Average Error: 30.2 → 0.5
Time: 7.5s
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 r64597 = 1.0;
        double r64598 = x;
        double r64599 = cos(r64598);
        double r64600 = r64597 - r64599;
        double r64601 = sin(r64598);
        double r64602 = r64600 / r64601;
        return r64602;
}
↓
double f(double x) {
        double r64603 = x;
        double r64604 = -0.024280922068190693;
        bool r64605 = r64603 <= r64604;
        double r64606 = 1.0;
        double r64607 = sin(r64603);
        double r64608 = r64606 / r64607;
        double r64609 = cos(r64603);
        double r64610 = r64609 / r64607;
        double r64611 = r64608 - r64610;
        double r64612 = 0.021088117470294238;
        bool r64613 = r64603 <= r64612;
        double r64614 = 0.041666666666666664;
        double r64615 = 3.0;
        double r64616 = pow(r64603, r64615);
        double r64617 = 0.004166666666666667;
        double r64618 = 5.0;
        double r64619 = pow(r64603, r64618);
        double r64620 = 0.5;
        double r64621 = r64620 * r64603;
        double r64622 = fma(r64617, r64619, r64621);
        double r64623 = fma(r64614, r64616, r64622);
        double r64624 = r64606 - r64609;
        double r64625 = r64607 * r64624;
        double r64626 = r64607 * r64607;
        double r64627 = r64625 / r64626;
        double r64628 = r64613 ? r64623 : r64627;
        double r64629 = r64605 ? r64611 : r64628;
        return r64629;
}

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)))