Average Error: 30.2 → 0.5
Time: 7.2s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0242809220681906925976001332401210675016:\\ \;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\ \mathbf{elif}\;x \le 0.02108811747029423755428467757155885919929:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot \left(1 - \cos x\right)}{\sin x \cdot \sin x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0242809220681906925976001332401210675016:\\
\;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\

\mathbf{elif}\;x \le 0.02108811747029423755428467757155885919929:\\
\;\;\;\;\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{\sin x \cdot \left(1 - \cos x\right)}{\sin x \cdot \sin x}\\

\end{array}
double f(double x) {
        double r58173 = 1.0;
        double r58174 = x;
        double r58175 = cos(r58174);
        double r58176 = r58173 - r58175;
        double r58177 = sin(r58174);
        double r58178 = r58176 / r58177;
        return r58178;
}
↓
double f(double x) {
        double r58179 = x;
        double r58180 = -0.024280922068190693;
        bool r58181 = r58179 <= r58180;
        double r58182 = 1.0;
        double r58183 = sin(r58179);
        double r58184 = r58182 / r58183;
        double r58185 = cos(r58179);
        double r58186 = r58185 / r58183;
        double r58187 = r58184 - r58186;
        double r58188 = 0.021088117470294238;
        bool r58189 = r58179 <= r58188;
        double r58190 = 0.041666666666666664;
        double r58191 = 3.0;
        double r58192 = pow(r58179, r58191);
        double r58193 = 0.004166666666666667;
        double r58194 = 5.0;
        double r58195 = pow(r58179, r58194);
        double r58196 = 0.5;
        double r58197 = r58196 * r58179;
        double r58198 = fma(r58193, r58195, r58197);
        double r58199 = fma(r58190, r58192, r58198);
        double r58200 = r58182 - r58185;
        double r58201 = r58183 * r58200;
        double r58202 = r58183 * r58183;
        double r58203 = r58201 / r58202;
        double r58204 = r58189 ? r58199 : r58203;
        double r58205 = r58181 ? r58187 : r58204;
        return r58205;
}

Error

Bits error versus x

Target

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

Derivation

  1. Split input into 3 regimes

  2. if x < -0.024280922068190693

    1. Initial program 0.9

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

    3. Applied div-sub1.1

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

    if -0.024280922068190693 < x < 0.021088117470294238

    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 0.021088117470294238 < x

    1. Initial program 0.9

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

    3. Applied div-sub1.1

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

    5. Applied frac-sub1.1

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

    6. Simplified1.0

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0242809220681906925976001332401210675016:\\ \;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\ \mathbf{elif}\;x \le 0.02108811747029423755428467757155885919929:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot \left(1 - \cos x\right)}{\sin x \cdot \sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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)))