Average Error: 30.1 → 0.6
Time: 38.3s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.001608059772693871567958434631862019159598:\\ \;\;\;\;\frac{\frac{1 \cdot \left(1 \cdot 1\right) - \cos x \cdot \left(\cos x \cdot \cos x\right)}{\sin x}}{\left(\cos x \cdot \cos x + 1 \cdot \cos x\right) + 1 \cdot 1}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 7.540623703770628424457225097432555571686 \cdot 10^{-7}:\\ \;\;\;\;\left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right) + x \cdot \frac{1}{2}\right) + \frac{1}{240} \cdot {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.001608059772693871567958434631862019159598:\\
\;\;\;\;\frac{\frac{1 \cdot \left(1 \cdot 1\right) - \cos x \cdot \left(\cos x \cdot \cos x\right)}{\sin x}}{\left(\cos x \cdot \cos x + 1 \cdot \cos x\right) + 1 \cdot 1}\\

\mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 7.540623703770628424457225097432555571686 \cdot 10^{-7}:\\
\;\;\;\;\left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right) + x \cdot \frac{1}{2}\right) + \frac{1}{240} \cdot {x}^{5}\\

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

\end{array}
double f(double x) {
        double r2910872 = 1.0;
        double r2910873 = x;
        double r2910874 = cos(r2910873);
        double r2910875 = r2910872 - r2910874;
        double r2910876 = sin(r2910873);
        double r2910877 = r2910875 / r2910876;
        return r2910877;
}
↓
double f(double x) {
        double r2910878 = 1.0;
        double r2910879 = x;
        double r2910880 = cos(r2910879);
        double r2910881 = r2910878 - r2910880;
        double r2910882 = sin(r2910879);
        double r2910883 = r2910881 / r2910882;
        double r2910884 = -0.0016080597726938716;
        bool r2910885 = r2910883 <= r2910884;
        double r2910886 = r2910878 * r2910878;
        double r2910887 = r2910878 * r2910886;
        double r2910888 = r2910880 * r2910880;
        double r2910889 = r2910880 * r2910888;
        double r2910890 = r2910887 - r2910889;
        double r2910891 = r2910890 / r2910882;
        double r2910892 = r2910878 * r2910880;
        double r2910893 = r2910888 + r2910892;
        double r2910894 = r2910893 + r2910886;
        double r2910895 = r2910891 / r2910894;
        double r2910896 = 7.540623703770628e-07;
        bool r2910897 = r2910883 <= r2910896;
        double r2910898 = r2910879 * r2910879;
        double r2910899 = 0.041666666666666664;
        double r2910900 = r2910898 * r2910899;
        double r2910901 = r2910879 * r2910900;
        double r2910902 = 0.5;
        double r2910903 = r2910879 * r2910902;
        double r2910904 = r2910901 + r2910903;
        double r2910905 = 0.004166666666666667;
        double r2910906 = 5.0;
        double r2910907 = pow(r2910879, r2910906);
        double r2910908 = r2910905 * r2910907;
        double r2910909 = r2910904 + r2910908;
        double r2910910 = exp(r2910883);
        double r2910911 = log(r2910910);
        double r2910912 = r2910897 ? r2910909 : r2910911;
        double r2910913 = r2910885 ? r2910895 : r2910912;
        return r2910913;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.1
Target0
Herbie0.6
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (- 1.0 (cos x)) (sin x)) < -0.0016080597726938716

    1. Initial program 0.9

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

    3. Applied clear-num1.0

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{1 - \cos x}}}\]
    4. Using strategy rm

    5. Applied flip3--1.0

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

    6. Applied associate-/r/1.1

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

    7. Applied associate-/r*1.1

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

    8. Simplified1.0

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

    if -0.0016080597726938716 < (/ (- 1.0 (cos x)) (sin x)) < 7.540623703770628e-07

    1. Initial program 60.2

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

    2. Taylor expanded around 0 0

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

    3. Simplified0.0

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

    5. Applied distribute-lft-in0.0

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

    if 7.540623703770628e-07 < (/ (- 1.0 (cos x)) (sin x))

    1. Initial program 1.2

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

    3. Applied add-log-exp1.3

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.001608059772693871567958434631862019159598:\\ \;\;\;\;\frac{\frac{1 \cdot \left(1 \cdot 1\right) - \cos x \cdot \left(\cos x \cdot \cos x\right)}{\sin x}}{\left(\cos x \cdot \cos x + 1 \cdot \cos x\right) + 1 \cdot 1}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 7.540623703770628424457225097432555571686 \cdot 10^{-7}:\\ \;\;\;\;\left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right) + x \cdot \frac{1}{2}\right) + \frac{1}{240} \cdot {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 2

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

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