Average Error: 0.5 → 0.3
Time: 9.0s
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 r10639 = x;
        double r10640 = 1.0;
        double r10641 = r10639 - r10640;
        double r10642 = sqrt(r10641);
        double r10643 = sqrt(r10639);
        double r10644 = r10642 * r10643;
        return r10644;
}
↓
double f(double x) {
        double r10645 = x;
        double r10646 = 0.125;
        double r10647 = r10646 / r10645;
        double r10648 = 0.5;
        double r10649 = r10647 + r10648;
        double r10650 = r10645 - r10649;
        return r10650;
}

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 0.3

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1.0)) (sqrt x)))