Main:bigenough3 from C

Average Error: 29.5 → 0.3

Time: 6.1s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r494341 = x;
        double r494342 = 1.0;
        double r494343 = r494341 + r494342;
        double r494344 = sqrt(r494343);
        double r494345 = sqrt(r494341);
        double r494346 = r494344 - r494345;
        return r494346;
}

↓

double f(double x) {
        double r494347 = 1.0;
        double r494348 = 0.0;
        double r494349 = r494347 + r494348;
        double r494350 = x;
        double r494351 = r494350 + r494347;
        double r494352 = sqrt(r494351);
        double r494353 = sqrt(r494350);
        double r494354 = r494352 + r494353;
        double r494355 = r494349 / r494354;
        double r494356 = sqrt(r494355);
        double r494357 = r494356 * r494356;
        return r494357;
}

Error

Bits error versus x

Target

Original	29.5
Target	0.2
Herbie	0.3

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

Derivation

1. Initial program 29.5

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

3. Applied flip-- 29.2

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

4. Simplified 0.2

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

6. Applied add-sqr-sqrt 0.3

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

7. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

  :herbie-target
  (/ 1 (+ (sqrt (+ x 1)) (sqrt x)))

  (- (sqrt (+ x 1)) (sqrt x)))