Bouland and Aaronson, Equation (25)

Average Error: 0.2 → 0.1

Time: 47.6s

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 r67167709 = a;
        double r67167710 = r67167709 * r67167709;
        double r67167711 = b;
        double r67167712 = r67167711 * r67167711;
        double r67167713 = r67167710 + r67167712;
        double r67167714 = 2.0;
        double r67167715 = pow(r67167713, r67167714);
        double r67167716 = 4.0;
        double r67167717 = 1.0;
        double r67167718 = r67167717 + r67167709;
        double r67167719 = r67167710 * r67167718;
        double r67167720 = 3.0;
        double r67167721 = r67167720 * r67167709;
        double r67167722 = r67167717 - r67167721;
        double r67167723 = r67167712 * r67167722;
        double r67167724 = r67167719 + r67167723;
        double r67167725 = r67167716 * r67167724;
        double r67167726 = r67167715 + r67167725;
        double r67167727 = r67167726 - r67167717;
        return r67167727;
}

↓

double f(double a, double b) {
        double r67167728 = 4.0;
        double r67167729 = a;
        double r67167730 = r67167729 * r67167729;
        double r67167731 = 1.0;
        double r67167732 = r67167731 + r67167729;
        double r67167733 = r67167730 * r67167732;
        double r67167734 = b;
        double r67167735 = r67167734 * r67167734;
        double r67167736 = 3.0;
        double r67167737 = r67167729 * r67167736;
        double r67167738 = r67167731 - r67167737;
        double r67167739 = r67167735 * r67167738;
        double r67167740 = r67167733 + r67167739;
        double r67167741 = r67167728 * r67167740;
        double r67167742 = 2.0;
        double r67167743 = r67167730 * r67167742;
        double r67167744 = fma(r67167734, r67167734, r67167743);
        double r67167745 = r67167744 * r67167734;
        double r67167746 = pow(r67167729, r67167728);
        double r67167747 = fma(r67167734, r67167745, r67167746);
        double r67167748 = r67167741 + r67167747;
        double r67167749 = r67167748 - r67167731;
        return r67167749;
}

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

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

6. Applied pow1 0.2

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

7. Applied pow-sqr 0.2

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

8. Applied pow1 0.2

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

9. Applied pow-plus 0.2

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

10. Applied pow-prod-up 0.1

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

11. Simplified 0.1

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

12. 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 2019112 +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))