Average Error: 0.2 → 0.5
Time: 5.6s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(\sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)} \cdot \sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)}\right) \cdot \left(\sqrt[3]{\sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)} \cdot \sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)}} \cdot \sqrt[3]{\sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)}}\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\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 \left(\sqrt[3]{\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[3]{\sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)}}\right) - 1
double f(double a, double b) {
        double r377888 = a;
        double r377889 = r377888 * r377888;
        double r377890 = b;
        double r377891 = r377890 * r377890;
        double r377892 = r377889 + r377891;
        double r377893 = 2.0;
        double r377894 = pow(r377892, r377893);
        double r377895 = 4.0;
        double r377896 = r377895 * r377891;
        double r377897 = r377894 + r377896;
        double r377898 = 1.0;
        double r377899 = r377897 - r377898;
        return r377899;
}
↓
double f(double a, double b) {
        double r377900 = a;
        double r377901 = r377900 * r377900;
        double r377902 = b;
        double r377903 = r377902 * r377902;
        double r377904 = r377901 + r377903;
        double r377905 = 2.0;
        double r377906 = pow(r377904, r377905);
        double r377907 = 4.0;
        double r377908 = r377907 * r377903;
        double r377909 = r377906 + r377908;
        double r377910 = cbrt(r377909);
        double r377911 = r377910 * r377910;
        double r377912 = cbrt(r377911);
        double r377913 = cbrt(r377910);
        double r377914 = r377912 * r377913;
        double r377915 = r377911 * r377914;
        double r377916 = 1.0;
        double r377917 = r377915 - r377916;
        return r377917;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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-cbrt0.5

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

  5. Applied add-cube-cbrt0.5

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

  6. Applied cbrt-prod0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2020001 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))