Average Error: 19.7 → 0.4
Time: 16.0s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{x + 1}} \cdot \frac{\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{x}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{x + 1}} \cdot \frac{\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{x}}
double f(double x) {
        double r3848575 = 1.0;
        double r3848576 = x;
        double r3848577 = sqrt(r3848576);
        double r3848578 = r3848575 / r3848577;
        double r3848579 = r3848576 + r3848575;
        double r3848580 = sqrt(r3848579);
        double r3848581 = r3848575 / r3848580;
        double r3848582 = r3848578 - r3848581;
        return r3848582;
}
↓
double f(double x) {
        double r3848583 = 1.0;
        double r3848584 = x;
        double r3848585 = r3848584 + r3848583;
        double r3848586 = sqrt(r3848585);
        double r3848587 = sqrt(r3848584);
        double r3848588 = r3848586 + r3848587;
        double r3848589 = r3848583 / r3848588;
        double r3848590 = sqrt(r3848589);
        double r3848591 = r3848590 / r3848586;
        double r3848592 = r3848590 / r3848587;
        double r3848593 = r3848591 * r3848592;
        return r3848593;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Target

Original19.7
Target0.6
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.7

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

  3. Applied frac-sub19.7

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

  4. Simplified19.7

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

  6. Applied flip--19.4

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

  7. Simplified18.9

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

  8. Taylor expanded around 0 0.4

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

  10. Applied add-sqr-sqrt0.4

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

  11. Applied times-frac0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019163 
(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)))))