Average Error: 0.2 → 0.2
Time: 6.1s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)} \cdot \sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)} - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)} \cdot \sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)} - 1
double f(double a, double b) {
        double r202136 = a;
        double r202137 = r202136 * r202136;
        double r202138 = b;
        double r202139 = r202138 * r202138;
        double r202140 = r202137 + r202139;
        double r202141 = 2.0;
        double r202142 = pow(r202140, r202141);
        double r202143 = 4.0;
        double r202144 = r202143 * r202139;
        double r202145 = r202142 + r202144;
        double r202146 = 1.0;
        double r202147 = r202145 - r202146;
        return r202147;
}
↓
double f(double a, double b) {
        double r202148 = a;
        double r202149 = r202148 * r202148;
        double r202150 = b;
        double r202151 = r202150 * r202150;
        double r202152 = r202149 + r202151;
        double r202153 = 2.0;
        double r202154 = pow(r202152, r202153);
        double r202155 = 4.0;
        double r202156 = r202155 * r202151;
        double r202157 = r202154 + r202156;
        double r202158 = sqrt(r202157);
        double r202159 = r202158 * r202158;
        double r202160 = 1.0;
        double r202161 = r202159 - r202160;
        return r202161;
}

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

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2020036 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))