Bouland and Aaronson, Equation (25)

Average Error: 0.2 → 0.0

Time: 19.7s

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 r5643429 = a;
        double r5643430 = r5643429 * r5643429;
        double r5643431 = b;
        double r5643432 = r5643431 * r5643431;
        double r5643433 = r5643430 + r5643432;
        double r5643434 = 2.0;
        double r5643435 = pow(r5643433, r5643434);
        double r5643436 = 4.0;
        double r5643437 = 1.0;
        double r5643438 = r5643437 + r5643429;
        double r5643439 = r5643430 * r5643438;
        double r5643440 = 3.0;
        double r5643441 = r5643440 * r5643429;
        double r5643442 = r5643437 - r5643441;
        double r5643443 = r5643432 * r5643442;
        double r5643444 = r5643439 + r5643443;
        double r5643445 = r5643436 * r5643444;
        double r5643446 = r5643435 + r5643445;
        double r5643447 = r5643446 - r5643437;
        return r5643447;
}

↓

double f(double a, double b) {
        double r5643448 = 4.0;
        double r5643449 = a;
        double r5643450 = r5643449 * r5643449;
        double r5643451 = r5643450 * r5643449;
        double r5643452 = r5643448 * r5643451;
        double r5643453 = b;
        double r5643454 = r5643453 * r5643453;
        double r5643455 = r5643450 + r5643454;
        double r5643456 = -3.0;
        double r5643457 = r5643449 * r5643454;
        double r5643458 = r5643456 * r5643457;
        double r5643459 = r5643455 + r5643458;
        double r5643460 = r5643448 * r5643459;
        double r5643461 = -1.0;
        double r5643462 = sqrt(r5643455);
        double r5643463 = pow(r5643462, r5643448);
        double r5643464 = r5643461 + r5643463;
        double r5643465 = r5643460 + r5643464;
        double r5643466 = r5643452 + r5643465;
        return r5643466;
}

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))