Average Error: 0.2 → 0.2
Time: 20.5s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4, a \cdot a - \left(a \cdot a\right) \cdot a, \mathsf{fma}\left(\mathsf{fma}\left(a, 4, 12\right), b \cdot b, \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)\right)\]
\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(3 + a\right)\right)\right) - 1
↓
\mathsf{fma}\left(4, a \cdot a - \left(a \cdot a\right) \cdot a, \mathsf{fma}\left(\mathsf{fma}\left(a, 4, 12\right), b \cdot b, \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)\right)
double f(double a, double b) {
        double r8436043 = a;
        double r8436044 = r8436043 * r8436043;
        double r8436045 = b;
        double r8436046 = r8436045 * r8436045;
        double r8436047 = r8436044 + r8436046;
        double r8436048 = 2.0;
        double r8436049 = pow(r8436047, r8436048);
        double r8436050 = 4.0;
        double r8436051 = 1.0;
        double r8436052 = r8436051 - r8436043;
        double r8436053 = r8436044 * r8436052;
        double r8436054 = 3.0;
        double r8436055 = r8436054 + r8436043;
        double r8436056 = r8436046 * r8436055;
        double r8436057 = r8436053 + r8436056;
        double r8436058 = r8436050 * r8436057;
        double r8436059 = r8436049 + r8436058;
        double r8436060 = r8436059 - r8436051;
        return r8436060;
}
↓
double f(double a, double b) {
        double r8436061 = 4.0;
        double r8436062 = a;
        double r8436063 = r8436062 * r8436062;
        double r8436064 = r8436063 * r8436062;
        double r8436065 = r8436063 - r8436064;
        double r8436066 = 12.0;
        double r8436067 = fma(r8436062, r8436061, r8436066);
        double r8436068 = b;
        double r8436069 = r8436068 * r8436068;
        double r8436070 = fma(r8436062, r8436062, r8436069);
        double r8436071 = -1.0;
        double r8436072 = fma(r8436070, r8436070, r8436071);
        double r8436073 = fma(r8436067, r8436069, r8436072);
        double r8436074 = fma(r8436061, r8436065, r8436073);
        return r8436074;
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(4, a \cdot a - a \cdot \left(a \cdot a\right), \mathsf{fma}\left(\mathsf{fma}\left(a, 4, 12\right), b \cdot b, \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)\right)}\]

  3. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(4, a \cdot a - \left(a \cdot a\right) \cdot a, \mathsf{fma}\left(\mathsf{fma}\left(a, 4, 12\right), b \cdot b, \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)\right)\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))