2nthrt (problem 3.4.6)

Average Error: 29.4 → 18.8

Time: 29.9s

Precision: 64

• could not determine a ground truth for program body

  1. x = 2.0415539430385264e+110
  2. n = 3.213371914182865e-194

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -3.743722513168342 \cdot 10^{-08}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 4.1441434979757814 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log x}{x \cdot \left(n \cdot n\right)} + \mathsf{fma}\left(\frac{-1}{2}, \frac{\frac{\frac{1}{x}}{x}}{n}, \frac{\frac{1}{x}}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

C

double f(double x, double n) {
        double r2608201 = x;
        double r2608202 = 1.0;
        double r2608203 = r2608201 + r2608202;
        double r2608204 = n;
        double r2608205 = r2608202 / r2608204;
        double r2608206 = pow(r2608203, r2608205);
        double r2608207 = pow(r2608201, r2608205);
        double r2608208 = r2608206 - r2608207;
        return r2608208;
}

↓

double f(double x, double n) {
        double r2608209 = 1.0;
        double r2608210 = n;
        double r2608211 = r2608209 / r2608210;
        double r2608212 = -3.743722513168342e-08;
        bool r2608213 = r2608211 <= r2608212;
        double r2608214 = x;
        double r2608215 = r2608214 + r2608209;
        double r2608216 = pow(r2608215, r2608211);
        double r2608217 = cbrt(r2608214);
        double r2608218 = r2608217 * r2608217;
        double r2608219 = pow(r2608218, r2608211);
        double r2608220 = pow(r2608217, r2608211);
        double r2608221 = r2608219 * r2608220;
        double r2608222 = r2608216 - r2608221;
        double r2608223 = 4.1441434979757814e-11;
        bool r2608224 = r2608211 <= r2608223;
        double r2608225 = log(r2608214);
        double r2608226 = r2608210 * r2608210;
        double r2608227 = r2608214 * r2608226;
        double r2608228 = r2608225 / r2608227;
        double r2608229 = -0.5;
        double r2608230 = r2608209 / r2608214;
        double r2608231 = r2608230 / r2608214;
        double r2608232 = r2608231 / r2608210;
        double r2608233 = r2608230 / r2608210;
        double r2608234 = fma(r2608229, r2608232, r2608233);
        double r2608235 = r2608228 + r2608234;
        double r2608236 = log1p(r2608214);
        double r2608237 = r2608236 / r2608210;
        double r2608238 = exp(r2608237);
        double r2608239 = pow(r2608214, r2608211);
        double r2608240 = r2608238 - r2608239;
        double r2608241 = r2608224 ? r2608235 : r2608240;
        double r2608242 = r2608213 ? r2608222 : r2608241;
        return r2608242;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1 n) < -3.743722513168342e-08

   1. Initial program 0.8

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

   3. Applied sqr-pow 0.8

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

   5. Applied insert-posit16 0.9

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

   6. Simplified 0.9

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

   8. Applied add-cube-cbrt 0.9

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

   9. Applied unpow-prod-down 0.9

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

   10. Applied add-sqr-sqrt 0.9

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

   11. Applied prod-diff 0.9

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

   12. Simplified 0.9

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

   13. Taylor expanded around 0 0.9

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

   if -3.743722513168342e-08 < (/ 1 n) < 4.1441434979757814e-11

   1. Initial program 44.8

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

   3. Applied sqr-pow 44.8

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

   5. Applied insert-posit16 44.9

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

   6. Simplified 44.9

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

   7. Taylor expanded around inf 32.4

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

   8. Simplified 31.8

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

   if 4.1441434979757814e-11 < (/ 1 n)

   1. Initial program 25.4

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

   3. Applied add-exp-log 25.4

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

   4. Applied pow-exp 25.4

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

   5. Simplified 2.5

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

4. Final simplification 18.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -3.743722513168342 \cdot 10^{-08}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 4.1441434979757814 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log x}{x \cdot \left(n \cdot n\right)} + \mathsf{fma}\left(\frac{-1}{2}, \frac{\frac{\frac{1}{x}}{x}}{n}, \frac{\frac{1}{x}}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))