Average Error: 30.3 → 0.5
Time: 6.7s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02243905911353925516915630566927575273439:\\ \;\;\;\;\frac{1}{\frac{\sin x}{e^{\log \left(1 - \cos x\right)}}}\\ \mathbf{elif}\;x \le 0.02429226137720021799770719894695503171533:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin x} \cdot \left(1 - \cos x\right)\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02243905911353925516915630566927575273439:\\
\;\;\;\;\frac{1}{\frac{\sin x}{e^{\log \left(1 - \cos x\right)}}}\\

\mathbf{elif}\;x \le 0.02429226137720021799770719894695503171533:\\
\;\;\;\;\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{1}{\sin x} \cdot \left(1 - \cos x\right)\\

\end{array}
double f(double x) {
        double r40812 = 1.0;
        double r40813 = x;
        double r40814 = cos(r40813);
        double r40815 = r40812 - r40814;
        double r40816 = sin(r40813);
        double r40817 = r40815 / r40816;
        return r40817;
}
↓
double f(double x) {
        double r40818 = x;
        double r40819 = -0.022439059113539255;
        bool r40820 = r40818 <= r40819;
        double r40821 = 1.0;
        double r40822 = sin(r40818);
        double r40823 = 1.0;
        double r40824 = cos(r40818);
        double r40825 = r40823 - r40824;
        double r40826 = log(r40825);
        double r40827 = exp(r40826);
        double r40828 = r40822 / r40827;
        double r40829 = r40821 / r40828;
        double r40830 = 0.024292261377200218;
        bool r40831 = r40818 <= r40830;
        double r40832 = 0.041666666666666664;
        double r40833 = 3.0;
        double r40834 = pow(r40818, r40833);
        double r40835 = 0.004166666666666667;
        double r40836 = 5.0;
        double r40837 = pow(r40818, r40836);
        double r40838 = 0.5;
        double r40839 = r40838 * r40818;
        double r40840 = fma(r40835, r40837, r40839);
        double r40841 = fma(r40832, r40834, r40840);
        double r40842 = r40821 / r40822;
        double r40843 = r40842 * r40825;
        double r40844 = r40831 ? r40841 : r40843;
        double r40845 = r40820 ? r40829 : r40844;
        return r40845;
}

Error

Bits error versus x

Target

Original30.3
Target0.0
Herbie0.5
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.022439059113539255

    1. Initial program 1.0

      \[\frac{1 - \cos x}{\sin x}\]
    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 add-exp-log1.0

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

    if -0.022439059113539255 < x < 0.024292261377200218

    1. Initial program 60.0

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

    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 0.024292261377200218 < x

    1. Initial program 0.9

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

    3. Applied clear-num0.9

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

    5. Applied div-inv1.0

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

    6. Applied add-cube-cbrt1.0

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

    7. Applied times-frac1.0

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

    8. Simplified1.0

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

    9. Simplified1.0

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02243905911353925516915630566927575273439:\\ \;\;\;\;\frac{1}{\frac{\sin x}{e^{\log \left(1 - \cos x\right)}}}\\ \mathbf{elif}\;x \le 0.02429226137720021799770719894695503171533:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin x} \cdot \left(1 - \cos x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 +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)))