Average Error: 19.6 → 0.5
Time: 6.5s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1 \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\left(\left|\sqrt[3]{x + 1}\right| \cdot \sqrt{x}\right) \cdot \sqrt{\sqrt[3]{x + 1}}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1 \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\left(\left|\sqrt[3]{x + 1}\right| \cdot \sqrt{x}\right) \cdot \sqrt{\sqrt[3]{x + 1}}}
double f(double x) {
        double r169702 = 1.0;
        double r169703 = x;
        double r169704 = sqrt(r169703);
        double r169705 = r169702 / r169704;
        double r169706 = r169703 + r169702;
        double r169707 = sqrt(r169706);
        double r169708 = r169702 / r169707;
        double r169709 = r169705 - r169708;
        return r169709;
}
↓
double f(double x) {
        double r169710 = 1.0;
        double r169711 = x;
        double r169712 = r169711 + r169710;
        double r169713 = sqrt(r169712);
        double r169714 = sqrt(r169711);
        double r169715 = r169713 + r169714;
        double r169716 = r169710 / r169715;
        double r169717 = r169710 * r169716;
        double r169718 = cbrt(r169712);
        double r169719 = fabs(r169718);
        double r169720 = r169719 * r169714;
        double r169721 = sqrt(r169718);
        double r169722 = r169720 * r169721;
        double r169723 = r169717 / r169722;
        return r169723;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.6
Target0.7
Herbie0.5
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

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. Simplified19.1

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

  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 add-cube-cbrt0.5

    \[\leadsto \frac{1 \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x} \cdot \sqrt{\color{blue}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}}}\]

  11. Applied sqrt-prod0.5

    \[\leadsto \frac{1 \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x} \cdot \color{blue}{\left(\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt{\sqrt[3]{x + 1}}\right)}}\]

  12. Applied associate-*r*0.5

    \[\leadsto \frac{1 \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\color{blue}{\left(\sqrt{x} \cdot \sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right) \cdot \sqrt{\sqrt[3]{x + 1}}}}\]

  13. Simplified0.5

    \[\leadsto \frac{1 \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\color{blue}{\left(\left|\sqrt[3]{x + 1}\right| \cdot \sqrt{x}\right)} \cdot \sqrt{\sqrt[3]{x + 1}}}\]

  14. Final simplification0.5

    \[\leadsto \frac{1 \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\left(\left|\sqrt[3]{x + 1}\right| \cdot \sqrt{x}\right) \cdot \sqrt{\sqrt[3]{x + 1}}}\]

Reproduce

herbie shell --seed 2020036 
(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)))))