Average Error: 14.7 → 0.3
Time: 9.0m
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\left(\frac{1}{b \cdot a} \cdot \frac{\pi}{b - a}\right) \cdot \frac{\left(b - a\right) \cdot \frac{1}{2}}{a + b}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\left(\frac{1}{b \cdot a} \cdot \frac{\pi}{b - a}\right) \cdot \frac{\left(b - a\right) \cdot \frac{1}{2}}{a + b}
double f(double a, double b) {
        double r53187992 = atan2(1.0, 0.0);
        double r53187993 = 2.0;
        double r53187994 = r53187992 / r53187993;
        double r53187995 = 1.0;
        double r53187996 = b;
        double r53187997 = r53187996 * r53187996;
        double r53187998 = a;
        double r53187999 = r53187998 * r53187998;
        double r53188000 = r53187997 - r53187999;
        double r53188001 = r53187995 / r53188000;
        double r53188002 = r53187994 * r53188001;
        double r53188003 = r53187995 / r53187998;
        double r53188004 = r53187995 / r53187996;
        double r53188005 = r53188003 - r53188004;
        double r53188006 = r53188002 * r53188005;
        return r53188006;
}
↓
double f(double a, double b) {
        double r53188007 = 1.0;
        double r53188008 = b;
        double r53188009 = a;
        double r53188010 = r53188008 * r53188009;
        double r53188011 = r53188007 / r53188010;
        double r53188012 = atan2(1.0, 0.0);
        double r53188013 = r53188008 - r53188009;
        double r53188014 = r53188012 / r53188013;
        double r53188015 = r53188011 * r53188014;
        double r53188016 = 0.5;
        double r53188017 = r53188013 * r53188016;
        double r53188018 = r53188009 + r53188008;
        double r53188019 = r53188017 / r53188018;
        double r53188020 = r53188015 * r53188019;
        return r53188020;
}

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. Simplified14.7

    \[\leadsto \color{blue}{\frac{\frac{\pi}{b \cdot b - a \cdot a}}{\frac{2}{\frac{1}{a} - \frac{1}{b}}}}\]
  3. Using strategy rm

  4. Applied frac-sub14.7

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

  5. Applied associate-/r/14.7

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

  6. Applied difference-of-squares9.8

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

  7. Applied *-un-lft-identity9.8

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

  8. Applied times-frac9.1

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

  9. Applied times-frac0.3

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

  10. Simplified0.3

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

  12. Applied div-inv0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019120 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))