Average Error: 0.2 → 0.2
Time: 6.1s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(a \cdot a + b \cdot b\right)}^{2} - 1\right)\]
\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
↓
\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(a \cdot a + b \cdot b\right)}^{2} - 1\right)
double f(double a, double b) {
        double r196737 = a;
        double r196738 = r196737 * r196737;
        double r196739 = b;
        double r196740 = r196739 * r196739;
        double r196741 = r196738 + r196740;
        double r196742 = 2.0;
        double r196743 = pow(r196741, r196742);
        double r196744 = 4.0;
        double r196745 = 1.0;
        double r196746 = r196745 + r196737;
        double r196747 = r196738 * r196746;
        double r196748 = 3.0;
        double r196749 = r196748 * r196737;
        double r196750 = r196745 - r196749;
        double r196751 = r196740 * r196750;
        double r196752 = r196747 + r196751;
        double r196753 = r196744 * r196752;
        double r196754 = r196743 + r196753;
        double r196755 = r196754 - r196745;
        return r196755;
}
↓
double f(double a, double b) {
        double r196756 = 4.0;
        double r196757 = a;
        double r196758 = r196757 * r196757;
        double r196759 = 1.0;
        double r196760 = r196759 + r196757;
        double r196761 = b;
        double r196762 = r196761 * r196761;
        double r196763 = 3.0;
        double r196764 = r196763 * r196757;
        double r196765 = r196759 - r196764;
        double r196766 = r196762 * r196765;
        double r196767 = fma(r196758, r196760, r196766);
        double r196768 = r196758 + r196762;
        double r196769 = 2.0;
        double r196770 = pow(r196768, r196769);
        double r196771 = r196770 - r196759;
        double r196772 = fma(r196756, r196767, r196771);
        return r196772;
}

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. Simplified0.2

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2020027 +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))