Average Error: 30.2 → 1.3
Time: 8.0s
Precision: binary64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \leq -0.0760756754162463:\\ \;\;\;\;\frac{1}{\frac{\sin x}{1 - \cos x}}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \leq 2.542233451550557 \cdot 10^{-08}:\\ \;\;\;\;0.041666666666666664 \cdot {x}^{3} + \left(0.004166666666666667 \cdot {x}^{5} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \cos x\right) \cdot \frac{1}{\sin x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1 - \cos x}{\sin x} \leq -0.0760756754162463:\\
\;\;\;\;\frac{1}{\frac{\sin x}{1 - \cos x}}\\

\mathbf{elif}\;\frac{1 - \cos x}{\sin x} \leq 2.542233451550557 \cdot 10^{-08}:\\
\;\;\;\;0.041666666666666664 \cdot {x}^{3} + \left(0.004166666666666667 \cdot {x}^{5} + x \cdot 0.5\right)\\

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

\end{array}
double code(double x) {
	return (((double) (1.0 - ((double) cos(x)))) / ((double) sin(x)));
}
↓
double code(double x) {
	double VAR;
	if (((((double) (1.0 - ((double) cos(x)))) / ((double) sin(x))) <= -0.0760756754162463)) {
		VAR = (1.0 / (((double) sin(x)) / ((double) (1.0 - ((double) cos(x))))));
	} else {
		double VAR_1;
		if (((((double) (1.0 - ((double) cos(x)))) / ((double) sin(x))) <= 2.542233451550557e-08)) {
			VAR_1 = ((double) (((double) (0.041666666666666664 * ((double) pow(x, 3.0)))) + ((double) (((double) (0.004166666666666667 * ((double) pow(x, 5.0)))) + ((double) (x * 0.5))))));
		} else {
			VAR_1 = ((double) (((double) (1.0 - ((double) cos(x)))) * (1.0 / ((double) 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

Original30.2
Target0.0
Herbie1.3
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (- 1.0 (cos x)) (sin x)) < -0.076075675416246297

    1. Initial program 0.6

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

    3. Applied clear-num0.6

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

    if -0.076075675416246297 < (/ (- 1.0 (cos x)) (sin x)) < 2.542233451550557e-8

    1. Initial program 58.9

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

    2. Taylor expanded around 0 1.5

      \[\leadsto \color{blue}{0.041666666666666664 \cdot {x}^{3} + \left(0.004166666666666667 \cdot {x}^{5} + 0.5 \cdot x\right)}\]

    3. Simplified1.5

      \[\leadsto \color{blue}{0.041666666666666664 \cdot {x}^{3} + \left(0.004166666666666667 \cdot {x}^{5} + x \cdot 0.5\right)}\]

    if 2.542233451550557e-8 < (/ (- 1.0 (cos x)) (sin x))

    1. Initial program 1.5

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

    3. Applied div-inv1.5

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \leq -0.0760756754162463:\\ \;\;\;\;\frac{1}{\frac{\sin x}{1 - \cos x}}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \leq 2.542233451550557 \cdot 10^{-08}:\\ \;\;\;\;0.041666666666666664 \cdot {x}^{3} + \left(0.004166666666666667 \cdot {x}^{5} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \cos x\right) \cdot \frac{1}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020199 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

  :herbie-target
  (tan (/ x 2.0))

  (/ (- 1.0 (cos x)) (sin x)))