tanhf (example 3.4)

Average Error: 30.2 → 0.6

Time: 10.9min

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.02311169585439021:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \cos x \cdot \left(\cos x + 1\right)} \cdot \frac{1}{\sin x}\\ \mathbf{elif}\;x \le 0.0232251484276549437:\\ \;\;\;\;\frac{1}{2} \cdot x + \left(\frac{1}{24} \cdot {x}^{3} + \frac{1}{240} \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \cos x\right) \cdot \frac{1}{\log \left(e^{\sin x}\right)}\\ \end{array}\]

C

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

↓

double code(double x) {
	double VAR;
	if ((x <= -0.023111695854390213)) {
		VAR = ((double) ((((double) (((double) pow(1.0, 3.0)) - ((double) pow(((double) cos(x)), 3.0)))) / ((double) (((double) (1.0 * 1.0)) + ((double) (((double) cos(x)) * ((double) (((double) cos(x)) + 1.0))))))) * (1.0 / ((double) sin(x)))));
	} else {
		double VAR_1;
		if ((x <= 0.023225148427654944)) {
			VAR_1 = ((double) (((double) (0.5 * x)) + ((double) (((double) (0.041666666666666664 * ((double) pow(x, 3.0)))) + ((double) (0.004166666666666667 * ((double) pow(x, 5.0))))))));
		} else {
			VAR_1 = ((double) (((double) (1.0 - ((double) cos(x)))) * (1.0 / ((double) log(((double) exp(((double) sin(x)))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	30.2
Target	0.0
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if x < -0.02311169585439021

   1. Initial program 0.8

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

   3. Applied div-inv 0.9

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

   5. Applied flip3-- 1.0

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

   6. Simplified 1.0

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

   if -0.02311169585439021 < x < 0.0232251484276549437

   1. Initial program 59.7

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

   2. Taylor expanded around 0 0.0

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

   if 0.0232251484276549437 < x

   1. Initial program 0.9

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

   3. Applied div-inv 1.0

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

   5. Applied add-log-exp 1.2

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02311169585439021:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \cos x \cdot \left(\cos x + 1\right)} \cdot \frac{1}{\sin x}\\ \mathbf{elif}\;x \le 0.0232251484276549437:\\ \;\;\;\;\frac{1}{2} \cdot x + \left(\frac{1}{24} \cdot {x}^{3} + \frac{1}{240} \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \cos x\right) \cdot \frac{1}{\log \left(e^{\sin x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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