Average Error: 0.2 → 0.2
Time: 6.5s
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 r268732 = a;
        double r268733 = r268732 * r268732;
        double r268734 = b;
        double r268735 = r268734 * r268734;
        double r268736 = r268733 + r268735;
        double r268737 = 2.0;
        double r268738 = pow(r268736, r268737);
        double r268739 = 4.0;
        double r268740 = 1.0;
        double r268741 = r268740 + r268732;
        double r268742 = r268733 * r268741;
        double r268743 = 3.0;
        double r268744 = r268743 * r268732;
        double r268745 = r268740 - r268744;
        double r268746 = r268735 * r268745;
        double r268747 = r268742 + r268746;
        double r268748 = r268739 * r268747;
        double r268749 = r268738 + r268748;
        double r268750 = r268749 - r268740;
        return r268750;
}
↓
double f(double a, double b) {
        double r268751 = a;
        double r268752 = r268751 * r268751;
        double r268753 = b;
        double r268754 = r268753 * r268753;
        double r268755 = r268752 + r268754;
        double r268756 = 2.0;
        double r268757 = pow(r268755, r268756);
        double r268758 = 4.0;
        double r268759 = 1.0;
        double r268760 = r268759 + r268751;
        double r268761 = r268752 * r268760;
        double r268762 = 3.0;
        double r268763 = r268762 * r268751;
        double r268764 = r268759 - r268763;
        double r268765 = r268754 * r268764;
        double r268766 = r268761 + r268765;
        double r268767 = r268758 * r268766;
        double r268768 = r268757 + r268767;
        double r268769 = sqrt(r268768);
        double r268770 = r268769 * r268769;
        double r268771 = r268770 - r268759;
        return r268771;
}

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))