Average Error: 14.4 → 0.3
Time: 36.1s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{2} \cdot \frac{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b - a}}{b + a}\]
\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 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b - a}}{b + a}
double f(double a, double b) {
        double r80777 = atan2(1.0, 0.0);
        double r80778 = 2.0;
        double r80779 = r80777 / r80778;
        double r80780 = 1.0;
        double r80781 = b;
        double r80782 = r80781 * r80781;
        double r80783 = a;
        double r80784 = r80783 * r80783;
        double r80785 = r80782 - r80784;
        double r80786 = r80780 / r80785;
        double r80787 = r80779 * r80786;
        double r80788 = r80780 / r80783;
        double r80789 = r80780 / r80781;
        double r80790 = r80788 - r80789;
        double r80791 = r80787 * r80790;
        return r80791;
}
↓
double f(double a, double b) {
        double r80792 = atan2(1.0, 0.0);
        double r80793 = 2.0;
        double r80794 = r80792 / r80793;
        double r80795 = 1.0;
        double r80796 = a;
        double r80797 = r80795 / r80796;
        double r80798 = b;
        double r80799 = r80795 / r80798;
        double r80800 = r80797 - r80799;
        double r80801 = r80795 * r80800;
        double r80802 = r80798 - r80796;
        double r80803 = r80801 / r80802;
        double r80804 = r80794 * r80803;
        double r80805 = r80798 + r80796;
        double r80806 = r80804 / r80805;
        return r80806;
}

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 14.4

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

  3. Applied difference-of-squares9.3

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

  4. Applied *-un-lft-identity9.3

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

  5. Applied times-frac9.0

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

  6. Applied associate-*r*9.0

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

  7. Simplified8.9

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

  9. Applied associate-*l/8.9

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

  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. Using strategy rm

  12. Applied associate-*l*0.3

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

  13. Simplified0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))