Average Error: 30.1 → 0.7
Time: 8.2s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0107629324502667868:\\ \;\;\;\;\frac{\frac{{1}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1} - \frac{{\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 4.3298603204685633 \cdot 10^{-4}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{e^{\frac{1 - \cos x}{\sin x}}} \cdot \sqrt{e^{\frac{1 - \cos x}{\sin x}}}\right)\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0107629324502667868:\\
\;\;\;\;\frac{\frac{{1}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1} - \frac{{\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{\sin x}\\

\mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 4.3298603204685633 \cdot 10^{-4}:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\sqrt{e^{\frac{1 - \cos x}{\sin x}}} \cdot \sqrt{e^{\frac{1 - \cos x}{\sin x}}}\right)\\

\end{array}
double f(double x) {
        double r43513 = 1.0;
        double r43514 = x;
        double r43515 = cos(r43514);
        double r43516 = r43513 - r43515;
        double r43517 = sin(r43514);
        double r43518 = r43516 / r43517;
        return r43518;
}
↓
double f(double x) {
        double r43519 = 1.0;
        double r43520 = x;
        double r43521 = cos(r43520);
        double r43522 = r43519 - r43521;
        double r43523 = sin(r43520);
        double r43524 = r43522 / r43523;
        double r43525 = -0.010762932450266787;
        bool r43526 = r43524 <= r43525;
        double r43527 = 3.0;
        double r43528 = pow(r43519, r43527);
        double r43529 = r43521 + r43519;
        double r43530 = r43521 * r43529;
        double r43531 = r43519 * r43519;
        double r43532 = r43530 + r43531;
        double r43533 = r43528 / r43532;
        double r43534 = pow(r43521, r43527);
        double r43535 = r43534 / r43532;
        double r43536 = r43533 - r43535;
        double r43537 = r43536 / r43523;
        double r43538 = 0.00043298603204685633;
        bool r43539 = r43524 <= r43538;
        double r43540 = 0.041666666666666664;
        double r43541 = pow(r43520, r43527);
        double r43542 = r43540 * r43541;
        double r43543 = 0.004166666666666667;
        double r43544 = 5.0;
        double r43545 = pow(r43520, r43544);
        double r43546 = r43543 * r43545;
        double r43547 = 0.5;
        double r43548 = r43547 * r43520;
        double r43549 = r43546 + r43548;
        double r43550 = r43542 + r43549;
        double r43551 = exp(r43524);
        double r43552 = sqrt(r43551);
        double r43553 = r43552 * r43552;
        double r43554 = log(r43553);
        double r43555 = r43539 ? r43550 : r43554;
        double r43556 = r43526 ? r43537 : r43555;
        return r43556;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.1
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)) < -0.010762932450266787

    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 div-sub1.0

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

    if -0.010762932450266787 < (/ (- 1.0 (cos x)) (sin x)) < 0.00043298603204685633

    1. Initial program 59.8

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

    2. Taylor expanded around 0 0.3

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

    if 0.00043298603204685633 < (/ (- 1.0 (cos x)) (sin x))

    1. Initial program 1.0

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

    3. Applied add-log-exp1.0

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

    5. Applied add-sqr-sqrt1.3

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0107629324502667868:\\ \;\;\;\;\frac{\frac{{1}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1} - \frac{{\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 4.3298603204685633 \cdot 10^{-4}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{e^{\frac{1 - \cos x}{\sin x}}} \cdot \sqrt{e^{\frac{1 - \cos x}{\sin x}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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