Average Error: 19.6 → 0.4
Time: 32.4s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{\sqrt{\sqrt{x + 1}} \cdot \left(\sqrt{x} \cdot \sqrt{\sqrt{x + 1}}\right)}}{\sqrt{x} + \sqrt{x + 1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{\sqrt{\sqrt{x + 1}} \cdot \left(\sqrt{x} \cdot \sqrt{\sqrt{x + 1}}\right)}}{\sqrt{x} + \sqrt{x + 1}}
double f(double x) {
        double r6589201 = 1.0;
        double r6589202 = x;
        double r6589203 = sqrt(r6589202);
        double r6589204 = r6589201 / r6589203;
        double r6589205 = r6589202 + r6589201;
        double r6589206 = sqrt(r6589205);
        double r6589207 = r6589201 / r6589206;
        double r6589208 = r6589204 - r6589207;
        return r6589208;
}
↓
double f(double x) {
        double r6589209 = 1.0;
        double r6589210 = x;
        double r6589211 = r6589210 + r6589209;
        double r6589212 = sqrt(r6589211);
        double r6589213 = sqrt(r6589212);
        double r6589214 = sqrt(r6589210);
        double r6589215 = r6589214 * r6589213;
        double r6589216 = r6589213 * r6589215;
        double r6589217 = r6589209 / r6589216;
        double r6589218 = r6589214 + r6589212;
        double r6589219 = r6589217 / r6589218;
        return r6589219;
}

Error

Bits error versus x

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Your Program's Arguments

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

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

  6. Applied flip--19.4

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

  7. Applied associate-/l/19.4

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

  8. Simplified0.8

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

  10. Applied associate-/r*0.4

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

  12. Applied add-sqr-sqrt0.4

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

  13. Applied associate-*r*0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019119 
(FPCore (x)
  :name "2isqrt (example 3.6)"

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

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