Average Error: 29.5 → 0.3
Time: 5.2s
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 r445480 = x;
        double r445481 = 1.0;
        double r445482 = r445480 + r445481;
        double r445483 = sqrt(r445482);
        double r445484 = sqrt(r445480);
        double r445485 = r445483 - r445484;
        return r445485;
}
↓
double f(double x) {
        double r445486 = 1.0;
        double r445487 = x;
        double r445488 = r445487 + r445486;
        double r445489 = sqrt(r445488);
        double r445490 = sqrt(r445487);
        double r445491 = r445489 + r445490;
        double r445492 = r445486 / r445491;
        double r445493 = sqrt(r445492);
        double r445494 = r445493 * r445493;
        return r445494;
}

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 2020027 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

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

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