Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.7 → 0.5

Time: 5.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -6.8061860642039018 \cdot 10^{-50} \lor \neg \left(z \le 8.6210839703151076 \cdot 10^{-91}\right):\\ \;\;\;\;\frac{1}{\frac{y}{\sin y}} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z \cdot \frac{y}{\sin y}}{x}}\\ \end{array}\]

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 (((z <= -6.806186064203902e-50) || !(z <= 8.621083970315108e-91))) {
		VAR = ((double) (((double) (1.0 / ((double) (y / ((double) sin(y)))))) * ((double) (x / z))));
	} else {
		VAR = ((double) (1.0 / ((double) (((double) (z * ((double) (y / ((double) sin(y)))))) / x))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	2.7
Target	0.2
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 z < -6.806186064203902e-50 or 8.621083970315108e-91 < z

   1. Initial program 0.4

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

   3. Applied *-un-lft-identity 0.4

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

   4. Applied associate-*r* 0.4

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

   5. Applied associate-/l* 4.1

      \[\leadsto \color{blue}{\frac{x \cdot 1}{\frac{z}{\frac{\sin y}{y}}}}\]

   6. Simplified 4.1

      \[\leadsto \frac{x \cdot 1}{\color{blue}{z \cdot \frac{y}{\sin y}}}\]
   7. Using strategy rm

   8. Applied clear-num 4.6

      \[\leadsto \color{blue}{\frac{1}{\frac{z \cdot \frac{y}{\sin y}}{x \cdot 1}}}\]

   9. Simplified 4.6

      \[\leadsto \frac{1}{\color{blue}{\frac{z \cdot \frac{y}{\sin y}}{x}}}\]
   10. Using strategy rm

   11. Applied *-un-lft-identity 4.6

       \[\leadsto \frac{1}{\frac{z \cdot \frac{y}{\sin y}}{\color{blue}{1 \cdot x}}}\]

   12. Applied *-commutative 4.6

       \[\leadsto \frac{1}{\frac{\color{blue}{\frac{y}{\sin y} \cdot z}}{1 \cdot x}}\]

   13. Applied times-frac 1.0

       \[\leadsto \frac{1}{\color{blue}{\frac{\frac{y}{\sin y}}{1} \cdot \frac{z}{x}}}\]

   14. Applied add-sqr-sqrt 1.0

       \[\leadsto \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\frac{\frac{y}{\sin y}}{1} \cdot \frac{z}{x}}\]

   15. Applied times-frac 0.8

       \[\leadsto \color{blue}{\frac{\sqrt{1}}{\frac{\frac{y}{\sin y}}{1}} \cdot \frac{\sqrt{1}}{\frac{z}{x}}}\]

   16. Simplified 0.8

       \[\leadsto \color{blue}{\frac{1}{\frac{y}{\sin y}}} \cdot \frac{\sqrt{1}}{\frac{z}{x}}\]

   17. Simplified 0.5

       \[\leadsto \frac{1}{\frac{y}{\sin y}} \cdot \color{blue}{\frac{x}{z}}\]

   if -6.806186064203902e-50 < z < 8.621083970315108e-91

   1. Initial program 7.3

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

   3. Applied *-un-lft-identity 7.3

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

   4. Applied associate-*r* 7.3

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

   5. Applied associate-/l* 0.2

      \[\leadsto \color{blue}{\frac{x \cdot 1}{\frac{z}{\frac{\sin y}{y}}}}\]

   6. Simplified 0.2

      \[\leadsto \frac{x \cdot 1}{\color{blue}{z \cdot \frac{y}{\sin y}}}\]
   7. Using strategy rm

   8. Applied clear-num 0.5

      \[\leadsto \color{blue}{\frac{1}{\frac{z \cdot \frac{y}{\sin y}}{x \cdot 1}}}\]

   9. Simplified 0.5

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.8061860642039018 \cdot 10^{-50} \lor \neg \left(z \le 8.6210839703151076 \cdot 10^{-91}\right):\\ \;\;\;\;\frac{1}{\frac{y}{\sin y}} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z \cdot \frac{y}{\sin y}}{x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(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 (/ y (sin y)))) z) (if (< z 4.446702369113811e+64) (/ x (* z (/ y (sin y)))) (/ (* x (/ 1 (/ y (sin y)))) z)))

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