Average Error: 30.7 → 0.6
Time: 12.0s
Precision: binary64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.018075098629927108 \lor \neg \left(\frac{1 - \cos x}{\sin x} \le 0.0049859774648358061\right):\\ \;\;\;\;\frac{{e}^{\left(\log \left(1 - \cos x\right)\right)}}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot x + \left(\frac{1}{24} \cdot {x}^{3} + \frac{1}{240} \cdot {x}^{5}\right)\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.018075098629927108 \lor \neg \left(\frac{1 - \cos x}{\sin x} \le 0.0049859774648358061\right):\\
\;\;\;\;\frac{{e}^{\left(\log \left(1 - \cos x\right)\right)}}{\sin x}\\

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

\end{array}
double code(double x) {
	return (((double) (1.0 - ((double) cos(x)))) / ((double) sin(x)));
}
↓
double code(double x) {
	double VAR;
	if ((((((double) (1.0 - ((double) cos(x)))) / ((double) sin(x))) <= -0.018075098629927108) || !((((double) (1.0 - ((double) cos(x)))) / ((double) sin(x))) <= 0.004985977464835806))) {
		VAR = (((double) pow(((double) M_E), ((double) log(((double) (1.0 - ((double) cos(x)))))))) / ((double) sin(x)));
	} else {
		VAR = ((double) (((double) (0.5 * x)) + ((double) (((double) (0.041666666666666664 * ((double) pow(x, 3.0)))) + ((double) (0.004166666666666667 * ((double) pow(x, 5.0))))))));
	}
	return VAR;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.7
Target0.0
Herbie0.6
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- 1.0 (cos x)) (sin x)) < -0.018075098629927108 or 0.0049859774648358061 < (/ (- 1.0 (cos x)) (sin 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 pow10.9

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

    6. Applied log-pow0.9

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

    7. Applied exp-prod0.9

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

    8. Simplified0.9

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

    if -0.018075098629927108 < (/ (- 1.0 (cos x)) (sin x)) < 0.0049859774648358061

    1. Initial program 59.5

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

    2. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + \left(\frac{1}{24} \cdot {x}^{3} + \frac{1}{240} \cdot {x}^{5}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.018075098629927108 \lor \neg \left(\frac{1 - \cos x}{\sin x} \le 0.0049859774648358061\right):\\ \;\;\;\;\frac{{e}^{\left(\log \left(1 - \cos x\right)\right)}}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot x + \left(\frac{1}{24} \cdot {x}^{3} + \frac{1}{240} \cdot {x}^{5}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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