Average Error: 30.1 → 1.0
Time: 21.7s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.04371460353231055445677455395525612402707:\\ \;\;\;\;\frac{e^{\log \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}}{\sin x \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 6.2937074645496283672486959037684073337 \cdot 10^{-5}:\\ \;\;\;\;{x}^{5} \cdot \frac{1}{240} + \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{{e}^{\left(\log \left(1 - \cos x\right)\right)}}{\sin x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.04371460353231055445677455395525612402707:\\
\;\;\;\;\frac{e^{\log \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}}{\sin x \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}\\

\mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 6.2937074645496283672486959037684073337 \cdot 10^{-5}:\\
\;\;\;\;{x}^{5} \cdot \frac{1}{240} + \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x\right)\right) \cdot x\\

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

\end{array}
double f(double x) {
        double r59844 = 1.0;
        double r59845 = x;
        double r59846 = cos(r59845);
        double r59847 = r59844 - r59846;
        double r59848 = sin(r59845);
        double r59849 = r59847 / r59848;
        return r59849;
}
↓
double f(double x) {
        double r59850 = 1.0;
        double r59851 = x;
        double r59852 = cos(r59851);
        double r59853 = r59850 - r59852;
        double r59854 = sin(r59851);
        double r59855 = r59853 / r59854;
        double r59856 = -0.043714603532310554;
        bool r59857 = r59855 <= r59856;
        double r59858 = 3.0;
        double r59859 = pow(r59850, r59858);
        double r59860 = pow(r59852, r59858);
        double r59861 = r59859 - r59860;
        double r59862 = log(r59861);
        double r59863 = exp(r59862);
        double r59864 = r59850 + r59852;
        double r59865 = r59852 * r59864;
        double r59866 = r59850 * r59850;
        double r59867 = r59865 + r59866;
        double r59868 = r59854 * r59867;
        double r59869 = r59863 / r59868;
        double r59870 = 6.293707464549628e-05;
        bool r59871 = r59855 <= r59870;
        double r59872 = 5.0;
        double r59873 = pow(r59851, r59872);
        double r59874 = 0.004166666666666667;
        double r59875 = r59873 * r59874;
        double r59876 = 0.5;
        double r59877 = 0.041666666666666664;
        double r59878 = r59877 * r59851;
        double r59879 = r59851 * r59878;
        double r59880 = r59876 + r59879;
        double r59881 = r59880 * r59851;
        double r59882 = r59875 + r59881;
        double r59883 = exp(1.0);
        double r59884 = log(r59853);
        double r59885 = pow(r59883, r59884);
        double r59886 = r59885 / r59854;
        double r59887 = r59871 ? r59882 : r59886;
        double r59888 = r59857 ? r59869 : r59887;
        return r59888;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.1
Target0.0
Herbie1.0
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (- 1.0 (cos x)) (sin x)) < -0.043714603532310554

    1. Initial program 0.7

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

    3. Applied add-exp-log0.7

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

    5. Applied flip3--0.8

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

    6. Applied log-div0.8

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

    7. Applied exp-diff0.8

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

    8. Applied associate-/l/0.8

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

    9. Simplified0.8

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

    if -0.043714603532310554 < (/ (- 1.0 (cos x)) (sin x)) < 6.293707464549628e-05

    1. Initial program 59.2

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

    2. Taylor expanded around 0 0.9

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

    3. Simplified0.9

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

    if 6.293707464549628e-05 < (/ (- 1.0 (cos x)) (sin x))

    1. Initial program 1.1

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

    3. Applied add-exp-log1.1

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

    5. Applied pow11.1

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

    6. Applied log-pow1.1

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

    7. Applied exp-prod1.2

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

    8. Simplified1.2

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

    10. Applied pow11.2

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.04371460353231055445677455395525612402707:\\ \;\;\;\;\frac{e^{\log \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}}{\sin x \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 6.2937074645496283672486959037684073337 \cdot 10^{-5}:\\ \;\;\;\;{x}^{5} \cdot \frac{1}{240} + \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{{e}^{\left(\log \left(1 - \cos x\right)\right)}}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 2

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

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