Average Error: 30.1 → 0.6
Time: 8.0s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0199905813077168142:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left(\frac{1 - \cos x}{\sin x}\right)}^{3}}\right)}^{3}}\\ \mathbf{elif}\;x \le 0.021452837467613652:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{1 - \cos x}}{\sin x} \cdot \sqrt{1 - \cos x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0199905813077168142:\\
\;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left(\frac{1 - \cos x}{\sin x}\right)}^{3}}\right)}^{3}}\\

\mathbf{elif}\;x \le 0.021452837467613652:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{1 - \cos x}}{\sin x} \cdot \sqrt{1 - \cos x}\\

\end{array}
double f(double x) {
        double r77802 = 1.0;
        double r77803 = x;
        double r77804 = cos(r77803);
        double r77805 = r77802 - r77804;
        double r77806 = sin(r77803);
        double r77807 = r77805 / r77806;
        return r77807;
}
↓
double f(double x) {
        double r77808 = x;
        double r77809 = -0.019990581307716814;
        bool r77810 = r77808 <= r77809;
        double r77811 = 1.0;
        double r77812 = cos(r77808);
        double r77813 = r77811 - r77812;
        double r77814 = sin(r77808);
        double r77815 = r77813 / r77814;
        double r77816 = 3.0;
        double r77817 = pow(r77815, r77816);
        double r77818 = cbrt(r77817);
        double r77819 = pow(r77818, r77816);
        double r77820 = cbrt(r77819);
        double r77821 = 0.02145283746761365;
        bool r77822 = r77808 <= r77821;
        double r77823 = 0.041666666666666664;
        double r77824 = pow(r77808, r77816);
        double r77825 = 0.004166666666666667;
        double r77826 = 5.0;
        double r77827 = pow(r77808, r77826);
        double r77828 = 0.5;
        double r77829 = r77828 * r77808;
        double r77830 = fma(r77825, r77827, r77829);
        double r77831 = fma(r77823, r77824, r77830);
        double r77832 = sqrt(r77813);
        double r77833 = r77832 / r77814;
        double r77834 = r77833 * r77832;
        double r77835 = r77822 ? r77831 : r77834;
        double r77836 = r77810 ? r77820 : r77835;
        return r77836;
}

Error

Bits error versus x

Target

Original30.1
Target0
Herbie0.6
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.019990581307716814

    1. Initial program 0.9

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

    3. Applied add-cbrt-cube1.1

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

    4. Applied add-cbrt-cube1.3

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

    5. Applied cbrt-undiv1.1

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

    6. Simplified1.1

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

    8. Applied add-cbrt-cube1.3

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

    9. Applied add-cbrt-cube1.4

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

    10. Applied cbrt-undiv1.3

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

    11. Simplified1.2

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

    if -0.019990581307716814 < x < 0.02145283746761365

    1. Initial program 59.8

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

    2. Taylor expanded around 0 0.0

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

    3. Simplified0.0

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

    if 0.02145283746761365 < x

    1. Initial program 0.9

      \[\frac{1 - \cos x}{\sin x}\]
    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 *-un-lft-identity1.0

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

    6. Applied add-sqr-sqrt1.2

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

    7. Applied times-frac1.2

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

    8. Applied exp-prod1.2

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

    9. Applied log-pow1.1

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

    10. Simplified1.1

      \[\leadsto \frac{\sqrt{1 - \cos x}}{\sin x} \cdot \color{blue}{\sqrt{1 - \cos x}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0199905813077168142:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left(\frac{1 - \cos x}{\sin x}\right)}^{3}}\right)}^{3}}\\ \mathbf{elif}\;x \le 0.021452837467613652:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{1 - \cos x}}{\sin x} \cdot \sqrt{1 - \cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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