Average Error: 30.4 → 0.6
Time: 7.4s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.001308568773005096318520767972870544326724:\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 4.957178312169810962668929968898134941213 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{\sqrt{\sin x}} \cdot \frac{\sqrt[3]{1}}{\sqrt{\sin x}}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.001308568773005096318520767972870544326724:\\
\;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\

\mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 4.957178312169810962668929968898134941213 \cdot 10^{-8}:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1 - \cos x}{\sqrt{\sin x}} \cdot \frac{\sqrt[3]{1}}{\sqrt{\sin x}}\\

\end{array}
double f(double x) {
        double r59122 = 1.0;
        double r59123 = x;
        double r59124 = cos(r59123);
        double r59125 = r59122 - r59124;
        double r59126 = sin(r59123);
        double r59127 = r59125 / r59126;
        return r59127;
}
↓
double f(double x) {
        double r59128 = 1.0;
        double r59129 = x;
        double r59130 = cos(r59129);
        double r59131 = r59128 - r59130;
        double r59132 = sin(r59129);
        double r59133 = r59131 / r59132;
        double r59134 = -0.0013085687730050963;
        bool r59135 = r59133 <= r59134;
        double r59136 = exp(r59133);
        double r59137 = log(r59136);
        double r59138 = 4.957178312169811e-08;
        bool r59139 = r59133 <= r59138;
        double r59140 = 0.041666666666666664;
        double r59141 = 3.0;
        double r59142 = pow(r59129, r59141);
        double r59143 = r59140 * r59142;
        double r59144 = 0.004166666666666667;
        double r59145 = 5.0;
        double r59146 = pow(r59129, r59145);
        double r59147 = r59144 * r59146;
        double r59148 = 0.5;
        double r59149 = r59148 * r59129;
        double r59150 = r59147 + r59149;
        double r59151 = r59143 + r59150;
        double r59152 = sqrt(r59132);
        double r59153 = r59131 / r59152;
        double r59154 = 1.0;
        double r59155 = cbrt(r59154);
        double r59156 = r59155 / r59152;
        double r59157 = r59153 * r59156;
        double r59158 = r59139 ? r59151 : r59157;
        double r59159 = r59135 ? r59137 : r59158;
        return r59159;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.4
Target0.0
Herbie0.6
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (- 1.0 (cos x)) (sin x)) < -0.0013085687730050963

    1. Initial program 0.9

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

    3. Applied add-log-exp1.1

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

    if -0.0013085687730050963 < (/ (- 1.0 (cos x)) (sin x)) < 4.957178312169811e-08

    1. Initial program 60.2

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

    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 4.957178312169811e-08 < (/ (- 1.0 (cos x)) (sin x))

    1. Initial program 1.3

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

    3. Applied add-exp-log1.3

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

    5. Applied div-inv1.3

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

    7. Applied add-sqr-sqrt1.5

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

    8. Applied add-cube-cbrt1.5

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

    9. Applied times-frac1.6

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

    10. Applied associate-*r*1.6

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

    11. Simplified1.5

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.001308568773005096318520767972870544326724:\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 4.957178312169810962668929968898134941213 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{\sqrt{\sin x}} \cdot \frac{\sqrt[3]{1}}{\sqrt{\sin x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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