2sqrt (example 3.1)

Average Error: 29.8 → 0.3

Time: 27.4s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r14744799 = x;
        double r14744800 = 1.0;
        double r14744801 = r14744799 + r14744800;
        double r14744802 = sqrt(r14744801);
        double r14744803 = sqrt(r14744799);
        double r14744804 = r14744802 - r14744803;
        return r14744804;
}

↓

double f(double x) {
        double r14744805 = 1.0;
        double r14744806 = x;
        double r14744807 = r14744806 + r14744805;
        double r14744808 = sqrt(r14744807);
        double r14744809 = sqrt(r14744806);
        double r14744810 = r14744808 + r14744809;
        double r14744811 = sqrt(r14744810);
        double r14744812 = r14744805 / r14744811;
        double r14744813 = r14744812 / r14744811;
        return r14744813;
}

Error

Bits error versus x

Target

Original	29.8
Target	0.2
Herbie	0.3

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

Derivation

1. Initial program 29.8

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

3. Applied flip-- 29.6

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

4. Taylor expanded around 0 0.2

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

6. Applied add-sqr-sqrt 0.4

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

7. Applied associate-/r* 0.3

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

8. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019104 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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