Average Error: 29.6 → 0.2
Time: 19.6s
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 r319733 = x;
        double r319734 = 1.0;
        double r319735 = r319733 + r319734;
        double r319736 = sqrt(r319735);
        double r319737 = sqrt(r319733);
        double r319738 = r319736 - r319737;
        return r319738;
}
↓
double f(double x) {
        double r319739 = 1.0;
        double r319740 = x;
        double r319741 = sqrt(r319740);
        double r319742 = r319740 + r319739;
        double r319743 = sqrt(r319742);
        double r319744 = r319741 + r319743;
        double r319745 = r319739 / r319744;
        double r319746 = sqrt(r319745);
        double r319747 = sqrt(r319739);
        double r319748 = r319746 * r319747;
        double r319749 = sqrt(r319744);
        double r319750 = r319748 / r319749;
        return r319750;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Target

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

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. Simplified0.2

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

  5. Simplified0.2

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

  7. Applied add-sqr-sqrt0.3

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

  9. Applied sqrt-div0.3

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

  10. Applied associate-*r/0.2

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

  11. 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 2019303 +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)))