Average Error: 29.6 → 1.4
Time: 11.2s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \le 1.63025788424420170485973358154296875 \cdot 10^{-7}:\\ \;\;\;\;\left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right) + \frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333148296162562473909929395 - \frac{0.1111111111111111049432054187491303309798}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{1} \cdot \left(x \cdot \left(0.3333333333333333148296162562473909929395 + 0.05555555555555555247160270937456516548991 \cdot x\right) + 1\right) - \left(\left(x \cdot x\right) \cdot \sqrt[3]{\frac{1}{{1}^{5}}}\right) \cdot \frac{1}{6}\right) - \sqrt[3]{x}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \le 1.63025788424420170485973358154296875 \cdot 10^{-7}:\\
\;\;\;\;\left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right) + \frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333148296162562473909929395 - \frac{0.1111111111111111049432054187491303309798}{x}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\sqrt[3]{1} \cdot \left(x \cdot \left(0.3333333333333333148296162562473909929395 + 0.05555555555555555247160270937456516548991 \cdot x\right) + 1\right) - \left(\left(x \cdot x\right) \cdot \sqrt[3]{\frac{1}{{1}^{5}}}\right) \cdot \frac{1}{6}\right) - \sqrt[3]{x}\\

\end{array}
double f(double x) {
        double r71360 = x;
        double r71361 = 1.0;
        double r71362 = r71360 + r71361;
        double r71363 = cbrt(r71362);
        double r71364 = cbrt(r71360);
        double r71365 = r71363 - r71364;
        return r71365;
}
↓
double f(double x) {
        double r71366 = x;
        double r71367 = 1.0;
        double r71368 = r71366 + r71367;
        double r71369 = cbrt(r71368);
        double r71370 = cbrt(r71366);
        double r71371 = r71369 - r71370;
        double r71372 = 1.6302578842442017e-07;
        bool r71373 = r71371 <= r71372;
        double r71374 = -1.0;
        double r71375 = cbrt(r71374);
        double r71376 = -r71366;
        double r71377 = cbrt(r71376);
        double r71378 = r71375 * r71377;
        double r71379 = r71370 - r71378;
        double r71380 = r71370 / r71366;
        double r71381 = 0.3333333333333333;
        double r71382 = 0.1111111111111111;
        double r71383 = r71382 / r71366;
        double r71384 = r71381 - r71383;
        double r71385 = r71380 * r71384;
        double r71386 = r71379 + r71385;
        double r71387 = cbrt(r71367);
        double r71388 = 0.05555555555555555;
        double r71389 = r71388 * r71366;
        double r71390 = r71381 + r71389;
        double r71391 = r71366 * r71390;
        double r71392 = 1.0;
        double r71393 = r71391 + r71392;
        double r71394 = r71387 * r71393;
        double r71395 = r71366 * r71366;
        double r71396 = 5.0;
        double r71397 = pow(r71367, r71396);
        double r71398 = r71392 / r71397;
        double r71399 = cbrt(r71398);
        double r71400 = r71395 * r71399;
        double r71401 = 0.16666666666666666;
        double r71402 = r71400 * r71401;
        double r71403 = r71394 - r71402;
        double r71404 = r71403 - r71370;
        double r71405 = r71373 ? r71386 : r71404;
        return r71405;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (- (cbrt (+ x 1.0)) (cbrt x)) < 1.6302578842442017e-07

    1. Initial program 60.9

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

    3. Applied add-cube-cbrt61.2

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

    4. Taylor expanded around -inf 64.0

      \[\leadsto \color{blue}{\left(0.3333333333333333148296162562473909929395 \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{x} + e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}\right) - \left(0.1111111111111111049432054187491303309798 \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{{x}^{2}} + {\left(-1 \cdot x\right)}^{\frac{1}{3}} \cdot \sqrt[3]{-1}\right)}\]

    5. Simplified0.6

      \[\leadsto \color{blue}{\left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right) + \frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333148296162562473909929395 - \frac{0.1111111111111111049432054187491303309798}{x}\right)}\]

    if 1.6302578842442017e-07 < (- (cbrt (+ x 1.0)) (cbrt x))

    1. Initial program 0.5

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

    3. Applied add-cube-cbrt0.6

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

    4. Taylor expanded around 0 33.1

      \[\leadsto \color{blue}{\left(0.05555555555555555247160270937456516548991 \cdot \left({x}^{2} \cdot {1}^{\frac{1}{3}}\right) + \left(0.3333333333333333148296162562473909929395 \cdot \left(x \cdot {1}^{\frac{1}{3}}\right) + {1}^{\frac{1}{3}}\right)\right) - \left(\frac{1}{6} \cdot \left({x}^{2} \cdot {\left(\frac{1}{{1}^{5}}\right)}^{\frac{1}{3}}\right) + {x}^{\frac{1}{3}}\right)}\]

    5. Simplified2.2

      \[\leadsto \color{blue}{\left(\sqrt[3]{1} \cdot \left(x \cdot \left(0.3333333333333333148296162562473909929395 + 0.05555555555555555247160270937456516548991 \cdot x\right) + 1\right) - \left(\left(x \cdot x\right) \cdot \sqrt[3]{\frac{1}{{1}^{5}}}\right) \cdot \frac{1}{6}\right) - \sqrt[3]{x}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \le 1.63025788424420170485973358154296875 \cdot 10^{-7}:\\ \;\;\;\;\left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right) + \frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333148296162562473909929395 - \frac{0.1111111111111111049432054187491303309798}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{1} \cdot \left(x \cdot \left(0.3333333333333333148296162562473909929395 + 0.05555555555555555247160270937456516548991 \cdot x\right) + 1\right) - \left(\left(x \cdot x\right) \cdot \sqrt[3]{\frac{1}{{1}^{5}}}\right) \cdot \frac{1}{6}\right) - \sqrt[3]{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))