Average Error: 19.8 → 0.5
Time: 5.4s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1 \cdot \left(\frac{\sqrt{1}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}} \cdot \frac{\sqrt{1}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}\right)}{\sqrt{x} \cdot \sqrt{x + 1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1 \cdot \left(\frac{\sqrt{1}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}} \cdot \frac{\sqrt{1}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}\right)}{\sqrt{x} \cdot \sqrt{x + 1}}
double f(double x) {
        double r143152 = 1.0;
        double r143153 = x;
        double r143154 = sqrt(r143153);
        double r143155 = r143152 / r143154;
        double r143156 = r143153 + r143152;
        double r143157 = sqrt(r143156);
        double r143158 = r143152 / r143157;
        double r143159 = r143155 - r143158;
        return r143159;
}
↓
double f(double x) {
        double r143160 = 1.0;
        double r143161 = sqrt(r143160);
        double r143162 = x;
        double r143163 = r143162 + r143160;
        double r143164 = sqrt(r143163);
        double r143165 = sqrt(r143162);
        double r143166 = r143164 + r143165;
        double r143167 = sqrt(r143166);
        double r143168 = r143161 / r143167;
        double r143169 = r143168 * r143168;
        double r143170 = r143160 * r143169;
        double r143171 = r143165 * r143164;
        double r143172 = r143170 / r143171;
        return r143172;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.8
Target0.7
Herbie0.5
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.8

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

  3. Applied frac-sub19.8

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

  4. Simplified19.8

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

  6. Applied flip--19.6

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

  7. Simplified19.2

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

  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-sqrt0.5

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

  11. Applied add-sqr-sqrt0.5

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

  12. Applied times-frac0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2020064 
(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)))))