Average Error: 0.2 → 0.0
Time: 1.0m
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\left(\left(b \cdot b\right) \cdot \left(3 + a\right) + \left(a \cdot a\right) \cdot \left(1 - a\right)\right) \cdot 4 - \left(1 - {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4}\right)\]
\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(b \cdot b\right) \cdot \left(3 + a\right) + \left(a \cdot a\right) \cdot \left(1 - a\right)\right) \cdot 4 - \left(1 - {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4}\right)
double f(double a, double b) {
        double r53529921 = a;
        double r53529922 = r53529921 * r53529921;
        double r53529923 = b;
        double r53529924 = r53529923 * r53529923;
        double r53529925 = r53529922 + r53529924;
        double r53529926 = 2.0;
        double r53529927 = pow(r53529925, r53529926);
        double r53529928 = 4.0;
        double r53529929 = 1.0;
        double r53529930 = r53529929 - r53529921;
        double r53529931 = r53529922 * r53529930;
        double r53529932 = 3.0;
        double r53529933 = r53529932 + r53529921;
        double r53529934 = r53529924 * r53529933;
        double r53529935 = r53529931 + r53529934;
        double r53529936 = r53529928 * r53529935;
        double r53529937 = r53529927 + r53529936;
        double r53529938 = r53529937 - r53529929;
        return r53529938;
}
↓
double f(double a, double b) {
        double r53529939 = b;
        double r53529940 = r53529939 * r53529939;
        double r53529941 = 3.0;
        double r53529942 = a;
        double r53529943 = r53529941 + r53529942;
        double r53529944 = r53529940 * r53529943;
        double r53529945 = r53529942 * r53529942;
        double r53529946 = 1.0;
        double r53529947 = r53529946 - r53529942;
        double r53529948 = r53529945 * r53529947;
        double r53529949 = r53529944 + r53529948;
        double r53529950 = 4.0;
        double r53529951 = r53529949 * r53529950;
        double r53529952 = r53529945 + r53529940;
        double r53529953 = sqrt(r53529952);
        double r53529954 = pow(r53529953, r53529950);
        double r53529955 = r53529946 - r53529954;
        double r53529956 = r53529951 - r53529955;
        return r53529956;
}

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. Simplified0.2

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied associate-*l*0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied cube-unmult0.1

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

  10. Applied pow10.1

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

  11. Applied pow-prod-up0.0

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

  12. Simplified0.0

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

  13. Final simplification0.0

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

Reproduce

herbie shell --seed 2019120 
(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))