Average Error: 14.2 → 0.3
Time: 15.4s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\pi}{2}}{b + a} \cdot 1}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}\]
\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}}{b + a} \cdot 1}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}
double f(double a, double b) {
        double r68470 = atan2(1.0, 0.0);
        double r68471 = 2.0;
        double r68472 = r68470 / r68471;
        double r68473 = 1.0;
        double r68474 = b;
        double r68475 = r68474 * r68474;
        double r68476 = a;
        double r68477 = r68476 * r68476;
        double r68478 = r68475 - r68477;
        double r68479 = r68473 / r68478;
        double r68480 = r68472 * r68479;
        double r68481 = r68473 / r68476;
        double r68482 = r68473 / r68474;
        double r68483 = r68481 - r68482;
        double r68484 = r68480 * r68483;
        return r68484;
}
↓
double f(double a, double b) {
        double r68485 = atan2(1.0, 0.0);
        double r68486 = 2.0;
        double r68487 = r68485 / r68486;
        double r68488 = b;
        double r68489 = a;
        double r68490 = r68488 + r68489;
        double r68491 = r68487 / r68490;
        double r68492 = 1.0;
        double r68493 = r68491 * r68492;
        double r68494 = r68488 - r68489;
        double r68495 = r68492 / r68489;
        double r68496 = r68492 / r68488;
        double r68497 = r68495 - r68496;
        double r68498 = r68494 / r68497;
        double r68499 = r68493 / r68498;
        return r68499;
}

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 14.2

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

  3. Applied difference-of-squares9.4

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

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

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

  5. Applied times-frac8.9

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

  6. Applied associate-*r*8.9

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

  7. Simplified8.9

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

  9. Applied associate-*r/8.8

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

  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 associate-/l*0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019350 +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))))