Average Error: 0.2 → 0.2
Time: 27.8s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, a + 1, \left(1 - a \cdot 3\right) \cdot \left(b \cdot b\right)\right), {\left(\mathsf{fma}\left(b, b, a \cdot a\right)\right)}^{2} - 1\right)\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1
↓
\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, a + 1, \left(1 - a \cdot 3\right) \cdot \left(b \cdot b\right)\right), {\left(\mathsf{fma}\left(b, b, a \cdot a\right)\right)}^{2} - 1\right)
double f(double a, double b) {
        double r5556685 = a;
        double r5556686 = r5556685 * r5556685;
        double r5556687 = b;
        double r5556688 = r5556687 * r5556687;
        double r5556689 = r5556686 + r5556688;
        double r5556690 = 2.0;
        double r5556691 = pow(r5556689, r5556690);
        double r5556692 = 4.0;
        double r5556693 = 1.0;
        double r5556694 = r5556693 + r5556685;
        double r5556695 = r5556686 * r5556694;
        double r5556696 = 3.0;
        double r5556697 = r5556696 * r5556685;
        double r5556698 = r5556693 - r5556697;
        double r5556699 = r5556688 * r5556698;
        double r5556700 = r5556695 + r5556699;
        double r5556701 = r5556692 * r5556700;
        double r5556702 = r5556691 + r5556701;
        double r5556703 = r5556702 - r5556693;
        return r5556703;
}
↓
double f(double a, double b) {
        double r5556704 = 4.0;
        double r5556705 = a;
        double r5556706 = r5556705 * r5556705;
        double r5556707 = 1.0;
        double r5556708 = r5556705 + r5556707;
        double r5556709 = 3.0;
        double r5556710 = r5556705 * r5556709;
        double r5556711 = r5556707 - r5556710;
        double r5556712 = b;
        double r5556713 = r5556712 * r5556712;
        double r5556714 = r5556711 * r5556713;
        double r5556715 = fma(r5556706, r5556708, r5556714);
        double r5556716 = fma(r5556712, r5556712, r5556706);
        double r5556717 = 2.0;
        double r5556718 = pow(r5556716, r5556717);
        double r5556719 = r5556718 - r5556707;
        double r5556720 = fma(r5556704, r5556715, r5556719);
        return r5556720;
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

    \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]

  2. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\mathsf{fma}\left(b, b, a \cdot a\right)\right)}^{2} - 1\right)}\]

  3. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, a + 1, \left(1 - a \cdot 3\right) \cdot \left(b \cdot b\right)\right), {\left(\mathsf{fma}\left(b, b, a \cdot a\right)\right)}^{2} - 1\right)\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))