cos2 (problem 3.4.1)

Average Error: 31.6 → 0.3

Time: 4.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.037855314436600819:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{\log \left(e^{1 - \cos x}\right)}{x}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.03032722926070993:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\log \left(e^{1 - \cos x}\right)}}{\sqrt{x}}}{\frac{x \cdot \sqrt{x}}{\sqrt{1 - \cos x}}}\\ \end{array}\]

C

double f(double x) {
        double r27964 = 1.0;
        double r27965 = x;
        double r27966 = cos(r27965);
        double r27967 = r27964 - r27966;
        double r27968 = r27965 * r27965;
        double r27969 = r27967 / r27968;
        return r27969;
}

↓

double f(double x) {
        double r27970 = x;
        double r27971 = -0.03785531443660082;
        bool r27972 = r27970 <= r27971;
        double r27973 = 1.0;
        double r27974 = cos(r27970);
        double r27975 = r27973 - r27974;
        double r27976 = exp(r27975);
        double r27977 = log(r27976);
        double r27978 = r27977 / r27970;
        double r27979 = r27978 / r27970;
        double r27980 = expm1(r27979);
        double r27981 = log1p(r27980);
        double r27982 = 0.03032722926070993;
        bool r27983 = r27970 <= r27982;
        double r27984 = 4.0;
        double r27985 = pow(r27970, r27984);
        double r27986 = 0.001388888888888889;
        double r27987 = 0.5;
        double r27988 = 0.041666666666666664;
        double r27989 = 2.0;
        double r27990 = pow(r27970, r27989);
        double r27991 = r27988 * r27990;
        double r27992 = r27987 - r27991;
        double r27993 = fma(r27985, r27986, r27992);
        double r27994 = sqrt(r27977);
        double r27995 = sqrt(r27970);
        double r27996 = r27994 / r27995;
        double r27997 = r27970 * r27995;
        double r27998 = sqrt(r27975);
        double r27999 = r27997 / r27998;
        double r28000 = r27996 / r27999;
        double r28001 = r27983 ? r27993 : r28000;
        double r28002 = r27972 ? r27981 : r28001;
        return r28002;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.03785531443660082

   1. Initial program 1.1

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

   3. Applied associate-/r* 0.5

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

   5. Applied add-log-exp 0.6

      \[\leadsto \frac{\frac{1 - \color{blue}{\log \left(e^{\cos x}\right)}}{x}}{x}\]

   6. Applied add-log-exp 0.6

      \[\leadsto \frac{\frac{\color{blue}{\log \left(e^{1}\right)} - \log \left(e^{\cos x}\right)}{x}}{x}\]

   7. Applied diff-log 0.7

      \[\leadsto \frac{\frac{\color{blue}{\log \left(\frac{e^{1}}{e^{\cos x}}\right)}}{x}}{x}\]

   8. Simplified 0.6

      \[\leadsto \frac{\frac{\log \color{blue}{\left(e^{1 - \cos x}\right)}}{x}}{x}\]
   9. Using strategy rm

   10. Applied log1p-expm1-u 0.6

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{\log \left(e^{1 - \cos x}\right)}{x}}{x}\right)\right)}\]

   if -0.03785531443660082 < x < 0.03032722926070993

   1. Initial program 62.2

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

   2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}}\]

   3. Simplified 0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)}\]

   if 0.03032722926070993 < x

   1. Initial program 0.9

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

   3. Applied associate-/r* 0.4

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

   5. Applied add-log-exp 0.5

      \[\leadsto \frac{\frac{1 - \color{blue}{\log \left(e^{\cos x}\right)}}{x}}{x}\]

   6. Applied add-log-exp 0.5

      \[\leadsto \frac{\frac{\color{blue}{\log \left(e^{1}\right)} - \log \left(e^{\cos x}\right)}{x}}{x}\]

   7. Applied diff-log 0.6

      \[\leadsto \frac{\frac{\color{blue}{\log \left(\frac{e^{1}}{e^{\cos x}}\right)}}{x}}{x}\]

   8. Simplified 0.5

      \[\leadsto \frac{\frac{\log \color{blue}{\left(e^{1 - \cos x}\right)}}{x}}{x}\]
   9. Using strategy rm

   10. Applied add-sqr-sqrt 0.7

       \[\leadsto \frac{\frac{\log \left(e^{1 - \cos x}\right)}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}}{x}\]

   11. Applied add-sqr-sqrt 0.7

       \[\leadsto \frac{\frac{\color{blue}{\sqrt{\log \left(e^{1 - \cos x}\right)} \cdot \sqrt{\log \left(e^{1 - \cos x}\right)}}}{\sqrt{x} \cdot \sqrt{x}}}{x}\]

   12. Applied times-frac 0.7

       \[\leadsto \frac{\color{blue}{\frac{\sqrt{\log \left(e^{1 - \cos x}\right)}}{\sqrt{x}} \cdot \frac{\sqrt{\log \left(e^{1 - \cos x}\right)}}{\sqrt{x}}}}{x}\]

   13. Applied associate-/l* 0.7

       \[\leadsto \color{blue}{\frac{\frac{\sqrt{\log \left(e^{1 - \cos x}\right)}}{\sqrt{x}}}{\frac{x}{\frac{\sqrt{\log \left(e^{1 - \cos x}\right)}}{\sqrt{x}}}}}\]

   14. Simplified 0.6

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.037855314436600819:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{\log \left(e^{1 - \cos x}\right)}{x}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.03032722926070993:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\log \left(e^{1 - \cos x}\right)}}{\sqrt{x}}}{\frac{x \cdot \sqrt{x}}{\sqrt{1 - \cos x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))