Average Error: 14.1 → 0.2
Time: 22.4s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\left(\frac{\pi}{2} \cdot 1\right) \cdot \frac{\left(b - a\right) \cdot 1}{b + a}}{a \cdot b}}{b - a}\]
\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{\left(\frac{\pi}{2} \cdot 1\right) \cdot \frac{\left(b - a\right) \cdot 1}{b + a}}{a \cdot b}}{b - a}
double f(double a, double b) {
        double r78612 = atan2(1.0, 0.0);
        double r78613 = 2.0;
        double r78614 = r78612 / r78613;
        double r78615 = 1.0;
        double r78616 = b;
        double r78617 = r78616 * r78616;
        double r78618 = a;
        double r78619 = r78618 * r78618;
        double r78620 = r78617 - r78619;
        double r78621 = r78615 / r78620;
        double r78622 = r78614 * r78621;
        double r78623 = r78615 / r78618;
        double r78624 = r78615 / r78616;
        double r78625 = r78623 - r78624;
        double r78626 = r78622 * r78625;
        return r78626;
}
↓
double f(double a, double b) {
        double r78627 = atan2(1.0, 0.0);
        double r78628 = 2.0;
        double r78629 = r78627 / r78628;
        double r78630 = 1.0;
        double r78631 = r78629 * r78630;
        double r78632 = b;
        double r78633 = a;
        double r78634 = r78632 - r78633;
        double r78635 = r78634 * r78630;
        double r78636 = r78632 + r78633;
        double r78637 = r78635 / r78636;
        double r78638 = r78631 * r78637;
        double r78639 = r78633 * r78632;
        double r78640 = r78638 / r78639;
        double r78641 = r78640 / r78634;
        return r78641;
}

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 14.1

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

  3. Applied difference-of-squares9.2

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

  4. Applied *-un-lft-identity9.2

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

  5. Applied times-frac8.7

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

  6. Applied associate-*r*8.7

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

  7. Simplified8.7

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

  9. Applied associate-*r/8.6

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

  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

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

  16. Applied div-inv0.3

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

  17. Applied associate-*l*0.3

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

  18. Simplified0.2

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

  19. Final simplification0.2

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

Reproduce

herbie shell --seed 2019303 +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))))