Main:bigenough3 from C

Average Error: 29.5 → 0.2

Time: 12.6s

Precision: 64

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

Math

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

↓

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

C

double f(double x) {
        double r385575 = x;
        double r385576 = 1.0;
        double r385577 = r385575 + r385576;
        double r385578 = sqrt(r385577);
        double r385579 = sqrt(r385575);
        double r385580 = r385578 - r385579;
        return r385580;
}

↓

double f(double x) {
        double r385581 = 1.0;
        double r385582 = x;
        double r385583 = sqrt(r385582);
        double r385584 = r385582 + r385581;
        double r385585 = sqrt(r385584);
        double r385586 = r385583 + r385585;
        double r385587 = r385581 / r385586;
        double r385588 = log1p(r385587);
        double r385589 = expm1(r385588);
        return r385589;
}

Error

Bits error versus x

Target

Original	29.5
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 29.5

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

3. Applied flip-- 29.3

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

4. Simplified 0.2

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

5. Simplified 0.2

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

7. Applied expm1-log1p-u 0.2

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

8. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019351 +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)))