Average Error: 0.2 → 0.2
Time: 26.4s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4, b \cdot b, {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\mathsf{fma}\left(4, b \cdot b, {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1
double f(double a, double b) {
        double r11438236 = a;
        double r11438237 = r11438236 * r11438236;
        double r11438238 = b;
        double r11438239 = r11438238 * r11438238;
        double r11438240 = r11438237 + r11438239;
        double r11438241 = 2.0;
        double r11438242 = pow(r11438240, r11438241);
        double r11438243 = 4.0;
        double r11438244 = r11438243 * r11438239;
        double r11438245 = r11438242 + r11438244;
        double r11438246 = 1.0;
        double r11438247 = r11438245 - r11438246;
        return r11438247;
}
↓
double f(double a, double b) {
        double r11438248 = 4.0;
        double r11438249 = b;
        double r11438250 = r11438249 * r11438249;
        double r11438251 = a;
        double r11438252 = fma(r11438251, r11438251, r11438250);
        double r11438253 = 2.0;
        double r11438254 = pow(r11438252, r11438253);
        double r11438255 = fma(r11438248, r11438250, r11438254);
        double r11438256 = 1.0;
        double r11438257 = r11438255 - r11438256;
        return r11438257;
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))