Average Error: 14.6 → 0.3
Time: 7.2s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\pi \cdot \left(\frac{1}{b + a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)}{2 \cdot \left(b - a\right)}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\pi \cdot \left(\frac{1}{b + a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)}{2 \cdot \left(b - a\right)}
double f(double a, double b) {
        double r46890 = atan2(1.0, 0.0);
        double r46891 = 2.0;
        double r46892 = r46890 / r46891;
        double r46893 = 1.0;
        double r46894 = b;
        double r46895 = r46894 * r46894;
        double r46896 = a;
        double r46897 = r46896 * r46896;
        double r46898 = r46895 - r46897;
        double r46899 = r46893 / r46898;
        double r46900 = r46892 * r46899;
        double r46901 = r46893 / r46896;
        double r46902 = r46893 / r46894;
        double r46903 = r46901 - r46902;
        double r46904 = r46900 * r46903;
        return r46904;
}
↓
double f(double a, double b) {
        double r46905 = atan2(1.0, 0.0);
        double r46906 = 1.0;
        double r46907 = b;
        double r46908 = a;
        double r46909 = r46907 + r46908;
        double r46910 = r46906 / r46909;
        double r46911 = r46906 / r46908;
        double r46912 = r46906 / r46907;
        double r46913 = r46911 - r46912;
        double r46914 = r46910 * r46913;
        double r46915 = r46905 * r46914;
        double r46916 = 2.0;
        double r46917 = r46907 - r46908;
        double r46918 = r46916 * r46917;
        double r46919 = r46915 / r46918;
        return r46919;
}

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 14.6

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  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 associate-/r*9.2

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

  6. Applied frac-times9.2

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

  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

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2020039 +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))))