Average Error: 30.2 → 0.7
Time: 7.1s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0190087476176216581:\\ \;\;\;\;\frac{1}{\frac{\sin x}{1 - \cos x}}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 4.47527012274922384 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\log \left(e^{1 - \cos x}\right)\right)}}{\sin x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0190087476176216581:\\
\;\;\;\;\frac{1}{\frac{\sin x}{1 - \cos x}}\\

\mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 4.47527012274922384 \cdot 10^{-5}:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\

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

\end{array}
double f(double x) {
        double r62615 = 1.0;
        double r62616 = x;
        double r62617 = cos(r62616);
        double r62618 = r62615 - r62617;
        double r62619 = sin(r62616);
        double r62620 = r62618 / r62619;
        return r62620;
}
↓
double f(double x) {
        double r62621 = 1.0;
        double r62622 = x;
        double r62623 = cos(r62622);
        double r62624 = r62621 - r62623;
        double r62625 = sin(r62622);
        double r62626 = r62624 / r62625;
        double r62627 = -0.019008747617621658;
        bool r62628 = r62626 <= r62627;
        double r62629 = 1.0;
        double r62630 = r62625 / r62624;
        double r62631 = r62629 / r62630;
        double r62632 = 4.475270122749224e-05;
        bool r62633 = r62626 <= r62632;
        double r62634 = 0.041666666666666664;
        double r62635 = 3.0;
        double r62636 = pow(r62622, r62635);
        double r62637 = r62634 * r62636;
        double r62638 = 0.004166666666666667;
        double r62639 = 5.0;
        double r62640 = pow(r62622, r62639);
        double r62641 = r62638 * r62640;
        double r62642 = 0.5;
        double r62643 = r62642 * r62622;
        double r62644 = r62641 + r62643;
        double r62645 = r62637 + r62644;
        double r62646 = exp(r62624);
        double r62647 = log(r62646);
        double r62648 = log(r62647);
        double r62649 = exp(r62648);
        double r62650 = r62649 / r62625;
        double r62651 = r62633 ? r62645 : r62650;
        double r62652 = r62628 ? r62631 : r62651;
        return r62652;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.2
Target0
Herbie0.7
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (- 1.0 (cos x)) (sin x)) < -0.019008747617621658

    1. Initial program 0.8

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

    3. Applied clear-num0.8

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

    if -0.019008747617621658 < (/ (- 1.0 (cos x)) (sin x)) < 4.475270122749224e-05

    1. Initial program 59.6

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

    2. Taylor expanded around 0 0.4

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

    if 4.475270122749224e-05 < (/ (- 1.0 (cos x)) (sin x))

    1. Initial program 1.0

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

    3. Applied add-exp-log1.0

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

    5. Applied add-log-exp1.2

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

    6. Applied add-log-exp1.2

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

    7. Applied diff-log1.4

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

    8. Simplified1.2

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0190087476176216581:\\ \;\;\;\;\frac{1}{\frac{\sin x}{1 - \cos x}}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 4.47527012274922384 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\log \left(e^{1 - \cos x}\right)\right)}}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

  :herbie-target
  (tan (/ x 2))

  (/ (- 1 (cos x)) (sin x)))