Average Error: 0.2 → 0.2
Time: 12.2s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(1 \cdot {a}^{2} + 3 \cdot {b}^{2}\right) - {a}^{3}\right)\right) - 1\]
\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(3 + a\right)\right)\right) - 1
↓
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(1 \cdot {a}^{2} + 3 \cdot {b}^{2}\right) - {a}^{3}\right)\right) - 1
double f(double a, double b) {
        double r340903 = a;
        double r340904 = r340903 * r340903;
        double r340905 = b;
        double r340906 = r340905 * r340905;
        double r340907 = r340904 + r340906;
        double r340908 = 2.0;
        double r340909 = pow(r340907, r340908);
        double r340910 = 4.0;
        double r340911 = 1.0;
        double r340912 = r340911 - r340903;
        double r340913 = r340904 * r340912;
        double r340914 = 3.0;
        double r340915 = r340914 + r340903;
        double r340916 = r340906 * r340915;
        double r340917 = r340913 + r340916;
        double r340918 = r340910 * r340917;
        double r340919 = r340909 + r340918;
        double r340920 = r340919 - r340911;
        return r340920;
}
↓
double f(double a, double b) {
        double r340921 = a;
        double r340922 = r340921 * r340921;
        double r340923 = b;
        double r340924 = r340923 * r340923;
        double r340925 = r340922 + r340924;
        double r340926 = 2.0;
        double r340927 = pow(r340925, r340926);
        double r340928 = 4.0;
        double r340929 = 1.0;
        double r340930 = 2.0;
        double r340931 = pow(r340921, r340930);
        double r340932 = r340929 * r340931;
        double r340933 = 3.0;
        double r340934 = pow(r340923, r340930);
        double r340935 = r340933 * r340934;
        double r340936 = r340932 + r340935;
        double r340937 = 3.0;
        double r340938 = pow(r340921, r340937);
        double r340939 = r340936 - r340938;
        double r340940 = r340928 * r340939;
        double r340941 = r340927 + r340940;
        double r340942 = r340941 - r340929;
        return r340942;
}

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. Using strategy rm

  3. Applied sub-neg0.2

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

  4. Applied distribute-rgt-in0.2

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

  5. Applied associate-+l+0.2

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

  6. Simplified0.2

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

  7. Taylor expanded around 0 0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2020043 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))