Average Error: 19.6 → 0.3
Time: 5.4s
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 r127565 = 1.0;
        double r127566 = x;
        double r127567 = sqrt(r127566);
        double r127568 = r127565 / r127567;
        double r127569 = r127566 + r127565;
        double r127570 = sqrt(r127569);
        double r127571 = r127565 / r127570;
        double r127572 = r127568 - r127571;
        return r127572;
}
↓
double f(double x) {
        double r127573 = 1.0;
        double r127574 = x;
        double r127575 = sqrt(r127574);
        double r127576 = r127573 / r127575;
        double r127577 = r127574 + r127573;
        double r127578 = sqrt(r127577);
        double r127579 = r127578 * r127575;
        double r127580 = r127579 + r127577;
        double r127581 = r127573 / r127580;
        double r127582 = r127576 * r127581;
        return r127582;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.6
Target0.7
Herbie0.3
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.6

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

  3. Applied frac-sub19.6

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

  4. Simplified19.6

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

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

  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 2020049 
(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)))))