tanhf (example 3.4)

Average Error: 30.3 → 0.7

Time: 7.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0130850919083989967:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 0.0:\\ \;\;\;\;\left(\frac{1}{24} \cdot {x}^{3} + \frac{1}{240} \cdot {x}^{5}\right) + \frac{1}{2} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{1 - \cos x}\right)}{\sin x}\\ \end{array}\]

C

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

↓

double code(double x) {
	double VAR;
	if ((((double) (((double) (1.0 - ((double) cos(x)))) / ((double) sin(x)))) <= -0.013085091908398997)) {
		VAR = ((double) (((double) (((double) (((double) pow(1.0, 3.0)) - ((double) pow(((double) cos(x)), 3.0)))) / ((double) (((double) (((double) cos(x)) * ((double) (((double) cos(x)) + 1.0)))) + ((double) (1.0 * 1.0)))))) / ((double) sin(x))));
	} else {
		double VAR_1;
		if ((((double) (((double) (1.0 - ((double) cos(x)))) / ((double) sin(x)))) <= 0.0)) {
			VAR_1 = ((double) (((double) (((double) (0.041666666666666664 * ((double) pow(x, 3.0)))) + ((double) (0.004166666666666667 * ((double) pow(x, 5.0)))))) + ((double) (0.5 * x))));
		} else {
			VAR_1 = ((double) (((double) log(((double) exp(((double) (1.0 - ((double) cos(x)))))))) / ((double) sin(x))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	30.3
Target	0.0
Herbie	0.7

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

Derivation

1. Split input into 3 regimes

2. if (/ (- 1.0 (cos x)) (sin x)) < -0.013085091908398997

   1. Initial program 0.8

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

   3. Applied flip3-- 0.9

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

   4. Simplified 0.9

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

   if -0.013085091908398997 < (/ (- 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. Using strategy rm

   4. Applied associate-+r+ 0.2

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

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

   1. Initial program 1.3

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

   3. Applied add-log-exp 1.5

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

   4. Applied add-log-exp 1.5

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

   5. Applied diff-log 1.7

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

   6. Simplified 1.5

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0130850919083989967:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 0.0:\\ \;\;\;\;\left(\frac{1}{24} \cdot {x}^{3} + \frac{1}{240} \cdot {x}^{5}\right) + \frac{1}{2} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{1 - \cos x}\right)}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020148 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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