Average Error: 29.6 → 0.3
Time: 9.9s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\left(\sqrt{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}} \cdot \sqrt{\sqrt{\sqrt[3]{x + 1}}}\right) \cdot \sqrt{\sqrt{x + 1}} + \sqrt{x}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\left(\sqrt{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}} \cdot \sqrt{\sqrt{\sqrt[3]{x + 1}}}\right) \cdot \sqrt{\sqrt{x + 1}} + \sqrt{x}}
double f(double x) {
        double r1361874 = x;
        double r1361875 = 1.0;
        double r1361876 = r1361874 + r1361875;
        double r1361877 = sqrt(r1361876);
        double r1361878 = sqrt(r1361874);
        double r1361879 = r1361877 - r1361878;
        return r1361879;
}
↓
double f(double x) {
        double r1361880 = 1.0;
        double r1361881 = x;
        double r1361882 = r1361881 + r1361880;
        double r1361883 = cbrt(r1361882);
        double r1361884 = r1361883 * r1361883;
        double r1361885 = sqrt(r1361884);
        double r1361886 = sqrt(r1361885);
        double r1361887 = sqrt(r1361883);
        double r1361888 = sqrt(r1361887);
        double r1361889 = r1361886 * r1361888;
        double r1361890 = sqrt(r1361882);
        double r1361891 = sqrt(r1361890);
        double r1361892 = r1361889 * r1361891;
        double r1361893 = sqrt(r1361881);
        double r1361894 = r1361892 + r1361893;
        double r1361895 = r1361880 / r1361894;
        return r1361895;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.6
Target0.2
Herbie0.3
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.6

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

  3. Applied flip--29.4

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

  4. Simplified28.9

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

  5. Taylor expanded around 0 0.2

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

  7. Applied add-sqr-sqrt0.2

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

  8. Applied sqrt-prod0.3

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

  10. Applied add-cube-cbrt0.3

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

  11. Applied sqrt-prod0.3

    \[\leadsto \frac{1}{\sqrt{\sqrt{x + 1}} \cdot \sqrt{\color{blue}{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt{\sqrt[3]{x + 1}}}} + \sqrt{x}}\]

  12. Applied sqrt-prod0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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