Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.6 → 0.9

Time: 4.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -1.551768073007693 \cdot 10^{100}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{y}{z} \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}}\\ \mathbf{elif}\;\cosh x \cdot \frac{y}{x} \le 2.5851757423207426 \cdot 10^{215}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right) \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x \cdot z}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double temp;
	if (((cosh(x) * (y / x)) <= -1.551768073007693e+100)) {
		temp = (1.0 / (x / ((y / z) * fma(exp(x), 0.5, (0.5 / exp(x))))));
	} else {
		double temp_1;
		if (((cosh(x) * (y / x)) <= 2.5851757423207426e+215)) {
			temp_1 = ((fma(exp(x), 0.5, (0.5 / exp(x))) * (y / x)) / z);
		} else {
			temp_1 = ((y * fma(exp(x), 0.5, (0.5 / exp(x)))) / (x * z));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.6
Target	0.4
Herbie	0.9

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

Derivation

1. Split input into 3 regimes

2. if (* (cosh x) (/ y x)) < -1.551768073007693e+100

   1. Initial program 17.1

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

   2. Taylor expanded around inf 2.3

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

   3. Simplified 2.8

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

   5. Applied associate-*r/ 2.7

      \[\leadsto \color{blue}{\frac{\frac{y}{z} \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}}\]
   6. Using strategy rm

   7. Applied clear-num 2.8

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

   if -1.551768073007693e+100 < (* (cosh x) (/ y x)) < 2.5851757423207426e+215

   1. Initial program 0.2

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

   2. Taylor expanded around inf 0.2

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

   3. Simplified 0.3

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

   if 2.5851757423207426e+215 < (* (cosh x) (/ y x))

   1. Initial program 32.0

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

   2. Taylor expanded around inf 0.9

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

   3. Simplified 0.7

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

   5. Applied associate-*r/ 0.7

      \[\leadsto \color{blue}{\frac{\frac{y}{z} \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}}\]
   6. Using strategy rm

   7. Applied associate-*l/ 0.7

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

   8. Applied associate-/l/ 0.9

      \[\leadsto \color{blue}{\frac{y \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x \cdot z}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -1.551768073007693 \cdot 10^{100}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{y}{z} \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}}\\ \mathbf{elif}\;\cosh x \cdot \frac{y}{x} \le 2.5851757423207426 \cdot 10^{215}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right) \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +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))