Average Error: 0.2 → 0.2
Time: 18.0s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\sqrt{\left(b \cdot b\right) \cdot 4 + {\left(a \cdot a + b \cdot b\right)}^{2}} \cdot \sqrt{\left(b \cdot b\right) \cdot 4 + {\left(a \cdot a + b \cdot b\right)}^{2}} - 1\]
\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 \sqrt{\left(b \cdot b\right) \cdot 4 + {\left(a \cdot a + b \cdot b\right)}^{2}} - 1
double f(double a, double b) {
        double r5190962 = a;
        double r5190963 = r5190962 * r5190962;
        double r5190964 = b;
        double r5190965 = r5190964 * r5190964;
        double r5190966 = r5190963 + r5190965;
        double r5190967 = 2.0;
        double r5190968 = pow(r5190966, r5190967);
        double r5190969 = 4.0;
        double r5190970 = r5190969 * r5190965;
        double r5190971 = r5190968 + r5190970;
        double r5190972 = 1.0;
        double r5190973 = r5190971 - r5190972;
        return r5190973;
}
↓
double f(double a, double b) {
        double r5190974 = b;
        double r5190975 = r5190974 * r5190974;
        double r5190976 = 4.0;
        double r5190977 = r5190975 * r5190976;
        double r5190978 = a;
        double r5190979 = r5190978 * r5190978;
        double r5190980 = r5190979 + r5190975;
        double r5190981 = 2.0;
        double r5190982 = pow(r5190980, r5190981);
        double r5190983 = r5190977 + r5190982;
        double r5190984 = sqrt(r5190983);
        double r5190985 = r5190984 * r5190984;
        double r5190986 = 1.0;
        double r5190987 = r5190985 - r5190986;
        return r5190987;
}

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-sqr-sqrt0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))