Average Error: 30.2 → 0.2
Time: 20.5s
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 r618149 = x;
        double r618150 = 1.0;
        double r618151 = r618149 + r618150;
        double r618152 = sqrt(r618151);
        double r618153 = sqrt(r618149);
        double r618154 = r618152 - r618153;
        return r618154;
}
↓
double f(double x) {
        double r618155 = 1.0;
        double r618156 = x;
        double r618157 = r618156 + r618155;
        double r618158 = sqrt(r618157);
        double r618159 = sqrt(r618156);
        double r618160 = r618158 + r618159;
        double r618161 = r618155 / r618160;
        return r618161;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Target

Original30.2
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.2

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

  3. Applied flip--29.9

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2020047 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

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

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