Average Error: 0.2 → 0.0
Time: 22.7s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(4 \cdot \left(b \cdot b\right) + {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4}\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\left(4 \cdot \left(b \cdot b\right) + {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4}\right) - 1
double f(double a, double b) {
        double r7212957 = a;
        double r7212958 = r7212957 * r7212957;
        double r7212959 = b;
        double r7212960 = r7212959 * r7212959;
        double r7212961 = r7212958 + r7212960;
        double r7212962 = 2.0;
        double r7212963 = pow(r7212961, r7212962);
        double r7212964 = 4.0;
        double r7212965 = r7212964 * r7212960;
        double r7212966 = r7212963 + r7212965;
        double r7212967 = 1.0;
        double r7212968 = r7212966 - r7212967;
        return r7212968;
}
↓
double f(double a, double b) {
        double r7212969 = 4.0;
        double r7212970 = b;
        double r7212971 = r7212970 * r7212970;
        double r7212972 = r7212969 * r7212971;
        double r7212973 = a;
        double r7212974 = r7212973 * r7212973;
        double r7212975 = r7212971 + r7212974;
        double r7212976 = sqrt(r7212975);
        double r7212977 = pow(r7212976, r7212969);
        double r7212978 = r7212972 + r7212977;
        double r7212979 = 1.0;
        double r7212980 = r7212978 - r7212979;
        return r7212980;
}

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

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

  2. Simplified0.2

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied associate-*l*0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied cube-unmult0.1

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

  9. Applied pow10.1

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

  10. Applied pow-prod-up0.0

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

  11. Simplified0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2019143 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))