Average Error: 29.7 → 0.2
Time: 11.6s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\sqrt{x + 1} + \sqrt{x}}
double f(double x) {
        double r391402 = x;
        double r391403 = 1.0;
        double r391404 = r391402 + r391403;
        double r391405 = sqrt(r391404);
        double r391406 = sqrt(r391402);
        double r391407 = r391405 - r391406;
        return r391407;
}
↓
double f(double x) {
        double r391408 = 1.0;
        double r391409 = x;
        double r391410 = r391409 + r391408;
        double r391411 = sqrt(r391410);
        double r391412 = sqrt(r391409);
        double r391413 = r391411 + r391412;
        double r391414 = r391408 / r391413;
        return r391414;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.7
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.7

    \[\sqrt{x + 1} - \sqrt{x}\]

  2. Simplified29.7

    \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}}\]
  3. Using strategy rm

  4. Applied flip--29.4

    \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}}\]

  5. Simplified0.2

    \[\leadsto \frac{\color{blue}{1 + 0}}{\sqrt{1 + x} + \sqrt{x}}\]

  6. Final simplification0.2

    \[\leadsto \frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x)
  :name "Main:bigenough3 from C"

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

  (- (sqrt (+ x 1.0)) (sqrt x)))