NMSE Section 6.1 mentioned, B

Average Error: 14.3 → 0.3

Time: 38.3s

Precision: 64

Math

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

↓

\[\frac{\frac{\sqrt{\frac{1}{2}}}{a} - \frac{1}{\frac{b}{\sqrt{\frac{1}{2}}}}}{b - a} \cdot \frac{\pi \cdot \sqrt{\frac{1}{2}}}{a + b}\]

C

double f(double a, double b) {
        double r1704864 = atan2(1.0, 0.0);
        double r1704865 = 2.0;
        double r1704866 = r1704864 / r1704865;
        double r1704867 = 1.0;
        double r1704868 = b;
        double r1704869 = r1704868 * r1704868;
        double r1704870 = a;
        double r1704871 = r1704870 * r1704870;
        double r1704872 = r1704869 - r1704871;
        double r1704873 = r1704867 / r1704872;
        double r1704874 = r1704866 * r1704873;
        double r1704875 = r1704867 / r1704870;
        double r1704876 = r1704867 / r1704868;
        double r1704877 = r1704875 - r1704876;
        double r1704878 = r1704874 * r1704877;
        return r1704878;
}

↓

double f(double a, double b) {
        double r1704879 = 0.5;
        double r1704880 = sqrt(r1704879);
        double r1704881 = a;
        double r1704882 = r1704880 / r1704881;
        double r1704883 = 1.0;
        double r1704884 = b;
        double r1704885 = r1704884 / r1704880;
        double r1704886 = r1704883 / r1704885;
        double r1704887 = r1704882 - r1704886;
        double r1704888 = r1704884 - r1704881;
        double r1704889 = r1704887 / r1704888;
        double r1704890 = atan2(1.0, 0.0);
        double r1704891 = r1704890 * r1704880;
        double r1704892 = r1704881 + r1704884;
        double r1704893 = r1704891 / r1704892;
        double r1704894 = r1704889 * r1704893;
        return r1704894;
}

Error

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.3

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

2. Simplified 9.8

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

4. Applied *-un-lft-identity 9.8

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

5. Applied div-inv 9.8

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

6. Applied times-frac 9.8

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

7. Applied *-un-lft-identity 9.8

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

8. Applied div-inv 9.8

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

9. Applied times-frac 9.8

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

10. Applied distribute-lft-out-- 9.8

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

11. Applied times-frac 0.3

    \[\leadsto \color{blue}{\frac{\frac{\pi}{1}}{a + b} \cdot \frac{\frac{\frac{1}{2}}{a} - \frac{\frac{1}{2}}{b}}{b - a}}\]

12. Simplified 0.3

    \[\leadsto \color{blue}{\frac{\pi}{a + b}} \cdot \frac{\frac{\frac{1}{2}}{a} - \frac{\frac{1}{2}}{b}}{b - a}\]

13. Simplified 0.3

    \[\leadsto \frac{\pi}{a + b} \cdot \color{blue}{\frac{\frac{\frac{1}{2}}{a} - \frac{\frac{1}{2}}{b}}{b - a}}\]
14. Using strategy rm

15. Applied *-un-lft-identity 0.3

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

16. Applied *-un-lft-identity 0.3

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

17. Applied add-sqr-sqrt 0.5

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

18. Applied times-frac 0.4

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

19. Applied *-un-lft-identity 0.4

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

20. Applied add-sqr-sqrt 0.5

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

21. Applied times-frac 0.4

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

22. Applied distribute-lft-out-- 0.4

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

23. Applied times-frac 0.4

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

24. Applied associate-*r* 0.4

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

25. Simplified 0.3

    \[\leadsto \color{blue}{\frac{\pi \cdot \sqrt{\frac{1}{2}}}{b + a}} \cdot \frac{\frac{\sqrt{\frac{1}{2}}}{a} - \frac{\sqrt{\frac{1}{2}}}{b}}{b - a}\]
26. Using strategy rm

27. Applied clear-num 0.3

    \[\leadsto \frac{\pi \cdot \sqrt{\frac{1}{2}}}{b + a} \cdot \frac{\frac{\sqrt{\frac{1}{2}}}{a} - \color{blue}{\frac{1}{\frac{b}{\sqrt{\frac{1}{2}}}}}}{b - a}\]

28. Final simplification 0.3

    \[\leadsto \frac{\frac{\sqrt{\frac{1}{2}}}{a} - \frac{1}{\frac{b}{\sqrt{\frac{1}{2}}}}}{b - a} \cdot \frac{\pi \cdot \sqrt{\frac{1}{2}}}{a + b}\]

Reproduce

herbie shell --seed 2019144 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))