Average Error: 0.2 → 0.0
Time: 1.1m
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left({\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4} - 1\right) + 4 \cdot \left(a \cdot \left(a \cdot a + a\right) + \left(b \cdot \left(1 - 3 \cdot a\right)\right) \cdot b\right)\]
\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(1 - 3 \cdot a\right)\right)\right) - 1
↓
\left({\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4} - 1\right) + 4 \cdot \left(a \cdot \left(a \cdot a + a\right) + \left(b \cdot \left(1 - 3 \cdot a\right)\right) \cdot b\right)
double f(double a, double b) {
        double r22128891 = a;
        double r22128892 = r22128891 * r22128891;
        double r22128893 = b;
        double r22128894 = r22128893 * r22128893;
        double r22128895 = r22128892 + r22128894;
        double r22128896 = 2.0;
        double r22128897 = pow(r22128895, r22128896);
        double r22128898 = 4.0;
        double r22128899 = 1.0;
        double r22128900 = r22128899 + r22128891;
        double r22128901 = r22128892 * r22128900;
        double r22128902 = 3.0;
        double r22128903 = r22128902 * r22128891;
        double r22128904 = r22128899 - r22128903;
        double r22128905 = r22128894 * r22128904;
        double r22128906 = r22128901 + r22128905;
        double r22128907 = r22128898 * r22128906;
        double r22128908 = r22128897 + r22128907;
        double r22128909 = r22128908 - r22128899;
        return r22128909;
}
↓
double f(double a, double b) {
        double r22128910 = a;
        double r22128911 = r22128910 * r22128910;
        double r22128912 = b;
        double r22128913 = r22128912 * r22128912;
        double r22128914 = r22128911 + r22128913;
        double r22128915 = sqrt(r22128914);
        double r22128916 = 4.0;
        double r22128917 = pow(r22128915, r22128916);
        double r22128918 = 1.0;
        double r22128919 = r22128917 - r22128918;
        double r22128920 = r22128911 + r22128910;
        double r22128921 = r22128910 * r22128920;
        double r22128922 = 3.0;
        double r22128923 = r22128922 * r22128910;
        double r22128924 = r22128918 - r22128923;
        double r22128925 = r22128912 * r22128924;
        double r22128926 = r22128925 * r22128912;
        double r22128927 = r22128921 + r22128926;
        double r22128928 = r22128916 * r22128927;
        double r22128929 = r22128919 + r22128928;
        return r22128929;
}

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

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied associate-*r*0.1

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

  7. Applied pow10.1

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

  8. Applied add-sqr-sqrt0.1

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

  9. Applied pow30.1

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

  10. Applied pow-prod-up0.0

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

  11. Simplified0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2019124 
(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))