Average Error: 29.5 → 0.6
Time: 7.1s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0201136423449840597:\\ \;\;\;\;\frac{\sqrt{\log \left(e^{1 - \cos x}\right)}}{\frac{\sin x}{\sqrt{1 - \cos x}}}\\ \mathbf{elif}\;x \le 0.0224978003073234255:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \cos x\right) \cdot \frac{1}{\sin x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0201136423449840597:\\
\;\;\;\;\frac{\sqrt{\log \left(e^{1 - \cos x}\right)}}{\frac{\sin x}{\sqrt{1 - \cos x}}}\\

\mathbf{elif}\;x \le 0.0224978003073234255:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\

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

\end{array}
double code(double x) {
	return ((1.0 - cos(x)) / sin(x));
}
↓
double code(double x) {
	double VAR;
	if ((x <= -0.02011364234498406)) {
		VAR = (sqrt(log(exp((1.0 - cos(x))))) / (sin(x) / sqrt((1.0 - cos(x)))));
	} else {
		double VAR_1;
		if ((x <= 0.022497800307323425)) {
			VAR_1 = ((0.041666666666666664 * pow(x, 3.0)) + ((0.004166666666666667 * pow(x, 5.0)) + (0.5 * x)));
		} else {
			VAR_1 = ((1.0 - cos(x)) * (1.0 / 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

Original29.5
Target0.0
Herbie0.6
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.02011364234498406

    1. Initial program 1.0

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

    3. Applied add-log-exp1.2

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

    4. Applied add-log-exp1.2

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

    5. Applied diff-log1.4

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

    6. Simplified1.2

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

    8. Applied add-sqr-sqrt1.4

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

    9. Applied associate-/l*1.3

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

    10. Simplified1.2

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

    if -0.02011364234498406 < x < 0.022497800307323425

    1. Initial program 59.8

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

    2. Taylor expanded around 0 0.0

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

    if 0.022497800307323425 < x

    1. Initial program 0.9

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

    3. Applied div-inv1.0

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0201136423449840597:\\ \;\;\;\;\frac{\sqrt{\log \left(e^{1 - \cos x}\right)}}{\frac{\sin x}{\sqrt{1 - \cos x}}}\\ \mathbf{elif}\;x \le 0.0224978003073234255:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \cos x\right) \cdot \frac{1}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020102 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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