Average Error: 29.3 → 0.2
Time: 11.0s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}
double f(double x) {
        double r481530 = x;
        double r481531 = 1.0;
        double r481532 = r481530 + r481531;
        double r481533 = sqrt(r481532);
        double r481534 = sqrt(r481530);
        double r481535 = r481533 - r481534;
        return r481535;
}
↓
double f(double x) {
        double r481536 = 1.0;
        double r481537 = 0.0;
        double r481538 = r481536 + r481537;
        double r481539 = x;
        double r481540 = r481539 + r481536;
        double r481541 = sqrt(r481540);
        double r481542 = sqrt(r481539);
        double r481543 = r481541 + r481542;
        double r481544 = r481538 / r481543;
        return r481544;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Target

Original29.3
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.3

    \[\sqrt{x + 1} - \sqrt{x}\]
  2. Using strategy rm

  3. Applied flip--29.1

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

  :herbie-target
  (/ 1 (+ (sqrt (+ x 1)) (sqrt x)))

  (- (sqrt (+ x 1)) (sqrt x)))