Average Error: 19.7 → 0.4
Time: 19.0s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1 \cdot \frac{1}{\sqrt{x} + \sqrt{x + 1}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1 \cdot \frac{1}{\sqrt{x} + \sqrt{x + 1}}}{\sqrt{x} \cdot \sqrt{x + 1}}
double f(double x) {
        double r9746153 = 1.0;
        double r9746154 = x;
        double r9746155 = sqrt(r9746154);
        double r9746156 = r9746153 / r9746155;
        double r9746157 = r9746154 + r9746153;
        double r9746158 = sqrt(r9746157);
        double r9746159 = r9746153 / r9746158;
        double r9746160 = r9746156 - r9746159;
        return r9746160;
}
↓
double f(double x) {
        double r9746161 = 1.0;
        double r9746162 = x;
        double r9746163 = sqrt(r9746162);
        double r9746164 = r9746162 + r9746161;
        double r9746165 = sqrt(r9746164);
        double r9746166 = r9746163 + r9746165;
        double r9746167 = r9746161 / r9746166;
        double r9746168 = r9746161 * r9746167;
        double r9746169 = r9746163 * r9746165;
        double r9746170 = r9746168 / r9746169;
        return r9746170;
}

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

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

  6. Applied flip--19.5

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

  7. Simplified19.1

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

  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 +-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

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

Reproduce

herbie shell --seed 2019174 
(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)))))