.305797093227607e-238) {
		tmp = (x * (sin(y) / y)) / z;
	} else if (((x * (sin(y) / y)) / z) <= 1.623749272138731e-26) {
		tmp = (sin(y) / y) * (x / z);
	} else {
		tmp = x / (z / (sin(y) / y));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	2.7
Target	0.3
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;z < -4.2173720203427147 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -3.3057970932276072e-238

   1. Initial program 0.4

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

   if -3.3057970932276072e-238 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 1.6237492721387311e-26

   1. Initial program 5.3

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

   3. Applied associate-/l*_binary64 4.0

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}}\]
   4. Using strategy rm

   5. Applied associate-/r/_binary64 0.1

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

   if 1.6237492721387311e-26 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 0.2

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

   3. Applied associate-/l*_binary64 0.3

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -3.305797093227607 \cdot 10^{-238}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{elif}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq 1.623749272138731 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020268 
(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))