Average Error: 29.9 → 0.5
Time: 20.7s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02408789324737697218337117988085083197802:\\ \;\;\;\;\frac{1}{\frac{\sin x}{1 - \cos x}}\\ \mathbf{elif}\;x \le 0.02070648332399807264869728840039897477254:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3} - \sqrt[3]{{\left({\left(\cos x\right)}^{3}\right)}^{3}}}{\sin x \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02408789324737697218337117988085083197802:\\
\;\;\;\;\frac{1}{\frac{\sin x}{1 - \cos x}}\\

\mathbf{elif}\;x \le 0.02070648332399807264869728840039897477254:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{{1}^{3} - \sqrt[3]{{\left({\left(\cos x\right)}^{3}\right)}^{3}}}{\sin x \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}\\

\end{array}
double f(double x) {
        double r68016 = 1.0;
        double r68017 = x;
        double r68018 = cos(r68017);
        double r68019 = r68016 - r68018;
        double r68020 = sin(r68017);
        double r68021 = r68019 / r68020;
        return r68021;
}
↓
double f(double x) {
        double r68022 = x;
        double r68023 = -0.024087893247376972;
        bool r68024 = r68022 <= r68023;
        double r68025 = 1.0;
        double r68026 = sin(r68022);
        double r68027 = 1.0;
        double r68028 = cos(r68022);
        double r68029 = r68027 - r68028;
        double r68030 = r68026 / r68029;
        double r68031 = r68025 / r68030;
        double r68032 = 0.020706483323998073;
        bool r68033 = r68022 <= r68032;
        double r68034 = 0.041666666666666664;
        double r68035 = 3.0;
        double r68036 = pow(r68022, r68035);
        double r68037 = r68034 * r68036;
        double r68038 = 0.004166666666666667;
        double r68039 = 5.0;
        double r68040 = pow(r68022, r68039);
        double r68041 = r68038 * r68040;
        double r68042 = 0.5;
        double r68043 = r68042 * r68022;
        double r68044 = r68041 + r68043;
        double r68045 = r68037 + r68044;
        double r68046 = pow(r68027, r68035);
        double r68047 = pow(r68028, r68035);
        double r68048 = pow(r68047, r68035);
        double r68049 = cbrt(r68048);
        double r68050 = r68046 - r68049;
        double r68051 = r68027 + r68028;
        double r68052 = r68028 * r68051;
        double r68053 = r68027 * r68027;
        double r68054 = r68052 + r68053;
        double r68055 = r68026 * r68054;
        double r68056 = r68050 / r68055;
        double r68057 = r68033 ? r68045 : r68056;
        double r68058 = r68024 ? r68031 : r68057;
        return r68058;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.0
Herbie0.5
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.024087893247376972

    1. Initial program 0.9

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

    3. Applied clear-num1.0

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

    if -0.024087893247376972 < x < 0.020706483323998073

    1. Initial program 59.8

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

    2. Taylor expanded around 0 0.0

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

    if 0.020706483323998073 < x

    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

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

    7. Applied add-cbrt-cube1.1

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

    8. Simplified1.1

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02408789324737697218337117988085083197802:\\ \;\;\;\;\frac{1}{\frac{\sin x}{1 - \cos x}}\\ \mathbf{elif}\;x \le 0.02070648332399807264869728840039897477254:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3} - \sqrt[3]{{\left({\left(\cos x\right)}^{3}\right)}^{3}}}{\sin x \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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