Average Error: 29.3 → 0.2
Time: 17.0s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[{\left(\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)\right)}^{\frac{-1}{2}}\]
\sqrt{x + 1} - \sqrt{x}
↓
{\left(\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)\right)}^{\frac{-1}{2}}
double f(double x) {
        double r3356320 = x;
        double r3356321 = 1.0;
        double r3356322 = r3356320 + r3356321;
        double r3356323 = sqrt(r3356322);
        double r3356324 = sqrt(r3356320);
        double r3356325 = r3356323 - r3356324;
        return r3356325;
}
↓
double f(double x) {
        double r3356326 = 1.0;
        double r3356327 = x;
        double r3356328 = r3356326 + r3356327;
        double r3356329 = sqrt(r3356328);
        double r3356330 = sqrt(r3356327);
        double r3356331 = r3356329 + r3356330;
        double r3356332 = r3356331 * r3356331;
        double r3356333 = -0.5;
        double r3356334 = pow(r3356332, r3356333);
        return r3356334;
}

Error

Bits error versus x

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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-sqrt0.3

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

  8. Applied inv-pow0.3

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

  9. Applied sqrt-pow10.3

    \[\leadsto \sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} \cdot \color{blue}{{\left(\sqrt{x + 1} + \sqrt{x}\right)}^{\left(\frac{-1}{2}\right)}}\]

  10. Applied inv-pow0.3

    \[\leadsto \sqrt{\color{blue}{{\left(\sqrt{x + 1} + \sqrt{x}\right)}^{-1}}} \cdot {\left(\sqrt{x + 1} + \sqrt{x}\right)}^{\left(\frac{-1}{2}\right)}\]

  11. Applied sqrt-pow10.3

    \[\leadsto \color{blue}{{\left(\sqrt{x + 1} + \sqrt{x}\right)}^{\left(\frac{-1}{2}\right)}} \cdot {\left(\sqrt{x + 1} + \sqrt{x}\right)}^{\left(\frac{-1}{2}\right)}\]

  12. Applied pow-prod-down0.2

    \[\leadsto \color{blue}{{\left(\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)\right)}^{\left(\frac{-1}{2}\right)}}\]

  13. Final simplification0.2

    \[\leadsto {\left(\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)\right)}^{\frac{-1}{2}}\]

Reproduce

herbie shell --seed 2019138 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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