Bouland and Aaronson, Equation (26)

Average Error: 0.2 → 0.3

Time: 27.9s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r8724770 = a;
        double r8724771 = r8724770 * r8724770;
        double r8724772 = b;
        double r8724773 = r8724772 * r8724772;
        double r8724774 = r8724771 + r8724773;
        double r8724775 = 2.0;
        double r8724776 = pow(r8724774, r8724775);
        double r8724777 = 4.0;
        double r8724778 = r8724777 * r8724773;
        double r8724779 = r8724776 + r8724778;
        double r8724780 = 1.0;
        double r8724781 = r8724779 - r8724780;
        return r8724781;
}

↓

double f(double a, double b) {
        double r8724782 = b;
        double r8724783 = r8724782 * r8724782;
        double r8724784 = 4.0;
        double r8724785 = r8724783 * r8724784;
        double r8724786 = a;
        double r8724787 = r8724786 * r8724786;
        double r8724788 = r8724787 + r8724783;
        double r8724789 = 2.0;
        double r8724790 = pow(r8724788, r8724789);
        double r8724791 = r8724785 + r8724790;
        double r8724792 = sqrt(r8724791);
        double r8724793 = cbrt(r8724791);
        double r8724794 = sqrt(r8724793);
        double r8724795 = r8724793 * r8724793;
        double r8724796 = sqrt(r8724795);
        double r8724797 = r8724794 * r8724796;
        double r8724798 = r8724792 * r8724797;
        double r8724799 = 1.0;
        double r8724800 = r8724798 - r8724799;
        return r8724800;
}

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-sqr-sqrt 0.2

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

5. Applied add-cube-cbrt 0.3

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

6. Applied sqrt-prod 0.3

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

7. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019174 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))