tanhf (example 3.4)

Average Error: 30.8 → 0.5

Time: 14.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.018586529511459986:\\ \;\;\;\;\frac{{1}^{3} - \cos x \cdot {\left(\cos x\right)}^{2}}{\left(1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)\right) \cdot \sin x}\\ \mathbf{elif}\;x \leq 0.019474268356794753:\\ \;\;\;\;0.041666666666666664 \cdot {x}^{3} + \left(0.004166666666666667 \cdot {x}^{5} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}}{\left(1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)\right) \cdot \sin x}\\ \end{array}\]

C

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

↓

double code(double x) {
	double VAR;
	if ((x <= -0.018586529511459986)) {
		VAR = (((double) (((double) pow(1.0, 3.0)) - ((double) (((double) cos(x)) * ((double) pow(((double) cos(x)), 2.0)))))) / ((double) (((double) (((double) (1.0 * 1.0)) + ((double) (((double) cos(x)) * ((double) (1.0 + ((double) cos(x)))))))) * ((double) sin(x)))));
	} else {
		double VAR_1;
		if ((x <= 0.019474268356794753)) {
			VAR_1 = ((double) (((double) (0.041666666666666664 * ((double) pow(x, 3.0)))) + ((double) (((double) (0.004166666666666667 * ((double) pow(x, 5.0)))) + ((double) (x * 0.5))))));
		} else {
			VAR_1 = (((double) exp(((double) log(((double) (((double) pow(1.0, 3.0)) - ((double) pow(((double) cos(x)), 3.0)))))))) / ((double) (((double) (((double) (1.0 * 1.0)) + ((double) (((double) cos(x)) * ((double) (1.0 + ((double) cos(x)))))))) * ((double) sin(x)))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	30.8
Target	0.0
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

2. if x < -0.0185865295114599864

   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 1.1

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

   7. Applied add-cube-cbrt 1.9

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

   8. Applied unpow-prod-down 1.9

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

   9. Simplified 1.3

      \[\leadsto \frac{{1}^{3} - \color{blue}{{\left(\cos x\right)}^{2}} \cdot {\left(\sqrt[3]{\cos x}\right)}^{3}}{\left(1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)\right) \cdot \sin x}\]

   10. Simplified 1.1

       \[\leadsto \frac{{1}^{3} - {\left(\cos x\right)}^{2} \cdot \color{blue}{\cos x}}{\left(1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)\right) \cdot \sin x}\]

   if -0.0185865295114599864 < x < 0.0194742683567947529

   1. Initial program 59.9

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

   2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{0.041666666666666664 \cdot {x}^{3} + \left(0.004166666666666667 \cdot {x}^{5} + 0.5 \cdot x\right)}\]

   3. Simplified 0.0

      \[\leadsto \color{blue}{0.041666666666666664 \cdot {x}^{3} + \left(0.004166666666666667 \cdot {x}^{5} + x \cdot 0.5\right)}\]

   if 0.0194742683567947529 < x

   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

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

   7. Applied add-exp-log 1.0

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.018586529511459986:\\ \;\;\;\;\frac{{1}^{3} - \cos x \cdot {\left(\cos x\right)}^{2}}{\left(1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)\right) \cdot \sin x}\\ \mathbf{elif}\;x \leq 0.019474268356794753:\\ \;\;\;\;0.041666666666666664 \cdot {x}^{3} + \left(0.004166666666666667 \cdot {x}^{5} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}}{\left(1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)\right) \cdot \sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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