Average Error: 20.2 → 0.3
Time: 13.4s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1 \cdot 1}{x + \sqrt{1 + x} \cdot \sqrt{x}}}{\sqrt{1 + x}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1 \cdot 1}{x + \sqrt{1 + x} \cdot \sqrt{x}}}{\sqrt{1 + x}}
double f(double x) {
        double r142666 = 1.0;
        double r142667 = x;
        double r142668 = sqrt(r142667);
        double r142669 = r142666 / r142668;
        double r142670 = r142667 + r142666;
        double r142671 = sqrt(r142670);
        double r142672 = r142666 / r142671;
        double r142673 = r142669 - r142672;
        return r142673;
}
↓
double f(double x) {
        double r142674 = 1.0;
        double r142675 = r142674 * r142674;
        double r142676 = x;
        double r142677 = r142674 + r142676;
        double r142678 = sqrt(r142677);
        double r142679 = sqrt(r142676);
        double r142680 = r142678 * r142679;
        double r142681 = r142676 + r142680;
        double r142682 = r142675 / r142681;
        double r142683 = r142682 / r142678;
        return r142683;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.2
Target0.7
Herbie0.3
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 20.2

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

  3. Applied frac-sub20.2

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

  4. Simplified20.2

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

  5. Simplified20.2

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

  7. Applied flip--20.0

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

  8. Simplified0.4

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

  10. Applied times-frac0.4

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

  11. Simplified0.4

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

  12. Simplified0.3

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

  14. Applied pow10.3

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

  15. Applied pow10.3

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

  16. Applied pow-prod-down0.3

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

  17. Simplified0.3

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

  18. Final simplification0.3

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

Reproduce

herbie shell --seed 2019194 
(FPCore (x)
  :name "2isqrt (example 3.6)"

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

  (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))