Main:bigenough3 from C

Average Error: 30.4 → 0.2

Time: 6.2s

Precision: 64

Math

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

↓

\[\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}\right)\right)\]

C

double f(double x) {
        double r571979 = x;
        double r571980 = 1.0;
        double r571981 = r571979 + r571980;
        double r571982 = sqrt(r571981);
        double r571983 = sqrt(r571979);
        double r571984 = r571982 - r571983;
        return r571984;
}

↓

double f(double x) {
        double r571985 = 1.0;
        double r571986 = 0.0;
        double r571987 = r571985 + r571986;
        double r571988 = x;
        double r571989 = r571988 + r571985;
        double r571990 = sqrt(r571989);
        double r571991 = sqrt(r571988);
        double r571992 = r571990 + r571991;
        double r571993 = r571987 / r571992;
        double r571994 = log1p(r571993);
        double r571995 = expm1(r571994);
        return r571995;
}

Error

Bits error versus x

Target

Original	30.4
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 30.4

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

3. Applied flip-- 30.2

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

4. Simplified 0.2

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

6. Applied expm1-log1p-u 0.2

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}\right)\right)}\]

7. Final simplification 0.2

   \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}\right)\right)\]

Reproduce

herbie shell --seed 2020018 +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)))