tanhf (example 3.4)

Average Error: 16.3 → 0.8

Time: 7.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.023247613038875889:\\ \;\;\;\;\frac{e^{\log \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}}{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot \sin x}\\ \mathbf{elif}\;x \le 0.020879752328029882:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\ \end{array}\]

C

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

↓

double code(double x) {
	double VAR;
	if ((x <= -0.02324761303887589)) {
		VAR = ((double) (((double) exp(((double) log(((double) (((double) pow(1.0, 3.0)) - ((double) pow(((double) cos(x)), 3.0)))))))) / ((double) (((double) (((double) (((double) cos(x)) * ((double) (((double) cos(x)) + 1.0)))) + ((double) (1.0 * 1.0)))) * ((double) sin(x))))));
	} else {
		double VAR_1;
		if ((x <= 0.020879752328029882)) {
			VAR_1 = ((double) (((double) (0.041666666666666664 * ((double) pow(x, 3.0)))) + ((double) (((double) (0.004166666666666667 * ((double) pow(x, 5.0)))) + ((double) (0.5 * x))))));
		} else {
			VAR_1 = ((double) (((double) (1.0 / ((double) sin(x)))) - ((double) (((double) cos(x)) / ((double) sin(x))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	16.3
Target	0.0
Herbie	0.8

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

Derivation

1. Split input into 3 regimes

2. if x < -0.023247613038875889

   1. Initial program 0.9

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

   3. Applied flip3-- 1.0

      \[\leadsto \frac{\color{blue}{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}}{\sin x}\]

   4. Applied associate-/l/ 1.0

      \[\leadsto \color{blue}{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x \cdot \left(1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)\right)}}\]

   5. Simplified 1.0

      \[\leadsto \frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\color{blue}{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot \sin x}}\]
   6. Using strategy rm

   7. Applied add-exp-log 1.0

      \[\leadsto \frac{\color{blue}{e^{\log \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}}}{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot \sin x}\]

   if -0.023247613038875889 < x < 0.020879752328029882

   1. Initial program 59.6

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

   2. Taylor expanded around 0 0.0

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

   if 0.020879752328029882 < x

   1. Initial program 0.9

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

   3. Applied div-sub 1.1

      \[\leadsto \color{blue}{\frac{1}{\sin x} - \frac{\cos x}{\sin x}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.023247613038875889:\\ \;\;\;\;\frac{e^{\log \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}}{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot \sin x}\\ \mathbf{elif}\;x \le 0.020879752328029882:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020157 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

  :herbie-target
  (tan (/ x 2.0))

  (/ (- 1.0 (cos x)) (sin x)))