Average Error: 29.7 → 0.2
Time: 13.6s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{\sqrt{\frac{1}{\sqrt{x} + \sqrt{x + 1}}} \cdot \sqrt{1}}{\sqrt{\sqrt{x} + \sqrt{x + 1}}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{\sqrt{\frac{1}{\sqrt{x} + \sqrt{x + 1}}} \cdot \sqrt{1}}{\sqrt{\sqrt{x} + \sqrt{x + 1}}}
double f(double x) {
        double r377781 = x;
        double r377782 = 1.0;
        double r377783 = r377781 + r377782;
        double r377784 = sqrt(r377783);
        double r377785 = sqrt(r377781);
        double r377786 = r377784 - r377785;
        return r377786;
}
↓
double f(double x) {
        double r377787 = 1.0;
        double r377788 = x;
        double r377789 = sqrt(r377788);
        double r377790 = r377788 + r377787;
        double r377791 = sqrt(r377790);
        double r377792 = r377789 + r377791;
        double r377793 = r377787 / r377792;
        double r377794 = sqrt(r377793);
        double r377795 = sqrt(r377787);
        double r377796 = r377794 * r377795;
        double r377797 = sqrt(r377792);
        double r377798 = r377796 / r377797;
        return r377798;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.7
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.7

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

  3. Applied flip--29.5

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

  4. Simplified0.2

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

  5. Simplified0.2

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

  7. Applied add-sqr-sqrt0.3

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

  8. Simplified0.3

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

  9. Simplified0.3

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

  11. Applied sqrt-div0.3

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

  12. Applied associate-*r/0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

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

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