Average Error: 30.3 → 0.7
Time: 7.9s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -2.76308119989467809 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{{1}^{3} - \log \left(e^{{\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \log \left(e^{\cos x + 1}\right) + 1 \cdot 1}}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le -0.0:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -2.76308119989467809 \cdot 10^{-4}:\\
\;\;\;\;\frac{\frac{{1}^{3} - \log \left(e^{{\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \log \left(e^{\cos x + 1}\right) + 1 \cdot 1}}{\sin x}\\

\mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le -0.0:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\

\end{array}
double f(double x) {
        double r65511 = 1.0;
        double r65512 = x;
        double r65513 = cos(r65512);
        double r65514 = r65511 - r65513;
        double r65515 = sin(r65512);
        double r65516 = r65514 / r65515;
        return r65516;
}
↓
double f(double x) {
        double r65517 = 1.0;
        double r65518 = x;
        double r65519 = cos(r65518);
        double r65520 = r65517 - r65519;
        double r65521 = sin(r65518);
        double r65522 = r65520 / r65521;
        double r65523 = -0.0002763081199894678;
        bool r65524 = r65522 <= r65523;
        double r65525 = 3.0;
        double r65526 = pow(r65517, r65525);
        double r65527 = pow(r65519, r65525);
        double r65528 = exp(r65527);
        double r65529 = log(r65528);
        double r65530 = r65526 - r65529;
        double r65531 = r65519 + r65517;
        double r65532 = exp(r65531);
        double r65533 = log(r65532);
        double r65534 = r65519 * r65533;
        double r65535 = r65517 * r65517;
        double r65536 = r65534 + r65535;
        double r65537 = r65530 / r65536;
        double r65538 = r65537 / r65521;
        double r65539 = -0.0;
        bool r65540 = r65522 <= r65539;
        double r65541 = 0.041666666666666664;
        double r65542 = pow(r65518, r65525);
        double r65543 = r65541 * r65542;
        double r65544 = 0.004166666666666667;
        double r65545 = 5.0;
        double r65546 = pow(r65518, r65545);
        double r65547 = r65544 * r65546;
        double r65548 = 0.5;
        double r65549 = r65548 * r65518;
        double r65550 = r65547 + r65549;
        double r65551 = r65543 + r65550;
        double r65552 = r65517 / r65521;
        double r65553 = r65519 / r65521;
        double r65554 = r65552 - r65553;
        double r65555 = r65540 ? r65551 : r65554;
        double r65556 = r65524 ? r65538 : r65555;
        return r65556;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.3
Target0.0
Herbie0.7
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (- 1.0 (cos x)) (sin x)) < -0.0002763081199894678

    1. Initial program 1.0

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

    3. Applied flip3--1.1

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

    4. Simplified1.2

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

    6. Applied add-log-exp1.2

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

    8. Applied add-log-exp1.2

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

    9. Applied add-log-exp1.2

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

    10. Applied sum-log1.2

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

    11. Simplified1.2

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

    if -0.0002763081199894678 < (/ (- 1.0 (cos x)) (sin x)) < -0.0

    1. Initial program 60.3

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

    2. Taylor expanded around 0 0.0

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

    if -0.0 < (/ (- 1.0 (cos x)) (sin x))

    1. Initial program 1.3

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

    3. Applied div-sub1.5

      \[\leadsto \color{blue}{\frac{1}{\sin x} - \frac{\cos x}{\sin x}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -2.76308119989467809 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{{1}^{3} - \log \left(e^{{\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \log \left(e^{\cos x + 1}\right) + 1 \cdot 1}}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le -0.0:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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