Average Error: 30.1 → 0.6
Time: 7.0s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0242859294884205483:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}\\ \mathbf{elif}\;x \le 0.0212898443058771626:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\sqrt{1 - \cos x}\right)}}{\frac{\sin x}{\sqrt{1 - \cos x}}}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0242859294884205483:\\
\;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}\\

\mathbf{elif}\;x \le 0.0212898443058771626:\\
\;\;\;\;\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{e^{\log \left(\sqrt{1 - \cos x}\right)}}{\frac{\sin x}{\sqrt{1 - \cos x}}}\\

\end{array}
double f(double x) {
        double r73138 = 1.0;
        double r73139 = x;
        double r73140 = cos(r73139);
        double r73141 = r73138 - r73140;
        double r73142 = sin(r73139);
        double r73143 = r73141 / r73142;
        return r73143;
}
↓
double f(double x) {
        double r73144 = x;
        double r73145 = -0.02428592948842055;
        bool r73146 = r73144 <= r73145;
        double r73147 = 1.0;
        double r73148 = 3.0;
        double r73149 = pow(r73147, r73148);
        double r73150 = cos(r73144);
        double r73151 = pow(r73150, r73148);
        double r73152 = r73149 - r73151;
        double r73153 = sin(r73144);
        double r73154 = r73152 / r73153;
        double r73155 = r73147 * r73147;
        double r73156 = r73150 * r73150;
        double r73157 = r73147 * r73150;
        double r73158 = r73156 + r73157;
        double r73159 = r73155 + r73158;
        double r73160 = r73154 / r73159;
        double r73161 = 0.021289844305877163;
        bool r73162 = r73144 <= r73161;
        double r73163 = 0.041666666666666664;
        double r73164 = pow(r73144, r73148);
        double r73165 = 0.004166666666666667;
        double r73166 = 5.0;
        double r73167 = pow(r73144, r73166);
        double r73168 = 0.5;
        double r73169 = r73168 * r73144;
        double r73170 = fma(r73165, r73167, r73169);
        double r73171 = fma(r73163, r73164, r73170);
        double r73172 = r73147 - r73150;
        double r73173 = sqrt(r73172);
        double r73174 = log(r73173);
        double r73175 = exp(r73174);
        double r73176 = r73153 / r73173;
        double r73177 = r73175 / r73176;
        double r73178 = r73162 ? r73171 : r73177;
        double r73179 = r73146 ? r73160 : r73178;
        return r73179;
}

Error

Bits error versus x

Target

Original30.1
Target0.0
Herbie0.6
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.02428592948842055

    1. Initial program 0.9

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

    3. Applied div-inv1.0

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

    5. Applied flip3--1.1

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

    6. Applied associate-*l/1.1

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

    7. Simplified1.0

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

    if -0.02428592948842055 < x < 0.021289844305877163

    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 0.021289844305877163 < x

    1. Initial program 0.9

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

    3. Applied add-exp-log0.9

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

    5. Applied add-sqr-sqrt1.2

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

    6. Applied log-prod1.2

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

    7. Applied exp-sum1.2

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

    8. Applied associate-/l*1.2

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

    9. Simplified1.2

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0242859294884205483:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}\\ \mathbf{elif}\;x \le 0.0212898443058771626:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\sqrt{1 - \cos x}\right)}}{\frac{\sin x}{\sqrt{1 - \cos x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 +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)))