Main:bigenough3 from C

Average Error: 29.5 → 0.3

Time: 10.8s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\sqrt{\frac{1 \cdot 1}{\left(\sqrt{x} + \sqrt{x + 1}\right) \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)}}\]

C

double f(double x) {
        double r421918 = x;
        double r421919 = 1.0;
        double r421920 = r421918 + r421919;
        double r421921 = sqrt(r421920);
        double r421922 = sqrt(r421918);
        double r421923 = r421921 - r421922;
        return r421923;
}

↓

double f(double x) {
        double r421924 = 1.0;
        double r421925 = r421924 * r421924;
        double r421926 = x;
        double r421927 = sqrt(r421926);
        double r421928 = r421926 + r421924;
        double r421929 = sqrt(r421928);
        double r421930 = r421927 + r421929;
        double r421931 = r421930 * r421930;
        double r421932 = r421925 / r421931;
        double r421933 = sqrt(r421932);
        return r421933;
}

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-- 29.4

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

4. Simplified 0.2

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

5. Simplified 0.2

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

7. Applied add-sqr-sqrt 0.3

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

8. Simplified 0.3

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

9. Simplified 0.3

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

11. Applied sqrt-unprod 0.2

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

12. Simplified 0.3

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

13. Final simplification 0.3

    \[\leadsto \sqrt{\frac{1 \cdot 1}{\left(\sqrt{x} + \sqrt{x + 1}\right) \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)}}\]

Reproduce

herbie shell --seed 2020046 +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)))