2cbrt (problem 3.3.4)

Average Error: 29.7 → 8.1

Time: 19.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.336948920563128703949701544603869465591 \cdot 10^{154}:\\ \;\;\;\;\log \left(\sqrt{e^{\sqrt[3]{x + 1} - \sqrt[3]{x}}}\right) + \left(\frac{\sqrt[3]{x}}{x} \cdot \left(0.1666666666666666574148081281236954964697 - \frac{0.05555555555555555247160270937456516548991}{x}\right) + \log \left(\sqrt{e^{\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}}}\right)\right)\\ \mathbf{elif}\;x \le -3865.382861149344535078853368759155273438:\\ \;\;\;\;0.3333333333333333148296162562473909929395 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \left(0.06172839506172839163511412152729462832212 \cdot \sqrt[3]{\frac{1}{{x}^{8}}} - \sqrt[3]{\frac{1}{{x}^{5}}} \cdot 0.1111111111111111049432054187491303309798\right)\\ \mathbf{elif}\;x \le 4.799156511993158112663217912917390517435 \cdot 10^{-8}:\\ \;\;\;\;\sqrt[3]{x + 1} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]

C

double f(double x) {
        double r58158 = x;
        double r58159 = 1.0;
        double r58160 = r58158 + r58159;
        double r58161 = cbrt(r58160);
        double r58162 = cbrt(r58158);
        double r58163 = r58161 - r58162;
        return r58163;
}

↓

double f(double x) {
        double r58164 = x;
        double r58165 = -1.3369489205631287e+154;
        bool r58166 = r58164 <= r58165;
        double r58167 = 1.0;
        double r58168 = r58164 + r58167;
        double r58169 = cbrt(r58168);
        double r58170 = cbrt(r58164);
        double r58171 = r58169 - r58170;
        double r58172 = exp(r58171);
        double r58173 = sqrt(r58172);
        double r58174 = log(r58173);
        double r58175 = r58170 / r58164;
        double r58176 = 0.16666666666666666;
        double r58177 = 0.05555555555555555;
        double r58178 = r58177 / r58164;
        double r58179 = r58176 - r58178;
        double r58180 = r58175 * r58179;
        double r58181 = -1.0;
        double r58182 = cbrt(r58181);
        double r58183 = -r58164;
        double r58184 = cbrt(r58183);
        double r58185 = r58182 * r58184;
        double r58186 = r58170 - r58185;
        double r58187 = exp(r58186);
        double r58188 = sqrt(r58187);
        double r58189 = log(r58188);
        double r58190 = r58180 + r58189;
        double r58191 = r58174 + r58190;
        double r58192 = -3865.3828611493445;
        bool r58193 = r58164 <= r58192;
        double r58194 = 0.3333333333333333;
        double r58195 = 1.0;
        double r58196 = 2.0;
        double r58197 = pow(r58164, r58196);
        double r58198 = r58195 / r58197;
        double r58199 = cbrt(r58198);
        double r58200 = r58194 * r58199;
        double r58201 = 0.06172839506172839;
        double r58202 = 8.0;
        double r58203 = pow(r58164, r58202);
        double r58204 = r58195 / r58203;
        double r58205 = cbrt(r58204);
        double r58206 = r58201 * r58205;
        double r58207 = 5.0;
        double r58208 = pow(r58164, r58207);
        double r58209 = r58195 / r58208;
        double r58210 = cbrt(r58209);
        double r58211 = 0.1111111111111111;
        double r58212 = r58210 * r58211;
        double r58213 = r58206 - r58212;
        double r58214 = r58200 + r58213;
        double r58215 = 4.799156511993158e-08;
        bool r58216 = r58164 <= r58215;
        double r58217 = r58169 + r58170;
        double r58218 = r58169 * r58217;
        double r58219 = 0.6666666666666666;
        double r58220 = pow(r58164, r58219);
        double r58221 = r58218 + r58220;
        double r58222 = r58167 / r58221;
        double r58223 = r58216 ? r58171 : r58222;
        double r58224 = r58193 ? r58214 : r58223;
        double r58225 = r58166 ? r58191 : r58224;
        return r58225;
}

Error

Bits error versus x

Derivation

1. Split input into 4 regimes

2. if x < -1.3369489205631287e+154

   1. Initial program 61.0

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

   3. Applied add-log-exp 64.0

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

   4. Applied add-log-exp 64.0

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

   5. Applied diff-log 64.0

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

   6. Simplified 61.0

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

   8. Applied add-sqr-sqrt 61.0

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

   9. Applied log-prod 61.0

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

   10. Taylor expanded around -inf 64.0

       \[\leadsto \log \left(\sqrt{e^{\sqrt[3]{x + 1} - \sqrt[3]{x}}}\right) + \color{blue}{\left(\left(0.1666666666666666574148081281236954964697 \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{x} + \log \left(\sqrt{e^{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)} - {\left(-1 \cdot x\right)}^{\frac{1}{3}} \cdot \sqrt[3]{-1}}}\right)\right) - 0.05555555555555555247160270937456516548991 \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{{x}^{2}}\right)}\]

   11. Simplified 52.0

       \[\leadsto \log \left(\sqrt{e^{\sqrt[3]{x + 1} - \sqrt[3]{x}}}\right) + \color{blue}{\left(\frac{\sqrt[3]{x}}{x} \cdot \left(0.1666666666666666574148081281236954964697 - \frac{0.05555555555555555247160270937456516548991}{x}\right) + \log \left(\sqrt{e^{\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}}}\right)\right)}\]

   if -1.3369489205631287e+154 < x < -3865.3828611493445

   1. Initial program 59.7

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

   3. Applied add-log-exp 62.9

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

   4. Applied add-log-exp 62.9

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

   5. Applied diff-log 62.9

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

   6. Simplified 59.7

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

   7. Taylor expanded around inf 28.6

      \[\leadsto \color{blue}{\left(0.3333333333333333148296162562473909929395 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.06172839506172839163511412152729462832212 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}}\]

   8. Simplified 0.9

      \[\leadsto \color{blue}{0.3333333333333333148296162562473909929395 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \left(0.06172839506172839163511412152729462832212 \cdot \sqrt[3]{\frac{1}{{x}^{8}}} - \sqrt[3]{\frac{1}{{x}^{5}}} \cdot 0.1111111111111111049432054187491303309798\right)}\]

   if -3865.3828611493445 < x < 4.799156511993158e-08

   1. Initial program 0.1

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

   3. Applied add-log-exp 0.1

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

   4. Applied add-log-exp 0.1

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

   5. Applied diff-log 0.1

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

   6. Simplified 0.1

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

   8. Applied add-sqr-sqrt 0.1

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

   10. Applied *-un-lft-identity 0.1

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

   11. Applied log-prod 0.1

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

   12. Simplified 0.1

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

   13. Simplified 0.1

       \[\leadsto 0 + \color{blue}{\left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right)}\]

   if 4.799156511993158e-08 < x

   1. Initial program 58.2

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

   3. Applied flip3-- 58.1

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

   4. Simplified 1.0

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

   5. Simplified 4.4

      \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 8.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.336948920563128703949701544603869465591 \cdot 10^{154}:\\ \;\;\;\;\log \left(\sqrt{e^{\sqrt[3]{x + 1} - \sqrt[3]{x}}}\right) + \left(\frac{\sqrt[3]{x}}{x} \cdot \left(0.1666666666666666574148081281236954964697 - \frac{0.05555555555555555247160270937456516548991}{x}\right) + \log \left(\sqrt{e^{\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}}}\right)\right)\\ \mathbf{elif}\;x \le -3865.382861149344535078853368759155273438:\\ \;\;\;\;0.3333333333333333148296162562473909929395 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \left(0.06172839506172839163511412152729462832212 \cdot \sqrt[3]{\frac{1}{{x}^{8}}} - \sqrt[3]{\frac{1}{{x}^{5}}} \cdot 0.1111111111111111049432054187491303309798\right)\\ \mathbf{elif}\;x \le 4.799156511993158112663217912917390517435 \cdot 10^{-8}:\\ \;\;\;\;\sqrt[3]{x + 1} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))