Average Error: 30.2 → 0.5
Time: 26.0s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02366290210440796099033988753035373520106:\\ \;\;\;\;\frac{\frac{1 \cdot \left(1 \cdot 1\right) - \log \left(e^{\cos x \cdot \cos x}\right) \cdot \cos x}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}{\sin x}\\ \mathbf{elif}\;x \le 0.02175021537778742269919618479434575419873:\\ \;\;\;\;x \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{24}\right)\right) + {x}^{5} \cdot \frac{1}{240}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 \cdot \left(1 \cdot 1\right) - \frac{1}{2} \cdot \cos x\right) - \left(\frac{1}{2} \cdot \cos \left(x \cdot 2\right)\right) \cdot \cos x}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}{\sin x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02366290210440796099033988753035373520106:\\
\;\;\;\;\frac{\frac{1 \cdot \left(1 \cdot 1\right) - \log \left(e^{\cos x \cdot \cos x}\right) \cdot \cos x}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}{\sin x}\\

\mathbf{elif}\;x \le 0.02175021537778742269919618479434575419873:\\
\;\;\;\;x \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{24}\right)\right) + {x}^{5} \cdot \frac{1}{240}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\left(1 \cdot \left(1 \cdot 1\right) - \frac{1}{2} \cdot \cos x\right) - \left(\frac{1}{2} \cdot \cos \left(x \cdot 2\right)\right) \cdot \cos x}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}{\sin x}\\

\end{array}
double f(double x) {
        double r3830615 = 1.0;
        double r3830616 = x;
        double r3830617 = cos(r3830616);
        double r3830618 = r3830615 - r3830617;
        double r3830619 = sin(r3830616);
        double r3830620 = r3830618 / r3830619;
        return r3830620;
}
↓
double f(double x) {
        double r3830621 = x;
        double r3830622 = -0.02366290210440796;
        bool r3830623 = r3830621 <= r3830622;
        double r3830624 = 1.0;
        double r3830625 = r3830624 * r3830624;
        double r3830626 = r3830624 * r3830625;
        double r3830627 = cos(r3830621);
        double r3830628 = r3830627 * r3830627;
        double r3830629 = exp(r3830628);
        double r3830630 = log(r3830629);
        double r3830631 = r3830630 * r3830627;
        double r3830632 = r3830626 - r3830631;
        double r3830633 = r3830624 * r3830627;
        double r3830634 = r3830628 + r3830633;
        double r3830635 = r3830625 + r3830634;
        double r3830636 = r3830632 / r3830635;
        double r3830637 = sin(r3830621);
        double r3830638 = r3830636 / r3830637;
        double r3830639 = 0.021750215377787423;
        bool r3830640 = r3830621 <= r3830639;
        double r3830641 = 0.5;
        double r3830642 = 0.041666666666666664;
        double r3830643 = r3830621 * r3830642;
        double r3830644 = r3830621 * r3830643;
        double r3830645 = r3830641 + r3830644;
        double r3830646 = r3830621 * r3830645;
        double r3830647 = 5.0;
        double r3830648 = pow(r3830621, r3830647);
        double r3830649 = 0.004166666666666667;
        double r3830650 = r3830648 * r3830649;
        double r3830651 = r3830646 + r3830650;
        double r3830652 = r3830641 * r3830627;
        double r3830653 = r3830626 - r3830652;
        double r3830654 = 2.0;
        double r3830655 = r3830621 * r3830654;
        double r3830656 = cos(r3830655);
        double r3830657 = r3830641 * r3830656;
        double r3830658 = r3830657 * r3830627;
        double r3830659 = r3830653 - r3830658;
        double r3830660 = r3830659 / r3830635;
        double r3830661 = r3830660 / r3830637;
        double r3830662 = r3830640 ? r3830651 : r3830661;
        double r3830663 = r3830623 ? r3830638 : r3830662;
        return r3830663;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.2
Target0.0
Herbie0.5
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.02366290210440796

    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

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

    6. Applied add-log-exp1.0

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

    if -0.02366290210440796 < x < 0.021750215377787423

    1. Initial program 59.9

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

    2. Taylor expanded around 0 0.0

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

    3. Simplified0.0

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

    if 0.021750215377787423 < x

    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

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

    6. Applied sqr-cos0.9

      \[\leadsto \frac{\frac{\left(1 \cdot 1\right) \cdot 1 - \cos x \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}{\sin x}\]

    7. Applied distribute-lft-in0.9

      \[\leadsto \frac{\frac{\left(1 \cdot 1\right) \cdot 1 - \color{blue}{\left(\cos x \cdot \frac{1}{2} + \cos x \cdot \left(\frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right)}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}{\sin x}\]

    8. Applied associate--r+0.9

      \[\leadsto \frac{\frac{\color{blue}{\left(\left(1 \cdot 1\right) \cdot 1 - \cos x \cdot \frac{1}{2}\right) - \cos x \cdot \left(\frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}{\sin x}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02366290210440796099033988753035373520106:\\ \;\;\;\;\frac{\frac{1 \cdot \left(1 \cdot 1\right) - \log \left(e^{\cos x \cdot \cos x}\right) \cdot \cos x}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}{\sin x}\\ \mathbf{elif}\;x \le 0.02175021537778742269919618479434575419873:\\ \;\;\;\;x \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{1}{24}\right)\right) + {x}^{5} \cdot \frac{1}{240}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 \cdot \left(1 \cdot 1\right) - \frac{1}{2} \cdot \cos x\right) - \left(\frac{1}{2} \cdot \cos \left(x \cdot 2\right)\right) \cdot \cos x}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 2

  :herbie-target
  (tan (/ x 2.0))

  (/ (- 1.0 (cos x)) (sin x)))