Average Error: 14.5 → 0.8
Time: 14.5s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{1 \cdot \left(\left(\frac{\pi}{2} \cdot 1\right) \cdot 1\right)}{\left(a \cdot b\right) \cdot \left(b + a\right)}\]
\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 \cdot \left(\left(\frac{\pi}{2} \cdot 1\right) \cdot 1\right)}{\left(a \cdot b\right) \cdot \left(b + a\right)}
double f(double a, double b) {
        double r44542 = atan2(1.0, 0.0);
        double r44543 = 2.0;
        double r44544 = r44542 / r44543;
        double r44545 = 1.0;
        double r44546 = b;
        double r44547 = r44546 * r44546;
        double r44548 = a;
        double r44549 = r44548 * r44548;
        double r44550 = r44547 - r44549;
        double r44551 = r44545 / r44550;
        double r44552 = r44544 * r44551;
        double r44553 = r44545 / r44548;
        double r44554 = r44545 / r44546;
        double r44555 = r44553 - r44554;
        double r44556 = r44552 * r44555;
        return r44556;
}
↓
double f(double a, double b) {
        double r44557 = 1.0;
        double r44558 = atan2(1.0, 0.0);
        double r44559 = 2.0;
        double r44560 = r44558 / r44559;
        double r44561 = 1.0;
        double r44562 = r44560 * r44561;
        double r44563 = r44562 * r44561;
        double r44564 = r44557 * r44563;
        double r44565 = a;
        double r44566 = b;
        double r44567 = r44565 * r44566;
        double r44568 = r44566 + r44565;
        double r44569 = r44567 * r44568;
        double r44570 = r44564 / r44569;
        return r44570;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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-squares9.6

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

  4. Applied *-un-lft-identity9.6

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

  5. Applied times-frac9.1

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

  6. Applied associate-*r*9.1

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

  7. Simplified9.1

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

  9. Applied associate-*r/9.0

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

  10. Applied associate-*l/0.3

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

  12. Applied frac-sub0.3

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

  13. Applied associate-*r/0.3

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

  14. Simplified0.3

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

  16. Applied associate-*l/0.3

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

  17. Applied associate-*l/0.3

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

  18. Final simplification0.8

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

Reproduce

herbie shell --seed 2019298 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))