Average Error: 14.1 → 0.3
Time: 4.9s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\left(\frac{\pi}{2} \cdot 1\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b + a}}{b - a}\]
\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{\left(\frac{\pi}{2} \cdot 1\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b + a}}{b - a}
double f(double a, double b) {
        double r40953 = atan2(1.0, 0.0);
        double r40954 = 2.0;
        double r40955 = r40953 / r40954;
        double r40956 = 1.0;
        double r40957 = b;
        double r40958 = r40957 * r40957;
        double r40959 = a;
        double r40960 = r40959 * r40959;
        double r40961 = r40958 - r40960;
        double r40962 = r40956 / r40961;
        double r40963 = r40955 * r40962;
        double r40964 = r40956 / r40959;
        double r40965 = r40956 / r40957;
        double r40966 = r40964 - r40965;
        double r40967 = r40963 * r40966;
        return r40967;
}
↓
double f(double a, double b) {
        double r40968 = atan2(1.0, 0.0);
        double r40969 = 2.0;
        double r40970 = r40968 / r40969;
        double r40971 = 1.0;
        double r40972 = r40970 * r40971;
        double r40973 = a;
        double r40974 = r40971 / r40973;
        double r40975 = b;
        double r40976 = r40971 / r40975;
        double r40977 = r40974 - r40976;
        double r40978 = r40972 * r40977;
        double r40979 = r40975 + r40973;
        double r40980 = r40978 / r40979;
        double r40981 = r40975 - r40973;
        double r40982 = r40980 / r40981;
        return r40982;
}

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 14.1

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

  3. Applied difference-of-squares9.2

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

  4. Applied *-un-lft-identity9.2

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

  5. Applied times-frac8.8

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

  6. Applied associate-*r*8.8

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

  7. Simplified8.7

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

  9. Applied associate-*r/8.7

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

  10. Applied associate-*l/0.3

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

  12. Applied associate-*l/0.3

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

  13. Applied associate-*l/0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2020018 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))