Average Error: 30.4 → 0.5
Time: 7.5s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.015504801539486723 \lor \neg \left(x \le 0.024127924279373096\right):\\ \;\;\;\;\frac{\frac{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}{\frac{\cos x \cdot \frac{{\left(\cos x\right)}^{2} - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1}{e^{\log \left(1 - \cos x\right)}}}}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.015504801539486723 \lor \neg \left(x \le 0.024127924279373096\right):\\
\;\;\;\;\frac{\frac{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}{\frac{\cos x \cdot \frac{{\left(\cos x\right)}^{2} - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1}{e^{\log \left(1 - \cos x\right)}}}}{\sin 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 r164340 = 1.0;
        double r164341 = x;
        double r164342 = cos(r164341);
        double r164343 = r164340 - r164342;
        double r164344 = sin(r164341);
        double r164345 = r164343 / r164344;
        return r164345;
}
↓
double f(double x) {
        double r164346 = x;
        double r164347 = -0.015504801539486723;
        bool r164348 = r164346 <= r164347;
        double r164349 = 0.024127924279373096;
        bool r164350 = r164346 <= r164349;
        double r164351 = !r164350;
        bool r164352 = r164348 || r164351;
        double r164353 = 1.0;
        double r164354 = r164353 * r164353;
        double r164355 = cos(r164346);
        double r164356 = r164355 * r164355;
        double r164357 = r164353 * r164355;
        double r164358 = r164356 + r164357;
        double r164359 = r164354 + r164358;
        double r164360 = 2.0;
        double r164361 = pow(r164355, r164360);
        double r164362 = r164361 - r164354;
        double r164363 = r164355 - r164353;
        double r164364 = r164362 / r164363;
        double r164365 = r164355 * r164364;
        double r164366 = r164365 + r164354;
        double r164367 = r164353 - r164355;
        double r164368 = log(r164367);
        double r164369 = exp(r164368);
        double r164370 = r164366 / r164369;
        double r164371 = r164359 / r164370;
        double r164372 = sin(r164346);
        double r164373 = r164371 / r164372;
        double r164374 = 0.041666666666666664;
        double r164375 = 3.0;
        double r164376 = pow(r164346, r164375);
        double r164377 = r164374 * r164376;
        double r164378 = 0.004166666666666667;
        double r164379 = 5.0;
        double r164380 = pow(r164346, r164379);
        double r164381 = r164378 * r164380;
        double r164382 = 0.5;
        double r164383 = r164382 * r164346;
        double r164384 = r164381 + r164383;
        double r164385 = r164377 + r164384;
        double r164386 = r164352 ? r164373 : r164385;
        return r164386;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.4
Target0.0
Herbie0.5
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.015504801539486723 or 0.024127924279373096 < x

    1. Initial program 0.9

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

    3. Applied flip3--1.0

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

    4. Simplified1.1

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

    6. Applied difference-cubes1.0

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

    7. Applied associate-/l*1.0

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

    9. Applied add-exp-log1.0

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

    11. Applied flip-+1.0

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

    12. Simplified1.0

      \[\leadsto \frac{\frac{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}{\frac{\cos x \cdot \frac{\color{blue}{{\left(\cos x\right)}^{2} - 1 \cdot 1}}{\cos x - 1} + 1 \cdot 1}{e^{\log \left(1 - \cos x\right)}}}}{\sin x}\]

    if -0.015504801539486723 < x < 0.024127924279373096

    1. Initial program 59.9

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

    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 simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.015504801539486723 \lor \neg \left(x \le 0.024127924279373096\right):\\ \;\;\;\;\frac{\frac{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}{\frac{\cos x \cdot \frac{{\left(\cos x\right)}^{2} - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1}{e^{\log \left(1 - \cos x\right)}}}}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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