Average Error: 30.0 → 0.5
Time: 7.5s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.022628612332283863:\\ \;\;\;\;\frac{{1}^{3} - \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos x\right)}^{3}\right)\right)}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(\cos x, \cos x, 1 \cdot \cos x\right)\right) \cdot \sin x}\\ \mathbf{elif}\;x \le 0.02286088640089938:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{2} \cdot \cos x}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(\cos x, \cos x, 1 \cdot \cos x\right)\right) \cdot \sin x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.022628612332283863:\\
\;\;\;\;\frac{{1}^{3} - \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos x\right)}^{3}\right)\right)}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(\cos x, \cos x, 1 \cdot \cos x\right)\right) \cdot \sin x}\\

\mathbf{elif}\;x \le 0.02286088640089938:\\
\;\;\;\;\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} - {\left(\cos x\right)}^{2} \cdot \cos x}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(\cos x, \cos x, 1 \cdot \cos x\right)\right) \cdot \sin x}\\

\end{array}
double f(double x) {
        double r46747 = 1.0;
        double r46748 = x;
        double r46749 = cos(r46748);
        double r46750 = r46747 - r46749;
        double r46751 = sin(r46748);
        double r46752 = r46750 / r46751;
        return r46752;
}
↓
double f(double x) {
        double r46753 = x;
        double r46754 = -0.022628612332283863;
        bool r46755 = r46753 <= r46754;
        double r46756 = 1.0;
        double r46757 = 3.0;
        double r46758 = pow(r46756, r46757);
        double r46759 = cos(r46753);
        double r46760 = pow(r46759, r46757);
        double r46761 = log1p(r46760);
        double r46762 = expm1(r46761);
        double r46763 = r46758 - r46762;
        double r46764 = r46756 * r46759;
        double r46765 = fma(r46759, r46759, r46764);
        double r46766 = fma(r46756, r46756, r46765);
        double r46767 = sin(r46753);
        double r46768 = r46766 * r46767;
        double r46769 = r46763 / r46768;
        double r46770 = 0.02286088640089938;
        bool r46771 = r46753 <= r46770;
        double r46772 = 0.041666666666666664;
        double r46773 = pow(r46753, r46757);
        double r46774 = 0.004166666666666667;
        double r46775 = 5.0;
        double r46776 = pow(r46753, r46775);
        double r46777 = 0.5;
        double r46778 = r46777 * r46753;
        double r46779 = fma(r46774, r46776, r46778);
        double r46780 = fma(r46772, r46773, r46779);
        double r46781 = 2.0;
        double r46782 = pow(r46759, r46781);
        double r46783 = r46782 * r46759;
        double r46784 = r46758 - r46783;
        double r46785 = r46784 / r46768;
        double r46786 = r46771 ? r46780 : r46785;
        double r46787 = r46755 ? r46769 : r46786;
        return r46787;
}

Error

Bits error versus x

Target

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

Derivation

  1. Split input into 3 regimes

  2. if x < -0.022628612332283863

    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

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

    7. Applied expm1-log1p-u1.1

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

    if -0.022628612332283863 < x < 0.02286088640089938

    1. Initial program 59.9

      \[\frac{1 - \cos x}{\sin x}\]

    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 0.02286088640089938 < 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

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

    7. Applied add-cube-cbrt1.8

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

    8. Applied unpow-prod-down1.8

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

    9. Simplified1.3

      \[\leadsto \frac{{1}^{3} - \color{blue}{{\left(\cos x\right)}^{2}} \cdot {\left(\sqrt[3]{\cos x}\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(\cos x, \cos x, 1 \cdot \cos x\right)\right) \cdot \sin x}\]

    10. Simplified1.0

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.022628612332283863:\\ \;\;\;\;\frac{{1}^{3} - \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos x\right)}^{3}\right)\right)}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(\cos x, \cos x, 1 \cdot \cos x\right)\right) \cdot \sin x}\\ \mathbf{elif}\;x \le 0.02286088640089938:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{2} \cdot \cos x}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(\cos x, \cos x, 1 \cdot \cos x\right)\right) \cdot \sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +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)))