Average Error: 14.5 → 0.2
Time: 7.8s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1}{2} \cdot \frac{\pi}{a \cdot b}}{b + a}\]
\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{1}{2} \cdot \frac{\pi}{a \cdot b}}{b + a}
double f(double a, double b) {
        double r38778 = atan2(1.0, 0.0);
        double r38779 = 2.0;
        double r38780 = r38778 / r38779;
        double r38781 = 1.0;
        double r38782 = b;
        double r38783 = r38782 * r38782;
        double r38784 = a;
        double r38785 = r38784 * r38784;
        double r38786 = r38783 - r38785;
        double r38787 = r38781 / r38786;
        double r38788 = r38780 * r38787;
        double r38789 = r38781 / r38784;
        double r38790 = r38781 / r38782;
        double r38791 = r38789 - r38790;
        double r38792 = r38788 * r38791;
        return r38792;
}
↓
double f(double a, double b) {
        double r38793 = 1.0;
        double r38794 = 2.0;
        double r38795 = r38793 / r38794;
        double r38796 = atan2(1.0, 0.0);
        double r38797 = a;
        double r38798 = b;
        double r38799 = r38797 * r38798;
        double r38800 = r38796 / r38799;
        double r38801 = r38795 * r38800;
        double r38802 = r38798 + r38797;
        double r38803 = r38801 / r38802;
        return r38803;
}

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-squares9.6

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

  4. Applied *-un-lft-identity9.6

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

  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. Simplified9.1

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

  9. Applied associate-*l/9.1

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

  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. Simplified0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2019304 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))