NMSE Section 6.1 mentioned, B

Average Error: 14.5 → 0.3

Time: 5.0s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r38512 = atan2(1.0, 0.0);
        double r38513 = 2.0;
        double r38514 = r38512 / r38513;
        double r38515 = 1.0;
        double r38516 = b;
        double r38517 = r38516 * r38516;
        double r38518 = a;
        double r38519 = r38518 * r38518;
        double r38520 = r38517 - r38519;
        double r38521 = r38515 / r38520;
        double r38522 = r38514 * r38521;
        double r38523 = r38515 / r38518;
        double r38524 = r38515 / r38516;
        double r38525 = r38523 - r38524;
        double r38526 = r38522 * r38525;
        return r38526;
}

↓

double f(double a, double b) {
        double r38527 = atan2(1.0, 0.0);
        double r38528 = 2.0;
        double r38529 = r38527 / r38528;
        double r38530 = 1.0;
        double r38531 = r38529 * r38530;
        double r38532 = b;
        double r38533 = a;
        double r38534 = r38532 + r38533;
        double r38535 = r38531 / r38534;
        double r38536 = sqrt(r38530);
        double r38537 = 1.0;
        double r38538 = r38536 / r38537;
        double r38539 = r38536 / r38533;
        double r38540 = r38536 / r38532;
        double r38541 = r38539 - r38540;
        double r38542 = r38538 * r38541;
        double r38543 = r38535 * r38542;
        double r38544 = r38532 - r38533;
        double r38545 = r38543 / r38544;
        return r38545;
}

Error

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.5

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
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 associate-/r* 9.0

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

6. Applied associate-*r/ 9.0

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

7. Applied associate-*l/ 0.3

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

9. Applied associate-*r/ 0.3

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

11. Applied *-un-lft-identity 0.3

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

12. Applied add-sqr-sqrt 0.3

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

13. Applied times-frac 0.3

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

14. Applied *-un-lft-identity 0.3

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

15. Applied add-sqr-sqrt 0.3

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

16. Applied times-frac 0.3

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

17. Applied distribute-lft-out-- 0.3

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

18. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020020 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))