Average Error: 0.2 → 0.2
Time: 46.6s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left(\mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), \left(4 \cdot a\right) \cdot \mathsf{fma}\left(a, a, a\right)\right) + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right)\right) - 1\]
\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(\mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), \left(4 \cdot a\right) \cdot \mathsf{fma}\left(a, a, a\right)\right) + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right)\right) - 1
double f(double a, double b) {
        double r5924771 = a;
        double r5924772 = r5924771 * r5924771;
        double r5924773 = b;
        double r5924774 = r5924773 * r5924773;
        double r5924775 = r5924772 + r5924774;
        double r5924776 = 2.0;
        double r5924777 = pow(r5924775, r5924776);
        double r5924778 = 4.0;
        double r5924779 = 1.0;
        double r5924780 = r5924779 + r5924771;
        double r5924781 = r5924772 * r5924780;
        double r5924782 = 3.0;
        double r5924783 = r5924782 * r5924771;
        double r5924784 = r5924779 - r5924783;
        double r5924785 = r5924774 * r5924784;
        double r5924786 = r5924781 + r5924785;
        double r5924787 = r5924778 * r5924786;
        double r5924788 = r5924777 + r5924787;
        double r5924789 = r5924788 - r5924779;
        return r5924789;
}
↓
double f(double a, double b) {
        double r5924790 = a;
        double r5924791 = b;
        double r5924792 = r5924791 * r5924791;
        double r5924793 = fma(r5924790, r5924790, r5924792);
        double r5924794 = 4.0;
        double r5924795 = r5924794 * r5924790;
        double r5924796 = fma(r5924790, r5924790, r5924790);
        double r5924797 = r5924795 * r5924796;
        double r5924798 = fma(r5924793, r5924793, r5924797);
        double r5924799 = 1.0;
        double r5924800 = 3.0;
        double r5924801 = r5924790 * r5924800;
        double r5924802 = r5924799 - r5924801;
        double r5924803 = r5924792 * r5924802;
        double r5924804 = r5924794 * r5924803;
        double r5924805 = r5924798 + r5924804;
        double r5924806 = r5924805 - r5924799;
        return r5924806;
}

Error

Bits error versus a

Bits error versus b

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 distribute-rgt-in0.2

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

  4. Applied associate-+r+0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), \left(4 \cdot a\right) \cdot \mathsf{fma}\left(a, a, a\right)\right) + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right)\right) - 1\]

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (+ 1 a)) (* (* b b) (- 1 (* 3 a)))))) 1))