Average Error: 30.2 → 0.5
Time: 8.0s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.024441885565023871 \lor \neg \left(x \le 0.021343474190141223\right):\\ \;\;\;\;\frac{e^{\log \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}}{\left(\cos x \cdot \frac{{\left(\cos x\right)}^{2} - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1\right) \cdot \sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.024441885565023871 \lor \neg \left(x \le 0.021343474190141223\right):\\
\;\;\;\;\frac{e^{\log \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}}{\left(\cos x \cdot \frac{{\left(\cos x\right)}^{2} - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1\right) \cdot \sin x}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\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 <= -0.02444188556502387) || !(x <= 0.021343474190141223))) {
		VAR = ((double) (((double) exp(((double) log(((double) (((double) pow(1.0, 3.0)) - ((double) pow(((double) cos(x)), 3.0)))))))) / ((double) (((double) (((double) (((double) cos(x)) * ((double) (((double) (((double) pow(((double) cos(x)), 2.0)) - ((double) (1.0 * 1.0)))) / ((double) (((double) cos(x)) - 1.0)))))) + ((double) (1.0 * 1.0)))) * ((double) sin(x))))));
	} else {
		VAR = ((double) (((double) (0.041666666666666664 * ((double) pow(x, 3.0)))) + ((double) (((double) (0.004166666666666667 * ((double) pow(x, 5.0)))) + ((double) (0.5 * x))))));
	}
	return VAR;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.2
Target0.0
Herbie0.5
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.02444188556502387 or 0.021343474190141223 < x

    1. Initial program 0.9

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

    3. Applied add-exp-log0.9

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

    5. Applied flip3--1.0

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

    6. Applied log-div1.0

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

    7. Applied exp-diff1.0

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

    8. Applied associate-/l/1.0

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

    9. Simplified1.0

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

    11. Applied flip-+1.0

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

    12. Simplified1.0

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

    if -0.02444188556502387 < x < 0.021343474190141223

    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. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.024441885565023871 \lor \neg \left(x \le 0.021343474190141223\right):\\ \;\;\;\;\frac{e^{\log \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}}{\left(\cos x \cdot \frac{{\left(\cos x\right)}^{2} - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1\right) \cdot \sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020130 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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