Average Error: 30.0 → 0.3
Time: 12.9s
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 r142677 = x;
        double r142678 = 1.0;
        double r142679 = r142677 + r142678;
        double r142680 = sqrt(r142679);
        double r142681 = sqrt(r142677);
        double r142682 = r142680 - r142681;
        return r142682;
}
↓
double f(double x) {
        double r142683 = 1.0;
        double r142684 = x;
        double r142685 = r142684 + r142683;
        double r142686 = sqrt(r142685);
        double r142687 = sqrt(r142686);
        double r142688 = r142687 * r142687;
        double r142689 = sqrt(r142684);
        double r142690 = r142688 + r142689;
        double r142691 = r142683 / r142690;
        return r142691;
}

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. Simplified30.0

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

  4. Applied flip--29.9

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

  5. Simplified0.2

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

  7. Applied add-sqr-sqrt0.2

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

  8. Applied sqrt-prod0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

  (- (sqrt (+ x 1.0)) (sqrt x)))