Average Error: 0.2 → 0.2
Time: 17.7s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\left(\mathsf{fma}\left(a \cdot a, \left(1 - a\right) \cdot 4, {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) + \left(\left(b \cdot b\right) \cdot \left(3 + a\right)\right) \cdot 4\right) - 1\]
\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(3 + a\right)\right)\right) - 1
↓
\left(\mathsf{fma}\left(a \cdot a, \left(1 - a\right) \cdot 4, {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) + \left(\left(b \cdot b\right) \cdot \left(3 + a\right)\right) \cdot 4\right) - 1
double f(double a, double b) {
        double r165109 = a;
        double r165110 = r165109 * r165109;
        double r165111 = b;
        double r165112 = r165111 * r165111;
        double r165113 = r165110 + r165112;
        double r165114 = 2.0;
        double r165115 = pow(r165113, r165114);
        double r165116 = 4.0;
        double r165117 = 1.0;
        double r165118 = r165117 - r165109;
        double r165119 = r165110 * r165118;
        double r165120 = 3.0;
        double r165121 = r165120 + r165109;
        double r165122 = r165112 * r165121;
        double r165123 = r165119 + r165122;
        double r165124 = r165116 * r165123;
        double r165125 = r165115 + r165124;
        double r165126 = r165125 - r165117;
        return r165126;
}
↓
double f(double a, double b) {
        double r165127 = a;
        double r165128 = r165127 * r165127;
        double r165129 = 1.0;
        double r165130 = r165129 - r165127;
        double r165131 = 4.0;
        double r165132 = r165130 * r165131;
        double r165133 = b;
        double r165134 = r165133 * r165133;
        double r165135 = fma(r165127, r165127, r165134);
        double r165136 = 2.0;
        double r165137 = pow(r165135, r165136);
        double r165138 = fma(r165128, r165132, r165137);
        double r165139 = 3.0;
        double r165140 = r165139 + r165127;
        double r165141 = r165134 * r165140;
        double r165142 = r165141 * r165131;
        double r165143 = r165138 + r165142;
        double r165144 = r165143 - r165129;
        return r165144;
}

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. Using strategy rm

  3. Applied distribute-rgt-in0.2

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

  4. Applied associate-+r+0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))