Average Error: 30.4 → 0.2
Time: 5.2s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{\frac{\sqrt{1}}{1}}{\frac{\sqrt{x + 1} + \sqrt{x}}{\sqrt{1}}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{\frac{\sqrt{1}}{1}}{\frac{\sqrt{x + 1} + \sqrt{x}}{\sqrt{1}}}
double f(double x) {
        double r513444 = x;
        double r513445 = 1.0;
        double r513446 = r513444 + r513445;
        double r513447 = sqrt(r513446);
        double r513448 = sqrt(r513444);
        double r513449 = r513447 - r513448;
        return r513449;
}
↓
double f(double x) {
        double r513450 = 1.0;
        double r513451 = sqrt(r513450);
        double r513452 = 1.0;
        double r513453 = r513451 / r513452;
        double r513454 = x;
        double r513455 = r513454 + r513450;
        double r513456 = sqrt(r513455);
        double r513457 = sqrt(r513454);
        double r513458 = r513456 + r513457;
        double r513459 = r513458 / r513451;
        double r513460 = r513453 / r513459;
        return r513460;
}

Error

Bits error versus x

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Results

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Target

Original30.4
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.4

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

  3. Applied flip--30.2

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

  4. Simplified0.2

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

  6. Applied add-sqr-sqrt0.4

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

  7. Applied associate-/r*0.3

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

  9. Applied *-un-lft-identity0.3

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

  10. Applied add-sqr-sqrt0.3

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

  11. Applied times-frac0.3

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

  12. Applied associate-/l*0.4

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

  13. Simplified0.2

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

  14. Final simplification0.2

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

Reproduce

herbie shell --seed 2020018 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

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

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