Average Error: 0.5 → 0.4
Time: 10.0s
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 r16076 = x;
        double r16077 = 1.0;
        double r16078 = r16076 - r16077;
        double r16079 = sqrt(r16078);
        double r16080 = sqrt(r16076);
        double r16081 = r16079 * r16080;
        return r16081;
}
↓
double f(double x) {
        double r16082 = x;
        double r16083 = 0.125;
        double r16084 = r16083 / r16082;
        double r16085 = 0.5;
        double r16086 = r16084 + r16085;
        double r16087 = r16082 - r16086;
        return r16087;
}

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 2020043 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))