tanhf (example 3.4)

Average Error: 29.5 → 0.6

Time: 8.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.0221624842952990557:\\ \;\;\;\;\frac{e^{\sqrt[3]{{\left(\log \left(1 - \cos x\right)\right)}^{2}} \cdot \sqrt[3]{\log \left(1 - \cos x\right)}}}{\sin x}\\ \mathbf{elif}\;x \le 0.0226084732357230804:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + x \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(\log \left(1 - \cos x\right)\right)} \cdot \frac{1}{\sin x}\\ \end{array}\]

C

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

↓

double code(double x) {
	double VAR;
	if ((x <= -0.022162484295299056)) {
		VAR = ((double) (((double) exp(((double) (((double) cbrt(((double) pow(((double) log(((double) (1.0 - ((double) cos(x)))))), 2.0)))) * ((double) cbrt(((double) log(((double) (1.0 - ((double) cos(x)))))))))))) / ((double) sin(x))));
	} else {
		double VAR_1;
		if ((x <= 0.02260847323572308)) {
			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) (((double) pow(((double) M_E), ((double) log(((double) (1.0 - ((double) cos(x)))))))) * ((double) (1.0 / ((double) sin(x))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	29.5
Target	0
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if x < -0.0221624842952990557

   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 add-cbrt-cube 1.1

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

   6. Simplified 1.1

      \[\leadsto \frac{e^{\sqrt[3]{\color{blue}{{\left(\log \left(1 - \cos x\right)\right)}^{3}}}}}{\sin x}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 1.4

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

   9. Applied unpow-prod-down 1.4

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

   10. Applied cbrt-prod 1.5

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

   11. Simplified 1.2

       \[\leadsto \frac{e^{\color{blue}{\sqrt[3]{{\left(\log \left(1 - \cos x\right)\right)}^{2}}} \cdot \sqrt[3]{{\left(\sqrt[3]{\log \left(1 - \cos x\right)}\right)}^{3}}}}{\sin x}\]

   12. Simplified 1.2

       \[\leadsto \frac{e^{\sqrt[3]{{\left(\log \left(1 - \cos x\right)\right)}^{2}} \cdot \color{blue}{\sqrt[3]{\log \left(1 - \cos x\right)}}}}{\sin x}\]

   if -0.0221624842952990557 < x < 0.0226084732357230804

   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

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

   if 0.0226084732357230804 < 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 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 div-inv 1.0

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0221624842952990557:\\ \;\;\;\;\frac{e^{\sqrt[3]{{\left(\log \left(1 - \cos x\right)\right)}^{2}} \cdot \sqrt[3]{\log \left(1 - \cos x\right)}}}{\sin x}\\ \mathbf{elif}\;x \le 0.0226084732357230804:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + x \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(\log \left(1 - \cos x\right)\right)} \cdot \frac{1}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020185 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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