Average Error: 14.6 → 0.3
Time: 5.8s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1 \cdot \left(\frac{\pi}{a} - \frac{\pi}{b}\right)}{b + a}}{2 \cdot \left(b - a\right)}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\frac{1 \cdot \left(\frac{\pi}{a} - \frac{\pi}{b}\right)}{b + a}}{2 \cdot \left(b - a\right)}
double f(double a, double b) {
        double r44539 = atan2(1.0, 0.0);
        double r44540 = 2.0;
        double r44541 = r44539 / r44540;
        double r44542 = 1.0;
        double r44543 = b;
        double r44544 = r44543 * r44543;
        double r44545 = a;
        double r44546 = r44545 * r44545;
        double r44547 = r44544 - r44546;
        double r44548 = r44542 / r44547;
        double r44549 = r44541 * r44548;
        double r44550 = r44542 / r44545;
        double r44551 = r44542 / r44543;
        double r44552 = r44550 - r44551;
        double r44553 = r44549 * r44552;
        return r44553;
}
↓
double f(double a, double b) {
        double r44554 = 1.0;
        double r44555 = atan2(1.0, 0.0);
        double r44556 = a;
        double r44557 = r44555 / r44556;
        double r44558 = b;
        double r44559 = r44555 / r44558;
        double r44560 = r44557 - r44559;
        double r44561 = r44554 * r44560;
        double r44562 = r44558 + r44556;
        double r44563 = r44561 / r44562;
        double r44564 = 2.0;
        double r44565 = r44558 - r44556;
        double r44566 = r44564 * r44565;
        double r44567 = r44563 / r44566;
        return r44567;
}

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 14.6

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

  3. Applied difference-of-squares9.7

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

  4. Applied associate-/r*9.1

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

  6. Applied frac-times9.1

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

  7. Applied associate-*l/0.3

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

  9. Applied associate-*r/0.3

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

  10. Applied associate-*l/0.3

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

  11. Taylor expanded around 0 0.3

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

  12. Simplified0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2020034 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))