tanhf (example 3.4)

Average Error: 30.1 → 0.7

Time: 8.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0107629324502667868:\\ \;\;\;\;\frac{\frac{{1}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1} - \frac{{\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 4.3298603204685633 \cdot 10^{-4}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{e^{\frac{1 - \cos x}{\sin x}}} \cdot \sqrt{e^{\frac{1 - \cos x}{\sin x}}}\right)\\ \end{array}\]

C

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

↓

double code(double x) {
	double temp;
	if ((((1.0 - cos(x)) / sin(x)) <= -0.010762932450266787)) {
		temp = (((pow(1.0, 3.0) / ((cos(x) * (cos(x) + 1.0)) + (1.0 * 1.0))) - (pow(cos(x), 3.0) / ((cos(x) * (cos(x) + 1.0)) + (1.0 * 1.0)))) / sin(x));
	} else {
		double temp_1;
		if ((((1.0 - cos(x)) / sin(x)) <= 0.00043298603204685633)) {
			temp_1 = ((0.041666666666666664 * pow(x, 3.0)) + ((0.004166666666666667 * pow(x, 5.0)) + (0.5 * x)));
		} else {
			temp_1 = log((sqrt(exp(((1.0 - cos(x)) / sin(x)))) * sqrt(exp(((1.0 - cos(x)) / sin(x))))));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Target

Original	30.1
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)) < -0.010762932450266787

   1. Initial program 0.9

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

   3. Applied flip3-- 1.0

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

   4. Simplified 1.0

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

   6. Applied div-sub 1.0

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

   if -0.010762932450266787 < (/ (- 1.0 (cos x)) (sin x)) < 0.00043298603204685633

   1. Initial program 59.8

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

   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 0.00043298603204685633 < (/ (- 1.0 (cos x)) (sin x))

   1. Initial program 1.0

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

   3. Applied add-log-exp 1.0

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

   5. Applied add-sqr-sqrt 1.3

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0107629324502667868:\\ \;\;\;\;\frac{\frac{{1}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1} - \frac{{\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 4.3298603204685633 \cdot 10^{-4}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{e^{\frac{1 - \cos x}{\sin x}}} \cdot \sqrt{e^{\frac{1 - \cos x}{\sin x}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

  :herbie-target
  (tan (/ x 2))

  (/ (- 1 (cos x)) (sin x)))