Average Error: 0.5 → 0.4
Time: 1.9s
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 r3103 = x;
        double r3104 = 1.0;
        double r3105 = r3103 - r3104;
        double r3106 = sqrt(r3105);
        double r3107 = sqrt(r3103);
        double r3108 = r3106 * r3107;
        return r3108;
}
↓
double f(double x) {
        double r3109 = x;
        double r3110 = 0.5;
        double r3111 = 0.125;
        double r3112 = 1.0;
        double r3113 = r3112 / r3109;
        double r3114 = r3111 * r3113;
        double r3115 = r3110 + r3114;
        double r3116 = r3109 - r3115;
        return r3116;
}

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 2020064 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))