Average Error: 0.2 → 0.2
Time: 5.7s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 + a\right)\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
\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(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 + a\right)\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1
double f(double a, double b) {
        double r120761 = a;
        double r120762 = r120761 * r120761;
        double r120763 = b;
        double r120764 = r120763 * r120763;
        double r120765 = r120762 + r120764;
        double r120766 = 2.0;
        double r120767 = pow(r120765, r120766);
        double r120768 = 4.0;
        double r120769 = 1.0;
        double r120770 = r120769 + r120761;
        double r120771 = r120762 * r120770;
        double r120772 = 3.0;
        double r120773 = r120772 * r120761;
        double r120774 = r120769 - r120773;
        double r120775 = r120764 * r120774;
        double r120776 = r120771 + r120775;
        double r120777 = r120768 * r120776;
        double r120778 = r120767 + r120777;
        double r120779 = r120778 - r120769;
        return r120779;
}
↓
double f(double a, double b) {
        double r120780 = a;
        double r120781 = r120780 * r120780;
        double r120782 = b;
        double r120783 = r120782 * r120782;
        double r120784 = r120781 + r120783;
        double r120785 = 2.0;
        double r120786 = pow(r120784, r120785);
        double r120787 = 4.0;
        double r120788 = 1.0;
        double r120789 = r120788 + r120780;
        double r120790 = r120780 * r120789;
        double r120791 = r120780 * r120790;
        double r120792 = 3.0;
        double r120793 = r120792 * r120780;
        double r120794 = r120788 - r120793;
        double r120795 = r120783 * r120794;
        double r120796 = r120791 + r120795;
        double r120797 = r120787 * r120796;
        double r120798 = r120786 + r120797;
        double r120799 = r120798 - r120788;
        return r120799;
}

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 associate-*l*0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2020056 
(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))