Average Error: 0.2 → 0.2
Time: 46.6s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[(\left((a \cdot a + \left(b \cdot b\right))_*\right) \cdot \left((a \cdot a + \left(b \cdot b\right))_*\right) + \left((\left(b \cdot b\right) \cdot 4 + -1)_*\right))_*\]
double f(double a, double b) {
        double r51791604 = a;
        double r51791605 = r51791604 * r51791604;
        double r51791606 = b;
        double r51791607 = r51791606 * r51791606;
        double r51791608 = r51791605 + r51791607;
        double r51791609 = 2.0;
        double r51791610 = pow(r51791608, r51791609);
        double r51791611 = 4.0;
        double r51791612 = r51791611 * r51791607;
        double r51791613 = r51791610 + r51791612;
        double r51791614 = 1.0;
        double r51791615 = r51791613 - r51791614;
        return r51791615;
}
↓
double f(double a, double b) {
        double r51791616 = a;
        double r51791617 = b;
        double r51791618 = r51791617 * r51791617;
        double r51791619 = fma(r51791616, r51791616, r51791618);
        double r51791620 = 4.0;
        double r51791621 = -1.0;
        double r51791622 = fma(r51791618, r51791620, r51791621);
        double r51791623 = fma(r51791619, r51791619, r51791622);
        return r51791623;
}


\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
(\left((a \cdot a + \left(b \cdot b\right))_*\right) \cdot \left((a \cdot a + \left(b \cdot b\right))_*\right) + \left((\left(b \cdot b\right) \cdot 4 + -1)_*\right))_*

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

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

  3. Final simplification0.2

    \[\leadsto (\left((a \cdot a + \left(b \cdot b\right))_*\right) \cdot \left((a \cdot a + \left(b \cdot b\right))_*\right) + \left((\left(b \cdot b\right) \cdot 4 + -1)_*\right))_*\]

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))