Average Error: 30.1 → 0.2
Time: 11.0s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{x} + \left|\sqrt{x + 1}\right|}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\sqrt{x} + \left|\sqrt{x + 1}\right|}
double f(double x) {
        double r528830 = x;
        double r528831 = 1.0;
        double r528832 = r528830 + r528831;
        double r528833 = sqrt(r528832);
        double r528834 = sqrt(r528830);
        double r528835 = r528833 - r528834;
        return r528835;
}
↓
double f(double x) {
        double r528836 = 1.0;
        double r528837 = x;
        double r528838 = sqrt(r528837);
        double r528839 = r528837 + r528836;
        double r528840 = sqrt(r528839);
        double r528841 = fabs(r528840);
        double r528842 = r528838 + r528841;
        double r528843 = r528836 / r528842;
        return r528843;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

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

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

  5. Simplified0.2

    \[\leadsto \frac{1 + 0}{\color{blue}{\sqrt{x} + \sqrt{x + 1}}}\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt0.2

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

  8. Applied rem-sqrt-square0.2

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

  9. Final simplification0.2

    \[\leadsto \frac{1}{\sqrt{x} + \left|\sqrt{x + 1}\right|}\]

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

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

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