Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.8 → 0.4

Time: 3.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1066615151038.62585:\\ \;\;\;\;\frac{\cosh x \cdot \left(-y\right)}{z \cdot \left(-x\right)}\\ \mathbf{elif}\;y \le 2.7108396115711274 \cdot 10^{26}:\\ \;\;\;\;\frac{\frac{y}{x}}{\frac{z}{\cosh x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{y \cdot \cosh x}{z}}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double VAR;
	if ((y <= -1066615151038.6259)) {
		VAR = ((cosh(x) * -y) / (z * -x));
	} else {
		double VAR_1;
		if ((y <= 2.7108396115711274e+26)) {
			VAR_1 = ((y / x) / (z / cosh(x)));
		} else {
			VAR_1 = (1.0 / (x / ((y * cosh(x)) / z)));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.8
Target	0.5
Herbie	0.4

\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if y < -1066615151038.6259

   1. Initial program 22.7

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

   3. Applied frac-2neg 22.7

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

   4. Applied associate-*r/ 22.7

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

   5. Applied associate-/l/ 0.3

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

   if -1066615151038.6259 < y < 2.7108396115711274e+26

   1. Initial program 0.4

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

   3. Applied frac-2neg 0.4

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

   4. Applied associate-*r/ 0.4

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

   5. Applied associate-/l/ 9.8

      \[\leadsto \color{blue}{\frac{\cosh x \cdot \left(-y\right)}{z \cdot \left(-x\right)}}\]
   6. Using strategy rm

   7. Applied clear-num 10.1

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

   8. Simplified 9.9

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

   10. Applied *-commutative 9.9

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

   12. Applied associate-/l* 9.9

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

   13. Applied associate-/r/ 0.7

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

   14. Applied associate-/r* 0.5

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

   15. Simplified 0.4

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

   if 2.7108396115711274e+26 < y

   1. Initial program 25.6

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

   3. Applied frac-2neg 25.6

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

   4. Applied associate-*r/ 25.6

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

   5. Applied associate-/l/ 0.3

      \[\leadsto \color{blue}{\frac{\cosh x \cdot \left(-y\right)}{z \cdot \left(-x\right)}}\]
   6. Using strategy rm

   7. Applied clear-num 0.4

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

   8. Simplified 0.3

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

   10. Applied *-commutative 0.3

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1066615151038.62585:\\ \;\;\;\;\frac{\cosh x \cdot \left(-y\right)}{z \cdot \left(-x\right)}\\ \mathbf{elif}\;y \le 2.7108396115711274 \cdot 10^{26}:\\ \;\;\;\;\frac{\frac{y}{x}}{\frac{z}{\cosh x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{y \cdot \cosh x}{z}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< y -4.618902267687042e-52) (* (/ (/ y z) x) (cosh x)) (if (< y 1.0385305359351529e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

  (/ (* (cosh x) (/ y x)) z))