Average Error: 0.5 → 0.4
Time: 1.7s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[x - \left(0.5 + 0.125 \cdot \frac{1}{x}\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
x - \left(0.5 + 0.125 \cdot \frac{1}{x}\right)
double f(double x) {
        double r1185 = x;
        double r1186 = 1.0;
        double r1187 = r1185 - r1186;
        double r1188 = sqrt(r1187);
        double r1189 = sqrt(r1185);
        double r1190 = r1188 * r1189;
        return r1190;
}
↓
double f(double x) {
        double r1191 = x;
        double r1192 = 0.5;
        double r1193 = 0.125;
        double r1194 = 1.0;
        double r1195 = r1194 / r1191;
        double r1196 = r1193 * r1195;
        double r1197 = r1192 + r1196;
        double r1198 = r1191 - r1197;
        return r1198;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

    \[\sqrt{x - 1} \cdot \sqrt{x}\]

  2. Taylor expanded around inf 0.4

    \[\leadsto \color{blue}{x - \left(0.5 + 0.125 \cdot \frac{1}{x}\right)}\]

  3. Final simplification0.4

    \[\leadsto x - \left(0.5 + 0.125 \cdot \frac{1}{x}\right)\]

Reproduce

herbie shell --seed 2020018 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))