Average Error: 30.1 → 0.2
Time: 12.9s
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 r25229825 = x;
        double r25229826 = 1.0;
        double r25229827 = r25229825 + r25229826;
        double r25229828 = sqrt(r25229827);
        double r25229829 = sqrt(r25229825);
        double r25229830 = r25229828 - r25229829;
        return r25229830;
}
↓
double f(double x) {
        double r25229831 = 1.0;
        double r25229832 = x;
        double r25229833 = r25229832 + r25229831;
        double r25229834 = sqrt(r25229833);
        double r25229835 = sqrt(r25229832);
        double r25229836 = r25229834 + r25229835;
        double r25229837 = r25229831 / r25229836;
        return r25229837;
}

Error

Bits error versus x

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Results

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Target

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

Derivation

  1. Initial program 30.1

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

  3. Applied flip--29.8

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019174 
(FPCore (x)
  :name "Main:bigenough3 from C"

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

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