Average Error: 29.8 → 0.6
Time: 15.7s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\sqrt{\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\sqrt{\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}}
double f(double x) {
        double r54329 = x;
        double r54330 = 1.0;
        double r54331 = r54329 + r54330;
        double r54332 = cbrt(r54331);
        double r54333 = cbrt(r54329);
        double r54334 = r54332 - r54333;
        return r54334;
}
↓
double f(double x) {
        double r54335 = 1.0;
        double r54336 = x;
        double r54337 = cbrt(r54336);
        double r54338 = r54336 + r54335;
        double r54339 = cbrt(r54338);
        double r54340 = r54339 + r54337;
        double r54341 = r54339 * r54339;
        double r54342 = fma(r54337, r54340, r54341);
        double r54343 = r54335 / r54342;
        double r54344 = sqrt(r54343);
        double r54345 = r54344 * r54344;
        return r54345;
}

Error

Bits error versus x

Derivation

  1. Initial program 29.8

    \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
  2. Using strategy rm

  3. Applied add-log-exp31.4

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

  4. Applied add-log-exp31.4

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

  5. Applied diff-log31.4

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

  6. Simplified29.8

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

  8. Applied flip3--29.8

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

  9. Simplified29.4

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

  10. Simplified29.4

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

  12. Applied add-sqr-sqrt29.4

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

  13. Applied exp-prod29.4

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

  14. Applied log-pow27.9

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

  15. Simplified0.6

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

  16. Final simplification0.6

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

Reproduce

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