Average Error: 30.4 → 0.5
Time: 7.7s
Precision: binary64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.024260903871319413:\\ \;\;\;\;\frac{e^{\log \left(\log \left(e^{1 - \cos x}\right)\right)}}{\sin x}\\ \mathbf{elif}\;x \le 0.0222724579727386039:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + x \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \cos x\right) \cdot \frac{1}{\sin x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.024260903871319413:\\
\;\;\;\;\frac{e^{\log \left(\log \left(e^{1 - \cos x}\right)\right)}}{\sin x}\\

\mathbf{elif}\;x \le 0.0222724579727386039:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + x \cdot \frac{1}{2}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(1 - \cos x\right) \cdot \frac{1}{\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 <= -0.024260903871319413)) {
		VAR = ((double) (((double) exp(((double) log(((double) log(((double) exp(((double) (1.0 - ((double) cos(x)))))))))))) / ((double) sin(x))));
	} else {
		double VAR_1;
		if ((x <= 0.022272457972738604)) {
			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)))) * ((double) (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

Original30.4
Target0.0
Herbie0.5
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.024260903871319413

    1. Initial program 0.9

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

    3. Applied add-exp-log0.9

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

    5. Applied add-log-exp1.1

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

    6. Applied add-log-exp1.1

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

    7. Applied diff-log1.3

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

    8. Simplified1.1

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

    if -0.024260903871319413 < x < 0.0222724579727386039

    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

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

    if 0.0222724579727386039 < x

    1. Initial program 0.9

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

    3. Applied div-inv1.0

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.024260903871319413:\\ \;\;\;\;\frac{e^{\log \left(\log \left(e^{1 - \cos x}\right)\right)}}{\sin x}\\ \mathbf{elif}\;x \le 0.0222724579727386039:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + x \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \cos x\right) \cdot \frac{1}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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