Average Error: 20.0 → 0.3
Time: 7.8s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x + 1} \cdot \sqrt{x} + \left(x + 1\right)}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x + 1} \cdot \sqrt{x} + \left(x + 1\right)}
double f(double x) {
        double r169791 = 1.0;
        double r169792 = x;
        double r169793 = sqrt(r169792);
        double r169794 = r169791 / r169793;
        double r169795 = r169792 + r169791;
        double r169796 = sqrt(r169795);
        double r169797 = r169791 / r169796;
        double r169798 = r169794 - r169797;
        return r169798;
}
↓
double f(double x) {
        double r169799 = 1.0;
        double r169800 = x;
        double r169801 = sqrt(r169800);
        double r169802 = r169799 / r169801;
        double r169803 = r169800 + r169799;
        double r169804 = sqrt(r169803);
        double r169805 = r169804 * r169801;
        double r169806 = r169805 + r169803;
        double r169807 = r169799 / r169806;
        double r169808 = r169802 * r169807;
        return r169808;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.0
Target0.6
Herbie0.3
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 20.0

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

  3. Applied frac-sub20.0

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

  4. Simplified20.0

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

  6. Applied flip--19.7

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

  7. Simplified19.3

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

  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 times-frac0.4

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 197574269 
(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)))))