Average Error: 20.1 → 0.4
Time: 17.0s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{1 \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}}{\left(\sqrt{x} \cdot \sqrt{\sqrt{x + 1}}\right) \cdot \sqrt{\sqrt{x + 1}}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{1 \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}}{\left(\sqrt{x} \cdot \sqrt{\sqrt{x + 1}}\right) \cdot \sqrt{\sqrt{x + 1}}}
double f(double x) {
        double r176239 = 1.0;
        double r176240 = x;
        double r176241 = sqrt(r176240);
        double r176242 = r176239 / r176241;
        double r176243 = r176240 + r176239;
        double r176244 = sqrt(r176243);
        double r176245 = r176239 / r176244;
        double r176246 = r176242 - r176245;
        return r176246;
}
↓
double f(double x) {
        double r176247 = 1.0;
        double r176248 = x;
        double r176249 = r176248 + r176247;
        double r176250 = sqrt(r176249);
        double r176251 = sqrt(r176248);
        double r176252 = r176250 + r176251;
        double r176253 = r176247 * r176252;
        double r176254 = r176247 / r176253;
        double r176255 = sqrt(r176250);
        double r176256 = r176251 * r176255;
        double r176257 = r176256 * r176255;
        double r176258 = r176254 / r176257;
        return r176258;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.1
Target0.7
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 20.1

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

  3. Applied frac-sub20.1

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

  5. Applied flip--19.8

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

  6. Simplified19.5

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

  7. Simplified19.5

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

  8. Taylor expanded around 0 0.4

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

  10. Applied add-sqr-sqrt0.4

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

  11. Applied sqrt-prod0.4

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

  12. Applied associate-*r*0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(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)))))