Bouland and Aaronson, Equation (25)

Average Error: 0.2 → 0.1

Time: 49.6s

Precision: 64

Math

\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]

↓

\[\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) + (b \cdot \left((b \cdot b + \left(\left(a \cdot a\right) \cdot 2\right))_* \cdot b\right) + \left({a}^{4}\right))_*\right) - 1\]

C

double f(double a, double b) {
        double r22878579 = a;
        double r22878580 = r22878579 * r22878579;
        double r22878581 = b;
        double r22878582 = r22878581 * r22878581;
        double r22878583 = r22878580 + r22878582;
        double r22878584 = 2.0;
        double r22878585 = pow(r22878583, r22878584);
        double r22878586 = 4.0;
        double r22878587 = 1.0;
        double r22878588 = r22878587 + r22878579;
        double r22878589 = r22878580 * r22878588;
        double r22878590 = 3.0;
        double r22878591 = r22878590 * r22878579;
        double r22878592 = r22878587 - r22878591;
        double r22878593 = r22878582 * r22878592;
        double r22878594 = r22878589 + r22878593;
        double r22878595 = r22878586 * r22878594;
        double r22878596 = r22878585 + r22878595;
        double r22878597 = r22878596 - r22878587;
        return r22878597;
}

↓

double f(double a, double b) {
        double r22878598 = 4.0;
        double r22878599 = a;
        double r22878600 = r22878599 * r22878599;
        double r22878601 = 1.0;
        double r22878602 = r22878601 + r22878599;
        double r22878603 = r22878600 * r22878602;
        double r22878604 = b;
        double r22878605 = r22878604 * r22878604;
        double r22878606 = 3.0;
        double r22878607 = r22878599 * r22878606;
        double r22878608 = r22878601 - r22878607;
        double r22878609 = r22878605 * r22878608;
        double r22878610 = r22878603 + r22878609;
        double r22878611 = r22878598 * r22878610;
        double r22878612 = 2.0;
        double r22878613 = r22878600 * r22878612;
        double r22878614 = fma(r22878604, r22878604, r22878613);
        double r22878615 = r22878614 * r22878604;
        double r22878616 = pow(r22878599, r22878598);
        double r22878617 = fma(r22878604, r22878615, r22878616);
        double r22878618 = r22878611 + r22878617;
        double r22878619 = r22878618 - r22878601;
        return r22878619;
}

Error

Bits error versus a

Bits error versus b

Derivation

1. Initial program 0.2

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]

2. Taylor expanded around 0 0.0

   \[\leadsto \left(\color{blue}{\left({b}^{4} + \left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + {a}^{4}\right)\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]

3. Simplified 0.2

   \[\leadsto \left(\color{blue}{(b \cdot \left(b \cdot (b \cdot b + \left(2 \cdot \left(a \cdot a\right)\right))_*\right) + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right))_*} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
4. Using strategy rm

5. Applied pow1 0.2

   \[\leadsto \left((b \cdot \left(b \cdot (b \cdot b + \left(2 \cdot \left(a \cdot a\right)\right))_*\right) + \left(\left(a \cdot a\right) \cdot \left(\color{blue}{{a}^{1}} \cdot a\right)\right))_* + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]

6. Applied pow-plus 0.2

   \[\leadsto \left((b \cdot \left(b \cdot (b \cdot b + \left(2 \cdot \left(a \cdot a\right)\right))_*\right) + \left(\left(a \cdot a\right) \cdot \color{blue}{{a}^{\left(1 + 1\right)}}\right))_* + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]

7. Applied pow2 0.2

   \[\leadsto \left((b \cdot \left(b \cdot (b \cdot b + \left(2 \cdot \left(a \cdot a\right)\right))_*\right) + \left(\color{blue}{{a}^{2}} \cdot {a}^{\left(1 + 1\right)}\right))_* + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]

8. Applied pow-prod-up 0.1

   \[\leadsto \left((b \cdot \left(b \cdot (b \cdot b + \left(2 \cdot \left(a \cdot a\right)\right))_*\right) + \color{blue}{\left({a}^{\left(2 + \left(1 + 1\right)\right)}\right)})_* + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]

9. Simplified 0.1

   \[\leadsto \left((b \cdot \left(b \cdot (b \cdot b + \left(2 \cdot \left(a \cdot a\right)\right))_*\right) + \left({a}^{\color{blue}{4}}\right))_* + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]

10. Final simplification 0.1

    \[\leadsto \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) + (b \cdot \left((b \cdot b + \left(\left(a \cdot a\right) \cdot 2\right))_* \cdot b\right) + \left({a}^{4}\right))_*\right) - 1\]

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (+ 1 a)) (* (* b b) (- 1 (* 3 a)))))) 1))