Average Error: 13.8 → 0.3
Time: 7.2s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi \cdot 1}{b + a}}{\frac{2 \cdot \left(b - a\right)}{\frac{1}{a} - \frac{1}{b}}}\]
\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{\pi \cdot 1}{b + a}}{\frac{2 \cdot \left(b - a\right)}{\frac{1}{a} - \frac{1}{b}}}
double f(double a, double b) {
        double r44687 = atan2(1.0, 0.0);
        double r44688 = 2.0;
        double r44689 = r44687 / r44688;
        double r44690 = 1.0;
        double r44691 = b;
        double r44692 = r44691 * r44691;
        double r44693 = a;
        double r44694 = r44693 * r44693;
        double r44695 = r44692 - r44694;
        double r44696 = r44690 / r44695;
        double r44697 = r44689 * r44696;
        double r44698 = r44690 / r44693;
        double r44699 = r44690 / r44691;
        double r44700 = r44698 - r44699;
        double r44701 = r44697 * r44700;
        return r44701;
}
↓
double f(double a, double b) {
        double r44702 = atan2(1.0, 0.0);
        double r44703 = 1.0;
        double r44704 = r44702 * r44703;
        double r44705 = b;
        double r44706 = a;
        double r44707 = r44705 + r44706;
        double r44708 = r44704 / r44707;
        double r44709 = 2.0;
        double r44710 = r44705 - r44706;
        double r44711 = r44709 * r44710;
        double r44712 = r44703 / r44706;
        double r44713 = r44703 / r44705;
        double r44714 = r44712 - r44713;
        double r44715 = r44711 / r44714;
        double r44716 = r44708 / r44715;
        return r44716;
}

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 13.8

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  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 associate-/r*8.9

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

  6. Applied frac-times8.9

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

  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-/l*0.3

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

  11. Applied associate-*r/0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2020003 +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))))