2cos (problem 3.3.5)

Average Error: 39.9 → 15.6

Time: 6.5s

Precision: 64

Math

\[\cos \left(x + \varepsilon\right) - \cos x\]

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.39699580404506896243528445638072145972 \cdot 10^{-12} \lor \neg \left(\varepsilon \le 3.588703897944783975981375382800948881101 \cdot 10^{-9}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\right)\right) + \mathsf{fma}\left(-\cos x, 1, \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({\varepsilon}^{3} \cdot \frac{1}{24} - \mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right) + \mathsf{fma}\left(-\cos x, 1, \cos x\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r68120 = x;
        double r68121 = eps;
        double r68122 = r68120 + r68121;
        double r68123 = cos(r68122);
        double r68124 = cos(r68120);
        double r68125 = r68123 - r68124;
        return r68125;
}

↓

double f(double x, double eps) {
        double r68126 = eps;
        double r68127 = -8.396995804045069e-12;
        bool r68128 = r68126 <= r68127;
        double r68129 = 3.588703897944784e-09;
        bool r68130 = r68126 <= r68129;
        double r68131 = !r68130;
        bool r68132 = r68128 || r68131;
        double r68133 = cos(r68126);
        double r68134 = x;
        double r68135 = cos(r68134);
        double r68136 = sin(r68134);
        double r68137 = sin(r68126);
        double r68138 = fma(r68136, r68137, r68135);
        double r68139 = expm1(r68138);
        double r68140 = log1p(r68139);
        double r68141 = -r68140;
        double r68142 = fma(r68133, r68135, r68141);
        double r68143 = -r68135;
        double r68144 = 1.0;
        double r68145 = fma(r68143, r68144, r68135);
        double r68146 = r68142 + r68145;
        double r68147 = 3.0;
        double r68148 = pow(r68126, r68147);
        double r68149 = 0.041666666666666664;
        double r68150 = r68148 * r68149;
        double r68151 = 0.5;
        double r68152 = fma(r68151, r68126, r68134);
        double r68153 = r68150 - r68152;
        double r68154 = r68126 * r68153;
        double r68155 = r68154 + r68145;
        double r68156 = r68132 ? r68146 : r68155;
        return r68156;
}

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 2 regimes

2. if eps < -8.396995804045069e-12 or 3.588703897944784e-09 < eps

   1. Initial program 30.6

      \[\cos \left(x + \varepsilon\right) - \cos x\]
   2. Using strategy rm

   3. Applied cos-sum 1.4

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 1.8

      \[\leadsto \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \color{blue}{\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \sqrt[3]{\cos x}}\]

   6. Applied add-sqr-sqrt 32.6

      \[\leadsto \color{blue}{\sqrt{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon} \cdot \sqrt{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon}} - \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \sqrt[3]{\cos x}\]

   7. Applied prod-diff 32.6

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

   8. Simplified 1.4

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

   9. Simplified 1.3

      \[\leadsto \mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right) + \color{blue}{\mathsf{fma}\left(-\cos x, 1, \cos x\right)}\]
   10. Using strategy rm

   11. Applied log1p-expm1-u 1.4

       \[\leadsto \mathsf{fma}\left(\cos \varepsilon, \cos x, -\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\right)}\right) + \mathsf{fma}\left(-\cos x, 1, \cos x\right)\]

   if -8.396995804045069e-12 < eps < 3.588703897944784e-09

   1. Initial program 49.8

      \[\cos \left(x + \varepsilon\right) - \cos x\]
   2. Using strategy rm

   3. Applied cos-sum 49.6

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 50.0

      \[\leadsto \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \color{blue}{\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \sqrt[3]{\cos x}}\]

   6. Applied add-sqr-sqrt 50.8

      \[\leadsto \color{blue}{\sqrt{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon} \cdot \sqrt{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon}} - \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \sqrt[3]{\cos x}\]

   7. Applied prod-diff 50.9

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

   8. Simplified 50.1

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

   9. Simplified 49.6

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

   10. Taylor expanded around 0 30.8

       \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)} + \mathsf{fma}\left(-\cos x, 1, \cos x\right)\]

   11. Simplified 30.8

       \[\leadsto \color{blue}{\varepsilon \cdot \left({\varepsilon}^{3} \cdot \frac{1}{24} - \mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)} + \mathsf{fma}\left(-\cos x, 1, \cos x\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 15.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.39699580404506896243528445638072145972 \cdot 10^{-12} \lor \neg \left(\varepsilon \le 3.588703897944783975981375382800948881101 \cdot 10^{-9}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\right)\right) + \mathsf{fma}\left(-\cos x, 1, \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({\varepsilon}^{3} \cdot \frac{1}{24} - \mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right) + \mathsf{fma}\left(-\cos x, 1, \cos x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))