Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.9 → 0.4

Time: 4.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -3.87412231438610406 \cdot 10^{56} \lor \neg \left(z \le 2033295.47898289864\right):\\ \;\;\;\;\left(\cosh x \cdot y\right) \cdot \frac{1}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double VAR;
	if (((z <= -3.874122314386104e+56) || !(z <= 2033295.4789828986))) {
		VAR = ((double) (((double) (((double) cosh(x)) * y)) * ((double) (1.0 / ((double) (x * z))))));
	} else {
		VAR = ((double) (((double) cosh(x)) * ((double) (((double) (y / z)) / x))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.9
Target	0.4
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 2 regimes

2. if z < -3.87412231438610406e56 or 2033295.47898289864 < z

   1. Initial program 12.9

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

   3. Applied *-un-lft-identity 12.9

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

   4. Applied times-frac 12.9

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

   5. Simplified 12.9

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

   6. Simplified 0.3

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

   8. Applied div-inv 0.4

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

   9. Applied associate-*r* 0.4

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

   if -3.87412231438610406e56 < z < 2033295.47898289864

   1. Initial program 0.8

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

   3. Applied *-un-lft-identity 0.8

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

   4. Applied times-frac 0.8

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

   5. Simplified 0.8

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

   6. Simplified 17.4

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

   8. Applied *-un-lft-identity 17.4

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

   9. Applied times-frac 0.6

      \[\leadsto \cosh x \cdot \color{blue}{\left(\frac{1}{x} \cdot \frac{y}{z}\right)}\]
   10. Using strategy rm

   11. Applied associate-*l/ 0.5

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

   12. Simplified 0.5

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.87412231438610406 \cdot 10^{56} \lor \neg \left(z \le 2033295.47898289864\right):\\ \;\;\;\;\left(\cosh x \cdot y\right) \cdot \frac{1}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020174 
(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))