Average Error: 0.2 → 0.8
Time: 4.2s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(\mathsf{fma}\left(2 \cdot {a}^{2}, {b}^{2}, {b}^{4}\right) + {a}^{4}\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\left(\mathsf{fma}\left(2 \cdot {a}^{2}, {b}^{2}, {b}^{4}\right) + {a}^{4}\right) - 1
double f(double a, double b) {
        double r466635 = a;
        double r466636 = r466635 * r466635;
        double r466637 = b;
        double r466638 = r466637 * r466637;
        double r466639 = r466636 + r466638;
        double r466640 = 2.0;
        double r466641 = pow(r466639, r466640);
        double r466642 = 4.0;
        double r466643 = r466642 * r466638;
        double r466644 = r466641 + r466643;
        double r466645 = 1.0;
        double r466646 = r466644 - r466645;
        return r466646;
}
↓
double f(double a, double b) {
        double r466647 = 2.0;
        double r466648 = a;
        double r466649 = pow(r466648, r466647);
        double r466650 = r466647 * r466649;
        double r466651 = b;
        double r466652 = pow(r466651, r466647);
        double r466653 = 4.0;
        double r466654 = pow(r466651, r466653);
        double r466655 = fma(r466650, r466652, r466654);
        double r466656 = pow(r466648, r466653);
        double r466657 = r466655 + r466656;
        double r466658 = 1.0;
        double r466659 = r466657 - r466658;
        return r466659;
}

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. Taylor expanded around inf 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2020059 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))