Average Error: 19.6 → 0.2
Time: 13.8s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{1 + x}} \cdot {x}^{\frac{-1}{2}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{1 + x}} \cdot {x}^{\frac{-1}{2}}
double f(double x) {
        double r1460709 = 1.0;
        double r1460710 = x;
        double r1460711 = sqrt(r1460710);
        double r1460712 = r1460709 / r1460711;
        double r1460713 = r1460710 + r1460709;
        double r1460714 = sqrt(r1460713);
        double r1460715 = r1460709 / r1460714;
        double r1460716 = r1460712 - r1460715;
        return r1460716;
}
↓
double f(double x) {
        double r1460717 = 1.0;
        double r1460718 = x;
        double r1460719 = r1460717 + r1460718;
        double r1460720 = sqrt(r1460719);
        double r1460721 = sqrt(r1460718);
        double r1460722 = r1460720 + r1460721;
        double r1460723 = r1460717 / r1460722;
        double r1460724 = r1460723 / r1460720;
        double r1460725 = -0.5;
        double r1460726 = pow(r1460718, r1460725);
        double r1460727 = r1460724 * r1460726;
        return r1460727;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.6
Target0.7
Herbie0.2
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.6

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

  3. Applied frac-sub19.5

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

  4. Simplified19.5

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

  6. Applied flip--19.3

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

  7. Simplified18.9

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

  9. Applied *-un-lft-identity18.9

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

  10. Applied *-un-lft-identity18.9

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

  11. Applied times-frac18.9

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

  12. Applied times-frac18.9

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

  13. Simplified18.9

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

  14. Simplified0.4

    \[\leadsto \frac{1}{\sqrt{x}} \cdot \color{blue}{\frac{\frac{1 + 0}{\sqrt{x} + \sqrt{x + 1}}}{\sqrt{x + 1}}}\]
  15. Using strategy rm

  16. Applied pow1/20.4

    \[\leadsto \frac{1}{\color{blue}{{x}^{\frac{1}{2}}}} \cdot \frac{\frac{1 + 0}{\sqrt{x} + \sqrt{x + 1}}}{\sqrt{x + 1}}\]

  17. Applied pow-flip0.2

    \[\leadsto \color{blue}{{x}^{\left(-\frac{1}{2}\right)}} \cdot \frac{\frac{1 + 0}{\sqrt{x} + \sqrt{x + 1}}}{\sqrt{x + 1}}\]

  18. Simplified0.2

    \[\leadsto {x}^{\color{blue}{\frac{-1}{2}}} \cdot \frac{\frac{1 + 0}{\sqrt{x} + \sqrt{x + 1}}}{\sqrt{x + 1}}\]

  19. Final simplification0.2

    \[\leadsto \frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{1 + x}} \cdot {x}^{\frac{-1}{2}}\]

Reproduce

herbie shell --seed 2019153 
(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)))))