Average Error: 29.9 → 0.3
Time: 9.4s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}} + \sqrt{x}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}} + \sqrt{x}}
double f(double x) {
        double r770263 = x;
        double r770264 = 1.0;
        double r770265 = r770263 + r770264;
        double r770266 = sqrt(r770265);
        double r770267 = sqrt(r770263);
        double r770268 = r770266 - r770267;
        return r770268;
}
↓
double f(double x) {
        double r770269 = 1.0;
        double r770270 = x;
        double r770271 = r770270 + r770269;
        double r770272 = sqrt(r770271);
        double r770273 = sqrt(r770272);
        double r770274 = r770273 * r770273;
        double r770275 = sqrt(r770270);
        double r770276 = r770274 + r770275;
        double r770277 = r770269 / r770276;
        return r770277;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.2
Herbie0.3
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.9

    \[\sqrt{x + 1} - \sqrt{x}\]
  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

    \[\leadsto \frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}}\]
  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-prod0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2020042 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

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

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