Average Error: 20.0 → 0.4
Time: 5.9s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1 \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\left(\sqrt{x} \cdot \sqrt{\sqrt{x + 1}}\right) \cdot \sqrt{\sqrt{x + 1}}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1 \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\left(\sqrt{x} \cdot \sqrt{\sqrt{x + 1}}\right) \cdot \sqrt{\sqrt{x + 1}}}
double f(double x) {
        double r125208 = 1.0;
        double r125209 = x;
        double r125210 = sqrt(r125209);
        double r125211 = r125208 / r125210;
        double r125212 = r125209 + r125208;
        double r125213 = sqrt(r125212);
        double r125214 = r125208 / r125213;
        double r125215 = r125211 - r125214;
        return r125215;
}
↓
double f(double x) {
        double r125216 = 1.0;
        double r125217 = x;
        double r125218 = r125217 + r125216;
        double r125219 = sqrt(r125218);
        double r125220 = sqrt(r125217);
        double r125221 = r125219 + r125220;
        double r125222 = r125216 / r125221;
        double r125223 = r125216 * r125222;
        double r125224 = sqrt(r125219);
        double r125225 = r125220 * r125224;
        double r125226 = r125225 * r125224;
        double r125227 = r125223 / r125226;
        return r125227;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.0
Target0.6
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 20.0

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

  3. Applied frac-sub20.0

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

  4. Simplified20.0

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

  6. Applied flip--19.8

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

  7. Simplified19.4

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

  8. Taylor expanded around 0 0.4

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

  10. Applied add-sqr-sqrt0.4

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

  11. Applied sqrt-prod0.4

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

  12. Applied associate-*r*0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2020001 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

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

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