2nthrt (problem 3.4.6)

Average Error: 29.0 → 20.1

Time: 55.5s

Precision: 64

• could not determine a ground truth for program body

  1. x = 1.6619849573952127e+169
  2. n = 4.950543945967426e-264

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.0006009208131666345:\\ \;\;\;\;\mathsf{fma}\left({\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, -{x}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 1.2644711251010697 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{\log x}{x \cdot \left(n \cdot n\right)} + \frac{\frac{-1}{2}}{x \cdot \left(x \cdot n\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 5.133358431559432 \cdot 10^{+137}:\\ \;\;\;\;\frac{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} \cdot e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} + {x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{e^{\left(\left(\frac{\mathsf{log1p}\left(x\right)}{n}\right)\right)}}, \sqrt{e^{\left(\left(\frac{\mathsf{log1p}\left(x\right)}{n}\right)\right)}}, -{x}^{\left(\frac{1}{n}\right)}\right)\\ \end{array}\]

C

double f(double x, double n) {
        double r2170290 = x;
        double r2170291 = 1.0;
        double r2170292 = r2170290 + r2170291;
        double r2170293 = n;
        double r2170294 = r2170291 / r2170293;
        double r2170295 = pow(r2170292, r2170294);
        double r2170296 = pow(r2170290, r2170294);
        double r2170297 = r2170295 - r2170296;
        return r2170297;
}

↓

double f(double x, double n) {
        double r2170298 = 1.0;
        double r2170299 = n;
        double r2170300 = r2170298 / r2170299;
        double r2170301 = -0.0006009208131666345;
        bool r2170302 = r2170300 <= r2170301;
        double r2170303 = x;
        double r2170304 = r2170303 + r2170298;
        double r2170305 = cbrt(r2170304);
        double r2170306 = r2170305 * r2170305;
        double r2170307 = pow(r2170306, r2170300);
        double r2170308 = pow(r2170305, r2170300);
        double r2170309 = pow(r2170303, r2170300);
        double r2170310 = -r2170309;
        double r2170311 = fma(r2170307, r2170308, r2170310);
        double r2170312 = 1.2644711251010697e-18;
        bool r2170313 = r2170300 <= r2170312;
        double r2170314 = r2170300 / r2170303;
        double r2170315 = log(r2170303);
        double r2170316 = r2170299 * r2170299;
        double r2170317 = r2170303 * r2170316;
        double r2170318 = r2170315 / r2170317;
        double r2170319 = -0.5;
        double r2170320 = r2170303 * r2170299;
        double r2170321 = r2170303 * r2170320;
        double r2170322 = r2170319 / r2170321;
        double r2170323 = r2170318 + r2170322;
        double r2170324 = r2170314 + r2170323;
        double r2170325 = 5.133358431559432e+137;
        bool r2170326 = r2170300 <= r2170325;
        double r2170327 = log1p(r2170303);
        double r2170328 = r2170327 / r2170299;
        double r2170329 = exp(r2170328);
        double r2170330 = r2170329 * r2170329;
        double r2170331 = r2170309 * r2170309;
        double r2170332 = r2170330 - r2170331;
        double r2170333 = r2170329 + r2170309;
        double r2170334 = r2170332 / r2170333;
        double r2170335 = /* ERROR: no posit support in C */;
        double r2170336 = /* ERROR: no posit support in C */;
        double r2170337 = exp(r2170336);
        double r2170338 = sqrt(r2170337);
        double r2170339 = fma(r2170338, r2170338, r2170310);
        double r2170340 = r2170326 ? r2170334 : r2170339;
        double r2170341 = r2170313 ? r2170324 : r2170340;
        double r2170342 = r2170302 ? r2170311 : r2170341;
        return r2170342;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 4 regimes

2. if (/ 1 n) < -0.0006009208131666345

   1. Initial program 0.3

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

   3. Applied add-cube-cbrt 0.3

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

   4. Applied unpow-prod-down 0.3

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

   5. Applied fma-neg 0.4

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

   if -0.0006009208131666345 < (/ 1 n) < 1.2644711251010697e-18

   1. Initial program 44.5

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

   3. Applied add-exp-log 44.5

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

   4. Simplified 44.5

      \[\leadsto e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]

   5. Taylor expanded around inf 32.3

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

   6. Simplified 31.7

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

   if 1.2644711251010697e-18 < (/ 1 n) < 5.133358431559432e+137

   1. Initial program 17.3

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

   3. Applied add-exp-log 17.4

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

   4. Simplified 8.5

      \[\leadsto e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
   5. Using strategy rm

   6. Applied flip-- 18.3

      \[\leadsto \color{blue}{\frac{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} \cdot e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} + {x}^{\left(\frac{1}{n}\right)}}}\]

   if 5.133358431559432e+137 < (/ 1 n)

   1. Initial program 35.9

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

   3. Applied add-exp-log 35.9

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

   4. Simplified 0.0

      \[\leadsto e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
   5. Using strategy rm

   6. Applied insert-posit16 9.2

      \[\leadsto e^{\color{blue}{\left(\left(\frac{\mathsf{log1p}\left(x\right)}{n}\right)\right)}} - {x}^{\left(\frac{1}{n}\right)}\]
   7. Using strategy rm

   8. Applied add-sqr-sqrt 9.2

      \[\leadsto \color{blue}{\sqrt{e^{\left(\left(\frac{\mathsf{log1p}\left(x\right)}{n}\right)\right)}} \cdot \sqrt{e^{\left(\left(\frac{\mathsf{log1p}\left(x\right)}{n}\right)\right)}}} - {x}^{\left(\frac{1}{n}\right)}\]

   9. Applied fma-neg 9.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{e^{\left(\left(\frac{\mathsf{log1p}\left(x\right)}{n}\right)\right)}}, \sqrt{e^{\left(\left(\frac{\mathsf{log1p}\left(x\right)}{n}\right)\right)}}, -{x}^{\left(\frac{1}{n}\right)}\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 20.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.0006009208131666345:\\ \;\;\;\;\mathsf{fma}\left({\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, -{x}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 1.2644711251010697 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{\log x}{x \cdot \left(n \cdot n\right)} + \frac{\frac{-1}{2}}{x \cdot \left(x \cdot n\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 5.133358431559432 \cdot 10^{+137}:\\ \;\;\;\;\frac{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} \cdot e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} + {x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{e^{\left(\left(\frac{\mathsf{log1p}\left(x\right)}{n}\right)\right)}}, \sqrt{e^{\left(\left(\frac{\mathsf{log1p}\left(x\right)}{n}\right)\right)}}, -{x}^{\left(\frac{1}{n}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))