Average Error: 14.5 → 0.3
Time: 8.6s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\left(\pi \cdot 1\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b + a}}{2 \cdot \left(b - a\right)}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\frac{\left(\pi \cdot 1\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b + a}}{2 \cdot \left(b - a\right)}
double f(double a, double b) {
        double r64322 = atan2(1.0, 0.0);
        double r64323 = 2.0;
        double r64324 = r64322 / r64323;
        double r64325 = 1.0;
        double r64326 = b;
        double r64327 = r64326 * r64326;
        double r64328 = a;
        double r64329 = r64328 * r64328;
        double r64330 = r64327 - r64329;
        double r64331 = r64325 / r64330;
        double r64332 = r64324 * r64331;
        double r64333 = r64325 / r64328;
        double r64334 = r64325 / r64326;
        double r64335 = r64333 - r64334;
        double r64336 = r64332 * r64335;
        return r64336;
}
↓
double f(double a, double b) {
        double r64337 = atan2(1.0, 0.0);
        double r64338 = 1.0;
        double r64339 = r64337 * r64338;
        double r64340 = a;
        double r64341 = r64338 / r64340;
        double r64342 = b;
        double r64343 = r64338 / r64342;
        double r64344 = r64341 - r64343;
        double r64345 = r64339 * r64344;
        double r64346 = r64342 + r64340;
        double r64347 = r64345 / r64346;
        double r64348 = 2.0;
        double r64349 = r64342 - r64340;
        double r64350 = r64348 * r64349;
        double r64351 = r64347 / r64350;
        return r64351;
}

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 14.5

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

  3. Applied difference-of-squares9.7

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

  4. Applied associate-/r*9.2

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

  6. Applied frac-times9.2

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

  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-*r/0.3

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

  10. Applied associate-*l/0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))