Average Error: 0.2 → 0.2
Time: 23.0s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(a \cdot a + b \cdot b, a \cdot a + b \cdot b, 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right) + \left(a \cdot a\right) \cdot \left(a + 1\right)\right)\right) - 1\]
\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(a \cdot a + b \cdot b, a \cdot a + b \cdot b, 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right) + \left(a \cdot a\right) \cdot \left(a + 1\right)\right)\right) - 1
double f(double a, double b) {
        double r2435759 = a;
        double r2435760 = r2435759 * r2435759;
        double r2435761 = b;
        double r2435762 = r2435761 * r2435761;
        double r2435763 = r2435760 + r2435762;
        double r2435764 = 2.0;
        double r2435765 = pow(r2435763, r2435764);
        double r2435766 = 4.0;
        double r2435767 = 1.0;
        double r2435768 = r2435767 + r2435759;
        double r2435769 = r2435760 * r2435768;
        double r2435770 = 3.0;
        double r2435771 = r2435770 * r2435759;
        double r2435772 = r2435767 - r2435771;
        double r2435773 = r2435762 * r2435772;
        double r2435774 = r2435769 + r2435773;
        double r2435775 = r2435766 * r2435774;
        double r2435776 = r2435765 + r2435775;
        double r2435777 = r2435776 - r2435767;
        return r2435777;
}
↓
double f(double a, double b) {
        double r2435778 = a;
        double r2435779 = r2435778 * r2435778;
        double r2435780 = b;
        double r2435781 = r2435780 * r2435780;
        double r2435782 = r2435779 + r2435781;
        double r2435783 = 4.0;
        double r2435784 = 1.0;
        double r2435785 = 3.0;
        double r2435786 = r2435785 * r2435778;
        double r2435787 = r2435784 - r2435786;
        double r2435788 = r2435781 * r2435787;
        double r2435789 = r2435778 + r2435784;
        double r2435790 = r2435779 * r2435789;
        double r2435791 = r2435788 + r2435790;
        double r2435792 = r2435783 * r2435791;
        double r2435793 = fma(r2435782, r2435782, r2435792);
        double r2435794 = r2435793 - r2435784;
        return r2435794;
}

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. Using strategy rm

  3. Applied unpow20.2

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

  4. Applied fma-def0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (+ 1 a)) (* (* b b) (- 1 (* 3 a)))))) 1))