Average Error: 0.2 → 0.2
Time: 22.8s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\mathsf{fma}\left(\left(4 \cdot b\right), b, \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(b, b, \left(a \cdot a\right)\right)\right), \left(\mathsf{fma}\left(b, b, \left(a \cdot a\right)\right)\right), -1\right)\right)\right)\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\mathsf{fma}\left(\left(4 \cdot b\right), b, \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(b, b, \left(a \cdot a\right)\right)\right), \left(\mathsf{fma}\left(b, b, \left(a \cdot a\right)\right)\right), -1\right)\right)\right)
double f(double a, double b) {
        double r8099633 = a;
        double r8099634 = r8099633 * r8099633;
        double r8099635 = b;
        double r8099636 = r8099635 * r8099635;
        double r8099637 = r8099634 + r8099636;
        double r8099638 = 2.0;
        double r8099639 = pow(r8099637, r8099638);
        double r8099640 = 4.0;
        double r8099641 = r8099640 * r8099636;
        double r8099642 = r8099639 + r8099641;
        double r8099643 = 1.0;
        double r8099644 = r8099642 - r8099643;
        return r8099644;
}
↓
double f(double a, double b) {
        double r8099645 = 4.0;
        double r8099646 = b;
        double r8099647 = r8099645 * r8099646;
        double r8099648 = a;
        double r8099649 = r8099648 * r8099648;
        double r8099650 = fma(r8099646, r8099646, r8099649);
        double r8099651 = -1.0;
        double r8099652 = fma(r8099650, r8099650, r8099651);
        double r8099653 = fma(r8099647, r8099646, r8099652);
        return r8099653;
}

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. Simplified0.2

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))