Average Error: 30.1 → 0.7
Time: 8.4s
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 r62677 = 1.0;
        double r62678 = x;
        double r62679 = cos(r62678);
        double r62680 = r62677 - r62679;
        double r62681 = sin(r62678);
        double r62682 = r62680 / r62681;
        return r62682;
}
↓
double f(double x) {
        double r62683 = 1.0;
        double r62684 = x;
        double r62685 = cos(r62684);
        double r62686 = r62683 - r62685;
        double r62687 = sin(r62684);
        double r62688 = r62686 / r62687;
        double r62689 = -0.010762932450266787;
        bool r62690 = r62688 <= r62689;
        double r62691 = 3.0;
        double r62692 = pow(r62683, r62691);
        double r62693 = r62685 + r62683;
        double r62694 = r62685 * r62693;
        double r62695 = r62683 * r62683;
        double r62696 = r62694 + r62695;
        double r62697 = r62692 / r62696;
        double r62698 = pow(r62685, r62691);
        double r62699 = r62698 / r62696;
        double r62700 = r62697 - r62699;
        double r62701 = r62700 / r62687;
        double r62702 = 0.00043298603204685633;
        bool r62703 = r62688 <= r62702;
        double r62704 = 0.041666666666666664;
        double r62705 = pow(r62684, r62691);
        double r62706 = r62704 * r62705;
        double r62707 = 0.004166666666666667;
        double r62708 = 5.0;
        double r62709 = pow(r62684, r62708);
        double r62710 = r62707 * r62709;
        double r62711 = 0.5;
        double r62712 = r62711 * r62684;
        double r62713 = r62710 + r62712;
        double r62714 = r62706 + r62713;
        double r62715 = exp(r62688);
        double r62716 = sqrt(r62715);
        double r62717 = r62716 * r62716;
        double r62718 = log(r62717);
        double r62719 = r62703 ? r62714 : r62718;
        double r62720 = r62690 ? r62701 : r62719;
        return r62720;
}

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)))