2nthrt (problem 3.4.6)

Average Error: 29.6 → 21.9

Time: 38.5s

Precision: 64

• could not determine a ground truth for program body

  1. x = 1.198070884819284e+71
  2. n = 4.977801970910924e-308

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2.201151584268497640689698496471230740497 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(-{\left(\sqrt[3]{{x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)}, {\left(\sqrt[3]{{x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}} \cdot \sqrt[3]{{x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)}, {\left(\sqrt[3]{{x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)} \cdot {\left(\sqrt[3]{{x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}} \cdot \sqrt[3]{{x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)}\right) + \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{{x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)} \cdot {\left(\sqrt[3]{{x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}} \cdot \sqrt[3]{{x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 2.828861285269525417292226755937066285203 \cdot 10^{-16}:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} - \frac{\frac{0.5}{n}}{x \cdot x}\right) - \frac{-1}{x} \cdot \frac{\log x}{n \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{e^{\log \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}\\ \end{array}\]

C

double f(double x, double n) {
        double r3002184 = x;
        double r3002185 = 1.0;
        double r3002186 = r3002184 + r3002185;
        double r3002187 = n;
        double r3002188 = r3002185 / r3002187;
        double r3002189 = pow(r3002186, r3002188);
        double r3002190 = pow(r3002184, r3002188);
        double r3002191 = r3002189 - r3002190;
        return r3002191;
}

↓

double f(double x, double n) {
        double r3002192 = 1.0;
        double r3002193 = n;
        double r3002194 = r3002192 / r3002193;
        double r3002195 = -2.2011515842684976e-12;
        bool r3002196 = r3002194 <= r3002195;
        double r3002197 = x;
        double r3002198 = cbrt(r3002194);
        double r3002199 = r3002198 * r3002198;
        double r3002200 = pow(r3002197, r3002199);
        double r3002201 = cbrt(r3002200);
        double r3002202 = pow(r3002201, r3002198);
        double r3002203 = -r3002202;
        double r3002204 = r3002201 * r3002201;
        double r3002205 = pow(r3002204, r3002198);
        double r3002206 = r3002202 * r3002205;
        double r3002207 = fma(r3002203, r3002205, r3002206);
        double r3002208 = r3002197 + r3002192;
        double r3002209 = pow(r3002208, r3002194);
        double r3002210 = r3002209 - r3002206;
        double r3002211 = r3002207 + r3002210;
        double r3002212 = 2.8288612852695254e-16;
        bool r3002213 = r3002194 <= r3002212;
        double r3002214 = r3002192 / r3002197;
        double r3002215 = r3002214 / r3002193;
        double r3002216 = 0.5;
        double r3002217 = r3002216 / r3002193;
        double r3002218 = r3002197 * r3002197;
        double r3002219 = r3002217 / r3002218;
        double r3002220 = r3002215 - r3002219;
        double r3002221 = -r3002192;
        double r3002222 = r3002221 / r3002197;
        double r3002223 = log(r3002197);
        double r3002224 = r3002193 * r3002193;
        double r3002225 = r3002223 / r3002224;
        double r3002226 = r3002222 * r3002225;
        double r3002227 = r3002220 - r3002226;
        double r3002228 = pow(r3002197, r3002194);
        double r3002229 = r3002209 - r3002228;
        double r3002230 = cbrt(r3002229);
        double r3002231 = r3002230 * r3002230;
        double r3002232 = sqrt(r3002209);
        double r3002233 = sqrt(r3002228);
        double r3002234 = r3002232 - r3002233;
        double r3002235 = log(r3002234);
        double r3002236 = exp(r3002235);
        double r3002237 = r3002233 + r3002232;
        double r3002238 = r3002236 * r3002237;
        double r3002239 = cbrt(r3002238);
        double r3002240 = r3002231 * r3002239;
        double r3002241 = r3002213 ? r3002227 : r3002240;
        double r3002242 = r3002196 ? r3002211 : r3002241;
        return r3002242;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -2.2011515842684976e-12

   1. Initial program 1.5

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

   3. Applied add-cube-cbrt 1.5

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

   4. Applied pow-unpow 1.5

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

   6. Applied add-cube-cbrt 1.5

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

   7. Applied unpow-prod-down 1.5

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

   8. Applied *-un-lft-identity 1.5

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

   9. Applied unpow-prod-down 1.5

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

   10. Applied prod-diff 1.5

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

   11. Simplified 1.5

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

   if -2.2011515842684976e-12 < (/ 1.0 n) < 2.8288612852695254e-16

   1. Initial program 45.2

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

   2. Taylor expanded around inf 32.1

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

   3. Simplified 31.5

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

   if 2.8288612852695254e-16 < (/ 1.0 n)

   1. Initial program 27.0

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

   3. Applied add-cube-cbrt 27.0

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

   5. Applied add-sqr-sqrt 27.0

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

   6. Applied add-sqr-sqrt 27.0

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

   7. Applied difference-of-squares 27.0

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

   9. Applied add-exp-log 27.0

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

4. Final simplification 21.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2.201151584268497640689698496471230740497 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(-{\left(\sqrt[3]{{x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)}, {\left(\sqrt[3]{{x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}} \cdot \sqrt[3]{{x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)}, {\left(\sqrt[3]{{x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)} \cdot {\left(\sqrt[3]{{x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}} \cdot \sqrt[3]{{x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)}\right) + \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{{x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)} \cdot {\left(\sqrt[3]{{x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}} \cdot \sqrt[3]{{x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 2.828861285269525417292226755937066285203 \cdot 10^{-16}:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} - \frac{\frac{0.5}{n}}{x \cdot x}\right) - \frac{-1}{x} \cdot \frac{\log x}{n \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{e^{\log \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))