Average Error: 14.6 → 0.3
Time: 24.7s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{2}}{a + b} \cdot \frac{1}{b - a}\]
\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{\frac{\pi}{a} - \frac{\pi}{b}}{2}}{a + b} \cdot \frac{1}{b - a}
double f(double a, double b) {
        double r1656519 = atan2(1.0, 0.0);
        double r1656520 = 2.0;
        double r1656521 = r1656519 / r1656520;
        double r1656522 = 1.0;
        double r1656523 = b;
        double r1656524 = r1656523 * r1656523;
        double r1656525 = a;
        double r1656526 = r1656525 * r1656525;
        double r1656527 = r1656524 - r1656526;
        double r1656528 = r1656522 / r1656527;
        double r1656529 = r1656521 * r1656528;
        double r1656530 = r1656522 / r1656525;
        double r1656531 = r1656522 / r1656523;
        double r1656532 = r1656530 - r1656531;
        double r1656533 = r1656529 * r1656532;
        return r1656533;
}
↓
double f(double a, double b) {
        double r1656534 = atan2(1.0, 0.0);
        double r1656535 = a;
        double r1656536 = r1656534 / r1656535;
        double r1656537 = b;
        double r1656538 = r1656534 / r1656537;
        double r1656539 = r1656536 - r1656538;
        double r1656540 = 2.0;
        double r1656541 = r1656539 / r1656540;
        double r1656542 = r1656535 + r1656537;
        double r1656543 = r1656541 / r1656542;
        double r1656544 = 1.0;
        double r1656545 = r1656537 - r1656535;
        double r1656546 = r1656544 / r1656545;
        double r1656547 = r1656543 * r1656546;
        return r1656547;
}

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 14.6

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

  2. Simplified14.5

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

  4. Applied difference-of-squares9.7

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

  5. Applied associate-/r*0.3

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

  6. Taylor expanded around 0 0.3

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

  7. Simplified0.3

    \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{2}}}{b + a}}{b - a}\]
  8. Using strategy rm

  9. Applied div-inv0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))