Main:bigenough3 from C

Average Error: 30.3 → 0.2

Time: 5.0s

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 r421594 = x;
        double r421595 = 1.0;
        double r421596 = r421594 + r421595;
        double r421597 = sqrt(r421596);
        double r421598 = sqrt(r421594);
        double r421599 = r421597 - r421598;
        return r421599;
}

↓

double f(double x) {
        double r421600 = 1.0;
        double r421601 = 0.0;
        double r421602 = r421600 + r421601;
        double r421603 = x;
        double r421604 = r421603 + r421600;
        double r421605 = sqrt(r421604);
        double r421606 = sqrt(r421603);
        double r421607 = r421605 + r421606;
        double r421608 = r421602 / r421607;
        double r421609 = log1p(r421608);
        double r421610 = expm1(r421609);
        return r421610;
}

Error

Bits error versus x

Target

Original	30.3
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 30.3

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

3. Applied flip-- 30.1

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

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 2020001 +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)))