Average Error: 0.2 → 0.2
Time: 5.6s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(a \cdot a + b \cdot b\right)}^{2} - 1\right)\]
\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 r312991 = a;
        double r312992 = r312991 * r312991;
        double r312993 = b;
        double r312994 = r312993 * r312993;
        double r312995 = r312992 + r312994;
        double r312996 = 2.0;
        double r312997 = pow(r312995, r312996);
        double r312998 = 4.0;
        double r312999 = 1.0;
        double r313000 = r312999 + r312991;
        double r313001 = r312992 * r313000;
        double r313002 = 3.0;
        double r313003 = r313002 * r312991;
        double r313004 = r312999 - r313003;
        double r313005 = r312994 * r313004;
        double r313006 = r313001 + r313005;
        double r313007 = r312998 * r313006;
        double r313008 = r312997 + r313007;
        double r313009 = r313008 - r312999;
        return r313009;
}
↓
double f(double a, double b) {
        double r313010 = 4.0;
        double r313011 = a;
        double r313012 = r313011 * r313011;
        double r313013 = 1.0;
        double r313014 = r313013 + r313011;
        double r313015 = b;
        double r313016 = r313015 * r313015;
        double r313017 = 3.0;
        double r313018 = r313017 * r313011;
        double r313019 = r313013 - r313018;
        double r313020 = r313016 * r313019;
        double r313021 = fma(r313012, r313014, r313020);
        double r313022 = r313012 + r313016;
        double r313023 = 2.0;
        double r313024 = pow(r313022, r313023);
        double r313025 = r313024 - r313013;
        double r313026 = fma(r313010, r313021, r313025);
        return r313026;
}

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 2020036 +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))