Bouland and Aaronson, Equation (26)

Average Error: 0.2 → 0.5

Time: 9.7s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r671 = a;
        double r672 = r671 * r671;
        double r673 = b;
        double r674 = r673 * r673;
        double r675 = r672 + r674;
        double r676 = 2.0;
        double r677 = pow(r675, r676);
        double r678 = 4.0;
        double r679 = r678 * r674;
        double r680 = r677 + r679;
        double r681 = 1.0;
        double r682 = r680 - r681;
        return r682;
}

↓

double f(double a, double b) {
        double r683 = a;
        double r684 = r683 * r683;
        double r685 = b;
        double r686 = r685 * r685;
        double r687 = r684 + r686;
        double r688 = 2.0;
        double r689 = pow(r687, r688);
        double r690 = 4.0;
        double r691 = r690 * r686;
        double r692 = r689 + r691;
        double r693 = 1.0;
        double r694 = r692 - r693;
        double r695 = cbrt(r694);
        double r696 = sqrt(r692);
        double r697 = sqrt(r693);
        double r698 = r696 + r697;
        double r699 = cbrt(r698);
        double r700 = r695 * r699;
        double r701 = r696 - r697;
        double r702 = cbrt(r701);
        double r703 = r700 * r702;
        double r704 = r703 * r695;
        return r704;
}

Error

Bits error versus a

Bits error versus b

Derivation

1. Initial program 0.2

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

3. Applied add-cube-cbrt 0.5

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

5. Applied add-sqr-sqrt 0.5

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

6. Applied add-sqr-sqrt 0.5

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

7. Applied difference-of-squares 0.5

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

8. Applied cbrt-prod 0.5

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

9. Applied associate-*r* 0.5

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

10. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020025 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))