Average Error: 30.2 → 0.5
Time: 7.4s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.023734893863066063:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{\sin x}\\ \mathbf{elif}\;x \le 0.0246210202064482027:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\log \left(e^{1 - \cos x}\right)\right)}}{\sin x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.023734893863066063:\\
\;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{\sin x}\\

\mathbf{elif}\;x \le 0.0246210202064482027:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{e^{\log \left(\log \left(e^{1 - \cos x}\right)\right)}}{\sin x}\\

\end{array}
double code(double x) {
	return ((1.0 - cos(x)) / sin(x));
}
↓
double code(double x) {
	double VAR;
	if ((x <= -0.023734893863066063)) {
		VAR = (((pow(1.0, 3.0) - pow(cos(x), 3.0)) / ((cos(x) * (cos(x) + 1.0)) + (1.0 * 1.0))) / sin(x));
	} else {
		double VAR_1;
		if ((x <= 0.024621020206448203)) {
			VAR_1 = ((0.041666666666666664 * pow(x, 3.0)) + ((0.004166666666666667 * pow(x, 5.0)) + (0.5 * x)));
		} else {
			VAR_1 = (exp(log(log(exp((1.0 - cos(x)))))) / sin(x));
		}
		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

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

Derivation

  1. Split input into 3 regimes

  2. if x < -0.023734893863066063

    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. Simplified1.0

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

    if -0.023734893863066063 < x < 0.024621020206448203

    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 0.024621020206448203 < 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 add-log-exp1.1

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

    6. Applied add-log-exp1.1

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

    7. Applied diff-log1.3

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

    8. Simplified1.1

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.023734893863066063:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{\sin x}\\ \mathbf{elif}\;x \le 0.0246210202064482027:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\log \left(e^{1 - \cos x}\right)\right)}}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020091 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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