Average Error: 14.3 → 0.3
Time: 8.7s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\pi \cdot \frac{1}{b + a}}{a \cdot b}}{2} \cdot 1}{1}\]
\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 \cdot \frac{1}{b + a}}{a \cdot b}}{2} \cdot 1}{1}
double f(double a, double b) {
        double r41562 = atan2(1.0, 0.0);
        double r41563 = 2.0;
        double r41564 = r41562 / r41563;
        double r41565 = 1.0;
        double r41566 = b;
        double r41567 = r41566 * r41566;
        double r41568 = a;
        double r41569 = r41568 * r41568;
        double r41570 = r41567 - r41569;
        double r41571 = r41565 / r41570;
        double r41572 = r41564 * r41571;
        double r41573 = r41565 / r41568;
        double r41574 = r41565 / r41566;
        double r41575 = r41573 - r41574;
        double r41576 = r41572 * r41575;
        return r41576;
}
↓
double f(double a, double b) {
        double r41577 = atan2(1.0, 0.0);
        double r41578 = 1.0;
        double r41579 = b;
        double r41580 = a;
        double r41581 = r41579 + r41580;
        double r41582 = r41578 / r41581;
        double r41583 = r41577 * r41582;
        double r41584 = r41580 * r41579;
        double r41585 = r41583 / r41584;
        double r41586 = 2.0;
        double r41587 = r41585 / r41586;
        double r41588 = r41587 * r41578;
        double r41589 = 1.0;
        double r41590 = r41588 / r41589;
        return r41590;
}

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. Using strategy rm

  3. Applied difference-of-squares9.9

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

  4. Applied associate-/r*9.4

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

  6. Applied frac-times9.4

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

  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 frac-sub0.4

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

  10. Applied associate-*r/0.3

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

  11. Simplified0.3

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

  13. Applied associate-/l*0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2019308 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))