Average Error: 0.1 → 0.1
Time: 9.5s
Precision: 64
\[\sqrt{1 + 2 \cdot x} + \sqrt{x}\]
\[\sqrt{2 \cdot x + 1} + \sqrt{x}\]
\sqrt{1 + 2 \cdot x} + \sqrt{x}
\sqrt{2 \cdot x + 1} + \sqrt{x}
double f(double x) {
        double r1047142 = 1.0;
        double r1047143 = 2.0;
        double r1047144 = x;
        double r1047145 = r1047143 * r1047144;
        double r1047146 = r1047142 + r1047145;
        double r1047147 = sqrt(r1047146);
        double r1047148 = sqrt(r1047144);
        double r1047149 = r1047147 + r1047148;
        return r1047149;
}

double f(double x) {
        double r1047150 = 2.0;
        double r1047151 = x;
        double r1047152 = r1047150 * r1047151;
        double r1047153 = 1.0;
        double r1047154 = r1047152 + r1047153;
        double r1047155 = sqrt(r1047154);
        double r1047156 = sqrt(r1047151);
        double r1047157 = r1047155 + r1047156;
        return r1047157;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.1

    \[\sqrt{1 + 2 \cdot x} + \sqrt{x}\]
  2. Using strategy rm
  3. Applied add-sqr-sqrt0.1

    \[\leadsto \sqrt{\color{blue}{\sqrt{1 + 2 \cdot x} \cdot \sqrt{1 + 2 \cdot x}}} + \sqrt{x}\]
  4. Applied sqrt-prod0.2

    \[\leadsto \color{blue}{\sqrt{\sqrt{1 + 2 \cdot x}} \cdot \sqrt{\sqrt{1 + 2 \cdot x}}} + \sqrt{x}\]
  5. Using strategy rm
  6. Applied *-un-lft-identity0.2

    \[\leadsto \sqrt{\color{blue}{1 \cdot \sqrt{1 + 2 \cdot x}}} \cdot \sqrt{\sqrt{1 + 2 \cdot x}} + \sqrt{x}\]
  7. Applied sqrt-prod0.2

    \[\leadsto \color{blue}{\left(\sqrt{1} \cdot \sqrt{\sqrt{1 + 2 \cdot x}}\right)} \cdot \sqrt{\sqrt{1 + 2 \cdot x}} + \sqrt{x}\]
  8. Applied associate-*l*0.2

    \[\leadsto \color{blue}{\sqrt{1} \cdot \left(\sqrt{\sqrt{1 + 2 \cdot x}} \cdot \sqrt{\sqrt{1 + 2 \cdot x}}\right)} + \sqrt{x}\]
  9. Simplified0.1

    \[\leadsto \sqrt{1} \cdot \color{blue}{\sqrt{2 \cdot x + 1}} + \sqrt{x}\]
  10. Final simplification0.1

    \[\leadsto \sqrt{2 \cdot x + 1} + \sqrt{x}\]

Reproduce

herbie shell --seed 1 
(FPCore (x)
  :name "sqrt(1+2x) + sqrt(x)"
  :precision binary64
  (+ (sqrt (+ 1 (* 2 x))) (sqrt x)))