Average Error: 29.3 → 0.2
Time: 15.0s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\sqrt{x + 1} + \sqrt{x}}{\sqrt[3]{1}}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\sqrt{x + 1} + \sqrt{x}}{\sqrt[3]{1}}}
double f(double x) {
        double r113509 = x;
        double r113510 = 1.0;
        double r113511 = r113509 + r113510;
        double r113512 = sqrt(r113511);
        double r113513 = sqrt(r113509);
        double r113514 = r113512 - r113513;
        return r113514;
}
↓
double f(double x) {
        double r113515 = 1.0;
        double r113516 = cbrt(r113515);
        double r113517 = r113516 * r113516;
        double r113518 = x;
        double r113519 = r113518 + r113515;
        double r113520 = sqrt(r113519);
        double r113521 = sqrt(r113518);
        double r113522 = r113520 + r113521;
        double r113523 = r113522 / r113516;
        double r113524 = r113517 / r113523;
        return r113524;
}

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 add-cube-cbrt0.2

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}} + \sqrt{x}}\]

  10. Applied associate-/l*0.2

    \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}} + \sqrt{x}}{\sqrt[3]{1}}}}\]

  11. Simplified0.2

    \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\color{blue}{\frac{\sqrt{x + 1} + \sqrt{x}}{\sqrt[3]{1}}}}\]

  12. Final simplification0.2

    \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\sqrt{x + 1} + \sqrt{x}}{\sqrt[3]{1}}}\]

Reproduce

herbie shell --seed 2019326 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

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

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