Average Error: 29.9 → 0.6
Time: 1.5m
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02234465016774620296780007322468009078875:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x \cdot \left(\cos x \cdot \cos x + \left(1 + \cos x\right) \cdot 1\right)}\\ \mathbf{elif}\;x \le 0.02382325057837424500672973692871892126277:\\ \;\;\;\;{x}^{5} \cdot \frac{1}{240} + x \cdot \left(x \cdot \left(\frac{1}{24} \cdot x\right) + \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02234465016774620296780007322468009078875:\\
\;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x \cdot \left(\cos x \cdot \cos x + \left(1 + \cos x\right) \cdot 1\right)}\\

\mathbf{elif}\;x \le 0.02382325057837424500672973692871892126277:\\
\;\;\;\;{x}^{5} \cdot \frac{1}{240} + x \cdot \left(x \cdot \left(\frac{1}{24} \cdot x\right) + \frac{1}{2}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\

\end{array}
double f(double x) {
        double r4216477 = 1.0;
        double r4216478 = x;
        double r4216479 = cos(r4216478);
        double r4216480 = r4216477 - r4216479;
        double r4216481 = sin(r4216478);
        double r4216482 = r4216480 / r4216481;
        return r4216482;
}
↓
double f(double x) {
        double r4216483 = x;
        double r4216484 = -0.022344650167746203;
        bool r4216485 = r4216483 <= r4216484;
        double r4216486 = 1.0;
        double r4216487 = 3.0;
        double r4216488 = pow(r4216486, r4216487);
        double r4216489 = cos(r4216483);
        double r4216490 = pow(r4216489, r4216487);
        double r4216491 = r4216488 - r4216490;
        double r4216492 = sin(r4216483);
        double r4216493 = r4216489 * r4216489;
        double r4216494 = r4216486 + r4216489;
        double r4216495 = r4216494 * r4216486;
        double r4216496 = r4216493 + r4216495;
        double r4216497 = r4216492 * r4216496;
        double r4216498 = r4216491 / r4216497;
        double r4216499 = 0.023823250578374245;
        bool r4216500 = r4216483 <= r4216499;
        double r4216501 = 5.0;
        double r4216502 = pow(r4216483, r4216501);
        double r4216503 = 0.004166666666666667;
        double r4216504 = r4216502 * r4216503;
        double r4216505 = 0.041666666666666664;
        double r4216506 = r4216505 * r4216483;
        double r4216507 = r4216483 * r4216506;
        double r4216508 = 0.5;
        double r4216509 = r4216507 + r4216508;
        double r4216510 = r4216483 * r4216509;
        double r4216511 = r4216504 + r4216510;
        double r4216512 = r4216486 / r4216492;
        double r4216513 = r4216489 / r4216492;
        double r4216514 = r4216512 - r4216513;
        double r4216515 = r4216500 ? r4216511 : r4216514;
        double r4216516 = r4216485 ? r4216498 : r4216515;
        return r4216516;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0
Herbie0.6
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.022344650167746203

    1. Initial program 0.9

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

    3. Applied flip3--1.0

      \[\leadsto \frac{\color{blue}{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}}{\sin x}\]

    4. Applied associate-/l/1.0

      \[\leadsto \color{blue}{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x \cdot \left(1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)\right)}}\]

    5. Simplified1.0

      \[\leadsto \frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\color{blue}{\left(\cos x \cdot \cos x + 1 \cdot \left(\cos x + 1\right)\right) \cdot \sin x}}\]

    if -0.022344650167746203 < x < 0.023823250578374245

    1. Initial program 59.8

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

    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

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

    if 0.023823250578374245 < x

    1. Initial program 0.9

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

    3. Applied div-sub1.2

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02234465016774620296780007322468009078875:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x \cdot \left(\cos x \cdot \cos x + \left(1 + \cos x\right) \cdot 1\right)}\\ \mathbf{elif}\;x \le 0.02382325057837424500672973692871892126277:\\ \;\;\;\;{x}^{5} \cdot \frac{1}{240} + x \cdot \left(x \cdot \left(\frac{1}{24} \cdot x\right) + \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 2

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

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