Average Error: 29.3 → 0.2
Time: 15.9s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\frac{\sqrt{x + 1} + \sqrt{x}}{1}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\frac{\sqrt{x + 1} + \sqrt{x}}{1}}
double f(double x) {
        double r318160 = x;
        double r318161 = 1.0;
        double r318162 = r318160 + r318161;
        double r318163 = sqrt(r318162);
        double r318164 = sqrt(r318160);
        double r318165 = r318163 - r318164;
        return r318165;
}
↓
double f(double x) {
        double r318166 = 1.0;
        double r318167 = x;
        double r318168 = 1.0;
        double r318169 = r318167 + r318168;
        double r318170 = sqrt(r318169);
        double r318171 = sqrt(r318167);
        double r318172 = r318170 + r318171;
        double r318173 = r318172 / r318168;
        double r318174 = r318166 / r318173;
        return r318174;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 29.3

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

  3. Applied flip--29.1

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

  4. Simplified0.2

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

  6. Applied add-sqr-sqrt0.2

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

  7. Applied sqrt-prod0.2

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

  9. Applied clear-num0.2

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

  10. Simplified0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2019326 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

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

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