Average Error: 30.0 → 0.6
Time: 7.2s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0206102587514720333:\\ \;\;\;\;\log \left(\sqrt[3]{{\left(e^{\frac{1 - \cos x}{\sin x}}\right)}^{3}}\right)\\ \mathbf{elif}\;x \le 0.022224521115662491:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin x} - \cos x \cdot \frac{1}{\sin x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0206102587514720333:\\
\;\;\;\;\log \left(\sqrt[3]{{\left(e^{\frac{1 - \cos x}{\sin x}}\right)}^{3}}\right)\\

\mathbf{elif}\;x \le 0.022224521115662491:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\sin x} - \cos x \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 <= -0.020610258751472033)) {
		VAR = log(cbrt(pow(exp(((1.0 - cos(x)) / sin(x))), 3.0)));
	} else {
		double VAR_1;
		if ((x <= 0.02222452111566249)) {
			VAR_1 = fma(0.041666666666666664, pow(x, 3.0), fma(0.004166666666666667, pow(x, 5.0), (0.5 * x)));
		} else {
			VAR_1 = ((1.0 / sin(x)) - (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

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

Derivation

  1. Split input into 3 regimes

  2. if x < -0.020610258751472033

    1. Initial program 0.9

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

    3. Applied add-log-exp1.0

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

    5. Applied add-cbrt-cube1.4

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

    6. Simplified1.4

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

    if -0.020610258751472033 < x < 0.02222452111566249

    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. Simplified0.0

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

    if 0.02222452111566249 < x

    1. Initial program 0.9

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

    3. Applied div-sub1.1

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

    5. Applied div-inv1.0

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0206102587514720333:\\ \;\;\;\;\log \left(\sqrt[3]{{\left(e^{\frac{1 - \cos x}{\sin x}}\right)}^{3}}\right)\\ \mathbf{elif}\;x \le 0.022224521115662491:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin x} - \cos x \cdot \frac{1}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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