Average Error: 19.7 → 0.4
Time: 1.6m
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{\sqrt{x + 1}}}{\sqrt{x} \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{\sqrt{x + 1}}}{\sqrt{x} \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}
double f(double x) {
        double r13871371 = 1.0;
        double r13871372 = x;
        double r13871373 = sqrt(r13871372);
        double r13871374 = r13871371 / r13871373;
        double r13871375 = r13871372 + r13871371;
        double r13871376 = sqrt(r13871375);
        double r13871377 = r13871371 / r13871376;
        double r13871378 = r13871374 - r13871377;
        return r13871378;
}
↓
double f(double x) {
        double r13871379 = 1.0;
        double r13871380 = x;
        double r13871381 = r13871380 + r13871379;
        double r13871382 = sqrt(r13871381);
        double r13871383 = r13871379 / r13871382;
        double r13871384 = sqrt(r13871380);
        double r13871385 = r13871382 + r13871384;
        double r13871386 = r13871384 * r13871385;
        double r13871387 = r13871383 / r13871386;
        return r13871387;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.7
Target0.6
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.7

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

  3. Applied frac-sub19.7

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

  4. Simplified19.7

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

  6. Applied flip--19.5

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

  7. Simplified0.4

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

  9. Applied div-inv0.4

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

  10. Applied times-frac0.4

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

  12. Applied *-un-lft-identity0.4

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

  13. Applied associate-*l*0.4

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2019128 
(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)))))