Average Error: 30.3 → 0.5
Time: 23.8s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.023254737060871069620482387563242809847:\\ \;\;\;\;\frac{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}{\frac{\sin x \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \left(1 + \cos x\right)\right)}{1 - \cos x}}\\ \mathbf{elif}\;x \le 0.02350362276191214974674359439177351305261:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3} - \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos x\right)}^{3}\right)\right)}{\sin x \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \frac{1 \cdot 1 - {\left(\cos x\right)}^{2}}{1 - \cos x}\right)}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.023254737060871069620482387563242809847:\\
\;\;\;\;\frac{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}{\frac{\sin x \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \left(1 + \cos x\right)\right)}{1 - \cos x}}\\

\mathbf{elif}\;x \le 0.02350362276191214974674359439177351305261:\\
\;\;\;\;\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}^{3} - \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos x\right)}^{3}\right)\right)}{\sin x \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \frac{1 \cdot 1 - {\left(\cos x\right)}^{2}}{1 - \cos x}\right)}\\

\end{array}
double f(double x) {
        double r63687 = 1.0;
        double r63688 = x;
        double r63689 = cos(r63688);
        double r63690 = r63687 - r63689;
        double r63691 = sin(r63688);
        double r63692 = r63690 / r63691;
        return r63692;
}
↓
double f(double x) {
        double r63693 = x;
        double r63694 = -0.02325473706087107;
        bool r63695 = r63693 <= r63694;
        double r63696 = 1.0;
        double r63697 = r63696 * r63696;
        double r63698 = cos(r63693);
        double r63699 = r63698 * r63698;
        double r63700 = r63696 * r63698;
        double r63701 = r63699 + r63700;
        double r63702 = r63697 + r63701;
        double r63703 = sin(r63693);
        double r63704 = r63696 + r63698;
        double r63705 = r63698 * r63704;
        double r63706 = fma(r63696, r63696, r63705);
        double r63707 = r63703 * r63706;
        double r63708 = r63696 - r63698;
        double r63709 = r63707 / r63708;
        double r63710 = r63702 / r63709;
        double r63711 = 0.02350362276191215;
        bool r63712 = r63693 <= r63711;
        double r63713 = 0.041666666666666664;
        double r63714 = 3.0;
        double r63715 = pow(r63693, r63714);
        double r63716 = 0.004166666666666667;
        double r63717 = 5.0;
        double r63718 = pow(r63693, r63717);
        double r63719 = 0.5;
        double r63720 = r63719 * r63693;
        double r63721 = fma(r63716, r63718, r63720);
        double r63722 = fma(r63713, r63715, r63721);
        double r63723 = pow(r63696, r63714);
        double r63724 = pow(r63698, r63714);
        double r63725 = log1p(r63724);
        double r63726 = expm1(r63725);
        double r63727 = r63723 - r63726;
        double r63728 = 2.0;
        double r63729 = pow(r63698, r63728);
        double r63730 = r63697 - r63729;
        double r63731 = r63730 / r63708;
        double r63732 = r63698 * r63731;
        double r63733 = fma(r63696, r63696, r63732);
        double r63734 = r63703 * r63733;
        double r63735 = r63727 / r63734;
        double r63736 = r63712 ? r63722 : r63735;
        double r63737 = r63695 ? r63710 : r63736;
        return r63737;
}

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 < -0.02325473706087107

    1. Initial program 0.9

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

    3. Applied flip3--1.0

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

    4. Applied associate-/l/1.0

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

    5. Simplified1.0

      \[\leadsto \frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\color{blue}{\sin x \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \left(1 + \cos x\right)\right)}}\]
    6. Using strategy rm

    7. Applied difference-cubes1.0

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

    8. Applied associate-/l*1.0

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

    if -0.02325473706087107 < x < 0.02350362276191215

    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 0.02350362276191215 < x

    1. Initial program 0.9

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

    3. Applied flip3--1.0

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

    4. Applied associate-/l/1.0

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

    5. Simplified1.0

      \[\leadsto \frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\color{blue}{\sin x \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \left(1 + \cos x\right)\right)}}\]
    6. Using strategy rm

    7. Applied expm1-log1p-u1.0

      \[\leadsto \frac{{1}^{3} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos x\right)}^{3}\right)\right)}}{\sin x \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \left(1 + \cos x\right)\right)}\]
    8. Using strategy rm

    9. Applied flip-+1.0

      \[\leadsto \frac{{1}^{3} - \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos x\right)}^{3}\right)\right)}{\sin x \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 - \cos x}}\right)}\]

    10. Simplified1.0

      \[\leadsto \frac{{1}^{3} - \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos x\right)}^{3}\right)\right)}{\sin x \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \frac{\color{blue}{1 \cdot 1 - {\left(\cos x\right)}^{2}}}{1 - \cos x}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.023254737060871069620482387563242809847:\\ \;\;\;\;\frac{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}{\frac{\sin x \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \left(1 + \cos x\right)\right)}{1 - \cos x}}\\ \mathbf{elif}\;x \le 0.02350362276191214974674359439177351305261:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3} - \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos x\right)}^{3}\right)\right)}{\sin x \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \frac{1 \cdot 1 - {\left(\cos x\right)}^{2}}{1 - \cos x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +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)))