Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.5 → 0.5

Time: 3.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -5.45981265466444877 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{\frac{z}{\cosh x \cdot y} \cdot x}\\ \mathbf{elif}\;y \le 18720211855765745700:\\ \;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{\frac{x}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{x}}{\frac{\frac{z}{e^{-1 \cdot x} + e^{x}}}{y}}\\ \end{array}\]

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 <= -5.459812654664449e-22)) {
		VAR = (1.0 / ((z / (cosh(x) * y)) * x));
	} else {
		double VAR_1;
		if ((y <= 1.8720211855765746e+19)) {
			VAR_1 = (((0.5 * (exp((-1.0 * x)) + exp(x))) / (x / y)) / z);
		} else {
			VAR_1 = ((0.5 / x) / ((z / (exp((-1.0 * x)) + exp(x))) / y));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.5
Target	0.4
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 y < -5.459812654664449e-22

   1. Initial program 18.1

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

   3. Applied div-inv 18.2

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

   4. Applied associate-*r* 18.3

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

   6. Applied clear-num 18.3

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

   7. Simplified 0.7

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

   if -5.459812654664449e-22 < y < 1.8720211855765746e+19

   1. Initial program 0.4

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

   2. Taylor expanded around inf 0.4

      \[\leadsto \frac{\color{blue}{\frac{y \cdot \left(\frac{1}{2} \cdot e^{x} + \frac{1}{2} \cdot e^{-x}\right)}{x}}}{z}\]

   3. Simplified 0.5

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{\frac{x}{y}}}}{z}\]

   if 1.8720211855765746e+19 < y

   1. Initial program 22.7

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

   2. Taylor expanded around inf 22.8

      \[\leadsto \frac{\color{blue}{\frac{y \cdot \left(\frac{1}{2} \cdot e^{x} + \frac{1}{2} \cdot e^{-x}\right)}{x}}}{z}\]

   3. Simplified 22.8

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{\frac{x}{y}}}}{z}\]
   4. Using strategy rm

   5. Applied div-inv 22.8

      \[\leadsto \frac{\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{\color{blue}{x \cdot \frac{1}{y}}}}{z}\]

   6. Applied times-frac 22.9

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{x} \cdot \frac{e^{-1 \cdot x} + e^{x}}{\frac{1}{y}}}}{z}\]

   7. Applied associate-/l* 0.6

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2}}{x}}{\frac{z}{\frac{e^{-1 \cdot x} + e^{x}}{\frac{1}{y}}}}}\]

   8. Simplified 0.4

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -5.45981265466444877 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{\frac{z}{\cosh x \cdot y} \cdot x}\\ \mathbf{elif}\;y \le 18720211855765745700:\\ \;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{\frac{x}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{x}}{\frac{\frac{z}{e^{-1 \cdot x} + e^{x}}}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020103 
(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))