Average Error: 0.5 → 0.4
Time: 12.1s
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 r12248 = x;
        double r12249 = 1.0;
        double r12250 = r12248 - r12249;
        double r12251 = sqrt(r12250);
        double r12252 = sqrt(r12248);
        double r12253 = r12251 * r12252;
        return r12253;
}
↓
double f(double x) {
        double r12254 = x;
        double r12255 = 0.125;
        double r12256 = r12255 / r12254;
        double r12257 = 0.5;
        double r12258 = r12256 + r12257;
        double r12259 = r12254 - r12258;
        return r12259;
}

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

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

  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 2019199 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1.0)) (sqrt x)))