Average Error: 30.1 → 0.2
Time: 15.6s
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 r437323 = x;
        double r437324 = 1.0;
        double r437325 = r437323 + r437324;
        double r437326 = sqrt(r437325);
        double r437327 = sqrt(r437323);
        double r437328 = r437326 - r437327;
        return r437328;
}
↓
double f(double x) {
        double r437329 = 1.0;
        double r437330 = x;
        double r437331 = r437330 + r437329;
        double r437332 = sqrt(r437331);
        double r437333 = sqrt(r437330);
        double r437334 = r437332 + r437333;
        double r437335 = r437329 / r437334;
        return r437335;
}

Error

Bits error versus x

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Results

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Target

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

Derivation

  1. Initial program 30.1

    \[\sqrt{x + 1} - \sqrt{x}\]

  2. Simplified30.1

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

  4. Applied flip--29.8

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

  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 2019174 
(FPCore (x)
  :name "Main:bigenough3 from C"

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

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