Average Error: 14.1 → 0.3
Time: 18.6s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{1 \cdot \frac{\frac{\pi}{2}}{a + b}}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{1 \cdot \frac{\frac{\pi}{2}}{a + b}}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}
double f(double a, double b) {
        double r2315547 = atan2(1.0, 0.0);
        double r2315548 = 2.0;
        double r2315549 = r2315547 / r2315548;
        double r2315550 = 1.0;
        double r2315551 = b;
        double r2315552 = r2315551 * r2315551;
        double r2315553 = a;
        double r2315554 = r2315553 * r2315553;
        double r2315555 = r2315552 - r2315554;
        double r2315556 = r2315550 / r2315555;
        double r2315557 = r2315549 * r2315556;
        double r2315558 = r2315550 / r2315553;
        double r2315559 = r2315550 / r2315551;
        double r2315560 = r2315558 - r2315559;
        double r2315561 = r2315557 * r2315560;
        return r2315561;
}
↓
double f(double a, double b) {
        double r2315562 = 1.0;
        double r2315563 = atan2(1.0, 0.0);
        double r2315564 = 2.0;
        double r2315565 = r2315563 / r2315564;
        double r2315566 = a;
        double r2315567 = b;
        double r2315568 = r2315566 + r2315567;
        double r2315569 = r2315565 / r2315568;
        double r2315570 = r2315562 * r2315569;
        double r2315571 = r2315567 - r2315566;
        double r2315572 = r2315562 / r2315566;
        double r2315573 = r2315562 / r2315567;
        double r2315574 = r2315572 - r2315573;
        double r2315575 = r2315571 / r2315574;
        double r2315576 = r2315570 / r2315575;
        return r2315576;
}

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-frac8.8

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

  6. Applied associate-*r*8.8

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

  7. Simplified8.8

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

  9. Applied associate-*r/8.7

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

  10. Applied associate-*l/0.3

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

  12. Applied associate-/l*0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019172 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))