tanhf (example 3.4)

Average Error: 30.0 → 0.7

Time: 8.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0128247688256291088:\\ \;\;\;\;\frac{e^{\sqrt[3]{{\left(\log \left({1}^{3} - {\left(\cos x\right)}^{3}\right)\right)}^{3}} - \log \left(1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)\right)}}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le -0.0:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \end{array}\]

C

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

↓

double code(double x) {
	double VAR;
	if ((((1.0 - cos(x)) / sin(x)) <= -0.012824768825629109)) {
		VAR = (exp((cbrt(pow(log((pow(1.0, 3.0) - pow(cos(x), 3.0))), 3.0)) - log(((1.0 * 1.0) + ((cos(x) * cos(x)) + (1.0 * cos(x))))))) / sin(x));
	} else {
		double VAR_1;
		if ((((1.0 - cos(x)) / sin(x)) <= -0.0)) {
			VAR_1 = ((0.041666666666666664 * pow(x, 3.0)) + ((0.004166666666666667 * pow(x, 5.0)) + (0.5 * x)));
		} else {
			VAR_1 = log(exp(((1.0 - cos(x)) / sin(x))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	30.0
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)) < -0.012824768825629109

   1. Initial program 0.8

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

   3. Applied add-exp-log 0.8

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

   5. Applied flip3-- 0.9

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

   6. Applied log-div 0.9

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

   8. Applied add-cbrt-cube 1.0

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

   9. Simplified 1.0

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

   if -0.012824768825629109 < (/ (- 1.0 (cos x)) (sin x)) < -0.0

   1. Initial program 60.1

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

   2. Taylor expanded around 0 0.1

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

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

   1. Initial program 1.4

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

   3. Applied add-log-exp 1.5

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0128247688256291088:\\ \;\;\;\;\frac{e^{\sqrt[3]{{\left(\log \left({1}^{3} - {\left(\cos x\right)}^{3}\right)\right)}^{3}} - \log \left(1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)\right)}}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le -0.0:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020103 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

  :herbie-target
  (tan (/ x 2))

  (/ (- 1 (cos x)) (sin x)))