Average Error: 0.2 → 0.2
Time: 6.4s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} \cdot \sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\]
\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
↓
\sqrt{{\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)} \cdot \sqrt{{\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)} - 1
double f(double a, double b) {
        double r124759 = a;
        double r124760 = r124759 * r124759;
        double r124761 = b;
        double r124762 = r124761 * r124761;
        double r124763 = r124760 + r124762;
        double r124764 = 2.0;
        double r124765 = pow(r124763, r124764);
        double r124766 = 4.0;
        double r124767 = 1.0;
        double r124768 = r124767 + r124759;
        double r124769 = r124760 * r124768;
        double r124770 = 3.0;
        double r124771 = r124770 * r124759;
        double r124772 = r124767 - r124771;
        double r124773 = r124762 * r124772;
        double r124774 = r124769 + r124773;
        double r124775 = r124766 * r124774;
        double r124776 = r124765 + r124775;
        double r124777 = r124776 - r124767;
        return r124777;
}
↓
double f(double a, double b) {
        double r124778 = a;
        double r124779 = r124778 * r124778;
        double r124780 = b;
        double r124781 = r124780 * r124780;
        double r124782 = r124779 + r124781;
        double r124783 = 2.0;
        double r124784 = pow(r124782, r124783);
        double r124785 = 4.0;
        double r124786 = 1.0;
        double r124787 = r124786 + r124778;
        double r124788 = r124779 * r124787;
        double r124789 = 3.0;
        double r124790 = r124789 * r124778;
        double r124791 = r124786 - r124790;
        double r124792 = r124781 * r124791;
        double r124793 = r124788 + r124792;
        double r124794 = r124785 * r124793;
        double r124795 = r124784 + r124794;
        double r124796 = sqrt(r124795);
        double r124797 = r124796 * r124796;
        double r124798 = r124797 - r124786;
        return r124798;
}

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-sqr-sqrt0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(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))