Average Error: 0.2 → 0.2
Time: 19.3s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left(\sqrt{\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot \left(\left(1 - a \cdot 3\right) \cdot b\right) + a \cdot \left(\left(a + 1\right) \cdot a\right)\right)}} \cdot \sqrt{\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot \left(\left(1 - a \cdot 3\right) \cdot b\right) + a \cdot \left(\left(a + 1\right) \cdot a\right)\right)}}\right) \cdot \sqrt{4 \cdot \left(\left(1 - a \cdot 3\right) \cdot {b}^{2} + a \cdot \left(\left(a + 1\right) \cdot a\right)\right) + {\left({b}^{2} + a \cdot a\right)}^{2}} - 1\]
\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(\sqrt{\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot \left(\left(1 - a \cdot 3\right) \cdot b\right) + a \cdot \left(\left(a + 1\right) \cdot a\right)\right)}} \cdot \sqrt{\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot \left(\left(1 - a \cdot 3\right) \cdot b\right) + a \cdot \left(\left(a + 1\right) \cdot a\right)\right)}}\right) \cdot \sqrt{4 \cdot \left(\left(1 - a \cdot 3\right) \cdot {b}^{2} + a \cdot \left(\left(a + 1\right) \cdot a\right)\right) + {\left({b}^{2} + a \cdot a\right)}^{2}} - 1
double f(double a, double b) {
        double r141763 = a;
        double r141764 = r141763 * r141763;
        double r141765 = b;
        double r141766 = r141765 * r141765;
        double r141767 = r141764 + r141766;
        double r141768 = 2.0;
        double r141769 = pow(r141767, r141768);
        double r141770 = 4.0;
        double r141771 = 1.0;
        double r141772 = r141771 + r141763;
        double r141773 = r141764 * r141772;
        double r141774 = 3.0;
        double r141775 = r141774 * r141763;
        double r141776 = r141771 - r141775;
        double r141777 = r141766 * r141776;
        double r141778 = r141773 + r141777;
        double r141779 = r141770 * r141778;
        double r141780 = r141769 + r141779;
        double r141781 = r141780 - r141771;
        return r141781;
}
↓
double f(double a, double b) {
        double r141782 = a;
        double r141783 = r141782 * r141782;
        double r141784 = b;
        double r141785 = r141784 * r141784;
        double r141786 = r141783 + r141785;
        double r141787 = 2.0;
        double r141788 = pow(r141786, r141787);
        double r141789 = 4.0;
        double r141790 = 1.0;
        double r141791 = 3.0;
        double r141792 = r141782 * r141791;
        double r141793 = r141790 - r141792;
        double r141794 = r141793 * r141784;
        double r141795 = r141784 * r141794;
        double r141796 = r141782 + r141790;
        double r141797 = r141796 * r141782;
        double r141798 = r141782 * r141797;
        double r141799 = r141795 + r141798;
        double r141800 = r141789 * r141799;
        double r141801 = r141788 + r141800;
        double r141802 = sqrt(r141801);
        double r141803 = sqrt(r141802);
        double r141804 = r141803 * r141803;
        double r141805 = 2.0;
        double r141806 = pow(r141784, r141805);
        double r141807 = r141793 * r141806;
        double r141808 = r141807 + r141798;
        double r141809 = r141789 * r141808;
        double r141810 = r141806 + r141783;
        double r141811 = pow(r141810, r141787);
        double r141812 = r141809 + r141811;
        double r141813 = sqrt(r141812);
        double r141814 = r141804 * r141813;
        double r141815 = r141814 - r141790;
        return r141815;
}

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. Simplified0.2

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

  5. Simplified0.2

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

  7. Applied add-sqr-sqrt0.2

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

  8. Applied sqrt-prod0.2

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

  9. Simplified0.2

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

  10. Simplified0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2019196 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))