NMSE Section 6.1 mentioned, B

Average Error: 14.3 → 0.2

Time: 23.3s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r59577 = atan2(1.0, 0.0);
        double r59578 = 2.0;
        double r59579 = r59577 / r59578;
        double r59580 = 1.0;
        double r59581 = b;
        double r59582 = r59581 * r59581;
        double r59583 = a;
        double r59584 = r59583 * r59583;
        double r59585 = r59582 - r59584;
        double r59586 = r59580 / r59585;
        double r59587 = r59579 * r59586;
        double r59588 = r59580 / r59583;
        double r59589 = r59580 / r59581;
        double r59590 = r59588 - r59589;
        double r59591 = r59587 * r59590;
        return r59591;
}

↓

double f(double a, double b) {
        double r59592 = atan2(1.0, 0.0);
        double r59593 = 2.0;
        double r59594 = r59592 / r59593;
        double r59595 = b;
        double r59596 = a;
        double r59597 = r59595 + r59596;
        double r59598 = r59594 / r59597;
        double r59599 = 1.0;
        double r59600 = r59598 * r59599;
        double r59601 = r59596 * r59595;
        double r59602 = r59599 * r59601;
        double r59603 = r59600 / r59602;
        return r59603;
}

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))))