Average Error: 20.1 → 0.4
Time: 20.2s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1 \cdot \frac{1}{\sqrt{x} + \sqrt{x + 1}}}{\sqrt{x + 1} \cdot \sqrt{x}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1 \cdot \frac{1}{\sqrt{x} + \sqrt{x + 1}}}{\sqrt{x + 1} \cdot \sqrt{x}}
double f(double x) {
        double r6078941 = 1.0;
        double r6078942 = x;
        double r6078943 = sqrt(r6078942);
        double r6078944 = r6078941 / r6078943;
        double r6078945 = r6078942 + r6078941;
        double r6078946 = sqrt(r6078945);
        double r6078947 = r6078941 / r6078946;
        double r6078948 = r6078944 - r6078947;
        return r6078948;
}
↓
double f(double x) {
        double r6078949 = 1.0;
        double r6078950 = x;
        double r6078951 = sqrt(r6078950);
        double r6078952 = r6078950 + r6078949;
        double r6078953 = sqrt(r6078952);
        double r6078954 = r6078951 + r6078953;
        double r6078955 = r6078949 / r6078954;
        double r6078956 = r6078949 * r6078955;
        double r6078957 = r6078953 * r6078951;
        double r6078958 = r6078956 / r6078957;
        return r6078958;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.1
Target0.6
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 20.1

    \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
  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--19.9

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

  7. Simplified19.4

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

  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 +-commutative0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019192 
(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)))))