Average Error: 0.2 → 0.2
Time: 24.7s
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 r10105553 = a;
        double r10105554 = r10105553 * r10105553;
        double r10105555 = b;
        double r10105556 = r10105555 * r10105555;
        double r10105557 = r10105554 + r10105556;
        double r10105558 = 2.0;
        double r10105559 = pow(r10105557, r10105558);
        double r10105560 = 4.0;
        double r10105561 = r10105560 * r10105556;
        double r10105562 = r10105559 + r10105561;
        double r10105563 = 1.0;
        double r10105564 = r10105562 - r10105563;
        return r10105564;
}
↓
double f(double a, double b) {
        double r10105565 = 4.0;
        double r10105566 = b;
        double r10105567 = r10105566 * r10105566;
        double r10105568 = a;
        double r10105569 = fma(r10105568, r10105568, r10105567);
        double r10105570 = 2.0;
        double r10105571 = pow(r10105569, r10105570);
        double r10105572 = fma(r10105565, r10105567, r10105571);
        double r10105573 = 1.0;
        double r10105574 = r10105572 - r10105573;
        return r10105574;
}

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 2019171 +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))