Average Error: 30.3 → 0.5
Time: 6.9s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.023228222164187143 \lor \neg \left(x \le 0.0219772417603219007\right):\\ \;\;\;\;\frac{1}{\sin x \cdot \frac{1}{1 - \cos x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.023228222164187143 \lor \neg \left(x \le 0.0219772417603219007\right):\\
\;\;\;\;\frac{1}{\sin x \cdot \frac{1}{1 - \cos x}}\\

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

\end{array}
double code(double x) {
	return ((1.0 - cos(x)) / sin(x));
}
↓
double code(double x) {
	double VAR;
	if (((x <= -0.023228222164187143) || !(x <= 0.0219772417603219))) {
		VAR = (1.0 / (sin(x) * (1.0 / (1.0 - cos(x)))));
	} else {
		VAR = fma(0.041666666666666664, pow(x, 3.0), fma(0.004166666666666667, pow(x, 5.0), (0.5 * x)));
	}
	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
Herbie0.5
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.023228222164187143 or 0.0219772417603219 < x

    1. Initial program 0.9

      \[\frac{1 - \cos x}{\sin x}\]
    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 div-inv1.0

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

    if -0.023228222164187143 < x < 0.0219772417603219

    1. Initial program 60.0

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

    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

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.023228222164187143 \lor \neg \left(x \le 0.0219772417603219007\right):\\ \;\;\;\;\frac{1}{\sin x \cdot \frac{1}{1 - \cos x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020079 +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)))