Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.7 → 0.5

Time: 4.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -3.9969049673634803 \cdot 10^{173}:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{elif}\;\cosh x \cdot \frac{y}{x} \le 1.274519611088763 \cdot 10^{192}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \end{array}\]

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 (((cosh(x) * (y / x)) <= -3.9969049673634803e+173)) {
		VAR = (cosh(x) * (y / (x * z)));
	} else {
		double VAR_1;
		if (((cosh(x) * (y / x)) <= 1.274519611088763e+192)) {
			VAR_1 = ((cosh(x) * (y / x)) / z);
		} else {
			VAR_1 = ((cosh(x) / x) * (y / z));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.7
Target	0.5
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

2. if (* (cosh x) (/ y x)) < -3.9969049673634803e+173

   1. Initial program 24.7

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

   3. Applied *-un-lft-identity 24.7

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

   4. Applied times-frac 24.7

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

   5. Simplified 24.7

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

   6. Simplified 1.1

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

   if -3.9969049673634803e+173 < (* (cosh x) (/ y x)) < 1.274519611088763e+192

   1. Initial program 0.2

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

   if 1.274519611088763e+192 < (* (cosh x) (/ y x))

   1. Initial program 26.9

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

   3. Applied *-un-lft-identity 26.9

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

   4. Applied times-frac 26.8

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

   5. Simplified 26.8

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

   6. Simplified 1.3

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

   8. Applied *-un-lft-identity 1.3

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

   9. Applied times-frac 1.2

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

   10. Applied associate-*r* 1.2

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

   11. Simplified 1.2

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -3.9969049673634803 \cdot 10^{173}:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{elif}\;\cosh x \cdot \frac{y}{x} \le 1.274519611088763 \cdot 10^{192}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(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))