Average Error: 31.0 → 0.6
Time: 7.1s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.014503396267940102:\\ \;\;\;\;\frac{\frac{1}{\sin x} \cdot \frac{1}{\sin x} - \frac{\cos x}{\sin x} \cdot \frac{\cos x}{\sin x}}{\frac{1}{\sin x} + \frac{\cos x}{\sin x}}\\ \mathbf{elif}\;x \le 0.024683183633800634:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{1 - \cos x}{\sin x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.014503396267940102:\\
\;\;\;\;\frac{\frac{1}{\sin x} \cdot \frac{1}{\sin x} - \frac{\cos x}{\sin x} \cdot \frac{\cos x}{\sin x}}{\frac{1}{\sin x} + \frac{\cos x}{\sin x}}\\

\mathbf{elif}\;x \le 0.024683183633800634:\\
\;\;\;\;\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}:\\
\;\;\;\;1 \cdot \frac{1 - \cos x}{\sin x}\\

\end{array}
double code(double x) {
	return ((double) (((double) (1.0 - ((double) cos(x)))) / ((double) sin(x))));
}
↓
double code(double x) {
	double VAR;
	if ((x <= -0.014503396267940102)) {
		VAR = ((double) (((double) (((double) (((double) (1.0 / ((double) sin(x)))) * ((double) (1.0 / ((double) sin(x)))))) - ((double) (((double) (((double) cos(x)) / ((double) sin(x)))) * ((double) (((double) cos(x)) / ((double) sin(x)))))))) / ((double) (((double) (1.0 / ((double) sin(x)))) + ((double) (((double) cos(x)) / ((double) sin(x))))))));
	} else {
		double VAR_1;
		if ((x <= 0.024683183633800634)) {
			VAR_1 = ((double) fma(0.041666666666666664, ((double) pow(x, 3.0)), ((double) fma(0.004166666666666667, ((double) pow(x, 5.0)), ((double) (0.5 * x))))));
		} else {
			VAR_1 = ((double) (1.0 * ((double) (((double) (1.0 - ((double) cos(x)))) / ((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

Original31.0
Target0.0
Herbie0.6
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.014503396267940102

    1. Initial program 0.9

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

    3. Applied div-sub1.2

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

    5. Applied flip--1.6

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

    if -0.014503396267940102 < x < 0.024683183633800634

    1. Initial program 59.9

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

    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 0.024683183633800634 < x

    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 pow11.0

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

    6. Applied log-pow1.0

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

    7. Simplified0.9

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.014503396267940102:\\ \;\;\;\;\frac{\frac{1}{\sin x} \cdot \frac{1}{\sin x} - \frac{\cos x}{\sin x} \cdot \frac{\cos x}{\sin x}}{\frac{1}{\sin x} + \frac{\cos x}{\sin x}}\\ \mathbf{elif}\;x \le 0.024683183633800634:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{1 - \cos x}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 +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)))