Average Error: 0.2 → 0.2
Time: 22.6s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(\left(a + 3\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(1 - a\right)\right) \cdot 4\right) - 1\]
\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(3 + a\right)\right)\right) - 1
↓
\left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(\left(a + 3\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(1 - a\right)\right) \cdot 4\right) - 1
double f(double a, double b) {
        double r7948840 = a;
        double r7948841 = r7948840 * r7948840;
        double r7948842 = b;
        double r7948843 = r7948842 * r7948842;
        double r7948844 = r7948841 + r7948843;
        double r7948845 = 2.0;
        double r7948846 = pow(r7948844, r7948845);
        double r7948847 = 4.0;
        double r7948848 = 1.0;
        double r7948849 = r7948848 - r7948840;
        double r7948850 = r7948841 * r7948849;
        double r7948851 = 3.0;
        double r7948852 = r7948851 + r7948840;
        double r7948853 = r7948843 * r7948852;
        double r7948854 = r7948850 + r7948853;
        double r7948855 = r7948847 * r7948854;
        double r7948856 = r7948846 + r7948855;
        double r7948857 = r7948856 - r7948848;
        return r7948857;
}
↓
double f(double a, double b) {
        double r7948858 = a;
        double r7948859 = r7948858 * r7948858;
        double r7948860 = b;
        double r7948861 = r7948860 * r7948860;
        double r7948862 = r7948859 + r7948861;
        double r7948863 = 2.0;
        double r7948864 = pow(r7948862, r7948863);
        double r7948865 = 3.0;
        double r7948866 = r7948858 + r7948865;
        double r7948867 = r7948866 * r7948861;
        double r7948868 = 1.0;
        double r7948869 = r7948868 - r7948858;
        double r7948870 = r7948859 * r7948869;
        double r7948871 = r7948867 + r7948870;
        double r7948872 = 4.0;
        double r7948873 = r7948871 * r7948872;
        double r7948874 = r7948864 + r7948873;
        double r7948875 = r7948874 - r7948868;
        return r7948875;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

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

  2. Final simplification0.2

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

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))