tanhf (example 3.4)

Average Error: 29.5 → 0.8

Time: 7.3s

Precision: 64

Math

\[\frac{1 - \cos x}{\sin x}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0154574244505038347:\\ \;\;\;\;\left(1 - \cos x\right) \cdot \frac{1}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 0.0:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 - \cos x}{\sin x}\right)\right)\\ \end{array}\]

C

double code(double x) {
	return ((1.0 - cos(x)) / sin(x));
}

↓

double code(double x) {
	double VAR;
	if ((((1.0 - cos(x)) / sin(x)) <= -0.015457424450503835)) {
		VAR = ((1.0 - cos(x)) * (1.0 / sin(x)));
	} else {
		double VAR_1;
		if ((((1.0 - cos(x)) / sin(x)) <= 0.0)) {
			VAR_1 = fma(0.041666666666666664, pow(x, 3.0), fma(0.004166666666666667, pow(x, 5.0), (0.5 * x)));
		} else {
			VAR_1 = log1p(expm1(((1.0 - cos(x)) / sin(x))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	29.5
Target	0.0
Herbie	0.8

\[\tan \left(\frac{x}{2}\right)\]

Derivation

1. Split input into 3 regimes

2. if (/ (- 1.0 (cos x)) (sin x)) < -0.015457424450503835

   1. Initial program 0.8

      \[\frac{1 - \cos x}{\sin x}\]
   2. Using strategy rm

   3. Applied div-inv 0.9

      \[\leadsto \color{blue}{\left(1 - \cos x\right) \cdot \frac{1}{\sin x}}\]

   if -0.015457424450503835 < (/ (- 1.0 (cos x)) (sin x)) < 0.0

   1. Initial program 60.0

      \[\frac{1 - \cos x}{\sin x}\]

   2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)}\]

   3. Simplified 0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)}\]

   if 0.0 < (/ (- 1.0 (cos x)) (sin x))

   1. Initial program 1.7

      \[\frac{1 - \cos x}{\sin x}\]
   2. Using strategy rm

   3. Applied log1p-expm1-u 1.8

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 - \cos x}{\sin x}\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0154574244505038347:\\ \;\;\;\;\left(1 - \cos x\right) \cdot \frac{1}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 0.0:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 - \cos x}{\sin x}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020102 +o rules:numerics
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

  :herbie-target
  (tan (/ x 2))

  (/ (- 1 (cos x)) (sin x)))