tanhf (example 3.4)

Average Error: 29.7 → 1.1

Time: 8.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.046654090225008896:\\ \;\;\;\;\frac{\sqrt{\log \left(e^{1 - \cos x}\right)}}{\frac{\sin x}{\sqrt{1 - \cos x}}}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 2.51416832555365565 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{1 - \cos x}\right)}{\sin x}\\ \end{array}\]

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)))) <= -0.046654090225008896)) {
		VAR = ((double) (((double) sqrt(((double) log(((double) exp(((double) (1.0 - ((double) cos(x)))))))))) / ((double) (((double) sin(x)) / ((double) sqrt(((double) (1.0 - ((double) cos(x))))))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (1.0 - ((double) cos(x)))) / ((double) sin(x)))) <= 2.5141683255536557e-06)) {
			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) log(((double) exp(((double) (1.0 - ((double) cos(x)))))))) / ((double) sin(x))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	29.7
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)) < -0.046654090225008896

   1. Initial program 0.7

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

   3. Applied add-log-exp 0.9

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

   4. Applied add-log-exp 0.9

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

   5. Applied diff-log 1.1

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

   6. Simplified 0.9

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

   8. Applied add-sqr-sqrt 1.1

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

   9. Applied associate-/l* 1.1

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

   10. Simplified 1.0

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

   if -0.046654090225008896 < (/ (- 1.0 (cos x)) (sin x)) < 2.51416832555365565e-6

   1. Initial program 59.3

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

   2. Taylor expanded around 0 0.9

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

   if 2.51416832555365565e-6 < (/ (- 1.0 (cos x)) (sin x))

   1. Initial program 1.2

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

   3. Applied add-log-exp 1.4

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

   4. Applied add-log-exp 1.4

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

   5. Applied diff-log 1.6

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

   6. Simplified 1.4

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

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.046654090225008896:\\ \;\;\;\;\frac{\sqrt{\log \left(e^{1 - \cos x}\right)}}{\frac{\sin x}{\sqrt{1 - \cos x}}}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 2.51416832555365565 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{1 - \cos x}\right)}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020175 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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