tanhf (example 3.4)

Average Error: 30.5 → 1.1

Time: 8.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0019486776866956498:\\ \;\;\;\;\frac{\frac{{1}^{3} - \sqrt[3]{{\left(\cos x\right)}^{9}}}{\cos x \cdot \frac{\cos x \cdot \cos x - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1}}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 0.0:\\ \;\;\;\;x \cdot \left(\frac{1}{2} + x\right) + \frac{1}{24} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{1 - \cos x}{\sin x}\right)}^{3}}\\ \end{array}\]

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)))) <= -0.0019486776866956498)) {
		VAR = ((double) (((double) (((double) (((double) pow(1.0, 3.0)) - ((double) cbrt(((double) pow(((double) cos(x)), 9.0)))))) / ((double) (((double) (((double) cos(x)) * ((double) (((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)))) <= 0.0)) {
			VAR_1 = ((double) (((double) (x * ((double) (0.5 + x)))) + ((double) (0.041666666666666664 * ((double) pow(x, 3.0))))));
		} else {
			VAR_1 = ((double) cbrt(((double) pow(((double) (((double) (1.0 - ((double) cos(x)))) / ((double) sin(x)))), 3.0))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	30.5
Target	0.0
Herbie	1.1

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

Derivation

1. Split input into 3 regimes

2. if (/ (- 1.0 (cos x)) (sin x)) < -0.0019486776866956498

   1. Initial program 1.0

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

   3. Applied flip3-- 1.1

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

   4. Simplified 1.1

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

   6. Applied add-cbrt-cube 1.1

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

   7. Simplified 1.1

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

   9. Applied pow-pow 1.1

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

   10. Simplified 1.1

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

   12. Applied flip-+ 1.1

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

   if -0.0019486776866956498 < (/ (- 1.0 (cos x)) (sin x)) < 0.0

   1. Initial program 60.2

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

   2. Taylor expanded around 0 0.7

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

   3. Simplified 0.7

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

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

   1. Initial program 1.6

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

   3. Applied add-cbrt-cube 1.8

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

   4. Applied add-cbrt-cube 1.9

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

   5. Applied cbrt-undiv 1.8

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

   6. Simplified 1.8

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

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0019486776866956498:\\ \;\;\;\;\frac{\frac{{1}^{3} - \sqrt[3]{{\left(\cos x\right)}^{9}}}{\cos x \cdot \frac{\cos x \cdot \cos x - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1}}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 0.0:\\ \;\;\;\;x \cdot \left(\frac{1}{2} + x\right) + \frac{1}{24} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{1 - \cos x}{\sin x}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020147 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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