2nthrt (problem 3.4.6)

Average Error: 29.4 → 23.2

Time: 1.1m

Precision: 64

• could not determine a ground truth for program body

  1. x = 9.137530169746875e+200
  2. n = 1.4011542472758516e-71

Math

\[{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1.0}{n} \le -6.88847562666282 \cdot 10^{-20}:\\ \;\;\;\;\sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}} \cdot \left(\sqrt[3]{\log \left(\frac{e^{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)}}}{\sqrt{e^{{x}^{\left(\frac{1.0}{n}\right)}}}}\right) + \left(-\log \left(\sqrt{e^{{x}^{\left(\frac{1.0}{n}\right)}}}\right)\right)} \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\right)\\ \mathbf{elif}\;\frac{1.0}{n} \le 1.8205393655820836 \cdot 10^{-32}:\\ \;\;\;\;\left(\frac{\frac{1.0}{x}}{n} - \frac{0.5}{n \cdot \left(x \cdot x\right)}\right) + \frac{\log x \cdot 1.0}{n \cdot \left(x \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\log \left(\frac{e^{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)}}}{e^{{x}^{\left(\frac{1.0}{n}\right)}}}\right)} \cdot \sqrt[3]{e^{\log \left(\log \left(e^{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\right)\right)}}\right) \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\\ \end{array}\]

C

double f(double x, double n) {
        double r3849121 = x;
        double r3849122 = 1.0;
        double r3849123 = r3849121 + r3849122;
        double r3849124 = n;
        double r3849125 = r3849122 / r3849124;
        double r3849126 = pow(r3849123, r3849125);
        double r3849127 = pow(r3849121, r3849125);
        double r3849128 = r3849126 - r3849127;
        return r3849128;
}

↓

double f(double x, double n) {
        double r3849129 = 1.0;
        double r3849130 = n;
        double r3849131 = r3849129 / r3849130;
        double r3849132 = -6.88847562666282e-20;
        bool r3849133 = r3849131 <= r3849132;
        double r3849134 = x;
        double r3849135 = r3849134 + r3849129;
        double r3849136 = pow(r3849135, r3849131);
        double r3849137 = pow(r3849134, r3849131);
        double r3849138 = r3849136 - r3849137;
        double r3849139 = cbrt(r3849138);
        double r3849140 = exp(r3849136);
        double r3849141 = exp(r3849137);
        double r3849142 = sqrt(r3849141);
        double r3849143 = r3849140 / r3849142;
        double r3849144 = log(r3849143);
        double r3849145 = log(r3849142);
        double r3849146 = -r3849145;
        double r3849147 = r3849144 + r3849146;
        double r3849148 = cbrt(r3849147);
        double r3849149 = r3849148 * r3849139;
        double r3849150 = r3849139 * r3849149;
        double r3849151 = 1.8205393655820836e-32;
        bool r3849152 = r3849131 <= r3849151;
        double r3849153 = r3849129 / r3849134;
        double r3849154 = r3849153 / r3849130;
        double r3849155 = 0.5;
        double r3849156 = r3849134 * r3849134;
        double r3849157 = r3849130 * r3849156;
        double r3849158 = r3849155 / r3849157;
        double r3849159 = r3849154 - r3849158;
        double r3849160 = log(r3849134);
        double r3849161 = r3849160 * r3849129;
        double r3849162 = r3849134 * r3849130;
        double r3849163 = r3849130 * r3849162;
        double r3849164 = r3849161 / r3849163;
        double r3849165 = r3849159 + r3849164;
        double r3849166 = r3849140 / r3849141;
        double r3849167 = log(r3849166);
        double r3849168 = cbrt(r3849167);
        double r3849169 = exp(r3849138);
        double r3849170 = log(r3849169);
        double r3849171 = log(r3849170);
        double r3849172 = exp(r3849171);
        double r3849173 = cbrt(r3849172);
        double r3849174 = r3849168 * r3849173;
        double r3849175 = r3849174 * r3849139;
        double r3849176 = r3849152 ? r3849165 : r3849175;
        double r3849177 = r3849133 ? r3849150 : r3849176;
        return r3849177;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -6.88847562666282e-20

   1. Initial program 2.4

      \[{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 2.4

      \[\leadsto \color{blue}{\left(\sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}}\]
   4. Using strategy rm

   5. Applied add-log-exp 2.6

      \[\leadsto \left(\sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - \color{blue}{\log \left(e^{{x}^{\left(\frac{1.0}{n}\right)}}\right)}} \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\]

   6. Applied add-log-exp 2.6

      \[\leadsto \left(\sqrt[3]{\color{blue}{\log \left(e^{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)}}\right)} - \log \left(e^{{x}^{\left(\frac{1.0}{n}\right)}}\right)} \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\]

   7. Applied diff-log 2.5

      \[\leadsto \left(\sqrt[3]{\color{blue}{\log \left(\frac{e^{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)}}}{e^{{x}^{\left(\frac{1.0}{n}\right)}}}\right)}} \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\]
   8. Using strategy rm

   9. Applied add-sqr-sqrt 2.6

      \[\leadsto \left(\sqrt[3]{\log \left(\frac{e^{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)}}}{\color{blue}{\sqrt{e^{{x}^{\left(\frac{1.0}{n}\right)}}} \cdot \sqrt{e^{{x}^{\left(\frac{1.0}{n}\right)}}}}}\right)} \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\]

   10. Applied *-un-lft-identity 2.6

       \[\leadsto \left(\sqrt[3]{\log \left(\frac{\color{blue}{1 \cdot e^{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)}}}}{\sqrt{e^{{x}^{\left(\frac{1.0}{n}\right)}}} \cdot \sqrt{e^{{x}^{\left(\frac{1.0}{n}\right)}}}}\right)} \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\]

   11. Applied times-frac 2.6

       \[\leadsto \left(\sqrt[3]{\log \color{blue}{\left(\frac{1}{\sqrt{e^{{x}^{\left(\frac{1.0}{n}\right)}}}} \cdot \frac{e^{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)}}}{\sqrt{e^{{x}^{\left(\frac{1.0}{n}\right)}}}}\right)}} \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\]

   12. Applied log-prod 2.6

       \[\leadsto \left(\sqrt[3]{\color{blue}{\log \left(\frac{1}{\sqrt{e^{{x}^{\left(\frac{1.0}{n}\right)}}}}\right) + \log \left(\frac{e^{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)}}}{\sqrt{e^{{x}^{\left(\frac{1.0}{n}\right)}}}}\right)}} \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\]

   13. Simplified 2.6

       \[\leadsto \left(\sqrt[3]{\color{blue}{\left(-\log \left(\sqrt{e^{{x}^{\left(\frac{1.0}{n}\right)}}}\right)\right)} + \log \left(\frac{e^{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)}}}{\sqrt{e^{{x}^{\left(\frac{1.0}{n}\right)}}}}\right)} \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\]

   if -6.88847562666282e-20 < (/ 1.0 n) < 1.8205393655820836e-32

   1. Initial program 44.1

      \[{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}\]

   2. Taylor expanded around inf 33.1

      \[\leadsto \color{blue}{1.0 \cdot \frac{1}{x \cdot n} - \left(1.0 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}} + 0.5 \cdot \frac{1}{{x}^{2} \cdot n}\right)}\]

   3. Simplified 32.6

      \[\leadsto \color{blue}{\left(\frac{\frac{1.0}{x}}{n} - \frac{0.5}{n \cdot \left(x \cdot x\right)}\right) + \frac{1.0 \cdot \log x}{\left(n \cdot x\right) \cdot n}}\]

   if 1.8205393655820836e-32 < (/ 1.0 n)

   1. Initial program 29.2

      \[{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 29.2

      \[\leadsto \color{blue}{\left(\sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}}\]
   4. Using strategy rm

   5. Applied add-log-exp 29.2

      \[\leadsto \left(\sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - \color{blue}{\log \left(e^{{x}^{\left(\frac{1.0}{n}\right)}}\right)}} \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\]

   6. Applied add-log-exp 29.3

      \[\leadsto \left(\sqrt[3]{\color{blue}{\log \left(e^{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)}}\right)} - \log \left(e^{{x}^{\left(\frac{1.0}{n}\right)}}\right)} \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\]

   7. Applied diff-log 29.3

      \[\leadsto \left(\sqrt[3]{\color{blue}{\log \left(\frac{e^{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)}}}{e^{{x}^{\left(\frac{1.0}{n}\right)}}}\right)}} \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\]
   8. Using strategy rm

   9. Applied add-log-exp 29.3

      \[\leadsto \left(\sqrt[3]{\log \left(\frac{e^{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)}}}{e^{{x}^{\left(\frac{1.0}{n}\right)}}}\right)} \cdot \sqrt[3]{\color{blue}{\log \left(e^{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\]
   10. Using strategy rm

   11. Applied add-exp-log 29.3

       \[\leadsto \left(\sqrt[3]{\log \left(\frac{e^{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)}}}{e^{{x}^{\left(\frac{1.0}{n}\right)}}}\right)} \cdot \sqrt[3]{\color{blue}{e^{\log \left(\log \left(e^{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\right)\right)}}}\right) \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 23.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1.0}{n} \le -6.88847562666282 \cdot 10^{-20}:\\ \;\;\;\;\sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}} \cdot \left(\sqrt[3]{\log \left(\frac{e^{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)}}}{\sqrt{e^{{x}^{\left(\frac{1.0}{n}\right)}}}}\right) + \left(-\log \left(\sqrt{e^{{x}^{\left(\frac{1.0}{n}\right)}}}\right)\right)} \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\right)\\ \mathbf{elif}\;\frac{1.0}{n} \le 1.8205393655820836 \cdot 10^{-32}:\\ \;\;\;\;\left(\frac{\frac{1.0}{x}}{n} - \frac{0.5}{n \cdot \left(x \cdot x\right)}\right) + \frac{\log x \cdot 1.0}{n \cdot \left(x \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\log \left(\frac{e^{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)}}}{e^{{x}^{\left(\frac{1.0}{n}\right)}}}\right)} \cdot \sqrt[3]{e^{\log \left(\log \left(e^{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\right)\right)}}\right) \cdot \sqrt[3]{{\left(x + 1.0\right)}^{\left(\frac{1.0}{n}\right)} - {x}^{\left(\frac{1.0}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))