Average Error: 19.7 → 0.4
Time: 2.5m
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{\sqrt{x}}}{\frac{\sqrt{x + 1} + \sqrt{x}}{\frac{1}{\sqrt{x + 1}}}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{\sqrt{x}}}{\frac{\sqrt{x + 1} + \sqrt{x}}{\frac{1}{\sqrt{x + 1}}}}
double f(double x) {
        double r12272539 = 1.0;
        double r12272540 = x;
        double r12272541 = sqrt(r12272540);
        double r12272542 = r12272539 / r12272541;
        double r12272543 = r12272540 + r12272539;
        double r12272544 = sqrt(r12272543);
        double r12272545 = r12272539 / r12272544;
        double r12272546 = r12272542 - r12272545;
        return r12272546;
}
↓
double f(double x) {
        double r12272547 = 1.0;
        double r12272548 = x;
        double r12272549 = sqrt(r12272548);
        double r12272550 = r12272547 / r12272549;
        double r12272551 = r12272548 + r12272547;
        double r12272552 = sqrt(r12272551);
        double r12272553 = r12272552 + r12272549;
        double r12272554 = r12272547 / r12272552;
        double r12272555 = r12272553 / r12272554;
        double r12272556 = r12272550 / r12272555;
        return r12272556;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.7
Target0.7
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

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

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

  6. Applied flip--19.5

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

  7. Applied associate-/l/19.5

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

  8. Simplified0.8

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

  10. Applied associate-/r*0.4

    \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}}{\sqrt{x + 1} + \sqrt{x}}}\]
  11. Using strategy rm

  12. Applied *-un-lft-identity0.4

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

  13. Applied times-frac0.4

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

  14. Applied associate-/l*0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (x)
  :name "2isqrt (example 3.6)"

  :herbie-target
  (/ 1 (+ (* (+ x 1) (sqrt x)) (* x (sqrt (+ x 1)))))

  (- (/ 1 (sqrt x)) (/ 1 (sqrt (+ x 1)))))