Average Error: 29.8 → 0.2
Time: 12.0s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{1}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{1}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}
double f(double x) {
        double r378496 = x;
        double r378497 = 1.0;
        double r378498 = r378496 + r378497;
        double r378499 = sqrt(r378498);
        double r378500 = sqrt(r378496);
        double r378501 = r378499 - r378500;
        return r378501;
}
↓
double f(double x) {
        double r378502 = 1.0;
        double r378503 = x;
        double r378504 = r378503 + r378502;
        double r378505 = sqrt(r378504);
        double r378506 = sqrt(r378503);
        double r378507 = r378505 + r378506;
        double r378508 = r378502 / r378507;
        double r378509 = sqrt(r378508);
        double r378510 = sqrt(r378502);
        double r378511 = r378509 * r378510;
        double r378512 = sqrt(r378507);
        double r378513 = r378511 / r378512;
        return r378513;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Target

Original29.8
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.8

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

  3. Applied flip--29.5

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

  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 sqrt-div0.3

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

  9. Applied associate-*r/0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019208 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

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

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