Average Error: 29.2 → 0.6
Time: 11.5s
Precision: binary64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.022949180550174302:\\ \;\;\;\;\frac{\sqrt{1 - \cos x} \cdot \sqrt{1 - \cos x}}{\sin x}\\ \mathbf{elif}\;x \le 0.0213211679511081213:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x}}{1 \cdot 1 + \frac{{\left(\cos x\right)}^{2} \cdot \left({\left(\cos x\right)}^{2} - 1 \cdot 1\right)}{\cos x \cdot \left(\cos x - 1\right)}}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.022949180550174302:\\
\;\;\;\;\frac{\sqrt{1 - \cos x} \cdot \sqrt{1 - \cos x}}{\sin x}\\

\mathbf{elif}\;x \le 0.0213211679511081213:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x}}{1 \cdot 1 + \frac{{\left(\cos x\right)}^{2} \cdot \left({\left(\cos x\right)}^{2} - 1 \cdot 1\right)}{\cos x \cdot \left(\cos x - 1\right)}}\\

\end{array}
double code(double x) {
	return ((double) (((double) (1.0 - ((double) cos(x)))) / ((double) sin(x))));
}
↓
double code(double x) {
	double VAR;
	if ((x <= -0.0229491805501743)) {
		VAR = ((double) (((double) (((double) sqrt(((double) (1.0 - ((double) cos(x)))))) * ((double) sqrt(((double) (1.0 - ((double) cos(x)))))))) / ((double) sin(x))));
	} else {
		double VAR_1;
		if ((x <= 0.02132116795110812)) {
			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) (((double) (((double) (((double) pow(1.0, 3.0)) - ((double) pow(((double) cos(x)), 3.0)))) / ((double) sin(x)))) / ((double) (((double) (1.0 * 1.0)) + ((double) (((double) (((double) pow(((double) cos(x)), 2.0)) * ((double) (((double) pow(((double) cos(x)), 2.0)) - ((double) (1.0 * 1.0)))))) / ((double) (((double) cos(x)) * ((double) (((double) cos(x)) - 1.0))))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.2
Target0
Herbie0.6
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.022949180550174302

    1. Initial program 0.9

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

    3. Applied add-sqr-sqrt1.1

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

    if -0.022949180550174302 < x < 0.0213211679511081213

    1. Initial program 59.9

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

    2. Taylor expanded around 0 0.0

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

    if 0.0213211679511081213 < x

    1. Initial program 1.0

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

    3. Applied div-inv1.0

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

    5. Applied flip3--1.1

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

    6. Applied associate-*l/1.1

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

    7. Simplified1.1

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

    9. Applied flip-+1.1

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

    10. Simplified1.1

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

    11. Simplified1.1

      \[\leadsto \frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x}}{1 \cdot 1 + \frac{{\left(\cos x\right)}^{2} \cdot \left({\left(\cos x\right)}^{2} - 1 \cdot 1\right)}{\color{blue}{\cos x \cdot \left(\cos x - 1\right)}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.022949180550174302:\\ \;\;\;\;\frac{\sqrt{1 - \cos x} \cdot \sqrt{1 - \cos x}}{\sin x}\\ \mathbf{elif}\;x \le 0.0213211679511081213:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x}}{1 \cdot 1 + \frac{{\left(\cos x\right)}^{2} \cdot \left({\left(\cos x\right)}^{2} - 1 \cdot 1\right)}{\cos x \cdot \left(\cos x - 1\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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