Average Error: 0.2 → 0.2
Time: 5.8s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4 \cdot b, b, {\left(a \cdot a + b \cdot b\right)}^{2} - 1\right)\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\mathsf{fma}\left(4 \cdot b, b, {\left(a \cdot a + b \cdot b\right)}^{2} - 1\right)
double f(double a, double b) {
        double r372367 = a;
        double r372368 = r372367 * r372367;
        double r372369 = b;
        double r372370 = r372369 * r372369;
        double r372371 = r372368 + r372370;
        double r372372 = 2.0;
        double r372373 = pow(r372371, r372372);
        double r372374 = 4.0;
        double r372375 = r372374 * r372370;
        double r372376 = r372373 + r372375;
        double r372377 = 1.0;
        double r372378 = r372376 - r372377;
        return r372378;
}
↓
double f(double a, double b) {
        double r372379 = 4.0;
        double r372380 = b;
        double r372381 = r372379 * r372380;
        double r372382 = a;
        double r372383 = r372382 * r372382;
        double r372384 = r372380 * r372380;
        double r372385 = r372383 + r372384;
        double r372386 = 2.0;
        double r372387 = pow(r372385, r372386);
        double r372388 = 1.0;
        double r372389 = r372387 - r372388;
        double r372390 = fma(r372381, r372380, r372389);
        return r372390;
}

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

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))