Average Error: 29.5 → 0.3
Time: 5.8s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}
double f(double x) {
        double r600585 = x;
        double r600586 = 1.0;
        double r600587 = r600585 + r600586;
        double r600588 = sqrt(r600587);
        double r600589 = sqrt(r600585);
        double r600590 = r600588 - r600589;
        return r600590;
}
↓
double f(double x) {
        double r600591 = 1.0;
        double r600592 = x;
        double r600593 = r600592 + r600591;
        double r600594 = sqrt(r600593);
        double r600595 = sqrt(r600592);
        double r600596 = r600594 + r600595;
        double r600597 = r600591 / r600596;
        double r600598 = sqrt(r600597);
        double r600599 = r600598 * r600598;
        return r600599;
}

Error

Bits error versus x

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Results

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Target

Original29.5
Target0.2
Herbie0.3
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.5

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

  3. Applied flip--29.2

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

  4. Simplified0.2

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

  6. Applied add-sqr-sqrt0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2020089 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

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

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