Average Error: 0.2 → 0.2
Time: 15.7s
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 r4206766 = a;
        double r4206767 = r4206766 * r4206766;
        double r4206768 = b;
        double r4206769 = r4206768 * r4206768;
        double r4206770 = r4206767 + r4206769;
        double r4206771 = 2.0;
        double r4206772 = pow(r4206770, r4206771);
        double r4206773 = 4.0;
        double r4206774 = r4206773 * r4206769;
        double r4206775 = r4206772 + r4206774;
        double r4206776 = 1.0;
        double r4206777 = r4206775 - r4206776;
        return r4206777;
}
↓
double f(double a, double b) {
        double r4206778 = b;
        double r4206779 = r4206778 * r4206778;
        double r4206780 = 4.0;
        double r4206781 = r4206779 * r4206780;
        double r4206782 = a;
        double r4206783 = r4206782 * r4206782;
        double r4206784 = r4206783 + r4206779;
        double r4206785 = 2.0;
        double r4206786 = pow(r4206784, r4206785);
        double r4206787 = r4206781 + r4206786;
        double r4206788 = sqrt(r4206787);
        double r4206789 = r4206788 * r4206788;
        double r4206790 = 1.0;
        double r4206791 = r4206789 - r4206790;
        return r4206791;
}

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 2019154 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))