Average Error: 15.0 → 0.4
Time: 18.6s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\frac{\pi}{2} \cdot \left(\left(b - a\right) \cdot 1\right)}{\frac{b + a}{1}}}{a \cdot b}}{b - a}\]
\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{\frac{\frac{\pi}{2} \cdot \left(\left(b - a\right) \cdot 1\right)}{\frac{b + a}{1}}}{a \cdot b}}{b - a}
double f(double a, double b) {
        double r2463933 = atan2(1.0, 0.0);
        double r2463934 = 2.0;
        double r2463935 = r2463933 / r2463934;
        double r2463936 = 1.0;
        double r2463937 = b;
        double r2463938 = r2463937 * r2463937;
        double r2463939 = a;
        double r2463940 = r2463939 * r2463939;
        double r2463941 = r2463938 - r2463940;
        double r2463942 = r2463936 / r2463941;
        double r2463943 = r2463935 * r2463942;
        double r2463944 = r2463936 / r2463939;
        double r2463945 = r2463936 / r2463937;
        double r2463946 = r2463944 - r2463945;
        double r2463947 = r2463943 * r2463946;
        return r2463947;
}
↓
double f(double a, double b) {
        double r2463948 = atan2(1.0, 0.0);
        double r2463949 = 2.0;
        double r2463950 = r2463948 / r2463949;
        double r2463951 = b;
        double r2463952 = a;
        double r2463953 = r2463951 - r2463952;
        double r2463954 = 1.0;
        double r2463955 = r2463953 * r2463954;
        double r2463956 = r2463950 * r2463955;
        double r2463957 = r2463951 + r2463952;
        double r2463958 = r2463957 / r2463954;
        double r2463959 = r2463956 / r2463958;
        double r2463960 = r2463952 * r2463951;
        double r2463961 = r2463959 / r2463960;
        double r2463962 = r2463961 / r2463953;
        return r2463962;
}

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 15.0

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

  3. Applied difference-of-squares10.0

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

  4. Applied associate-/r*9.4

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

  6. Applied associate-*r/9.4

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

  7. Applied associate-*l/0.3

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

  9. Applied associate-*r/0.3

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

  11. Applied frac-sub0.3

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

  12. Applied associate-*r/0.3

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

  13. Simplified0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019171 
(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))))