Average Error: 14.7 → 0.3
Time: 5.7s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{2} \cdot \frac{1}{a \cdot b}}{b + a}\]
\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{\pi}{2} \cdot \frac{1}{a \cdot b}}{b + a}
double f(double a, double b) {
        double r45771 = atan2(1.0, 0.0);
        double r45772 = 2.0;
        double r45773 = r45771 / r45772;
        double r45774 = 1.0;
        double r45775 = b;
        double r45776 = r45775 * r45775;
        double r45777 = a;
        double r45778 = r45777 * r45777;
        double r45779 = r45776 - r45778;
        double r45780 = r45774 / r45779;
        double r45781 = r45773 * r45780;
        double r45782 = r45774 / r45777;
        double r45783 = r45774 / r45775;
        double r45784 = r45782 - r45783;
        double r45785 = r45781 * r45784;
        return r45785;
}
↓
double f(double a, double b) {
        double r45786 = atan2(1.0, 0.0);
        double r45787 = 2.0;
        double r45788 = r45786 / r45787;
        double r45789 = 1.0;
        double r45790 = a;
        double r45791 = b;
        double r45792 = r45790 * r45791;
        double r45793 = r45789 / r45792;
        double r45794 = r45788 * r45793;
        double r45795 = r45791 + r45790;
        double r45796 = r45794 / r45795;
        return r45796;
}

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 14.7

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  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-frac9.3

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

  6. Applied associate-*r*9.3

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

  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*0.3

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

  11. Applied associate-*l/0.3

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

  12. Taylor expanded around 0 0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2020002 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))