Average Error: 0.2 → 0.2
Time: 6.4s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} \cdot \sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\]
\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
↓
\sqrt{{\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)} \cdot \sqrt{{\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)} - 1
double f(double a, double b) {
        double r400025 = a;
        double r400026 = r400025 * r400025;
        double r400027 = b;
        double r400028 = r400027 * r400027;
        double r400029 = r400026 + r400028;
        double r400030 = 2.0;
        double r400031 = pow(r400029, r400030);
        double r400032 = 4.0;
        double r400033 = 1.0;
        double r400034 = r400033 + r400025;
        double r400035 = r400026 * r400034;
        double r400036 = 3.0;
        double r400037 = r400036 * r400025;
        double r400038 = r400033 - r400037;
        double r400039 = r400028 * r400038;
        double r400040 = r400035 + r400039;
        double r400041 = r400032 * r400040;
        double r400042 = r400031 + r400041;
        double r400043 = r400042 - r400033;
        return r400043;
}
↓
double f(double a, double b) {
        double r400044 = a;
        double r400045 = r400044 * r400044;
        double r400046 = b;
        double r400047 = r400046 * r400046;
        double r400048 = r400045 + r400047;
        double r400049 = 2.0;
        double r400050 = pow(r400048, r400049);
        double r400051 = 4.0;
        double r400052 = 1.0;
        double r400053 = r400052 + r400044;
        double r400054 = r400045 * r400053;
        double r400055 = 3.0;
        double r400056 = r400055 * r400044;
        double r400057 = r400052 - r400056;
        double r400058 = r400047 * r400057;
        double r400059 = r400054 + r400058;
        double r400060 = r400051 * r400059;
        double r400061 = r400050 + r400060;
        double r400062 = sqrt(r400061);
        double r400063 = r400062 * r400062;
        double r400064 = r400063 - r400052;
        return r400064;
}

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. Using strategy rm

  3. Applied add-sqr-sqrt0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (+ 1 a)) (* (* b b) (- 1 (* 3 a)))))) 1))