tanhf (example 3.4)

Average Error: 30.2 → 0.5

Time: 7.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.024735187037949399:\\ \;\;\;\;\frac{{e}^{\left(\log \left(\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(\cos x, 1 + \cos x, 1 \cdot 1\right)}\right)\right)}}{\sin x}\\ \mathbf{elif}\;x \le 0.0203308535179141901:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{{e}^{\left(\log \left(1 - \cos x\right)\right)}}{\sin x}}\right)\\ \end{array}\]

C

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

↓

double code(double x) {
	double VAR;
	if ((x <= -0.0247351870379494)) {
		VAR = ((double) (((double) pow(((double) M_E), ((double) log(((double) (((double) (((double) pow(1.0, 3.0)) - ((double) pow(((double) cos(x)), 3.0)))) / ((double) fma(((double) cos(x)), ((double) (1.0 + ((double) cos(x)))), ((double) (1.0 * 1.0)))))))))) / ((double) sin(x))));
	} else {
		double VAR_1;
		if ((x <= 0.02033085351791419)) {
			VAR_1 = ((double) fma(0.041666666666666664, ((double) pow(x, 3.0)), ((double) fma(0.004166666666666667, ((double) pow(x, 5.0)), ((double) (0.5 * x))))));
		} else {
			VAR_1 = ((double) log(((double) exp(((double) (((double) pow(((double) M_E), ((double) log(((double) (1.0 - ((double) cos(x)))))))) / ((double) sin(x))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	30.2
Target	0.0
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

2. if x < -0.0247351870379494

   1. Initial program 0.9

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

   3. Applied add-exp-log 0.9

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

   5. Applied pow1 0.9

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

   6. Applied log-pow 0.9

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

   7. Applied exp-prod 1.0

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

   8. Simplified 1.0

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

   10. Applied flip3-- 1.1

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

   11. Simplified 1.1

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

   if -0.0247351870379494 < x < 0.02033085351791419

   1. Initial program 59.9

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

   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 0.02033085351791419 < x

   1. Initial program 0.9

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

   3. Applied add-exp-log 0.9

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

   5. Applied pow1 0.9

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

   6. Applied log-pow 0.9

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

   7. Applied exp-prod 0.9

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

   8. Simplified 0.9

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

   10. Applied add-log-exp 1.1

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.024735187037949399:\\ \;\;\;\;\frac{{e}^{\left(\log \left(\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(\cos x, 1 + \cos x, 1 \cdot 1\right)}\right)\right)}}{\sin x}\\ \mathbf{elif}\;x \le 0.0203308535179141901:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{{e}^{\left(\log \left(1 - \cos x\right)\right)}}{\sin x}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020120 +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)))