Average Error: 29.9 → 0.5
Time: 21.0s
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 r60125 = 1.0;
        double r60126 = x;
        double r60127 = cos(r60126);
        double r60128 = r60125 - r60127;
        double r60129 = sin(r60126);
        double r60130 = r60128 / r60129;
        return r60130;
}
↓
double f(double x) {
        double r60131 = x;
        double r60132 = -0.024087893247376972;
        bool r60133 = r60131 <= r60132;
        double r60134 = 1.0;
        double r60135 = sin(r60131);
        double r60136 = 1.0;
        double r60137 = cos(r60131);
        double r60138 = r60136 - r60137;
        double r60139 = r60135 / r60138;
        double r60140 = r60134 / r60139;
        double r60141 = 0.020706483323998073;
        bool r60142 = r60131 <= r60141;
        double r60143 = 0.041666666666666664;
        double r60144 = 3.0;
        double r60145 = pow(r60131, r60144);
        double r60146 = r60143 * r60145;
        double r60147 = 0.004166666666666667;
        double r60148 = 5.0;
        double r60149 = pow(r60131, r60148);
        double r60150 = r60147 * r60149;
        double r60151 = 0.5;
        double r60152 = r60151 * r60131;
        double r60153 = r60150 + r60152;
        double r60154 = r60146 + r60153;
        double r60155 = pow(r60136, r60144);
        double r60156 = pow(r60137, r60144);
        double r60157 = pow(r60156, r60144);
        double r60158 = cbrt(r60157);
        double r60159 = r60155 - r60158;
        double r60160 = r60136 + r60137;
        double r60161 = r60137 * r60160;
        double r60162 = r60136 * r60136;
        double r60163 = r60161 + r60162;
        double r60164 = r60135 * r60163;
        double r60165 = r60159 / r60164;
        double r60166 = r60142 ? r60154 : r60165;
        double r60167 = r60133 ? r60140 : r60166;
        return r60167;
}

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)))