Average Error: 0.2 → 0.2
Time: 5.5s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)} \cdot \sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)} - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\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
double f(double a, double b) {
        double r439998 = a;
        double r439999 = r439998 * r439998;
        double r440000 = b;
        double r440001 = r440000 * r440000;
        double r440002 = r439999 + r440001;
        double r440003 = 2.0;
        double r440004 = pow(r440002, r440003);
        double r440005 = 4.0;
        double r440006 = r440005 * r440001;
        double r440007 = r440004 + r440006;
        double r440008 = 1.0;
        double r440009 = r440007 - r440008;
        return r440009;
}
↓
double f(double a, double b) {
        double r440010 = a;
        double r440011 = r440010 * r440010;
        double r440012 = b;
        double r440013 = r440012 * r440012;
        double r440014 = r440011 + r440013;
        double r440015 = 2.0;
        double r440016 = pow(r440014, r440015);
        double r440017 = 4.0;
        double r440018 = r440017 * r440013;
        double r440019 = r440016 + r440018;
        double r440020 = sqrt(r440019);
        double r440021 = r440020 * r440020;
        double r440022 = 1.0;
        double r440023 = r440021 - r440022;
        return r440023;
}

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

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

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))