Average Error: 30.3 → 0.2
Time: 17.2s
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 r29923936 = x;
        double r29923937 = 1.0;
        double r29923938 = r29923936 + r29923937;
        double r29923939 = sqrt(r29923938);
        double r29923940 = sqrt(r29923936);
        double r29923941 = r29923939 - r29923940;
        return r29923941;
}
↓
double f(double x) {
        double r29923942 = 1.0;
        double r29923943 = x;
        double r29923944 = r29923943 + r29923942;
        double r29923945 = sqrt(r29923944);
        double r29923946 = sqrt(r29923943);
        double r29923947 = r29923945 + r29923946;
        double r29923948 = r29923942 / r29923947;
        return r29923948;
}

Error

Bits error versus x

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Results

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Target

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

Derivation

  1. Initial program 30.3

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

  3. Applied flip--30.1

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019171 
(FPCore (x)
  :name "Main:bigenough3 from C"

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

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