Average Error: 19.7 → 0.3
Time: 10.5s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1 \cdot 1}{x + \sqrt{x} \cdot \sqrt{x + 1}}}{\sqrt{x + 1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1 \cdot 1}{x + \sqrt{x} \cdot \sqrt{x + 1}}}{\sqrt{x + 1}}
double f(double x) {
        double r160952 = 1.0;
        double r160953 = x;
        double r160954 = sqrt(r160953);
        double r160955 = r160952 / r160954;
        double r160956 = r160953 + r160952;
        double r160957 = sqrt(r160956);
        double r160958 = r160952 / r160957;
        double r160959 = r160955 - r160958;
        return r160959;
}
↓
double f(double x) {
        double r160960 = 1.0;
        double r160961 = r160960 * r160960;
        double r160962 = x;
        double r160963 = sqrt(r160962);
        double r160964 = r160962 + r160960;
        double r160965 = sqrt(r160964);
        double r160966 = r160963 * r160965;
        double r160967 = r160962 + r160966;
        double r160968 = r160961 / r160967;
        double r160969 = r160968 / r160965;
        return r160969;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.7
Target0.7
Herbie0.3
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

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

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

  6. Applied flip--19.4

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

  7. Simplified19.0

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

  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 associate-/r*0.4

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2020043 
(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)))))