Average Error: 0.5 → 0.4
Time: 3.7s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[x - \left(\frac{0.125}{x} + 0.5\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
x - \left(\frac{0.125}{x} + 0.5\right)
double f(double x) {
        double r1279 = x;
        double r1280 = 1.0;
        double r1281 = r1279 - r1280;
        double r1282 = sqrt(r1281);
        double r1283 = sqrt(r1279);
        double r1284 = r1282 * r1283;
        return r1284;
}
↓
double f(double x) {
        double r1285 = x;
        double r1286 = 0.125;
        double r1287 = r1286 / r1285;
        double r1288 = 0.5;
        double r1289 = r1287 + r1288;
        double r1290 = r1285 - r1289;
        return r1290;
}

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. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))