Average Error: 19.5 → 0.5
Time: 5.9s
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 r110029 = 1.0;
        double r110030 = x;
        double r110031 = sqrt(r110030);
        double r110032 = r110029 / r110031;
        double r110033 = r110030 + r110029;
        double r110034 = sqrt(r110033);
        double r110035 = r110029 / r110034;
        double r110036 = r110032 - r110035;
        return r110036;
}
↓
double f(double x) {
        double r110037 = 1.0;
        double r110038 = x;
        double r110039 = r110038 + r110037;
        double r110040 = sqrt(r110039);
        double r110041 = sqrt(r110038);
        double r110042 = r110040 + r110041;
        double r110043 = r110037 / r110042;
        double r110044 = r110037 * r110043;
        double r110045 = cbrt(r110039);
        double r110046 = fabs(r110045);
        double r110047 = r110046 * r110041;
        double r110048 = sqrt(r110045);
        double r110049 = r110047 * r110048;
        double r110050 = r110044 / r110049;
        return r110050;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 19.5

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

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

  6. Applied flip--19.3

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

  7. Simplified18.9

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

  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 2020089 
(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)))))