Average Error: 19.3 → 0.4
Time: 12.8s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{x + 1}}{\left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right) \cdot x}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{x + 1}}{\left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right) \cdot x}
double f(double x) {
        double r1944160 = 1.0;
        double r1944161 = x;
        double r1944162 = sqrt(r1944161);
        double r1944163 = r1944160 / r1944162;
        double r1944164 = r1944161 + r1944160;
        double r1944165 = sqrt(r1944164);
        double r1944166 = r1944160 / r1944165;
        double r1944167 = r1944163 - r1944166;
        return r1944167;
}
↓
double f(double x) {
        double r1944168 = 1.0;
        double r1944169 = x;
        double r1944170 = r1944169 + r1944168;
        double r1944171 = r1944168 / r1944170;
        double r1944172 = sqrt(r1944169);
        double r1944173 = r1944168 / r1944172;
        double r1944174 = sqrt(r1944170);
        double r1944175 = r1944168 / r1944174;
        double r1944176 = r1944173 + r1944175;
        double r1944177 = r1944176 * r1944169;
        double r1944178 = r1944171 / r1944177;
        return r1944178;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.3
Target0.6
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.3

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

  3. Applied flip--19.3

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

  5. Applied frac-times24.6

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

  6. Applied frac-times19.4

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

  7. Applied frac-sub19.2

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

  8. Simplified18.8

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

  9. Simplified18.7

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

  10. Taylor expanded around 0 5.5

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

  12. Applied *-un-lft-identity5.5

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

  13. Applied times-frac5.2

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

  14. Applied associate-/l*0.5

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

  15. Simplified0.4

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

  16. Final simplification0.4

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

Reproduce

herbie shell --seed 2019156 
(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)))))