NMSE Section 6.1 mentioned, B

Average Error: 14.1 → 0.3

Time: 19.9s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r1564982 = atan2(1.0, 0.0);
        double r1564983 = 2.0;
        double r1564984 = r1564982 / r1564983;
        double r1564985 = 1.0;
        double r1564986 = b;
        double r1564987 = r1564986 * r1564986;
        double r1564988 = a;
        double r1564989 = r1564988 * r1564988;
        double r1564990 = r1564987 - r1564989;
        double r1564991 = r1564985 / r1564990;
        double r1564992 = r1564984 * r1564991;
        double r1564993 = r1564985 / r1564988;
        double r1564994 = r1564985 / r1564986;
        double r1564995 = r1564993 - r1564994;
        double r1564996 = r1564992 * r1564995;
        return r1564996;
}

↓

double f(double a, double b) {
        double r1564997 = 0.5;
        double r1564998 = a;
        double r1564999 = b;
        double r1565000 = r1564998 * r1564999;
        double r1565001 = r1564997 / r1565000;
        double r1565002 = atan2(1.0, 0.0);
        double r1565003 = r1564999 + r1564998;
        double r1565004 = r1565002 / r1565003;
        double r1565005 = r1565001 * r1565004;
        return r1565005;
}

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 0.3

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

4. Applied *-un-lft-identity 0.3

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

5. Applied *-un-lft-identity 0.3

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

6. Applied div-inv 0.3

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

7. Applied times-frac 0.3

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

8. Applied distribute-rgt-neg-in 0.3

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

9. Applied *-un-lft-identity 0.3

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

10. Applied div-inv 0.3

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

11. Applied times-frac 0.3

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

12. Applied distribute-lft-out 0.3

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

13. Applied times-frac 0.3

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

14. Simplified 0.3

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

15. Simplified 0.3

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

16. Taylor expanded around 0 0.3

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

17. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019158 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))