2nthrt (problem 3.4.6)

Average Error: 29.4 → 18.9

Time: 38.4s

Precision: 64

• could not determine a ground truth for program body

  1. x = 5.8115194724836035e+243
  2. n = 7.435582231179137e-82

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -781173098.4534194:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{-\frac{\frac{\log x}{x}}{n}}{n}\right) - \frac{\frac{1}{2}}{\left(x \cdot n\right) \cdot x}\\ \mathbf{elif}\;n \le -2.9363061398401 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right), \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right), \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right) + \sqrt[3]{\left({\left(\sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left({\left(\sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left({\left(\sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)}\\ \mathbf{elif}\;n \le 16414365.079812422:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{-\frac{\frac{\log x}{x}}{n}}{n}\right) - \frac{\frac{1}{2}}{\left(x \cdot n\right) \cdot x}\\ \end{array}\]

C

double f(double x, double n) {
        double r2161152 = x;
        double r2161153 = 1.0;
        double r2161154 = r2161152 + r2161153;
        double r2161155 = n;
        double r2161156 = r2161153 / r2161155;
        double r2161157 = pow(r2161154, r2161156);
        double r2161158 = pow(r2161152, r2161156);
        double r2161159 = r2161157 - r2161158;
        return r2161159;
}

↓

double f(double x, double n) {
        double r2161160 = n;
        double r2161161 = -781173098.4534194;
        bool r2161162 = r2161160 <= r2161161;
        double r2161163 = 1.0;
        double r2161164 = r2161163 / r2161160;
        double r2161165 = x;
        double r2161166 = r2161164 / r2161165;
        double r2161167 = log(r2161165);
        double r2161168 = r2161167 / r2161165;
        double r2161169 = r2161168 / r2161160;
        double r2161170 = -r2161169;
        double r2161171 = r2161170 / r2161160;
        double r2161172 = r2161166 - r2161171;
        double r2161173 = 0.5;
        double r2161174 = r2161165 * r2161160;
        double r2161175 = r2161174 * r2161165;
        double r2161176 = r2161173 / r2161175;
        double r2161177 = r2161172 - r2161176;
        double r2161178 = -2.9363061398401e-310;
        bool r2161179 = r2161160 <= r2161178;
        double r2161180 = 2.0;
        double r2161181 = r2161164 / r2161180;
        double r2161182 = pow(r2161165, r2161181);
        double r2161183 = -r2161182;
        double r2161184 = r2161182 * r2161182;
        double r2161185 = fma(r2161183, r2161182, r2161184);
        double r2161186 = r2161163 + r2161165;
        double r2161187 = cbrt(r2161186);
        double r2161188 = pow(r2161187, r2161164);
        double r2161189 = r2161187 * r2161187;
        double r2161190 = pow(r2161189, r2161164);
        double r2161191 = r2161188 * r2161190;
        double r2161192 = r2161191 - r2161184;
        double r2161193 = r2161192 * r2161192;
        double r2161194 = r2161192 * r2161193;
        double r2161195 = cbrt(r2161194);
        double r2161196 = r2161185 + r2161195;
        double r2161197 = 16414365.079812422;
        bool r2161198 = r2161160 <= r2161197;
        double r2161199 = log1p(r2161165);
        double r2161200 = r2161199 / r2161160;
        double r2161201 = exp(r2161200);
        double r2161202 = pow(r2161165, r2161164);
        double r2161203 = r2161201 - r2161202;
        double r2161204 = r2161198 ? r2161203 : r2161177;
        double r2161205 = r2161179 ? r2161196 : r2161204;
        double r2161206 = r2161162 ? r2161177 : r2161205;
        return r2161206;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if n < -781173098.4534194 or 16414365.079812422 < n

   1. Initial program 44.8

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

   2. Taylor expanded around inf 32.7

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

   3. Simplified 32.0

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

   if -781173098.4534194 < n < -2.9363061398401e-310

   1. Initial program 0.9

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

   3. Applied sqr-pow 0.9

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

   4. Applied add-cube-cbrt 0.9

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

   5. Applied unpow-prod-down 0.9

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

   6. Applied prod-diff 0.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left({\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}\right), \left({\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}\right), \left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right) + \mathsf{fma}\left(\left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right), \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right), \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)}\]
   7. Using strategy rm

   8. Applied add-cbrt-cube 1.0

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

   9. Simplified 1.0

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

   if -2.9363061398401e-310 < n < 16414365.079812422

   1. Initial program 25.0

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

   3. Applied add-exp-log 25.0

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

   4. Applied pow-exp 25.0

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

   5. Simplified 1.9

      \[\leadsto e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 18.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -781173098.4534194:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{-\frac{\frac{\log x}{x}}{n}}{n}\right) - \frac{\frac{1}{2}}{\left(x \cdot n\right) \cdot x}\\ \mathbf{elif}\;n \le -2.9363061398401 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right), \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right), \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right) + \sqrt[3]{\left({\left(\sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left({\left(\sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left({\left(\sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)}\\ \mathbf{elif}\;n \le 16414365.079812422:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{-\frac{\frac{\log x}{x}}{n}}{n}\right) - \frac{\frac{1}{2}}{\left(x \cdot n\right) \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))