NMSE Section 6.1 mentioned, B

Average Error: 14.3 → 0.3

Time: 8.9s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r53683 = atan2(1.0, 0.0);
        double r53684 = 2.0;
        double r53685 = r53683 / r53684;
        double r53686 = 1.0;
        double r53687 = b;
        double r53688 = r53687 * r53687;
        double r53689 = a;
        double r53690 = r53689 * r53689;
        double r53691 = r53688 - r53690;
        double r53692 = r53686 / r53691;
        double r53693 = r53685 * r53692;
        double r53694 = r53686 / r53689;
        double r53695 = r53686 / r53687;
        double r53696 = r53694 - r53695;
        double r53697 = r53693 * r53696;
        return r53697;
}

↓

double f(double a, double b) {
        double r53698 = atan2(1.0, 0.0);
        double r53699 = 2.0;
        double r53700 = r53698 / r53699;
        double r53701 = b;
        double r53702 = a;
        double r53703 = r53701 + r53702;
        double r53704 = r53700 / r53703;
        double r53705 = 1.0;
        double r53706 = r53704 * r53705;
        double r53707 = r53702 * r53701;
        double r53708 = r53705 / r53707;
        double r53709 = r53706 * r53708;
        return r53709;
}

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 9.6

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

4. Applied *-un-lft-identity 9.6

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

5. Applied times-frac 9.2

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

6. Applied associate-*r* 9.1

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

7. Simplified 9.1

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

9. Applied associate-*r/ 9.1

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

10. Applied associate-*l/ 0.3

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

12. Applied *-un-lft-identity 0.3

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

13. Applied times-frac 0.3

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

14. Simplified 0.3

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

15. Taylor expanded around 0 0.3

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

16. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019356 +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))))