Average Error: 0.2 → 0.2
Time: 26.0s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4 \cdot b, b, {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\mathsf{fma}\left(4 \cdot b, b, {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1
double f(double a, double b) {
        double r283537 = a;
        double r283538 = r283537 * r283537;
        double r283539 = b;
        double r283540 = r283539 * r283539;
        double r283541 = r283538 + r283540;
        double r283542 = 2.0;
        double r283543 = pow(r283541, r283542);
        double r283544 = 4.0;
        double r283545 = r283544 * r283540;
        double r283546 = r283543 + r283545;
        double r283547 = 1.0;
        double r283548 = r283546 - r283547;
        return r283548;
}
↓
double f(double a, double b) {
        double r283549 = 4.0;
        double r283550 = b;
        double r283551 = r283549 * r283550;
        double r283552 = a;
        double r283553 = r283550 * r283550;
        double r283554 = fma(r283552, r283552, r283553);
        double r283555 = 2.0;
        double r283556 = pow(r283554, r283555);
        double r283557 = fma(r283551, r283550, r283556);
        double r283558 = 1.0;
        double r283559 = r283557 - r283558;
        return r283559;
}

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

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))