Average Error: 30.0 → 0.3
Time: 13.2s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{x} + \sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\sqrt{x} + \sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}}}
double f(double x) {
        double r351649 = x;
        double r351650 = 1.0;
        double r351651 = r351649 + r351650;
        double r351652 = sqrt(r351651);
        double r351653 = sqrt(r351649);
        double r351654 = r351652 - r351653;
        return r351654;
}
↓
double f(double x) {
        double r351655 = 1.0;
        double r351656 = x;
        double r351657 = sqrt(r351656);
        double r351658 = r351656 + r351655;
        double r351659 = sqrt(r351658);
        double r351660 = sqrt(r351659);
        double r351661 = r351660 * r351660;
        double r351662 = r351657 + r351661;
        double r351663 = r351655 / r351662;
        return r351663;
}

Error

Bits error versus x

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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--29.8

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

  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-sqrt0.2

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

  8. Applied sqrt-prod0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019235 +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)))