Average Error: 30.6 → 0.3
Time: 6.5s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\sqrt{\frac{\frac{1}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}} \cdot \sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\sqrt{\frac{\frac{1}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}} \cdot \sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}
double f(double x) {
        double r683862 = x;
        double r683863 = 1.0;
        double r683864 = r683862 + r683863;
        double r683865 = sqrt(r683864);
        double r683866 = sqrt(r683862);
        double r683867 = r683865 - r683866;
        return r683867;
}
↓
double f(double x) {
        double r683868 = 1.0;
        double r683869 = x;
        double r683870 = r683869 + r683868;
        double r683871 = sqrt(r683870);
        double r683872 = sqrt(r683869);
        double r683873 = r683871 + r683872;
        double r683874 = sqrt(r683873);
        double r683875 = r683868 / r683874;
        double r683876 = r683875 / r683874;
        double r683877 = sqrt(r683876);
        double r683878 = r683868 / r683873;
        double r683879 = sqrt(r683878);
        double r683880 = r683877 * r683879;
        return r683880;
}

Error

Bits error versus x

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Results

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Target

Original30.6
Target0.2
Herbie0.3
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.6

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

  3. Applied flip--30.4

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

  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. Using strategy rm

  8. Applied add-sqr-sqrt0.4

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

  9. Applied associate-/r*0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2020025 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

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

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