NMSE Section 6.1 mentioned, B

Average Error: 14.6 → 0.3

Time: 28.7s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r1775558 = atan2(1.0, 0.0);
        double r1775559 = 2.0;
        double r1775560 = r1775558 / r1775559;
        double r1775561 = 1.0;
        double r1775562 = b;
        double r1775563 = r1775562 * r1775562;
        double r1775564 = a;
        double r1775565 = r1775564 * r1775564;
        double r1775566 = r1775563 - r1775565;
        double r1775567 = r1775561 / r1775566;
        double r1775568 = r1775560 * r1775567;
        double r1775569 = r1775561 / r1775564;
        double r1775570 = r1775561 / r1775562;
        double r1775571 = r1775569 - r1775570;
        double r1775572 = r1775568 * r1775571;
        return r1775572;
}

↓

double f(double a, double b) {
        double r1775573 = 0.5;
        double r1775574 = a;
        double r1775575 = r1775573 / r1775574;
        double r1775576 = -0.5;
        double r1775577 = b;
        double r1775578 = r1775576 / r1775577;
        double r1775579 = r1775575 + r1775578;
        double r1775580 = r1775577 - r1775574;
        double r1775581 = r1775579 / r1775580;
        double r1775582 = r1775577 + r1775574;
        double r1775583 = r1775581 / r1775582;
        double r1775584 = atan2(1.0, 0.0);
        double r1775585 = r1775583 * r1775584;
        return r1775585;
}

Error

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.6

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

2. Simplified 0.2

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

4. Applied *-un-lft-identity 0.2

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

5. Applied *-un-lft-identity 0.2

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

6. Applied *-un-lft-identity 0.2

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

7. Applied div-inv 0.2

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

8. Applied distribute-rgt-neg-in 0.2

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

9. Applied times-frac 0.3

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

10. Applied *-un-lft-identity 0.3

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

11. Applied div-inv 0.3

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

12. Applied times-frac 0.3

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

13. Applied distribute-lft-out 0.3

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

14. Applied times-frac 0.3

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

15. Applied times-frac 0.3

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

16. Simplified 0.3

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

17. Simplified 0.3

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

18. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019163 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))