Average Error: 19.6 → 0.4
Time: 16.2s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1 \cdot \frac{1 + \left(x - x\right)}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + 1} \cdot \sqrt{x}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1 \cdot \frac{1 + \left(x - x\right)}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + 1} \cdot \sqrt{x}}
double f(double x) {
        double r92475 = 1.0;
        double r92476 = x;
        double r92477 = sqrt(r92476);
        double r92478 = r92475 / r92477;
        double r92479 = r92476 + r92475;
        double r92480 = sqrt(r92479);
        double r92481 = r92475 / r92480;
        double r92482 = r92478 - r92481;
        return r92482;
}
↓
double f(double x) {
        double r92483 = 1.0;
        double r92484 = x;
        double r92485 = r92484 - r92484;
        double r92486 = r92483 + r92485;
        double r92487 = r92484 + r92483;
        double r92488 = sqrt(r92487);
        double r92489 = sqrt(r92484);
        double r92490 = r92488 + r92489;
        double r92491 = r92486 / r92490;
        double r92492 = r92483 * r92491;
        double r92493 = r92488 * r92489;
        double r92494 = r92492 / r92493;
        return r92494;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 19.6

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

  3. Applied frac-sub19.6

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

  4. Simplified19.6

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

  5. Simplified19.6

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

  7. Applied flip--19.5

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019195 
(FPCore (x)
  :name "2isqrt (example 3.6)"

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

  (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))