Average Error: 30.0 → 0.3
Time: 11.1s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}} + \sqrt{x}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}} + \sqrt{x}}
double f(double x) {
        double r412803 = x;
        double r412804 = 1.0;
        double r412805 = r412803 + r412804;
        double r412806 = sqrt(r412805);
        double r412807 = sqrt(r412803);
        double r412808 = r412806 - r412807;
        return r412808;
}
↓
double f(double x) {
        double r412809 = 1.0;
        double r412810 = x;
        double r412811 = r412810 + r412809;
        double r412812 = sqrt(r412811);
        double r412813 = sqrt(r412812);
        double r412814 = r412813 * r412813;
        double r412815 = sqrt(r412810);
        double r412816 = r412814 + r412815;
        double r412817 = r412809 / r412816;
        return r412817;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.0
Target0.2
Herbie0.3
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.0

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

  3. Applied flip--29.7

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

  4. Simplified0.2

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

  6. Applied add-sqr-sqrt0.2

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

  7. Applied sqrt-prod0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2020045 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

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

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