tanhf (example 3.4)

Average Error: 30.1 → 0.6

Time: 6.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0017051463579065111:\\ \;\;\;\;\frac{1}{\sin x} - \cos x \cdot \frac{1}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 1.093818087626018 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left({e}^{\left(\frac{1 - \cos x}{\sin x}\right)}\right)\\ \end{array}\]

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)))) <= -0.0017051463579065111)) {
		VAR = ((double) (((double) (1.0 / ((double) sin(x)))) - ((double) (((double) cos(x)) * ((double) (1.0 / ((double) sin(x))))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (1.0 - ((double) cos(x)))) / ((double) sin(x)))) <= 1.093818087626018e-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) log(((double) pow(((double) M_E), ((double) (((double) (1.0 - ((double) cos(x)))) / ((double) sin(x))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	30.1
Target	0.0
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if (/ (- 1.0 (cos x)) (sin x)) < -0.0017051463579065111

   1. Initial program 0.9

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

   3. Applied div-sub 1.1

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

   5. Applied div-inv 1.1

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

   if -0.0017051463579065111 < (/ (- 1.0 (cos x)) (sin x)) < 1.093818087626018e-05

   1. Initial program 60.0

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

   2. Taylor expanded around 0 0.0

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

   if 1.093818087626018e-05 < (/ (- 1.0 (cos x)) (sin x))

   1. Initial program 1.0

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

   3. Applied div-sub 1.3

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

   5. Applied div-inv 1.2

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

   7. Applied add-log-exp 1.5

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

   8. Applied add-log-exp 1.5

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

   9. Applied diff-log 1.6

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

   10. Simplified 1.3

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0017051463579065111:\\ \;\;\;\;\frac{1}{\sin x} - \cos x \cdot \frac{1}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 1.093818087626018 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left({e}^{\left(\frac{1 - \cos x}{\sin x}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020124 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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