Average Error: 20.0 → 0.3
Time: 15.6s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{\sqrt{x}}}{\frac{\sqrt{x + 1} \cdot \sqrt{x} + \left(x + 1\right)}{1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{\sqrt{x}}}{\frac{\sqrt{x + 1} \cdot \sqrt{x} + \left(x + 1\right)}{1}}
double f(double x) {
        double r124235 = 1.0;
        double r124236 = x;
        double r124237 = sqrt(r124236);
        double r124238 = r124235 / r124237;
        double r124239 = r124236 + r124235;
        double r124240 = sqrt(r124239);
        double r124241 = r124235 / r124240;
        double r124242 = r124238 - r124241;
        return r124242;
}
↓
double f(double x) {
        double r124243 = 1.0;
        double r124244 = x;
        double r124245 = sqrt(r124244);
        double r124246 = r124243 / r124245;
        double r124247 = r124244 + r124243;
        double r124248 = sqrt(r124247);
        double r124249 = r124248 * r124245;
        double r124250 = r124249 + r124247;
        double r124251 = r124250 / r124243;
        double r124252 = r124246 / r124251;
        return r124252;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.0
Target0.7
Herbie0.3
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 20.0

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

  3. Applied frac-sub19.9

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

  4. Simplified19.9

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

  6. Applied flip--19.7

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

  7. Simplified19.3

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

  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 simplification0.3

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

Reproduce

herbie shell --seed 2019298 
(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)))))