Average Error: 0.2 → 0.2
Time: 4.3s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left({\left({\left(a \cdot a + b \cdot b\right)}^{\frac{1}{2}} \cdot {\left(a \cdot a + b \cdot b\right)}^{\frac{1}{2}}\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\left({\left({\left(a \cdot a + b \cdot b\right)}^{\frac{1}{2}} \cdot {\left(a \cdot a + b \cdot b\right)}^{\frac{1}{2}}\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
double f(double a, double b) {
        double r207927 = a;
        double r207928 = r207927 * r207927;
        double r207929 = b;
        double r207930 = r207929 * r207929;
        double r207931 = r207928 + r207930;
        double r207932 = 2.0;
        double r207933 = pow(r207931, r207932);
        double r207934 = 4.0;
        double r207935 = r207934 * r207930;
        double r207936 = r207933 + r207935;
        double r207937 = 1.0;
        double r207938 = r207936 - r207937;
        return r207938;
}
↓
double f(double a, double b) {
        double r207939 = a;
        double r207940 = r207939 * r207939;
        double r207941 = b;
        double r207942 = r207941 * r207941;
        double r207943 = r207940 + r207942;
        double r207944 = 0.5;
        double r207945 = pow(r207943, r207944);
        double r207946 = r207945 * r207945;
        double r207947 = 2.0;
        double r207948 = pow(r207946, r207947);
        double r207949 = 4.0;
        double r207950 = r207949 * r207942;
        double r207951 = r207948 + r207950;
        double r207952 = 1.0;
        double r207953 = r207951 - r207952;
        return r207953;
}

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. Using strategy rm

  3. Applied add-sqr-sqrt0.2

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

  4. Simplified0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019318 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))