tanhf (example 3.4)

Average Error: 30.2 → 0.8

Time: 7.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0315680071226008205:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\frac{\cos x \cdot \left(\cos x \cdot \cos x - 1 \cdot 1\right)}{\cos x - 1} + 1 \cdot 1\right) \cdot \sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 1.05832862745451739 \cdot 10^{-4}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3} - \log \left(e^{{\left(\cos x\right)}^{3}}\right)}{\left(\cos x \cdot \sqrt[3]{{\left(\cos x + 1\right)}^{3}} + 1 \cdot 1\right) \cdot \sin x}\\ \end{array}\]

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)))) <= -0.03156800712260082)) {
		VAR = ((double) (((double) (((double) pow(1.0, 3.0)) - ((double) pow(((double) cos(x)), 3.0)))) / ((double) (((double) (((double) (((double) (((double) cos(x)) * ((double) (((double) (((double) cos(x)) * ((double) cos(x)))) - ((double) (1.0 * 1.0)))))) / ((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)))) <= 0.00010583286274545174)) {
			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) (((double) pow(1.0, 3.0)) - ((double) log(((double) exp(((double) pow(((double) cos(x)), 3.0)))))))) / ((double) (((double) (((double) (((double) cos(x)) * ((double) cbrt(((double) pow(((double) (((double) cos(x)) + 1.0)), 3.0)))))) + ((double) (1.0 * 1.0)))) * ((double) sin(x))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	30.2
Target	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)) < -0.0315680071226008205

   1. Initial program 0.7

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

   3. Applied flip3-- 0.8

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

   4. Applied associate-/l/ 0.9

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

   5. Simplified 0.9

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

   7. Applied flip-+ 0.9

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

   8. Applied associate-*r/ 0.9

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

   if -0.0315680071226008205 < (/ (- 1.0 (cos x)) (sin x)) < 1.05832862745451739e-4

   1. Initial program 59.4

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

   2. Taylor expanded around 0 0.6

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

   if 1.05832862745451739e-4 < (/ (- 1.0 (cos x)) (sin x))

   1. Initial program 1.1

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

   3. Applied flip3-- 1.2

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

   4. Applied associate-/l/ 1.2

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

   5. Simplified 1.2

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

   7. Applied add-cbrt-cube 1.2

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

   8. Simplified 1.2

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

   10. Applied add-log-exp 1.3

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0315680071226008205:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\frac{\cos x \cdot \left(\cos x \cdot \cos x - 1 \cdot 1\right)}{\cos x - 1} + 1 \cdot 1\right) \cdot \sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 1.05832862745451739 \cdot 10^{-4}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3} - \log \left(e^{{\left(\cos x\right)}^{3}}\right)}{\left(\cos x \cdot \sqrt[3]{{\left(\cos x + 1\right)}^{3}} + 1 \cdot 1\right) \cdot \sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020150 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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