Bouland and Aaronson, Equation (24)

Average Error: 0.2 → 0.5

Time: 7.2s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r225534 = a;
        double r225535 = r225534 * r225534;
        double r225536 = b;
        double r225537 = r225536 * r225536;
        double r225538 = r225535 + r225537;
        double r225539 = 2.0;
        double r225540 = pow(r225538, r225539);
        double r225541 = 4.0;
        double r225542 = 1.0;
        double r225543 = r225542 - r225534;
        double r225544 = r225535 * r225543;
        double r225545 = 3.0;
        double r225546 = r225545 + r225534;
        double r225547 = r225537 * r225546;
        double r225548 = r225544 + r225547;
        double r225549 = r225541 * r225548;
        double r225550 = r225540 + r225549;
        double r225551 = r225550 - r225542;
        return r225551;
}

↓

double f(double a, double b) {
        double r225552 = a;
        double r225553 = r225552 * r225552;
        double r225554 = b;
        double r225555 = r225554 * r225554;
        double r225556 = r225553 + r225555;
        double r225557 = 2.0;
        double r225558 = pow(r225556, r225557);
        double r225559 = 4.0;
        double r225560 = 1.0;
        double r225561 = r225560 - r225552;
        double r225562 = r225553 * r225561;
        double r225563 = 3.0;
        double r225564 = r225563 + r225552;
        double r225565 = r225555 * r225564;
        double r225566 = r225562 + r225565;
        double r225567 = r225559 * r225566;
        double r225568 = r225558 + r225567;
        double r225569 = cbrt(r225568);
        double r225570 = r225569 * r225569;
        double r225571 = cbrt(r225570);
        double r225572 = r225569 * r225571;
        double r225573 = cbrt(r225569);
        double r225574 = r225572 * r225573;
        double r225575 = r225574 * r225569;
        double r225576 = r225575 - r225560;
        return r225576;
}

Error

Bits error versus a

Bits error versus b

Derivation

1. Initial program 0.2

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

3. Applied add-cube-cbrt 0.5

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

5. Applied add-cube-cbrt 0.5

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

6. Applied cbrt-prod 0.5

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

7. Applied associate-*r* 0.5

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

8. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019353 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))