Average Error: 0.5 → 0.4
Time: 11.3s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[x - \left(0.5 + \frac{0.125}{x}\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
x - \left(0.5 + \frac{0.125}{x}\right)
double f(double x) {
        double r16151 = x;
        double r16152 = 1.0;
        double r16153 = r16151 - r16152;
        double r16154 = sqrt(r16153);
        double r16155 = sqrt(r16151);
        double r16156 = r16154 * r16155;
        return r16156;
}
↓
double f(double x) {
        double r16157 = x;
        double r16158 = 0.5;
        double r16159 = 0.125;
        double r16160 = r16159 / r16157;
        double r16161 = r16158 + r16160;
        double r16162 = r16157 - r16161;
        return r16162;
}

Error

Bits error versus x

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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

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019196 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1.0)) (sqrt x)))