Average Error: 30.4 → 0.5
Time: 14.7s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0227880187083014697224658817731324234046 \lor \neg \left(x \le 0.02338074252416355927608471176881721476093\right):\\ \;\;\;\;\frac{e^{\log \left(1 - \cos x\right)}}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0227880187083014697224658817731324234046 \lor \neg \left(x \le 0.02338074252416355927608471176881721476093\right):\\
\;\;\;\;\frac{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 r56394 = 1.0;
        double r56395 = x;
        double r56396 = cos(r56395);
        double r56397 = r56394 - r56396;
        double r56398 = sin(r56395);
        double r56399 = r56397 / r56398;
        return r56399;
}
↓
double f(double x) {
        double r56400 = x;
        double r56401 = -0.02278801870830147;
        bool r56402 = r56400 <= r56401;
        double r56403 = 0.02338074252416356;
        bool r56404 = r56400 <= r56403;
        double r56405 = !r56404;
        bool r56406 = r56402 || r56405;
        double r56407 = 1.0;
        double r56408 = cos(r56400);
        double r56409 = r56407 - r56408;
        double r56410 = log(r56409);
        double r56411 = exp(r56410);
        double r56412 = sin(r56400);
        double r56413 = r56411 / r56412;
        double r56414 = 0.041666666666666664;
        double r56415 = 3.0;
        double r56416 = pow(r56400, r56415);
        double r56417 = r56414 * r56416;
        double r56418 = 0.004166666666666667;
        double r56419 = 5.0;
        double r56420 = pow(r56400, r56419);
        double r56421 = r56418 * r56420;
        double r56422 = 0.5;
        double r56423 = r56422 * r56400;
        double r56424 = r56421 + r56423;
        double r56425 = r56417 + r56424;
        double r56426 = r56406 ? r56413 : r56425;
        return r56426;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 2 regimes

  2. if x < -0.02278801870830147 or 0.02338074252416356 < x

    1. Initial program 0.9

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

    3. Applied add-log-exp1.1

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

    4. Applied add-log-exp1.1

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

    5. Applied diff-log1.3

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

    6. Simplified1.1

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

    8. Applied add-exp-log1.1

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

    9. Simplified0.9

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

    if -0.02278801870830147 < x < 0.02338074252416356

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0227880187083014697224658817731324234046 \lor \neg \left(x \le 0.02338074252416355927608471176881721476093\right):\\ \;\;\;\;\frac{e^{\log \left(1 - \cos x\right)}}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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