Average Error: 30.2 → 0.5
Time: 7.0s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.01849382956573876121697530550136434612796 \lor \neg \left(x \le 0.02173189858796714321598209096464415779337\right):\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\frac{\cos x \cdot \left(\cos x \cdot \cos x - 1 \cdot 1\right)}{\cos x - 1} + 1 \cdot 1}}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.01849382956573876121697530550136434612796 \lor \neg \left(x \le 0.02173189858796714321598209096464415779337\right):\\
\;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\frac{\cos x \cdot \left(\cos x \cdot \cos x - 1 \cdot 1\right)}{\cos x - 1} + 1 \cdot 1}}{\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 r113388 = 1.0;
        double r113389 = x;
        double r113390 = cos(r113389);
        double r113391 = r113388 - r113390;
        double r113392 = sin(r113389);
        double r113393 = r113391 / r113392;
        return r113393;
}
↓
double f(double x) {
        double r113394 = x;
        double r113395 = -0.01849382956573876;
        bool r113396 = r113394 <= r113395;
        double r113397 = 0.021731898587967143;
        bool r113398 = r113394 <= r113397;
        double r113399 = !r113398;
        bool r113400 = r113396 || r113399;
        double r113401 = 1.0;
        double r113402 = 3.0;
        double r113403 = pow(r113401, r113402);
        double r113404 = cos(r113394);
        double r113405 = pow(r113404, r113402);
        double r113406 = r113403 - r113405;
        double r113407 = r113404 * r113404;
        double r113408 = r113401 * r113401;
        double r113409 = r113407 - r113408;
        double r113410 = r113404 * r113409;
        double r113411 = r113404 - r113401;
        double r113412 = r113410 / r113411;
        double r113413 = r113412 + r113408;
        double r113414 = r113406 / r113413;
        double r113415 = sin(r113394);
        double r113416 = r113414 / r113415;
        double r113417 = 0.041666666666666664;
        double r113418 = pow(r113394, r113402);
        double r113419 = r113417 * r113418;
        double r113420 = 0.004166666666666667;
        double r113421 = 5.0;
        double r113422 = pow(r113394, r113421);
        double r113423 = r113420 * r113422;
        double r113424 = 0.5;
        double r113425 = r113424 * r113394;
        double r113426 = r113423 + r113425;
        double r113427 = r113419 + r113426;
        double r113428 = r113400 ? r113416 : r113427;
        return r113428;
}

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 2 regimes

  2. if x < -0.01849382956573876 or 0.021731898587967143 < 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. Simplified1.0

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

    6. Applied flip-+1.0

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

    7. Applied associate-*r/1.0

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

    if -0.01849382956573876 < x < 0.021731898587967143

    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 simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.01849382956573876121697530550136434612796 \lor \neg \left(x \le 0.02173189858796714321598209096464415779337\right):\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\frac{\cos x \cdot \left(\cos x \cdot \cos x - 1 \cdot 1\right)}{\cos x - 1} + 1 \cdot 1}}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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