tanhf (example 3.4)

Average Error: 30.5 → 0.5

Time: 7.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.023427378097045405:\\ \;\;\;\;\frac{\log \left(e^{1 - \cos x}\right)}{\sin x}\\ \mathbf{elif}\;x \le 0.023613340017627771:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(\cos x, \log \left(e^{1 + \cos x}\right), 1 \cdot 1\right)}}{\sin x}\\ \end{array}\]

C

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

↓

double code(double x) {
	double VAR;
	if ((x <= -0.023427378097045405)) {
		VAR = (log(exp((1.0 - cos(x)))) / sin(x));
	} else {
		double VAR_1;
		if ((x <= 0.02361334001762777)) {
			VAR_1 = fma(0.041666666666666664, pow(x, 3.0), fma(0.004166666666666667, pow(x, 5.0), (0.5 * x)));
		} else {
			VAR_1 = (((pow(1.0, 3.0) - pow(cos(x), 3.0)) / fma(cos(x), log(exp((1.0 + cos(x)))), (1.0 * 1.0))) / sin(x));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	30.5
Target	0.0
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

2. if x < -0.023427378097045405

   1. Initial program 0.9

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

   3. Applied add-log-exp 1.1

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

   4. Applied add-log-exp 1.1

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

   5. Applied diff-log 1.3

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

   6. Simplified 1.1

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

   if -0.023427378097045405 < x < 0.02361334001762777

   1. Initial program 59.8

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

   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

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

   if 0.02361334001762777 < x

   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

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

   6. Applied add-log-exp 1.0

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

   7. Applied add-log-exp 1.0

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

   8. Applied sum-log 1.0

      \[\leadsto \frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(\cos x, \color{blue}{\log \left(e^{1} \cdot e^{\cos x}\right)}, 1 \cdot 1\right)}}{\sin x}\]

   9. Simplified 1.0

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.023427378097045405:\\ \;\;\;\;\frac{\log \left(e^{1 - \cos x}\right)}{\sin x}\\ \mathbf{elif}\;x \le 0.023613340017627771:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(\cos x, \log \left(e^{1 + \cos x}\right), 1 \cdot 1\right)}}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020092 +o rules:numerics
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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