Average Error: 29.6 → 0.3
Time: 9.4s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{x} + \left(\sqrt{\sqrt{\sqrt[3]{x + 1}}} \cdot \sqrt{\left|\sqrt[3]{x + 1}\right|}\right) \cdot \sqrt{\sqrt{x + 1}}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\sqrt{x} + \left(\sqrt{\sqrt{\sqrt[3]{x + 1}}} \cdot \sqrt{\left|\sqrt[3]{x + 1}\right|}\right) \cdot \sqrt{\sqrt{x + 1}}}
double f(double x) {
        double r2378721 = x;
        double r2378722 = 1.0;
        double r2378723 = r2378721 + r2378722;
        double r2378724 = sqrt(r2378723);
        double r2378725 = sqrt(r2378721);
        double r2378726 = r2378724 - r2378725;
        return r2378726;
}
↓
double f(double x) {
        double r2378727 = 1.0;
        double r2378728 = x;
        double r2378729 = sqrt(r2378728);
        double r2378730 = r2378728 + r2378727;
        double r2378731 = cbrt(r2378730);
        double r2378732 = sqrt(r2378731);
        double r2378733 = sqrt(r2378732);
        double r2378734 = fabs(r2378731);
        double r2378735 = sqrt(r2378734);
        double r2378736 = r2378733 * r2378735;
        double r2378737 = sqrt(r2378730);
        double r2378738 = sqrt(r2378737);
        double r2378739 = r2378736 * r2378738;
        double r2378740 = r2378729 + r2378739;
        double r2378741 = r2378727 / r2378740;
        return r2378741;
}

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

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

  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

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

  11. Applied sqrt-prod0.3

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

  12. Applied sqrt-prod0.3

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

  13. Simplified0.3

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

  14. Final simplification0.3

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

Reproduce

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

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

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