tanhf (example 3.4)

Average Error: 30.2 → 0.7

Time: 8.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.00558028084745742104:\\ \;\;\;\;\left(\left(1 - \cos x\right) \cdot \frac{1 + \cos x}{\sin x}\right) \cdot \frac{1}{1 + \cos x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le -0.0:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + x \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin x}{1 - \cos x}}\\ \end{array}\]

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)))) <= -0.005580280847457421)) {
		VAR = ((double) (((double) (((double) (1.0 - ((double) cos(x)))) * ((double) (((double) (1.0 + ((double) cos(x)))) / ((double) sin(x)))))) * ((double) (1.0 / ((double) (1.0 + ((double) cos(x))))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (1.0 - ((double) cos(x)))) / ((double) sin(x)))) <= -0.0)) {
			VAR_1 = ((double) (((double) (0.041666666666666664 * ((double) pow(x, 3.0)))) + ((double) (((double) (0.004166666666666667 * ((double) pow(x, 5.0)))) + ((double) (x * 0.5))))));
		} else {
			VAR_1 = ((double) (1.0 / ((double) (((double) sin(x)) / ((double) (1.0 - ((double) cos(x))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	30.2
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)) < -0.00558028084745742104

   1. Initial program 0.9

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

   3. Applied clear-num 0.9

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{1 - \cos x}}}\]
   4. Using strategy rm

   5. Applied flip-- 1.4

      \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}}\]

   6. Applied associate-/r/ 1.4

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

   7. Applied add-sqr-sqrt 1.4

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

   8. Applied times-frac 1.5

      \[\leadsto \color{blue}{\frac{\sqrt{1}}{\frac{\sin x}{1 \cdot 1 - \cos x \cdot \cos x}} \cdot \frac{\sqrt{1}}{1 + \cos x}}\]

   9. Simplified 1.0

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

   10. Simplified 1.0

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

   if -0.00558028084745742104 < (/ (- 1.0 (cos x)) (sin x)) < -0.0

   1. Initial program 60.2

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

   2. Taylor expanded around 0 0.0

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

   3. Simplified 0.0

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

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

   1. Initial program 1.7

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

   3. Applied clear-num 1.7

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{1 - \cos x}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.00558028084745742104:\\ \;\;\;\;\left(\left(1 - \cos x\right) \cdot \frac{1 + \cos x}{\sin x}\right) \cdot \frac{1}{1 + \cos x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le -0.0:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + x \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin x}{1 - \cos x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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