Average Error: 20.0 → 0.4
Time: 18.0s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{\sqrt{x + 1} \cdot \sqrt{x}}}{\sqrt{x + 1} \cdot 1 + 1 \cdot \sqrt{x}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{\sqrt{x + 1} \cdot \sqrt{x}}}{\sqrt{x + 1} \cdot 1 + 1 \cdot \sqrt{x}}
double f(double x) {
        double r6984339 = 1.0;
        double r6984340 = x;
        double r6984341 = sqrt(r6984340);
        double r6984342 = r6984339 / r6984341;
        double r6984343 = r6984340 + r6984339;
        double r6984344 = sqrt(r6984343);
        double r6984345 = r6984339 / r6984344;
        double r6984346 = r6984342 - r6984345;
        return r6984346;
}
↓
double f(double x) {
        double r6984347 = 1.0;
        double r6984348 = x;
        double r6984349 = r6984348 + r6984347;
        double r6984350 = sqrt(r6984349);
        double r6984351 = sqrt(r6984348);
        double r6984352 = r6984350 * r6984351;
        double r6984353 = r6984347 / r6984352;
        double r6984354 = r6984350 * r6984347;
        double r6984355 = r6984347 * r6984351;
        double r6984356 = r6984354 + r6984355;
        double r6984357 = r6984353 / r6984356;
        return r6984357;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.0
Target0.7
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

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. Using strategy rm

  5. Applied flip--19.8

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

  6. Applied associate-/l/19.8

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

  7. Taylor expanded around 0 0.8

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

  9. Applied associate-/r*0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019169 
(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)))))