Average Error: 30.6 → 0.6
Time: 7.9s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0224723957348866195:\\ \;\;\;\;\frac{1}{\frac{\sin x}{1 \cdot 1 - \cos x \cdot \cos x} \cdot \left(1 + \cos x\right)}\\ \mathbf{elif}\;x \le 0.0255308872267180458:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin x}{\log \left(e^{1 - \cos x}\right)}}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0224723957348866195:\\
\;\;\;\;\frac{1}{\frac{\sin x}{1 \cdot 1 - \cos x \cdot \cos x} \cdot \left(1 + \cos x\right)}\\

\mathbf{elif}\;x \le 0.0255308872267180458:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\

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

\end{array}
double code(double x) {
	return ((1.0 - cos(x)) / sin(x));
}
↓
double code(double x) {
	double temp;
	if ((x <= -0.02247239573488662)) {
		temp = (1.0 / ((sin(x) / ((1.0 * 1.0) - (cos(x) * cos(x)))) * (1.0 + cos(x))));
	} else {
		double temp_1;
		if ((x <= 0.025530887226718046)) {
			temp_1 = ((0.041666666666666664 * pow(x, 3.0)) + ((0.004166666666666667 * pow(x, 5.0)) + (0.5 * x)));
		} else {
			temp_1 = (1.0 / (sin(x) / log(exp((1.0 - cos(x))))));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 3 regimes

  2. if x < -0.02247239573488662

    1. Initial program 1.0

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

    3. Applied clear-num1.0

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{1 - \cos x}}}\]
    4. Using strategy rm

    5. Applied flip--1.4

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

    6. Applied associate-/r/1.4

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

    if -0.02247239573488662 < x < 0.025530887226718046

    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 0.025530887226718046 < x

    1. Initial program 0.9

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

    3. Applied clear-num0.9

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{1 - \cos x}}}\]
    4. Using strategy rm

    5. Applied add-log-exp1.1

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

    6. Applied add-log-exp1.1

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

    7. Applied diff-log1.3

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

    8. Simplified1.1

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0224723957348866195:\\ \;\;\;\;\frac{1}{\frac{\sin x}{1 \cdot 1 - \cos x \cdot \cos x} \cdot \left(1 + \cos x\right)}\\ \mathbf{elif}\;x \le 0.0255308872267180458:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin x}{\log \left(e^{1 - \cos x}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020053 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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