Average Error: 14.1 → 0.3
Time: 22.2s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\left(\frac{1}{a + b} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right) \cdot \frac{\pi}{2 \cdot \left(b - a\right)}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\left(\frac{1}{a + b} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right) \cdot \frac{\pi}{2 \cdot \left(b - a\right)}
double f(double a, double b) {
        double r2671998 = atan2(1.0, 0.0);
        double r2671999 = 2.0;
        double r2672000 = r2671998 / r2671999;
        double r2672001 = 1.0;
        double r2672002 = b;
        double r2672003 = r2672002 * r2672002;
        double r2672004 = a;
        double r2672005 = r2672004 * r2672004;
        double r2672006 = r2672003 - r2672005;
        double r2672007 = r2672001 / r2672006;
        double r2672008 = r2672000 * r2672007;
        double r2672009 = r2672001 / r2672004;
        double r2672010 = r2672001 / r2672002;
        double r2672011 = r2672009 - r2672010;
        double r2672012 = r2672008 * r2672011;
        return r2672012;
}
↓
double f(double a, double b) {
        double r2672013 = 1.0;
        double r2672014 = a;
        double r2672015 = b;
        double r2672016 = r2672014 + r2672015;
        double r2672017 = r2672013 / r2672016;
        double r2672018 = r2672013 / r2672014;
        double r2672019 = r2672013 / r2672015;
        double r2672020 = r2672018 - r2672019;
        double r2672021 = r2672017 * r2672020;
        double r2672022 = atan2(1.0, 0.0);
        double r2672023 = 2.0;
        double r2672024 = r2672015 - r2672014;
        double r2672025 = r2672023 * r2672024;
        double r2672026 = r2672022 / r2672025;
        double r2672027 = r2672021 * r2672026;
        return r2672027;
}

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-squares9.3

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

  4. Applied associate-/r*8.8

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

  6. Applied frac-times8.7

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

  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

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

  11. Applied div-inv0.3

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

  12. Applied times-frac0.3

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

  13. Simplified0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2019170 
(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))))