Average Error: 14.7 → 4.8
Time: 52.4s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\mathsf{fma}\left(\left(\frac{\pi}{b - a} \cdot \frac{1}{b + a}\right), \left(\frac{-1}{b}\right), \left(\frac{\frac{1}{b + a}}{\frac{1}{\sqrt{\pi}} \cdot \frac{a}{\frac{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}{b - a}}}\right)\right)}{2}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\mathsf{fma}\left(\left(\frac{\pi}{b - a} \cdot \frac{1}{b + a}\right), \left(\frac{-1}{b}\right), \left(\frac{\frac{1}{b + a}}{\frac{1}{\sqrt{\pi}} \cdot \frac{a}{\frac{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}{b - a}}}\right)\right)}{2}
double f(double a, double b) {
        double r1947821 = atan2(1.0, 0.0);
        double r1947822 = 2.0;
        double r1947823 = r1947821 / r1947822;
        double r1947824 = 1.0;
        double r1947825 = b;
        double r1947826 = r1947825 * r1947825;
        double r1947827 = a;
        double r1947828 = r1947827 * r1947827;
        double r1947829 = r1947826 - r1947828;
        double r1947830 = r1947824 / r1947829;
        double r1947831 = r1947823 * r1947830;
        double r1947832 = r1947824 / r1947827;
        double r1947833 = r1947824 / r1947825;
        double r1947834 = r1947832 - r1947833;
        double r1947835 = r1947831 * r1947834;
        return r1947835;
}
↓
double f(double a, double b) {
        double r1947836 = atan2(1.0, 0.0);
        double r1947837 = b;
        double r1947838 = a;
        double r1947839 = r1947837 - r1947838;
        double r1947840 = r1947836 / r1947839;
        double r1947841 = 1.0;
        double r1947842 = r1947837 + r1947838;
        double r1947843 = r1947841 / r1947842;
        double r1947844 = r1947840 * r1947843;
        double r1947845 = -1.0;
        double r1947846 = r1947845 / r1947837;
        double r1947847 = sqrt(r1947836);
        double r1947848 = r1947841 / r1947847;
        double r1947849 = sqrt(r1947847);
        double r1947850 = r1947849 * r1947849;
        double r1947851 = r1947850 / r1947839;
        double r1947852 = r1947838 / r1947851;
        double r1947853 = r1947848 * r1947852;
        double r1947854 = r1947843 / r1947853;
        double r1947855 = fma(r1947844, r1947846, r1947854);
        double r1947856 = 2.0;
        double r1947857 = r1947855 / r1947856;
        return r1947857;
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.7

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  2. Simplified14.7

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\frac{\pi}{b \cdot b - a \cdot a}\right), \left(\frac{-1}{b}\right), \left(\frac{\frac{\pi}{b \cdot b - a \cdot a}}{a}\right)\right)}{2}}\]
  3. Using strategy rm

  4. Applied difference-of-squares14.7

    \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{\pi}{b \cdot b - a \cdot a}\right), \left(\frac{-1}{b}\right), \left(\frac{\frac{\pi}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}}}{a}\right)\right)}{2}\]

  5. Applied *-un-lft-identity14.7

    \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{\pi}{b \cdot b - a \cdot a}\right), \left(\frac{-1}{b}\right), \left(\frac{\frac{\color{blue}{1 \cdot \pi}}{\left(b + a\right) \cdot \left(b - a\right)}}{a}\right)\right)}{2}\]

  6. Applied times-frac14.4

    \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{\pi}{b \cdot b - a \cdot a}\right), \left(\frac{-1}{b}\right), \left(\frac{\color{blue}{\frac{1}{b + a} \cdot \frac{\pi}{b - a}}}{a}\right)\right)}{2}\]

  7. Applied associate-/l*10.0

    \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{\pi}{b \cdot b - a \cdot a}\right), \left(\frac{-1}{b}\right), \color{blue}{\left(\frac{\frac{1}{b + a}}{\frac{a}{\frac{\pi}{b - a}}}\right)}\right)}{2}\]
  8. Using strategy rm

  9. Applied difference-of-squares5.1

    \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{\pi}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}}\right), \left(\frac{-1}{b}\right), \left(\frac{\frac{1}{b + a}}{\frac{a}{\frac{\pi}{b - a}}}\right)\right)}{2}\]

  10. Applied *-un-lft-identity5.1

    \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{\color{blue}{1 \cdot \pi}}{\left(b + a\right) \cdot \left(b - a\right)}\right), \left(\frac{-1}{b}\right), \left(\frac{\frac{1}{b + a}}{\frac{a}{\frac{\pi}{b - a}}}\right)\right)}{2}\]

  11. Applied times-frac4.8

    \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{b + a} \cdot \frac{\pi}{b - a}\right)}, \left(\frac{-1}{b}\right), \left(\frac{\frac{1}{b + a}}{\frac{a}{\frac{\pi}{b - a}}}\right)\right)}{2}\]
  12. Using strategy rm

  13. Applied *-un-lft-identity4.8

    \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{b + a} \cdot \frac{\pi}{b - a}\right), \left(\frac{-1}{b}\right), \left(\frac{\frac{1}{b + a}}{\frac{a}{\frac{\pi}{\color{blue}{1 \cdot \left(b - a\right)}}}}\right)\right)}{2}\]

  14. Applied add-sqr-sqrt5.0

    \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{b + a} \cdot \frac{\pi}{b - a}\right), \left(\frac{-1}{b}\right), \left(\frac{\frac{1}{b + a}}{\frac{a}{\frac{\color{blue}{\sqrt{\pi} \cdot \sqrt{\pi}}}{1 \cdot \left(b - a\right)}}}\right)\right)}{2}\]

  15. Applied times-frac4.9

    \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{b + a} \cdot \frac{\pi}{b - a}\right), \left(\frac{-1}{b}\right), \left(\frac{\frac{1}{b + a}}{\frac{a}{\color{blue}{\frac{\sqrt{\pi}}{1} \cdot \frac{\sqrt{\pi}}{b - a}}}}\right)\right)}{2}\]

  16. Applied *-un-lft-identity4.9

    \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{b + a} \cdot \frac{\pi}{b - a}\right), \left(\frac{-1}{b}\right), \left(\frac{\frac{1}{b + a}}{\frac{\color{blue}{1 \cdot a}}{\frac{\sqrt{\pi}}{1} \cdot \frac{\sqrt{\pi}}{b - a}}}\right)\right)}{2}\]

  17. Applied times-frac4.8

    \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{b + a} \cdot \frac{\pi}{b - a}\right), \left(\frac{-1}{b}\right), \left(\frac{\frac{1}{b + a}}{\color{blue}{\frac{1}{\frac{\sqrt{\pi}}{1}} \cdot \frac{a}{\frac{\sqrt{\pi}}{b - a}}}}\right)\right)}{2}\]

  18. Simplified4.8

    \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{b + a} \cdot \frac{\pi}{b - a}\right), \left(\frac{-1}{b}\right), \left(\frac{\frac{1}{b + a}}{\color{blue}{\frac{1}{\sqrt{\pi}}} \cdot \frac{a}{\frac{\sqrt{\pi}}{b - a}}}\right)\right)}{2}\]
  19. Using strategy rm

  20. Applied add-sqr-sqrt4.8

    \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{b + a} \cdot \frac{\pi}{b - a}\right), \left(\frac{-1}{b}\right), \left(\frac{\frac{1}{b + a}}{\frac{1}{\sqrt{\pi}} \cdot \frac{a}{\frac{\sqrt{\color{blue}{\sqrt{\pi} \cdot \sqrt{\pi}}}}{b - a}}}\right)\right)}{2}\]

  21. Applied sqrt-prod4.8

    \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{b + a} \cdot \frac{\pi}{b - a}\right), \left(\frac{-1}{b}\right), \left(\frac{\frac{1}{b + a}}{\frac{1}{\sqrt{\pi}} \cdot \frac{a}{\frac{\color{blue}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}}{b - a}}}\right)\right)}{2}\]

  22. Final simplification4.8

    \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{\pi}{b - a} \cdot \frac{1}{b + a}\right), \left(\frac{-1}{b}\right), \left(\frac{\frac{1}{b + a}}{\frac{1}{\sqrt{\pi}} \cdot \frac{a}{\frac{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}{b - a}}}\right)\right)}{2}\]

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))