NMSE Section 6.1 mentioned, B

Average Error: 14.3 → 0.2

Time: 23.7s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r59905 = atan2(1.0, 0.0);
        double r59906 = 2.0;
        double r59907 = r59905 / r59906;
        double r59908 = 1.0;
        double r59909 = b;
        double r59910 = r59909 * r59909;
        double r59911 = a;
        double r59912 = r59911 * r59911;
        double r59913 = r59910 - r59912;
        double r59914 = r59908 / r59913;
        double r59915 = r59907 * r59914;
        double r59916 = r59908 / r59911;
        double r59917 = r59908 / r59909;
        double r59918 = r59916 - r59917;
        double r59919 = r59915 * r59918;
        return r59919;
}

↓

double f(double a, double b) {
        double r59920 = atan2(1.0, 0.0);
        double r59921 = 2.0;
        double r59922 = r59920 / r59921;
        double r59923 = b;
        double r59924 = a;
        double r59925 = r59923 + r59924;
        double r59926 = r59922 / r59925;
        double r59927 = 1.0;
        double r59928 = r59926 * r59927;
        double r59929 = r59924 * r59923;
        double r59930 = r59927 * r59929;
        double r59931 = r59928 / r59930;
        return r59931;
}

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. Using strategy rm

3. Applied difference-of-squares 9.4

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

4. Applied *-un-lft-identity 9.4

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

5. Applied times-frac 9.0

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

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 8.9

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

9. Applied frac-sub 8.9

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

10. Applied associate-*r/ 8.9

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

11. Applied frac-times 0.8

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

13. Applied associate-/l* 0.7

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

14. Simplified 0.7

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

15. Taylor expanded around 0 0.2

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

16. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019323 +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))))