Average Error: 30.2 → 0.5
Time: 20.9s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02763831198473910497059868873748200712726:\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \mathbf{elif}\;x \le 0.02784500067586437310751534823793917894363:\\ \;\;\;\;\left(0.008333333333333331482961625624739099293947 \cdot {x}^{5} + x \cdot \left(1 - \left(x \cdot x\right) \cdot 0.1666666666666666574148081281236954964697\right)\right) \cdot \frac{1}{1 + \cos x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \cos x} \cdot \frac{\left(1 + \cos x\right) \cdot \left(1 - \cos x\right)}{\sin x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02763831198473910497059868873748200712726:\\
\;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\

\mathbf{elif}\;x \le 0.02784500067586437310751534823793917894363:\\
\;\;\;\;\left(0.008333333333333331482961625624739099293947 \cdot {x}^{5} + x \cdot \left(1 - \left(x \cdot x\right) \cdot 0.1666666666666666574148081281236954964697\right)\right) \cdot \frac{1}{1 + \cos x}\\

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

\end{array}
double f(double x) {
        double r3974690 = 1.0;
        double r3974691 = x;
        double r3974692 = cos(r3974691);
        double r3974693 = r3974690 - r3974692;
        double r3974694 = sin(r3974691);
        double r3974695 = r3974693 / r3974694;
        return r3974695;
}
↓
double f(double x) {
        double r3974696 = x;
        double r3974697 = -0.027638311984739105;
        bool r3974698 = r3974696 <= r3974697;
        double r3974699 = 1.0;
        double r3974700 = cos(r3974696);
        double r3974701 = r3974699 - r3974700;
        double r3974702 = sin(r3974696);
        double r3974703 = r3974701 / r3974702;
        double r3974704 = exp(r3974703);
        double r3974705 = log(r3974704);
        double r3974706 = 0.027845000675864373;
        bool r3974707 = r3974696 <= r3974706;
        double r3974708 = 0.008333333333333331;
        double r3974709 = 5.0;
        double r3974710 = pow(r3974696, r3974709);
        double r3974711 = r3974708 * r3974710;
        double r3974712 = r3974696 * r3974696;
        double r3974713 = 0.16666666666666666;
        double r3974714 = r3974712 * r3974713;
        double r3974715 = r3974699 - r3974714;
        double r3974716 = r3974696 * r3974715;
        double r3974717 = r3974711 + r3974716;
        double r3974718 = 1.0;
        double r3974719 = r3974699 + r3974700;
        double r3974720 = r3974718 / r3974719;
        double r3974721 = r3974717 * r3974720;
        double r3974722 = r3974719 * r3974701;
        double r3974723 = r3974722 / r3974702;
        double r3974724 = r3974720 * r3974723;
        double r3974725 = r3974707 ? r3974721 : r3974724;
        double r3974726 = r3974698 ? r3974705 : r3974725;
        return r3974726;
}

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 < -0.027638311984739105

    1. Initial program 0.8

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

    3. Applied add-log-exp1.0

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

    if -0.027638311984739105 < x < 0.027845000675864373

    1. Initial program 59.9

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

    3. Applied clear-num59.9

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{1 - \cos x}}}\]
    4. Using strategy rm

    5. Applied flip--59.9

      \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}}\]

    6. Applied associate-/r/59.9

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

    7. Applied add-cube-cbrt59.9

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

    8. Applied times-frac59.9

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\sin x}{1 \cdot 1 - \cos x \cdot \cos x}} \cdot \frac{\sqrt[3]{1}}{1 + \cos x}}\]

    9. Simplified59.9

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

    10. Simplified59.9

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

    11. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(\left(1 \cdot x + 0.008333333333333331482961625624739099293947 \cdot {x}^{5}\right) - 0.1666666666666666574148081281236954964697 \cdot {x}^{3}\right)} \cdot \frac{1}{1 + \cos x}\]

    12. Simplified0.0

      \[\leadsto \color{blue}{\left({x}^{5} \cdot 0.008333333333333331482961625624739099293947 + x \cdot \left(1 - \left(x \cdot x\right) \cdot 0.1666666666666666574148081281236954964697\right)\right)} \cdot \frac{1}{1 + \cos x}\]

    if 0.027845000675864373 < x

    1. Initial program 0.9

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

    3. Applied clear-num1.0

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{1 - \cos x}}}\]
    4. Using strategy rm

    5. Applied flip--1.4

      \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}}\]

    6. Applied associate-/r/1.4

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

    7. Applied add-cube-cbrt1.4

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

    8. Applied times-frac1.4

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\sin x}{1 \cdot 1 - \cos x \cdot \cos x}} \cdot \frac{\sqrt[3]{1}}{1 + \cos x}}\]

    9. Simplified1.0

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

    10. Simplified1.0

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02763831198473910497059868873748200712726:\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \mathbf{elif}\;x \le 0.02784500067586437310751534823793917894363:\\ \;\;\;\;\left(0.008333333333333331482961625624739099293947 \cdot {x}^{5} + x \cdot \left(1 - \left(x \cdot x\right) \cdot 0.1666666666666666574148081281236954964697\right)\right) \cdot \frac{1}{1 + \cos x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \cos x} \cdot \frac{\left(1 + \cos x\right) \cdot \left(1 - \cos x\right)}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 2

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

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