Main:bigenough3 from C

Average Error: 30.2 → 0.2

Time: 9.4s

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 r539881 = x;
        double r539882 = 1.0;
        double r539883 = r539881 + r539882;
        double r539884 = sqrt(r539883);
        double r539885 = sqrt(r539881);
        double r539886 = r539884 - r539885;
        return r539886;
}

↓

double f(double x) {
        double r539887 = 1.0;
        double r539888 = x;
        double r539889 = cbrt(r539888);
        double r539890 = fabs(r539889);
        double r539891 = sqrt(r539890);
        double r539892 = sqrt(r539889);
        double r539893 = sqrt(r539892);
        double r539894 = r539891 * r539893;
        double r539895 = sqrt(r539888);
        double r539896 = sqrt(r539895);
        double r539897 = r539888 + r539887;
        double r539898 = sqrt(r539897);
        double r539899 = fma(r539894, r539896, r539898);
        double r539900 = r539887 / r539899;
        return r539900;
}

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

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

5. Simplified 0.2

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

7. Applied add-sqr-sqrt 0.2

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

8. Applied sqrt-prod 0.3

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

9. Applied fma-def 0.2

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

11. Applied add-cube-cbrt 0.2

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

12. Applied sqrt-prod 0.2

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

13. Applied sqrt-prod 0.2

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

14. Simplified 0.2

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

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)))