Average Error: 14.3 → 0.2
Time: 40.5s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\pi}{2}}{a + b}}{a \cdot \left(b - a\right)} - \frac{\frac{\frac{\frac{\pi}{2}}{a + b}}{b}}{b - a}\]
\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{\frac{\pi}{2}}{a + b}}{a \cdot \left(b - a\right)} - \frac{\frac{\frac{\frac{\pi}{2}}{a + b}}{b}}{b - a}
double f(double a, double b) {
        double r1488537 = atan2(1.0, 0.0);
        double r1488538 = 2.0;
        double r1488539 = r1488537 / r1488538;
        double r1488540 = 1.0;
        double r1488541 = b;
        double r1488542 = r1488541 * r1488541;
        double r1488543 = a;
        double r1488544 = r1488543 * r1488543;
        double r1488545 = r1488542 - r1488544;
        double r1488546 = r1488540 / r1488545;
        double r1488547 = r1488539 * r1488546;
        double r1488548 = r1488540 / r1488543;
        double r1488549 = r1488540 / r1488541;
        double r1488550 = r1488548 - r1488549;
        double r1488551 = r1488547 * r1488550;
        return r1488551;
}
↓
double f(double a, double b) {
        double r1488552 = atan2(1.0, 0.0);
        double r1488553 = 2.0;
        double r1488554 = r1488552 / r1488553;
        double r1488555 = a;
        double r1488556 = b;
        double r1488557 = r1488555 + r1488556;
        double r1488558 = r1488554 / r1488557;
        double r1488559 = r1488556 - r1488555;
        double r1488560 = r1488555 * r1488559;
        double r1488561 = r1488558 / r1488560;
        double r1488562 = r1488558 / r1488556;
        double r1488563 = r1488562 / r1488559;
        double r1488564 = r1488561 - r1488563;
        return r1488564;
}

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 14.3

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

  2. Simplified9.0

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

  4. Applied div-inv9.1

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

  5. Applied associate-/l*4.6

    \[\leadsto \frac{\frac{\frac{\frac{\pi}{2}}{a + b}}{b - a}}{a} - \color{blue}{\frac{\frac{\frac{\pi}{2}}{a + b}}{\frac{b}{\frac{1}{b - a}}}}\]
  6. Using strategy rm

  7. Applied associate-/l/0.3

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

  9. Applied div-inv0.3

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

  10. Simplified0.2

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

  12. Applied associate-/r*0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))