Average Error: 0.2 → 0.2
Time: 24.9s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1\]
\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
↓
\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(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1
double f(double a, double b) {
        double r213971 = a;
        double r213972 = r213971 * r213971;
        double r213973 = b;
        double r213974 = r213973 * r213973;
        double r213975 = r213972 + r213974;
        double r213976 = 2.0;
        double r213977 = pow(r213975, r213976);
        double r213978 = 4.0;
        double r213979 = 1.0;
        double r213980 = r213979 + r213971;
        double r213981 = r213972 * r213980;
        double r213982 = 3.0;
        double r213983 = r213982 * r213971;
        double r213984 = r213979 - r213983;
        double r213985 = r213974 * r213984;
        double r213986 = r213981 + r213985;
        double r213987 = r213978 * r213986;
        double r213988 = r213977 + r213987;
        double r213989 = r213988 - r213979;
        return r213989;
}
↓
double f(double a, double b) {
        double r213990 = 4.0;
        double r213991 = a;
        double r213992 = r213991 * r213991;
        double r213993 = 1.0;
        double r213994 = r213993 + r213991;
        double r213995 = b;
        double r213996 = r213995 * r213995;
        double r213997 = 3.0;
        double r213998 = r213997 * r213991;
        double r213999 = r213993 - r213998;
        double r214000 = r213996 * r213999;
        double r214001 = fma(r213992, r213994, r214000);
        double r214002 = fma(r213991, r213991, r213996);
        double r214003 = 2.0;
        double r214004 = pow(r214002, r214003);
        double r214005 = fma(r213990, r214001, r214004);
        double r214006 = r214005 - r213993;
        return r214006;
}

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

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019303 +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))