Average Error: 0.2 → 0.2
Time: 18.7s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
double f(double a, double b) {
        double r3688740 = a;
        double r3688741 = r3688740 * r3688740;
        double r3688742 = b;
        double r3688743 = r3688742 * r3688742;
        double r3688744 = r3688741 + r3688743;
        double r3688745 = 2.0;
        double r3688746 = pow(r3688744, r3688745);
        double r3688747 = 4.0;
        double r3688748 = r3688747 * r3688743;
        double r3688749 = r3688746 + r3688748;
        double r3688750 = 1.0;
        double r3688751 = r3688749 - r3688750;
        return r3688751;
}
↓
double f(double a, double b) {
        double r3688752 = a;
        double r3688753 = b;
        double r3688754 = r3688753 * r3688753;
        double r3688755 = fma(r3688752, r3688752, r3688754);
        double r3688756 = 2.0;
        double r3688757 = pow(r3688755, r3688756);
        double r3688758 = 4.0;
        double r3688759 = r3688758 * r3688754;
        double r3688760 = r3688757 + r3688759;
        double r3688761 = 1.0;
        double r3688762 = r3688760 - r3688761;
        return r3688762;
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

    \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
  2. Using strategy rm

  3. Applied fma-def0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))