NMSE Section 6.1 mentioned, B

Average Error: 14.2 → 0.2

Time: 15.0s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double a, double b) {
        double r66916 = atan2(1.0, 0.0);
        double r66917 = 2.0;
        double r66918 = r66916 / r66917;
        double r66919 = 1.0;
        double r66920 = b;
        double r66921 = r66920 * r66920;
        double r66922 = a;
        double r66923 = r66922 * r66922;
        double r66924 = r66921 - r66923;
        double r66925 = r66919 / r66924;
        double r66926 = r66918 * r66925;
        double r66927 = r66919 / r66922;
        double r66928 = r66919 / r66920;
        double r66929 = r66927 - r66928;
        double r66930 = r66926 * r66929;
        return r66930;
}

↓

double f(double a, double b) {
        double r66931 = atan2(1.0, 0.0);
        double r66932 = 2.0;
        double r66933 = r66931 / r66932;
        double r66934 = b;
        double r66935 = a;
        double r66936 = r66934 + r66935;
        double r66937 = r66933 / r66936;
        double r66938 = 1.0;
        double r66939 = r66938 * r66938;
        double r66940 = r66937 * r66939;
        double r66941 = r66935 * r66934;
        double r66942 = r66940 / r66941;
        double r66943 = sqrt(r66938);
        double r66944 = r66943 / r66934;
        double r66945 = -r66944;
        double r66946 = r66943 * r66943;
        double r66947 = r66946 / r66934;
        double r66948 = fma(r66945, r66943, r66947);
        double r66949 = r66934 - r66935;
        double r66950 = r66938 / r66949;
        double r66951 = r66950 * r66937;
        double r66952 = r66948 * r66951;
        double r66953 = r66942 + r66952;
        return r66953;
}

Error

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.2

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

3. Applied difference-of-squares 9.5

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

4. Applied *-un-lft-identity 9.5

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

5. Applied times-frac 9.1

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

6. Applied associate-*r* 9.0

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

7. Simplified 9.0

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

9. Applied *-un-lft-identity 9.0

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

10. Applied add-sqr-sqrt 9.0

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

11. Applied times-frac 9.0

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

12. Applied *-un-lft-identity 9.0

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

13. Applied add-cube-cbrt 9.0

    \[\leadsto \left(\frac{\frac{\pi}{2}}{b + a} \cdot \frac{1}{b - a}\right) \cdot \left(\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot a} - \frac{\sqrt{1}}{1} \cdot \frac{\sqrt{1}}{b}\right)\]

14. Applied times-frac 9.0

    \[\leadsto \left(\frac{\frac{\pi}{2}}{b + a} \cdot \frac{1}{b - a}\right) \cdot \left(\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{a}} - \frac{\sqrt{1}}{1} \cdot \frac{\sqrt{1}}{b}\right)\]

15. Applied prod-diff 9.0

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

16. Applied distribute-lft-in 9.0

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

17. Simplified 0.3

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

18. Simplified 0.3

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

19. Taylor expanded around 0 0.3

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

20. Simplified 0.3

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

22. Applied associate-*l/ 0.3

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

23. Applied associate-*r/ 0.2

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

24. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))