Average Error: 30.1 → 0.6
Time: 8.0s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0199905813077168142 \lor \neg \left(x \le 0.021452837467613652\right):\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left(\frac{1 - \cos x}{\sin x}\right)}^{3}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0199905813077168142 \lor \neg \left(x \le 0.021452837467613652\right):\\
\;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left(\frac{1 - \cos x}{\sin x}\right)}^{3}}\right)}^{3}}\\

\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 r60288 = 1.0;
        double r60289 = x;
        double r60290 = cos(r60289);
        double r60291 = r60288 - r60290;
        double r60292 = sin(r60289);
        double r60293 = r60291 / r60292;
        return r60293;
}
↓
double f(double x) {
        double r60294 = x;
        double r60295 = -0.019990581307716814;
        bool r60296 = r60294 <= r60295;
        double r60297 = 0.02145283746761365;
        bool r60298 = r60294 <= r60297;
        double r60299 = !r60298;
        bool r60300 = r60296 || r60299;
        double r60301 = 1.0;
        double r60302 = cos(r60294);
        double r60303 = r60301 - r60302;
        double r60304 = sin(r60294);
        double r60305 = r60303 / r60304;
        double r60306 = 3.0;
        double r60307 = pow(r60305, r60306);
        double r60308 = cbrt(r60307);
        double r60309 = pow(r60308, r60306);
        double r60310 = cbrt(r60309);
        double r60311 = 0.041666666666666664;
        double r60312 = pow(r60294, r60306);
        double r60313 = r60311 * r60312;
        double r60314 = 0.004166666666666667;
        double r60315 = 5.0;
        double r60316 = pow(r60294, r60315);
        double r60317 = r60314 * r60316;
        double r60318 = 0.5;
        double r60319 = r60318 * r60294;
        double r60320 = r60317 + r60319;
        double r60321 = r60313 + r60320;
        double r60322 = r60300 ? r60310 : r60321;
        return r60322;
}

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 2 regimes

  2. if x < -0.019990581307716814 or 0.02145283746761365 < x

    1. Initial program 0.9

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

    3. Applied add-cbrt-cube1.1

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

    4. Applied add-cbrt-cube1.3

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

    5. Applied cbrt-undiv1.1

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

    6. Simplified1.2

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

    8. Applied add-cbrt-cube1.3

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

    9. Applied add-cbrt-cube1.4

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

    10. Applied cbrt-undiv1.3

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

    11. Simplified1.2

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

    if -0.019990581307716814 < x < 0.02145283746761365

    1. Initial program 59.8

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

    2. Taylor expanded around 0 0.0

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0199905813077168142 \lor \neg \left(x \le 0.021452837467613652\right):\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left(\frac{1 - \cos x}{\sin x}\right)}^{3}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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