Average Error: 19.8 → 0.4
Time: 36.6s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{\frac{1}{\sqrt{x + 1} \cdot \sqrt{x}}}{\sqrt{\sqrt{x} + \sqrt{x + 1}}}}{\sqrt{\sqrt{x} + \sqrt{x + 1}}}\]
double f(double x) {
        double r6635496 = 1.0;
        double r6635497 = x;
        double r6635498 = sqrt(r6635497);
        double r6635499 = r6635496 / r6635498;
        double r6635500 = r6635497 + r6635496;
        double r6635501 = sqrt(r6635500);
        double r6635502 = r6635496 / r6635501;
        double r6635503 = r6635499 - r6635502;
        return r6635503;
}
↓
double f(double x) {
        double r6635504 = 1.0;
        double r6635505 = x;
        double r6635506 = r6635505 + r6635504;
        double r6635507 = sqrt(r6635506);
        double r6635508 = sqrt(r6635505);
        double r6635509 = r6635507 * r6635508;
        double r6635510 = r6635504 / r6635509;
        double r6635511 = r6635508 + r6635507;
        double r6635512 = sqrt(r6635511);
        double r6635513 = r6635510 / r6635512;
        double r6635514 = r6635513 / r6635512;
        return r6635514;
}


\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{\frac{1}{\sqrt{x + 1} \cdot \sqrt{x}}}{\sqrt{\sqrt{x} + \sqrt{x + 1}}}}{\sqrt{\sqrt{x} + \sqrt{x + 1}}}

Error

Bits error versus x

Target

Original19.8
Target0.7
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.8

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

  3. Applied frac-sub19.8

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

  4. Simplified19.8

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

  6. Applied flip--19.6

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

  7. Applied associate-/l/19.6

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

  8. Simplified0.8

    \[\leadsto \frac{\color{blue}{1}}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}\]
  9. Using strategy rm

  10. Applied associate-/r*0.4

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

  12. Applied add-sqr-sqrt0.4

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

  13. Applied associate-/r*0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019102 
(FPCore (x)
  :name "2isqrt (example 3.6)"

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

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