tanhf (example 3.4)

Average Error: 30.3 → 0.7

Time: 10.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.019140663594110131:\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 4.8226349012260431 \cdot 10^{-4}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{1 - \cos x}{\sin x}\right)}^{3}}\\ \end{array}\]

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)))) <= -0.01914066359411013)) {
		VAR = ((double) log(((double) exp(((double) (((double) (1.0 - ((double) cos(x)))) / ((double) sin(x))))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (1.0 - ((double) cos(x)))) / ((double) sin(x)))) <= 0.0004822634901226043)) {
			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) 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	30.3
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)) < -0.019140663594110131

   1. Initial program 0.8

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

   3. Applied add-log-exp 0.9

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

   if -0.019140663594110131 < (/ (- 1.0 (cos x)) (sin x)) < 4.8226349012260431e-4

   1. Initial program 59.7

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

   2. Taylor expanded around 0 0.3

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

   if 4.8226349012260431e-4 < (/ (- 1.0 (cos x)) (sin x))

   1. Initial program 0.9

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

   3. Applied add-cbrt-cube 1.1

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

   4. Applied add-cbrt-cube 1.3

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

   5. Applied cbrt-undiv 1.1

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

   6. Simplified 1.1

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.019140663594110131:\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 4.8226349012260431 \cdot 10^{-4}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{1 - \cos x}{\sin x}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020169 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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