Average Error: 30.4 → 0.5
Time: 7.2s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02322762662824425991536791968883335357532:\\ \;\;\;\;\frac{{1}^{3} - \sqrt[3]{{\left({\left(\cos x\right)}^{3}\right)}^{3}}}{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot \sin x}\\ \mathbf{elif}\;x \le 0.02090434993213574621506722905905917286873:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3} - \log \left(e^{{\left(\cos x\right)}^{3}}\right)}{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot \sin x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02322762662824425991536791968883335357532:\\
\;\;\;\;\frac{{1}^{3} - \sqrt[3]{{\left({\left(\cos x\right)}^{3}\right)}^{3}}}{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot \sin x}\\

\mathbf{elif}\;x \le 0.02090434993213574621506722905905917286873:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{{1}^{3} - \log \left(e^{{\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 r54711 = 1.0;
        double r54712 = x;
        double r54713 = cos(r54712);
        double r54714 = r54711 - r54713;
        double r54715 = sin(r54712);
        double r54716 = r54714 / r54715;
        return r54716;
}
↓
double f(double x) {
        double r54717 = x;
        double r54718 = -0.02322762662824426;
        bool r54719 = r54717 <= r54718;
        double r54720 = 1.0;
        double r54721 = 3.0;
        double r54722 = pow(r54720, r54721);
        double r54723 = cos(r54717);
        double r54724 = pow(r54723, r54721);
        double r54725 = pow(r54724, r54721);
        double r54726 = cbrt(r54725);
        double r54727 = r54722 - r54726;
        double r54728 = r54723 + r54720;
        double r54729 = r54723 * r54728;
        double r54730 = r54720 * r54720;
        double r54731 = r54729 + r54730;
        double r54732 = sin(r54717);
        double r54733 = r54731 * r54732;
        double r54734 = r54727 / r54733;
        double r54735 = 0.020904349932135746;
        bool r54736 = r54717 <= r54735;
        double r54737 = 0.041666666666666664;
        double r54738 = pow(r54717, r54721);
        double r54739 = r54737 * r54738;
        double r54740 = 0.004166666666666667;
        double r54741 = 5.0;
        double r54742 = pow(r54717, r54741);
        double r54743 = r54740 * r54742;
        double r54744 = 0.5;
        double r54745 = r54744 * r54717;
        double r54746 = r54743 + r54745;
        double r54747 = r54739 + r54746;
        double r54748 = exp(r54724);
        double r54749 = log(r54748);
        double r54750 = r54722 - r54749;
        double r54751 = r54750 / r54733;
        double r54752 = r54736 ? r54747 : r54751;
        double r54753 = r54719 ? r54734 : r54752;
        return r54753;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.4
Target0.0
Herbie0.5
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.02322762662824426

    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. Applied associate-/l/1.0

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

    5. Simplified1.0

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

    7. Applied add-cbrt-cube1.1

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

    8. Simplified1.1

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

    if -0.02322762662824426 < x < 0.020904349932135746

    1. Initial program 59.7

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

    2. Taylor expanded around 0 0.0

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

    if 0.020904349932135746 < 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. Applied associate-/l/1.0

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

    5. Simplified1.0

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

    7. Applied add-log-exp1.1

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02322762662824425991536791968883335357532:\\ \;\;\;\;\frac{{1}^{3} - \sqrt[3]{{\left({\left(\cos x\right)}^{3}\right)}^{3}}}{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot \sin x}\\ \mathbf{elif}\;x \le 0.02090434993213574621506722905905917286873:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3} - \log \left(e^{{\left(\cos x\right)}^{3}}\right)}{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot \sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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