Average Error: 0.2 → 0.2
Time: 2.2s
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 r239525 = a;
        double r239526 = r239525 * r239525;
        double r239527 = b;
        double r239528 = r239527 * r239527;
        double r239529 = r239526 + r239528;
        double r239530 = 2.0;
        double r239531 = pow(r239529, r239530);
        double r239532 = 4.0;
        double r239533 = r239532 * r239528;
        double r239534 = r239531 + r239533;
        double r239535 = 1.0;
        double r239536 = r239534 - r239535;
        return r239536;
}
↓
double f(double a, double b) {
        double r239537 = 4.0;
        double r239538 = b;
        double r239539 = r239537 * r239538;
        double r239540 = a;
        double r239541 = r239540 * r239540;
        double r239542 = r239538 * r239538;
        double r239543 = r239541 + r239542;
        double r239544 = 2.0;
        double r239545 = pow(r239543, r239544);
        double r239546 = 1.0;
        double r239547 = r239545 - r239546;
        double r239548 = fma(r239539, r239538, r239547);
        return r239548;
}

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 2020024 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))