Average Error: 14.4 → 0.3
Time: 7.8m
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1}{\frac{1}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{\left(a + b\right) \cdot 2}}}}{b - a}\]
\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}{\frac{1}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{\left(a + b\right) \cdot 2}}}}{b - a}
double f(double a, double b) {
        double r19032395 = atan2(1.0, 0.0);
        double r19032396 = 2.0;
        double r19032397 = r19032395 / r19032396;
        double r19032398 = 1.0;
        double r19032399 = b;
        double r19032400 = r19032399 * r19032399;
        double r19032401 = a;
        double r19032402 = r19032401 * r19032401;
        double r19032403 = r19032400 - r19032402;
        double r19032404 = r19032398 / r19032403;
        double r19032405 = r19032397 * r19032404;
        double r19032406 = r19032398 / r19032401;
        double r19032407 = r19032398 / r19032399;
        double r19032408 = r19032406 - r19032407;
        double r19032409 = r19032405 * r19032408;
        return r19032409;
}
↓
double f(double a, double b) {
        double r19032410 = 1.0;
        double r19032411 = atan2(1.0, 0.0);
        double r19032412 = a;
        double r19032413 = r19032411 / r19032412;
        double r19032414 = b;
        double r19032415 = r19032411 / r19032414;
        double r19032416 = r19032413 - r19032415;
        double r19032417 = r19032412 + r19032414;
        double r19032418 = 2.0;
        double r19032419 = r19032417 * r19032418;
        double r19032420 = r19032416 / r19032419;
        double r19032421 = r19032410 / r19032420;
        double r19032422 = r19032410 / r19032421;
        double r19032423 = r19032414 - r19032412;
        double r19032424 = r19032422 / r19032423;
        return r19032424;
}

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 14.4

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

  2. Simplified14.3

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

  4. Applied difference-of-squares9.3

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

  5. Applied associate-/r*0.3

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

  6. Simplified0.3

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

  8. Applied *-un-lft-identity0.3

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

  9. Applied associate-/l*0.3

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

  11. Applied clear-num0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))