Average Error: 30.3 → 0.9
Time: 7.3s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0441131830391816929:\\ \;\;\;\;\frac{1}{\frac{\sin x}{1 - \cos x}}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 1.0999914160827158 \cdot 10^{-4}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \cos x\right)\right)}{\sin x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0441131830391816929:\\
\;\;\;\;\frac{1}{\frac{\sin x}{1 - \cos x}}\\

\mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 1.0999914160827158 \cdot 10^{-4}:\\
\;\;\;\;\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{\mathsf{expm1}\left(\mathsf{log1p}\left(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 ((((1.0 - cos(x)) / sin(x)) <= -0.04411318303918169)) {
		VAR = (1.0 / (sin(x) / (1.0 - cos(x))));
	} else {
		double VAR_1;
		if ((((1.0 - cos(x)) / sin(x)) <= 0.00010999914160827158)) {
			VAR_1 = fma(0.041666666666666664, pow(x, 3.0), fma(0.004166666666666667, pow(x, 5.0), (0.5 * x)));
		} else {
			VAR_1 = (expm1(log1p((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

Original30.3
Target0.0
Herbie0.9
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (- 1.0 (cos x)) (sin x)) < -0.04411318303918169

    1. Initial program 0.7

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

    3. Applied clear-num0.8

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

    if -0.04411318303918169 < (/ (- 1.0 (cos x)) (sin x)) < 0.00010999914160827158

    1. Initial program 59.1

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

    2. Taylor expanded around 0 0.9

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

    3. Simplified0.9

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

    if 0.00010999914160827158 < (/ (- 1.0 (cos x)) (sin x))

    1. Initial program 1.0

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

    3. Applied expm1-log1p-u1.1

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \cos x\right)\right)}}{\sin x}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0441131830391816929:\\ \;\;\;\;\frac{1}{\frac{\sin x}{1 - \cos x}}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 1.0999914160827158 \cdot 10^{-4}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \cos x\right)\right)}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +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)))