Average Error: 0.2 → 0.2
Time: 43.7s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4, \mathsf{fma}\left(1 - 3 \cdot a, b \cdot b, \left(a \cdot a\right) \cdot \left(a + 1\right)\right), {\left(\mathsf{fma}\left(b, b, a \cdot a\right)\right)}^{2}\right) - 1\]
\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(1 - 3 \cdot a, b \cdot b, \left(a \cdot a\right) \cdot \left(a + 1\right)\right), {\left(\mathsf{fma}\left(b, b, a \cdot a\right)\right)}^{2}\right) - 1
double f(double a, double b) {
        double r6144139 = a;
        double r6144140 = r6144139 * r6144139;
        double r6144141 = b;
        double r6144142 = r6144141 * r6144141;
        double r6144143 = r6144140 + r6144142;
        double r6144144 = 2.0;
        double r6144145 = pow(r6144143, r6144144);
        double r6144146 = 4.0;
        double r6144147 = 1.0;
        double r6144148 = r6144147 + r6144139;
        double r6144149 = r6144140 * r6144148;
        double r6144150 = 3.0;
        double r6144151 = r6144150 * r6144139;
        double r6144152 = r6144147 - r6144151;
        double r6144153 = r6144142 * r6144152;
        double r6144154 = r6144149 + r6144153;
        double r6144155 = r6144146 * r6144154;
        double r6144156 = r6144145 + r6144155;
        double r6144157 = r6144156 - r6144147;
        return r6144157;
}
↓
double f(double a, double b) {
        double r6144158 = 4.0;
        double r6144159 = 1.0;
        double r6144160 = 3.0;
        double r6144161 = a;
        double r6144162 = r6144160 * r6144161;
        double r6144163 = r6144159 - r6144162;
        double r6144164 = b;
        double r6144165 = r6144164 * r6144164;
        double r6144166 = r6144161 * r6144161;
        double r6144167 = r6144161 + r6144159;
        double r6144168 = r6144166 * r6144167;
        double r6144169 = fma(r6144163, r6144165, r6144168);
        double r6144170 = fma(r6144164, r6144164, r6144166);
        double r6144171 = 2.0;
        double r6144172 = pow(r6144170, r6144171);
        double r6144173 = fma(r6144158, r6144169, r6144172);
        double r6144174 = r6144173 - r6144159;
        return r6144174;
}

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

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))