2sqrt (example 3.1)

Average Error: 29.5 → 0.2

Time: 16.0s

Precision: 64

Math

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

↓

\[\frac{1}{\mathsf{fma}\left(\sqrt{\sqrt{x + 1}}, \sqrt{\sqrt{x + 1}}, \sqrt{x}\right)}\]

C

double f(double x) {
        double r6057534 = x;
        double r6057535 = 1.0;
        double r6057536 = r6057534 + r6057535;
        double r6057537 = sqrt(r6057536);
        double r6057538 = sqrt(r6057534);
        double r6057539 = r6057537 - r6057538;
        return r6057539;
}

↓

double f(double x) {
        double r6057540 = 1.0;
        double r6057541 = x;
        double r6057542 = r6057541 + r6057540;
        double r6057543 = sqrt(r6057542);
        double r6057544 = sqrt(r6057543);
        double r6057545 = sqrt(r6057541);
        double r6057546 = fma(r6057544, r6057544, r6057545);
        double r6057547 = r6057540 / r6057546;
        return r6057547;
}

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. Using strategy rm

6. Applied add-sqr-sqrt 0.2

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

7. Applied sqrt-prod 0.3

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

8. Applied fma-def 0.2

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

9. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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