Bouland and Aaronson, Equation (25)

Average Error: 0.2 → 0.0

Time: 20.5s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r9473658 = a;
        double r9473659 = r9473658 * r9473658;
        double r9473660 = b;
        double r9473661 = r9473660 * r9473660;
        double r9473662 = r9473659 + r9473661;
        double r9473663 = 2.0;
        double r9473664 = pow(r9473662, r9473663);
        double r9473665 = 4.0;
        double r9473666 = 1.0;
        double r9473667 = r9473666 + r9473658;
        double r9473668 = r9473659 * r9473667;
        double r9473669 = 3.0;
        double r9473670 = r9473669 * r9473658;
        double r9473671 = r9473666 - r9473670;
        double r9473672 = r9473661 * r9473671;
        double r9473673 = r9473668 + r9473672;
        double r9473674 = r9473665 * r9473673;
        double r9473675 = r9473664 + r9473674;
        double r9473676 = r9473675 - r9473666;
        return r9473676;
}

↓

double f(double a, double b) {
        double r9473677 = 4.0;
        double r9473678 = a;
        double r9473679 = r9473678 * r9473678;
        double r9473680 = r9473679 * r9473678;
        double r9473681 = r9473677 * r9473680;
        double r9473682 = b;
        double r9473683 = r9473682 * r9473682;
        double r9473684 = r9473679 + r9473683;
        double r9473685 = -3.0;
        double r9473686 = r9473678 * r9473683;
        double r9473687 = r9473685 * r9473686;
        double r9473688 = r9473684 + r9473687;
        double r9473689 = r9473677 * r9473688;
        double r9473690 = -1.0;
        double r9473691 = sqrt(r9473684);
        double r9473692 = pow(r9473691, r9473677);
        double r9473693 = r9473690 + r9473692;
        double r9473694 = r9473689 + r9473693;
        double r9473695 = r9473681 + r9473694;
        return r9473695;
}

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. Simplified 0.2

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

4. Applied add-sqr-sqrt 0.2

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

5. Applied associate-*r* 0.1

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

7. Applied pow1 0.1

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

8. Applied add-sqr-sqrt 0.1

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

9. Applied pow3 0.1

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

10. Applied pow-prod-up 0.0

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

11. Simplified 0.0

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

12. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019162 
(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))