2cbrt (problem 3.3.4)

Average Error: 29.8 → 0.6

Time: 15.7s

Precision: 64

Math

\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]

↓

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

C

double f(double x) {
        double r81656 = x;
        double r81657 = 1.0;
        double r81658 = r81656 + r81657;
        double r81659 = cbrt(r81658);
        double r81660 = cbrt(r81656);
        double r81661 = r81659 - r81660;
        return r81661;
}

↓

double f(double x) {
        double r81662 = 1.0;
        double r81663 = x;
        double r81664 = cbrt(r81663);
        double r81665 = r81663 + r81662;
        double r81666 = cbrt(r81665);
        double r81667 = r81664 + r81666;
        double r81668 = r81666 * r81666;
        double r81669 = fma(r81664, r81667, r81668);
        double r81670 = r81662 / r81669;
        double r81671 = sqrt(r81670);
        double r81672 = r81671 * r81671;
        return r81672;
}

Error

Bits error versus x

Derivation

1. Initial program 29.8

   \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]

2. Simplified 29.8

   \[\leadsto \color{blue}{\sqrt[3]{1 + x} - \sqrt[3]{x}}\]
3. Using strategy rm

4. Applied add-log-exp 31.4

   \[\leadsto \sqrt[3]{1 + x} - \color{blue}{\log \left(e^{\sqrt[3]{x}}\right)}\]

5. Applied add-log-exp 31.4

   \[\leadsto \color{blue}{\log \left(e^{\sqrt[3]{1 + x}}\right)} - \log \left(e^{\sqrt[3]{x}}\right)\]

6. Applied diff-log 31.4

   \[\leadsto \color{blue}{\log \left(\frac{e^{\sqrt[3]{1 + x}}}{e^{\sqrt[3]{x}}}\right)}\]

7. Simplified 29.8

   \[\leadsto \log \color{blue}{\left(e^{\sqrt[3]{1 + x} - \sqrt[3]{x}}\right)}\]
8. Using strategy rm

9. Applied flip3-- 29.8

   \[\leadsto \log \left(e^{\color{blue}{\frac{{\left(\sqrt[3]{1 + x}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{1 + x} \cdot \sqrt[3]{x}\right)}}}\right)\]

10. Simplified 29.4

    \[\leadsto \log \left(e^{\frac{\color{blue}{1 + 0}}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{1 + x} \cdot \sqrt[3]{x}\right)}}\right)\]

11. Simplified 29.4

    \[\leadsto \log \left(e^{\frac{1 + 0}{\color{blue}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, \sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}}}\right)\]
12. Using strategy rm

13. Applied add-sqr-sqrt 29.4

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

14. Simplified 29.4

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

15. Simplified 0.6

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

16. Final simplification 0.6

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1.0)) (cbrt x)))