Bouland and Aaronson, Equation (25)

Average Error: 0.2 → 0.5

Time: 8.1s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r141147 = a;
        double r141148 = r141147 * r141147;
        double r141149 = b;
        double r141150 = r141149 * r141149;
        double r141151 = r141148 + r141150;
        double r141152 = 2.0;
        double r141153 = pow(r141151, r141152);
        double r141154 = 4.0;
        double r141155 = 1.0;
        double r141156 = r141155 + r141147;
        double r141157 = r141148 * r141156;
        double r141158 = 3.0;
        double r141159 = r141158 * r141147;
        double r141160 = r141155 - r141159;
        double r141161 = r141150 * r141160;
        double r141162 = r141157 + r141161;
        double r141163 = r141154 * r141162;
        double r141164 = r141153 + r141163;
        double r141165 = r141164 - r141155;
        return r141165;
}

↓

double f(double a, double b) {
        double r141166 = a;
        double r141167 = r141166 * r141166;
        double r141168 = b;
        double r141169 = r141168 * r141168;
        double r141170 = r141167 + r141169;
        double r141171 = 2.0;
        double r141172 = 2.0;
        double r141173 = r141171 / r141172;
        double r141174 = pow(r141170, r141173);
        double r141175 = cbrt(r141174);
        double r141176 = r141175 * r141175;
        double r141177 = cbrt(r141176);
        double r141178 = cbrt(r141175);
        double r141179 = r141177 * r141178;
        double r141180 = pow(r141170, r141171);
        double r141181 = cbrt(r141180);
        double r141182 = r141181 * r141175;
        double r141183 = r141179 * r141182;
        double r141184 = r141183 * r141181;
        double r141185 = 4.0;
        double r141186 = 1.0;
        double r141187 = r141186 + r141166;
        double r141188 = r141167 * r141187;
        double r141189 = 3.0;
        double r141190 = r141189 * r141166;
        double r141191 = r141186 - r141190;
        double r141192 = r141169 * r141191;
        double r141193 = r141188 + r141192;
        double r141194 = r141185 * r141193;
        double r141195 = r141184 + r141194;
        double r141196 = r141195 - r141186;
        return r141196;
}

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. Using strategy rm

3. Applied add-cube-cbrt 0.5

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

5. Applied sqr-pow 0.5

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

6. Applied cbrt-prod 0.5

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

7. Applied associate-*l* 0.5

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

8. Simplified 0.5

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

10. Applied add-cube-cbrt 0.5

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

11. Applied cbrt-prod 0.5

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

12. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019352 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (+ 1 a)) (* (* b b) (- 1 (* 3 a)))))) 1))