Average Error: 0.2 → 0.2
Time: 16.0s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\mathsf{fma}\left({\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)}, {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)}, 4 \cdot \left(b \cdot b\right)\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\mathsf{fma}\left({\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)}, {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)}, 4 \cdot \left(b \cdot b\right)\right) - 1
double f(double a, double b) {
        double r210991 = a;
        double r210992 = r210991 * r210991;
        double r210993 = b;
        double r210994 = r210993 * r210993;
        double r210995 = r210992 + r210994;
        double r210996 = 2.0;
        double r210997 = pow(r210995, r210996);
        double r210998 = 4.0;
        double r210999 = r210998 * r210994;
        double r211000 = r210997 + r210999;
        double r211001 = 1.0;
        double r211002 = r211000 - r211001;
        return r211002;
}
↓
double f(double a, double b) {
        double r211003 = a;
        double r211004 = r211003 * r211003;
        double r211005 = b;
        double r211006 = r211005 * r211005;
        double r211007 = r211004 + r211006;
        double r211008 = 2.0;
        double r211009 = 2.0;
        double r211010 = r211008 / r211009;
        double r211011 = pow(r211007, r211010);
        double r211012 = 4.0;
        double r211013 = r211012 * r211006;
        double r211014 = fma(r211011, r211011, r211013);
        double r211015 = 1.0;
        double r211016 = r211014 - r211015;
        return r211016;
}

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. Using strategy rm

  3. Applied sqr-pow0.2

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

  4. Applied fma-def0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))