Average Error: 30.3 → 0.8
Time: 23.0s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.02115882893523894289766751342085626674816 \lor \neg \left(\frac{1 - \cos x}{\sin x} \le 7.646982874077240654179165799120276858503 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{1}{\frac{\sin x}{1 - \cos x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.02115882893523894289766751342085626674816 \lor \neg \left(\frac{1 - \cos x}{\sin x} \le 7.646982874077240654179165799120276858503 \cdot 10^{-8}\right):\\
\;\;\;\;\frac{1}{\frac{\sin x}{1 - \cos x}}\\

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

\end{array}
double f(double x) {
        double r54825 = 1.0;
        double r54826 = x;
        double r54827 = cos(r54826);
        double r54828 = r54825 - r54827;
        double r54829 = sin(r54826);
        double r54830 = r54828 / r54829;
        return r54830;
}
↓
double f(double x) {
        double r54831 = 1.0;
        double r54832 = x;
        double r54833 = cos(r54832);
        double r54834 = r54831 - r54833;
        double r54835 = sin(r54832);
        double r54836 = r54834 / r54835;
        double r54837 = -0.021158828935238943;
        bool r54838 = r54836 <= r54837;
        double r54839 = 7.64698287407724e-08;
        bool r54840 = r54836 <= r54839;
        double r54841 = !r54840;
        bool r54842 = r54838 || r54841;
        double r54843 = 1.0;
        double r54844 = r54835 / r54834;
        double r54845 = r54843 / r54844;
        double r54846 = 0.041666666666666664;
        double r54847 = 3.0;
        double r54848 = pow(r54832, r54847);
        double r54849 = r54846 * r54848;
        double r54850 = 0.004166666666666667;
        double r54851 = 5.0;
        double r54852 = pow(r54832, r54851);
        double r54853 = r54850 * r54852;
        double r54854 = 0.5;
        double r54855 = r54854 * r54832;
        double r54856 = r54853 + r54855;
        double r54857 = r54849 + r54856;
        double r54858 = r54842 ? r54845 : r54857;
        return r54858;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.3
Target0.0
Herbie0.8
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- 1.0 (cos x)) (sin x)) < -0.021158828935238943 or 7.64698287407724e-08 < (/ (- 1.0 (cos x)) (sin x))

    1. Initial program 1.0

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

    3. Applied clear-num1.1

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

    if -0.021158828935238943 < (/ (- 1.0 (cos x)) (sin x)) < 7.64698287407724e-08

    1. Initial program 59.8

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

    2. Taylor expanded around 0 0.5

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.02115882893523894289766751342085626674816 \lor \neg \left(\frac{1 - \cos x}{\sin x} \le 7.646982874077240654179165799120276858503 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{1}{\frac{\sin x}{1 - \cos x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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