tanhf (example 3.4)

Average Error: 30.1 → 0.8

Time: 8.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.033563598404528272:\\ \;\;\;\;\frac{\frac{{1}^{3} - \sqrt[3]{{\left({\left(\cos x\right)}^{3}\right)}^{3}}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 1.24398434498272872 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\frac{{1}^{3} - \log \left(e^{{\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}\right)}}{\sin x}\\ \end{array}\]

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)))) <= -0.03356359840452827)) {
		VAR = ((double) (((double) (((double) (((double) pow(1.0, 3.0)) - ((double) cbrt(((double) pow(((double) pow(((double) cos(x)), 3.0)), 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)))) <= 1.2439843449827287e-05)) {
			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) exp(((double) log(((double) (((double) (((double) pow(1.0, 3.0)) - ((double) log(((double) exp(((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))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	30.1
Target	0.0
Herbie	0.8

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

Derivation

1. Split input into 3 regimes

2. if (/ (- 1.0 (cos x)) (sin x)) < -0.033563598404528272

   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. Simplified 0.8

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

   6. Applied add-cbrt-cube 0.9

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

   7. Simplified 0.9

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

   if -0.033563598404528272 < (/ (- 1.0 (cos x)) (sin x)) < 1.24398434498272872e-5

   1. Initial program 59.6

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

   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 1.24398434498272872e-5 < (/ (- 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. Simplified 1.2

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

   6. Applied add-exp-log 1.2

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

   7. Applied add-exp-log 1.2

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

   8. Applied div-exp 1.3

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

   9. Simplified 1.2

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

   11. Applied add-log-exp 1.3

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.033563598404528272:\\ \;\;\;\;\frac{\frac{{1}^{3} - \sqrt[3]{{\left({\left(\cos x\right)}^{3}\right)}^{3}}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 1.24398434498272872 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\frac{{1}^{3} - \log \left(e^{{\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}\right)}}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020153 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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