2nthrt (problem 3.4.6)

Average Error: 29.1 → 22.6

Time: 15.2s

Precision: 64

• could not determine a ground truth for program body

  1. x = 1.3063160135701974e+105
  2. n = 1.297677126524779e-204

Math

\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;n \le -235.54342409600352:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{\log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right)}{x \cdot {n}^{2}} + \left(\frac{1}{x \cdot n} + \frac{\log \left({\left(\frac{1}{x}\right)}^{\frac{-2}{3}}\right)}{x \cdot {n}^{2}}\right), \frac{-0.5}{{x}^{2} \cdot n}\right) + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{elif}\;n \le 0.2317853494525819:\\ \;\;\;\;\mathsf{fma}\left({\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \end{array}\]

C

double f(double x, double n) {
        double r84175 = x;
        double r84176 = 1.0;
        double r84177 = r84175 + r84176;
        double r84178 = n;
        double r84179 = r84176 / r84178;
        double r84180 = pow(r84177, r84179);
        double r84181 = pow(r84175, r84179);
        double r84182 = r84180 - r84181;
        return r84182;
}

↓

double f(double x, double n) {
        double r84183 = n;
        double r84184 = -235.54342409600352;
        bool r84185 = r84183 <= r84184;
        double r84186 = 1.0;
        double r84187 = 1.0;
        double r84188 = x;
        double r84189 = r84187 / r84188;
        double r84190 = -0.3333333333333333;
        double r84191 = pow(r84189, r84190);
        double r84192 = log(r84191);
        double r84193 = 2.0;
        double r84194 = pow(r84183, r84193);
        double r84195 = r84188 * r84194;
        double r84196 = r84192 / r84195;
        double r84197 = r84188 * r84183;
        double r84198 = r84187 / r84197;
        double r84199 = -0.6666666666666666;
        double r84200 = pow(r84189, r84199);
        double r84201 = log(r84200);
        double r84202 = r84201 / r84195;
        double r84203 = r84198 + r84202;
        double r84204 = r84196 + r84203;
        double r84205 = 0.5;
        double r84206 = -r84205;
        double r84207 = pow(r84188, r84193);
        double r84208 = r84207 * r84183;
        double r84209 = r84206 / r84208;
        double r84210 = fma(r84186, r84204, r84209);
        double r84211 = cbrt(r84188);
        double r84212 = r84211 * r84211;
        double r84213 = r84186 / r84183;
        double r84214 = pow(r84212, r84213);
        double r84215 = pow(r84211, r84213);
        double r84216 = -r84215;
        double r84217 = r84216 + r84215;
        double r84218 = r84214 * r84217;
        double r84219 = r84210 + r84218;
        double r84220 = 0.2317853494525819;
        bool r84221 = r84183 <= r84220;
        double r84222 = r84188 + r84186;
        double r84223 = cbrt(r84222);
        double r84224 = r84223 * r84223;
        double r84225 = pow(r84224, r84213);
        double r84226 = pow(r84223, r84213);
        double r84227 = r84215 * r84214;
        double r84228 = -r84227;
        double r84229 = fma(r84225, r84226, r84228);
        double r84230 = r84229 + r84218;
        double r84231 = r84187 / r84208;
        double r84232 = log(r84189);
        double r84233 = r84232 / r84195;
        double r84234 = r84186 * r84233;
        double r84235 = fma(r84205, r84231, r84234);
        double r84236 = -r84235;
        double r84237 = fma(r84186, r84198, r84236);
        double r84238 = r84221 ? r84230 : r84237;
        double r84239 = r84185 ? r84219 : r84238;
        return r84239;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if n < -235.54342409600352

   1. Initial program 44.7

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 44.7

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

   4. Applied unpow-prod-down 44.8

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

   5. Applied add-cube-cbrt 44.8

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

   6. Applied unpow-prod-down 44.8

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

   7. Applied prod-diff 44.8

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

   8. Simplified 44.8

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

   9. Taylor expanded around inf 33.2

      \[\leadsto \color{blue}{\left(\left(1 \cdot \frac{\log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right)}{x \cdot {n}^{2}} + \left(1 \cdot \frac{1}{x \cdot n} + 1 \cdot \frac{\log \left({\left(\frac{1}{x}\right)}^{\frac{-2}{3}}\right)}{x \cdot {n}^{2}}\right)\right) - 0.5 \cdot \frac{1}{{x}^{2} \cdot n}\right)} + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\]

   10. Simplified 33.2

       \[\leadsto \color{blue}{\mathsf{fma}\left(1, \frac{\log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right)}{x \cdot {n}^{2}} + \left(\frac{1}{x \cdot n} + \frac{\log \left({\left(\frac{1}{x}\right)}^{\frac{-2}{3}}\right)}{x \cdot {n}^{2}}\right), \frac{-0.5}{{x}^{2} \cdot n}\right)} + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\]

   if -235.54342409600352 < n < 0.2317853494525819

   1. Initial program 7.6

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 7.6

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

   4. Applied unpow-prod-down 7.6

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

   5. Applied add-cube-cbrt 7.7

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

   6. Applied unpow-prod-down 7.7

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

   7. Applied prod-diff 7.7

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

   8. Simplified 7.7

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

   if 0.2317853494525819 < n

   1. Initial program 44.2

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]

   2. Taylor expanded around inf 33.4

      \[\leadsto \color{blue}{1 \cdot \frac{1}{x \cdot n} - \left(0.5 \cdot \frac{1}{{x}^{2} \cdot n} + 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)}\]

   3. Simplified 33.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 22.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -235.54342409600352:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{\log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right)}{x \cdot {n}^{2}} + \left(\frac{1}{x \cdot n} + \frac{\log \left({\left(\frac{1}{x}\right)}^{\frac{-2}{3}}\right)}{x \cdot {n}^{2}}\right), \frac{-0.5}{{x}^{2} \cdot n}\right) + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{elif}\;n \le 0.2317853494525819:\\ \;\;\;\;\mathsf{fma}\left({\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))