Average Error: 14.5 → 0.3
Time: 7.5s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\left(0.5 \cdot \frac{\pi}{a \cdot b}\right) \cdot \frac{1}{b + a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\left(0.5 \cdot \frac{\pi}{a \cdot b}\right) \cdot \frac{1}{b + a}
double f(double a, double b) {
        double r45707 = atan2(1.0, 0.0);
        double r45708 = 2.0;
        double r45709 = r45707 / r45708;
        double r45710 = 1.0;
        double r45711 = b;
        double r45712 = r45711 * r45711;
        double r45713 = a;
        double r45714 = r45713 * r45713;
        double r45715 = r45712 - r45714;
        double r45716 = r45710 / r45715;
        double r45717 = r45709 * r45716;
        double r45718 = r45710 / r45713;
        double r45719 = r45710 / r45711;
        double r45720 = r45718 - r45719;
        double r45721 = r45717 * r45720;
        return r45721;
}
↓
double f(double a, double b) {
        double r45722 = 0.5;
        double r45723 = atan2(1.0, 0.0);
        double r45724 = a;
        double r45725 = b;
        double r45726 = r45724 * r45725;
        double r45727 = r45723 / r45726;
        double r45728 = r45722 * r45727;
        double r45729 = 1.0;
        double r45730 = r45725 + r45724;
        double r45731 = r45729 / r45730;
        double r45732 = r45728 * r45731;
        return r45732;
}

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 14.5

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  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 *-un-lft-identity9.7

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

  5. Applied times-frac9.2

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

  6. Applied associate-*r*9.2

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

  7. Simplified9.2

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

  9. Applied associate-*l/9.2

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

  10. Applied associate-*l/0.3

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

  11. Taylor expanded around 0 0.2

    \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{\pi}{a \cdot b}}}{b + a}\]
  12. Using strategy rm

  13. Applied div-inv0.3

    \[\leadsto \color{blue}{\left(0.5 \cdot \frac{\pi}{a \cdot b}\right) \cdot \frac{1}{b + a}}\]

  14. Final simplification0.3

    \[\leadsto \left(0.5 \cdot \frac{\pi}{a \cdot b}\right) \cdot \frac{1}{b + a}\]

Reproduce

herbie shell --seed 2020021 +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))))