Average Error: 0.2 → 0.2
Time: 22.8s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(4, a, 4\right), a \cdot a, \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(a, -12, 4\right), \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)\right)\]
\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(1 - 3 \cdot a\right)\right)\right) - 1
↓
\mathsf{fma}\left(\mathsf{fma}\left(4, a, 4\right), a \cdot a, \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(a, -12, 4\right), \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 r8261175 = a;
        double r8261176 = r8261175 * r8261175;
        double r8261177 = b;
        double r8261178 = r8261177 * r8261177;
        double r8261179 = r8261176 + r8261178;
        double r8261180 = 2.0;
        double r8261181 = pow(r8261179, r8261180);
        double r8261182 = 4.0;
        double r8261183 = 1.0;
        double r8261184 = r8261183 + r8261175;
        double r8261185 = r8261176 * r8261184;
        double r8261186 = 3.0;
        double r8261187 = r8261186 * r8261175;
        double r8261188 = r8261183 - r8261187;
        double r8261189 = r8261178 * r8261188;
        double r8261190 = r8261185 + r8261189;
        double r8261191 = r8261182 * r8261190;
        double r8261192 = r8261181 + r8261191;
        double r8261193 = r8261192 - r8261183;
        return r8261193;
}
↓
double f(double a, double b) {
        double r8261194 = 4.0;
        double r8261195 = a;
        double r8261196 = fma(r8261194, r8261195, r8261194);
        double r8261197 = r8261195 * r8261195;
        double r8261198 = b;
        double r8261199 = r8261198 * r8261198;
        double r8261200 = -12.0;
        double r8261201 = fma(r8261195, r8261200, r8261194);
        double r8261202 = fma(r8261195, r8261195, r8261199);
        double r8261203 = -1.0;
        double r8261204 = fma(r8261202, r8261202, r8261203);
        double r8261205 = fma(r8261199, r8261201, r8261204);
        double r8261206 = fma(r8261196, r8261197, r8261205);
        return r8261206;
}

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

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (+ 1 a)) (* (* b b) (- 1 (* 3 a)))))) 1))