Average Error: 30.3 → 0.8
Time: 7.5s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.00408598860077935295:\\ \;\;\;\;\log \left(e^{\frac{1}{\sin x} - \frac{\cos x}{\sin x}}\right)\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le -0.0:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}}{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot \sin x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.00408598860077935295:\\
\;\;\;\;\log \left(e^{\frac{1}{\sin x} - \frac{\cos x}{\sin x}}\right)\\

\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{e^{\log \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}}{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot \sin x}\\

\end{array}
double f(double x) {
        double r66589 = 1.0;
        double r66590 = x;
        double r66591 = cos(r66590);
        double r66592 = r66589 - r66591;
        double r66593 = sin(r66590);
        double r66594 = r66592 / r66593;
        return r66594;
}
↓
double f(double x) {
        double r66595 = 1.0;
        double r66596 = x;
        double r66597 = cos(r66596);
        double r66598 = r66595 - r66597;
        double r66599 = sin(r66596);
        double r66600 = r66598 / r66599;
        double r66601 = -0.004085988600779353;
        bool r66602 = r66600 <= r66601;
        double r66603 = r66595 / r66599;
        double r66604 = r66597 / r66599;
        double r66605 = r66603 - r66604;
        double r66606 = exp(r66605);
        double r66607 = log(r66606);
        double r66608 = -0.0;
        bool r66609 = r66600 <= r66608;
        double r66610 = 0.041666666666666664;
        double r66611 = 3.0;
        double r66612 = pow(r66596, r66611);
        double r66613 = r66610 * r66612;
        double r66614 = 0.004166666666666667;
        double r66615 = 5.0;
        double r66616 = pow(r66596, r66615);
        double r66617 = r66614 * r66616;
        double r66618 = 0.5;
        double r66619 = r66618 * r66596;
        double r66620 = r66617 + r66619;
        double r66621 = r66613 + r66620;
        double r66622 = pow(r66595, r66611);
        double r66623 = pow(r66597, r66611);
        double r66624 = r66622 - r66623;
        double r66625 = log(r66624);
        double r66626 = exp(r66625);
        double r66627 = r66597 + r66595;
        double r66628 = r66597 * r66627;
        double r66629 = r66595 * r66595;
        double r66630 = r66628 + r66629;
        double r66631 = r66630 * r66599;
        double r66632 = r66626 / r66631;
        double r66633 = r66609 ? r66621 : r66632;
        double r66634 = r66602 ? r66607 : r66633;
        return r66634;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.3
Target0
Herbie0.8
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (- 1.0 (cos x)) (sin x)) < -0.004085988600779353

    1. Initial program 0.9

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

    3. Applied add-log-exp1.1

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

    5. Applied div-sub1.2

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

    if -0.004085988600779353 < (/ (- 1.0 (cos x)) (sin x)) < -0.0

    1. Initial program 60.1

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

    2. Taylor expanded around 0 0.1

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

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

    1. Initial program 1.5

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

    3. Applied flip3--1.6

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

    4. Applied associate-/l/1.6

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

    5. Simplified1.7

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

    7. Applied add-exp-log1.7

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.00408598860077935295:\\ \;\;\;\;\log \left(e^{\frac{1}{\sin x} - \frac{\cos x}{\sin x}}\right)\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le -0.0:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}}{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot \sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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