Average Error: 30.2 → 0.5
Time: 19.7s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.01962732699026020072308185149267956148833:\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \mathbf{elif}\;x \le 0.02449600219494943789677598999787733191624:\\ \;\;\;\;\left({x}^{5} \cdot \frac{1}{240} + \frac{1}{2} \cdot x\right) + \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{24}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \cos x} \cdot \frac{\left(1 + \cos x\right) \cdot \left(1 - \cos x\right)}{\sin x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.01962732699026020072308185149267956148833:\\
\;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\

\mathbf{elif}\;x \le 0.02449600219494943789677598999787733191624:\\
\;\;\;\;\left({x}^{5} \cdot \frac{1}{240} + \frac{1}{2} \cdot x\right) + \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{24}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{1 + \cos x} \cdot \frac{\left(1 + \cos x\right) \cdot \left(1 - \cos x\right)}{\sin x}\\

\end{array}
double f(double x) {
        double r2504279 = 1.0;
        double r2504280 = x;
        double r2504281 = cos(r2504280);
        double r2504282 = r2504279 - r2504281;
        double r2504283 = sin(r2504280);
        double r2504284 = r2504282 / r2504283;
        return r2504284;
}
↓
double f(double x) {
        double r2504285 = x;
        double r2504286 = -0.0196273269902602;
        bool r2504287 = r2504285 <= r2504286;
        double r2504288 = 1.0;
        double r2504289 = cos(r2504285);
        double r2504290 = r2504288 - r2504289;
        double r2504291 = sin(r2504285);
        double r2504292 = r2504290 / r2504291;
        double r2504293 = exp(r2504292);
        double r2504294 = log(r2504293);
        double r2504295 = 0.024496002194949438;
        bool r2504296 = r2504285 <= r2504295;
        double r2504297 = 5.0;
        double r2504298 = pow(r2504285, r2504297);
        double r2504299 = 0.004166666666666667;
        double r2504300 = r2504298 * r2504299;
        double r2504301 = 0.5;
        double r2504302 = r2504301 * r2504285;
        double r2504303 = r2504300 + r2504302;
        double r2504304 = r2504285 * r2504285;
        double r2504305 = r2504285 * r2504304;
        double r2504306 = 0.041666666666666664;
        double r2504307 = r2504305 * r2504306;
        double r2504308 = r2504303 + r2504307;
        double r2504309 = 1.0;
        double r2504310 = r2504288 + r2504289;
        double r2504311 = r2504309 / r2504310;
        double r2504312 = r2504310 * r2504290;
        double r2504313 = r2504312 / r2504291;
        double r2504314 = r2504311 * r2504313;
        double r2504315 = r2504296 ? r2504308 : r2504314;
        double r2504316 = r2504287 ? r2504294 : r2504315;
        return r2504316;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.2
Target0.0
Herbie0.5
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.0196273269902602

    1. Initial program 0.9

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

    3. Applied add-log-exp1.0

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

    if -0.0196273269902602 < x < 0.024496002194949438

    1. Initial program 60.0

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

    3. Applied clear-num60.0

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

    4. Taylor expanded around 0 0.0

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

    5. Simplified0.0

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

    if 0.024496002194949438 < x

    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 flip--1.4

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

    6. Applied associate-/r/1.4

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

    7. Applied *-un-lft-identity1.4

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

    8. Applied times-frac1.4

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

    9. Simplified1.1

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.01962732699026020072308185149267956148833:\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \mathbf{elif}\;x \le 0.02449600219494943789677598999787733191624:\\ \;\;\;\;\left({x}^{5} \cdot \frac{1}{240} + \frac{1}{2} \cdot x\right) + \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{24}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \cos x} \cdot \frac{\left(1 + \cos x\right) \cdot \left(1 - \cos x\right)}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 2

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

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