Average Error: 30.1 → 0.5
Time: 10.5s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.020168669770568547:\\ \;\;\;\;\frac{\sin x \cdot \left(1 - \cos x\right)}{\sin x \cdot \sin x}\\ \mathbf{elif}\;x \le 0.0257297152168081099:\\ \;\;\;\;0.041666666666666671 \cdot {x}^{3} + \left(0.00416666666666666661 \cdot {x}^{5} + 0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.020168669770568547:\\
\;\;\;\;\frac{\sin x \cdot \left(1 - \cos x\right)}{\sin x \cdot \sin x}\\

\mathbf{elif}\;x \le 0.0257297152168081099:\\
\;\;\;\;0.041666666666666671 \cdot {x}^{3} + \left(0.00416666666666666661 \cdot {x}^{5} + 0.5 \cdot x\right)\\

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

\end{array}
double f(double x) {
        double r89129 = 1.0;
        double r89130 = x;
        double r89131 = cos(r89130);
        double r89132 = r89129 - r89131;
        double r89133 = sin(r89130);
        double r89134 = r89132 / r89133;
        return r89134;
}
↓
double f(double x) {
        double r89135 = x;
        double r89136 = -0.020168669770568547;
        bool r89137 = r89135 <= r89136;
        double r89138 = sin(r89135);
        double r89139 = 1.0;
        double r89140 = cos(r89135);
        double r89141 = r89139 - r89140;
        double r89142 = r89138 * r89141;
        double r89143 = r89138 * r89138;
        double r89144 = r89142 / r89143;
        double r89145 = 0.02572971521680811;
        bool r89146 = r89135 <= r89145;
        double r89147 = 0.04166666666666667;
        double r89148 = 3.0;
        double r89149 = pow(r89135, r89148);
        double r89150 = r89147 * r89149;
        double r89151 = 0.004166666666666667;
        double r89152 = 5.0;
        double r89153 = pow(r89135, r89152);
        double r89154 = r89151 * r89153;
        double r89155 = 0.5;
        double r89156 = r89155 * r89135;
        double r89157 = r89154 + r89156;
        double r89158 = r89150 + r89157;
        double r89159 = r89139 / r89138;
        double r89160 = r89140 / r89138;
        double r89161 = r89159 - r89160;
        double r89162 = r89146 ? r89158 : r89161;
        double r89163 = r89137 ? r89144 : r89162;
        return r89163;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.1
Target0.0
Herbie0.5
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.020168669770568547

    1. Initial program 0.9

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

    3. Applied div-sub1.1

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

    5. Applied frac-sub1.0

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

    6. Simplified1.0

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

    if -0.020168669770568547 < x < 0.02572971521680811

    1. Initial program 59.8

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

    3. Applied div-sub59.8

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

    4. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{0.041666666666666671 \cdot {x}^{3} + \left(0.00416666666666666661 \cdot {x}^{5} + 0.5 \cdot x\right)}\]

    if 0.02572971521680811 < x

    1. Initial program 0.9

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

    3. Applied div-sub1.1

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.020168669770568547:\\ \;\;\;\;\frac{\sin x \cdot \left(1 - \cos x\right)}{\sin x \cdot \sin x}\\ \mathbf{elif}\;x \le 0.0257297152168081099:\\ \;\;\;\;0.041666666666666671 \cdot {x}^{3} + \left(0.00416666666666666661 \cdot {x}^{5} + 0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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