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 r21083211 = x;
        double r21083212 = 1.0;
        double r21083213 = r21083211 + r21083212;
        double r21083214 = sqrt(r21083213);
        double r21083215 = sqrt(r21083211);
        double r21083216 = r21083214 - r21083215;
        return r21083216;
}
↓
double f(double x) {
        double r21083217 = 1.0;
        double r21083218 = x;
        double r21083219 = r21083218 + r21083217;
        double r21083220 = sqrt(r21083219);
        double r21083221 = sqrt(r21083218);
        double r21083222 = r21083220 + r21083221;
        double r21083223 = r21083217 / r21083222;
        return r21083223;
}

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. Using strategy rm

  3. Applied flip--29.9

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

  4. Simplified29.4

    \[\leadsto \frac{\color{blue}{\left(1 + x\right) - x}}{\sqrt{x + 1} + \sqrt{x}}\]

  5. Simplified29.4

    \[\leadsto \frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}}\]

  6. Taylor expanded around 0 0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (x)
  :name "Main:bigenough3 from C"

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

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