Main:bigenough3 from C

Average Error: 30.2 → 0.2

Time: 8.2s

Precision: 64

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

Math

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

↓

\[\frac{1}{\mathsf{fma}\left(\sqrt{\left|\sqrt[3]{x}\right|} \cdot \sqrt{\sqrt{\sqrt[3]{x}}}, \sqrt{\sqrt{x}}, \sqrt{x + 1}\right)}\]

C

double f(double x) {
        double r441519 = x;
        double r441520 = 1.0;
        double r441521 = r441519 + r441520;
        double r441522 = sqrt(r441521);
        double r441523 = sqrt(r441519);
        double r441524 = r441522 - r441523;
        return r441524;
}

↓

double f(double x) {
        double r441525 = 1.0;
        double r441526 = x;
        double r441527 = cbrt(r441526);
        double r441528 = fabs(r441527);
        double r441529 = sqrt(r441528);
        double r441530 = sqrt(r441527);
        double r441531 = sqrt(r441530);
        double r441532 = r441529 * r441531;
        double r441533 = sqrt(r441526);
        double r441534 = sqrt(r441533);
        double r441535 = r441526 + r441525;
        double r441536 = sqrt(r441535);
        double r441537 = fma(r441532, r441534, r441536);
        double r441538 = r441525 / r441537;
        return r441538;
}

Error

Bits error versus x

Target

Original	30.2
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 30.2

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

3. Applied flip-- 29.9

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

4. Simplified 0.2

   \[\leadsto \frac{\color{blue}{0 + 1}}{\sqrt{x + 1} + \sqrt{x}}\]

5. Simplified 0.2

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

7. Applied add-sqr-sqrt 0.2

   \[\leadsto \frac{0 + 1}{\sqrt{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} + \sqrt{x + 1}}\]

8. Applied sqrt-prod 0.3

   \[\leadsto \frac{0 + 1}{\color{blue}{\sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{x}}} + \sqrt{x + 1}}\]

9. Applied fma-def 0.2

   \[\leadsto \frac{0 + 1}{\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt{x}}, \sqrt{\sqrt{x}}, \sqrt{x + 1}\right)}}\]
10. Using strategy rm

11. Applied add-cube-cbrt 0.2

    \[\leadsto \frac{0 + 1}{\mathsf{fma}\left(\sqrt{\sqrt{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}}, \sqrt{\sqrt{x}}, \sqrt{x + 1}\right)}\]

12. Applied sqrt-prod 0.2

    \[\leadsto \frac{0 + 1}{\mathsf{fma}\left(\sqrt{\color{blue}{\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt{\sqrt[3]{x}}}}, \sqrt{\sqrt{x}}, \sqrt{x + 1}\right)}\]

13. Applied sqrt-prod 0.2

    \[\leadsto \frac{0 + 1}{\mathsf{fma}\left(\color{blue}{\sqrt{\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \sqrt{\sqrt{\sqrt[3]{x}}}}, \sqrt{\sqrt{x}}, \sqrt{x + 1}\right)}\]

14. Simplified 0.2

    \[\leadsto \frac{0 + 1}{\mathsf{fma}\left(\color{blue}{\sqrt{\left|\sqrt[3]{x}\right|}} \cdot \sqrt{\sqrt{\sqrt[3]{x}}}, \sqrt{\sqrt{x}}, \sqrt{x + 1}\right)}\]

15. Final simplification 0.2

    \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt{\left|\sqrt[3]{x}\right|} \cdot \sqrt{\sqrt{\sqrt[3]{x}}}, \sqrt{\sqrt{x}}, \sqrt{x + 1}\right)}\]

Reproduce

herbie shell --seed 2020047 +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)))