Average Error: 0.2 → 0.7
Time: 11.4s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left({\left(\sqrt[3]{a \cdot a + b \cdot b} \cdot \sqrt[3]{a \cdot a + b \cdot b}\right)}^{2} \cdot {\left(\sqrt[3]{\sqrt{a \cdot a + b \cdot b}} \cdot \sqrt[3]{\sqrt{a \cdot a + b \cdot b}}\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
\left({\left(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(\sqrt[3]{a \cdot a + b \cdot b} \cdot \sqrt[3]{a \cdot a + b \cdot b}\right)}^{2} \cdot {\left(\sqrt[3]{\sqrt{a \cdot a + b \cdot b}} \cdot \sqrt[3]{\sqrt{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
double f(double a, double b) {
        double r156820 = a;
        double r156821 = r156820 * r156820;
        double r156822 = b;
        double r156823 = r156822 * r156822;
        double r156824 = r156821 + r156823;
        double r156825 = 2.0;
        double r156826 = pow(r156824, r156825);
        double r156827 = 4.0;
        double r156828 = 1.0;
        double r156829 = r156828 + r156820;
        double r156830 = r156821 * r156829;
        double r156831 = 3.0;
        double r156832 = r156831 * r156820;
        double r156833 = r156828 - r156832;
        double r156834 = r156823 * r156833;
        double r156835 = r156830 + r156834;
        double r156836 = r156827 * r156835;
        double r156837 = r156826 + r156836;
        double r156838 = r156837 - r156828;
        return r156838;
}
↓
double f(double a, double b) {
        double r156839 = a;
        double r156840 = r156839 * r156839;
        double r156841 = b;
        double r156842 = r156841 * r156841;
        double r156843 = r156840 + r156842;
        double r156844 = cbrt(r156843);
        double r156845 = r156844 * r156844;
        double r156846 = 2.0;
        double r156847 = pow(r156845, r156846);
        double r156848 = sqrt(r156843);
        double r156849 = cbrt(r156848);
        double r156850 = r156849 * r156849;
        double r156851 = pow(r156850, r156846);
        double r156852 = r156847 * r156851;
        double r156853 = 4.0;
        double r156854 = 1.0;
        double r156855 = r156854 + r156839;
        double r156856 = r156840 * r156855;
        double r156857 = 3.0;
        double r156858 = r156857 * r156839;
        double r156859 = r156854 - r156858;
        double r156860 = r156842 * r156859;
        double r156861 = r156856 + r156860;
        double r156862 = r156853 * r156861;
        double r156863 = r156852 + r156862;
        double r156864 = r156863 - r156854;
        return r156864;
}

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

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

  3. Applied add-cube-cbrt0.7

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

  4. Applied unpow-prod-down0.7

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

  6. Applied add-sqr-sqrt0.7

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

  7. Applied cbrt-prod0.7

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

  8. Final simplification0.7

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

Reproduce

herbie shell --seed 2020047 
(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))