Average Error: 13.8 → 0.3
Time: 8.8m
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1}{2}}{b + a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{b - a}\]
\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{1}{2}}{b + a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{b - a}
double f(double a, double b) {
        double r50971506 = atan2(1.0, 0.0);
        double r50971507 = 2.0;
        double r50971508 = r50971506 / r50971507;
        double r50971509 = 1.0;
        double r50971510 = b;
        double r50971511 = r50971510 * r50971510;
        double r50971512 = a;
        double r50971513 = r50971512 * r50971512;
        double r50971514 = r50971511 - r50971513;
        double r50971515 = r50971509 / r50971514;
        double r50971516 = r50971508 * r50971515;
        double r50971517 = r50971509 / r50971512;
        double r50971518 = r50971509 / r50971510;
        double r50971519 = r50971517 - r50971518;
        double r50971520 = r50971516 * r50971519;
        return r50971520;
}
↓
double f(double a, double b) {
        double r50971521 = 0.5;
        double r50971522 = b;
        double r50971523 = a;
        double r50971524 = r50971522 + r50971523;
        double r50971525 = r50971521 / r50971524;
        double r50971526 = atan2(1.0, 0.0);
        double r50971527 = r50971526 / r50971523;
        double r50971528 = r50971526 / r50971522;
        double r50971529 = r50971527 - r50971528;
        double r50971530 = r50971522 - r50971523;
        double r50971531 = r50971529 / r50971530;
        double r50971532 = r50971525 * r50971531;
        return r50971532;
}

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 13.8

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

  2. Simplified13.8

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

  4. Applied div-inv13.8

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

  5. Applied difference-of-squares9.4

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

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

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

  7. Applied times-frac9.0

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

  8. Applied times-frac0.3

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

  9. Simplified0.3

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

  10. Simplified0.3

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

  11. Taylor expanded around 0 0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019107 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))