Average Error: 0.2 → 0.0
Time: 35.6s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left({\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4} + \left(\left(-3 \cdot \left(b \cdot b\right) + \left(a + a \cdot a\right)\right) \cdot a + b \cdot b\right) \cdot 4\right) + -1\]
\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(1 - 3 \cdot a\right)\right)\right) - 1
↓
\left({\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4} + \left(\left(-3 \cdot \left(b \cdot b\right) + \left(a + a \cdot a\right)\right) \cdot a + b \cdot b\right) \cdot 4\right) + -1
double f(double a, double b) {
        double r9644959 = a;
        double r9644960 = r9644959 * r9644959;
        double r9644961 = b;
        double r9644962 = r9644961 * r9644961;
        double r9644963 = r9644960 + r9644962;
        double r9644964 = 2.0;
        double r9644965 = pow(r9644963, r9644964);
        double r9644966 = 4.0;
        double r9644967 = 1.0;
        double r9644968 = r9644967 + r9644959;
        double r9644969 = r9644960 * r9644968;
        double r9644970 = 3.0;
        double r9644971 = r9644970 * r9644959;
        double r9644972 = r9644967 - r9644971;
        double r9644973 = r9644962 * r9644972;
        double r9644974 = r9644969 + r9644973;
        double r9644975 = r9644966 * r9644974;
        double r9644976 = r9644965 + r9644975;
        double r9644977 = r9644976 - r9644967;
        return r9644977;
}
↓
double f(double a, double b) {
        double r9644978 = b;
        double r9644979 = r9644978 * r9644978;
        double r9644980 = a;
        double r9644981 = r9644980 * r9644980;
        double r9644982 = r9644979 + r9644981;
        double r9644983 = sqrt(r9644982);
        double r9644984 = 4.0;
        double r9644985 = pow(r9644983, r9644984);
        double r9644986 = -3.0;
        double r9644987 = r9644986 * r9644979;
        double r9644988 = r9644980 + r9644981;
        double r9644989 = r9644987 + r9644988;
        double r9644990 = r9644989 * r9644980;
        double r9644991 = r9644990 + r9644979;
        double r9644992 = r9644991 * r9644984;
        double r9644993 = r9644985 + r9644992;
        double r9644994 = -1.0;
        double r9644995 = r9644993 + r9644994;
        return r9644995;
}

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

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

  2. Simplified0.2

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied associate-*r*0.1

    \[\leadsto -1 + \left(\left(b \cdot b + a \cdot \left(\left(b \cdot b\right) \cdot -3 + \left(a \cdot a + a\right)\right)\right) \cdot 4 + \color{blue}{\left(\left(a \cdot a + b \cdot b\right) \cdot \sqrt{a \cdot a + b \cdot b}\right) \cdot \sqrt{a \cdot a + b \cdot b}}\right)\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt0.1

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

  8. Applied pow30.1

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

  9. Applied pow-plus0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019138 
(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))