Average Error: 29.7 → 0.2
Time: 18.8s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\sqrt{x + 1} + \sqrt{x}}
double f(double x) {
        double r418961 = x;
        double r418962 = 1.0;
        double r418963 = r418961 + r418962;
        double r418964 = sqrt(r418963);
        double r418965 = sqrt(r418961);
        double r418966 = r418964 - r418965;
        return r418966;
}
↓
double f(double x) {
        double r418967 = 1.0;
        double r418968 = x;
        double r418969 = r418968 + r418967;
        double r418970 = sqrt(r418969);
        double r418971 = sqrt(r418968);
        double r418972 = r418970 + r418971;
        double r418973 = r418967 / r418972;
        return r418973;
}

Error

Bits error versus x

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Results

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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. Simplified29.7

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

  4. Applied flip--29.4

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019196 
(FPCore (x)
  :name "Main:bigenough3 from C"

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

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