NMSE Section 6.1 mentioned, B

Average Error: 14.1 → 0.3

Time: 26.8s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r1050605 = atan2(1.0, 0.0);
        double r1050606 = 2.0;
        double r1050607 = r1050605 / r1050606;
        double r1050608 = 1.0;
        double r1050609 = b;
        double r1050610 = r1050609 * r1050609;
        double r1050611 = a;
        double r1050612 = r1050611 * r1050611;
        double r1050613 = r1050610 - r1050612;
        double r1050614 = r1050608 / r1050613;
        double r1050615 = r1050607 * r1050614;
        double r1050616 = r1050608 / r1050611;
        double r1050617 = r1050608 / r1050609;
        double r1050618 = r1050616 - r1050617;
        double r1050619 = r1050615 * r1050618;
        return r1050619;
}

↓

double f(double a, double b) {
        double r1050620 = 1.0;
        double r1050621 = a;
        double r1050622 = b;
        double r1050623 = r1050621 + r1050622;
        double r1050624 = r1050620 / r1050623;
        double r1050625 = atan2(1.0, 0.0);
        double r1050626 = r1050622 * r1050621;
        double r1050627 = r1050625 / r1050626;
        double r1050628 = 0.5;
        double r1050629 = r1050627 * r1050628;
        double r1050630 = r1050624 * r1050629;
        return r1050630;
}

Error

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.1

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

2. Simplified 9.5

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

4. Applied *-un-lft-identity 9.5

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

5. Applied *-un-lft-identity 9.5

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

6. Applied times-frac 9.5

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

7. Applied *-un-lft-identity 9.5

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

8. Applied *-un-lft-identity 9.5

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

9. Applied times-frac 9.5

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

10. Applied distribute-lft-out-- 9.5

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

11. Applied times-frac 0.3

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

12. Simplified 0.3

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

13. Taylor expanded around inf 0.3

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

14. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))