2nthrt (problem 3.4.6)

Average Error: 29.7 → 19.9

Time: 56.9s

Precision: 64

• could not determine a ground truth for program body

  1. x = 4.018212772523937e+152
  2. n = 1.5750708393838185e-127

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -5.374271620846741 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\frac{1}{x}}{x}}{n}, \frac{-1}{2}, \left(\frac{\frac{-2}{3}}{n \cdot n} + \frac{\frac{-1}{3}}{n \cdot n}\right) \cdot \frac{\log \left(\frac{1}{x}\right)}{x} + \frac{\frac{1}{n}}{x}\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)\\ \mathbf{elif}\;n \le 2.44541556257196 \cdot 10^{-310}:\\ \;\;\;\;\sqrt[3]{\left(\mathsf{fma}\left({\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\right) \cdot \mathsf{fma}\left({\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\right)\right) \cdot \mathsf{fma}\left({\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\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)\\ \mathbf{elif}\;n \le 2.054730747417339 \cdot 10^{+31}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\frac{1}{x}}{x}}{n}, \frac{-1}{2}, \left(\frac{\frac{-2}{3}}{n \cdot n} + \frac{\frac{-1}{3}}{n \cdot n}\right) \cdot \frac{\log \left(\frac{1}{x}\right)}{x} + \frac{\frac{1}{n}}{x}\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)\\ \end{array}\]

C

double f(double x, double n) {
        double r3930119 = x;
        double r3930120 = 1.0;
        double r3930121 = r3930119 + r3930120;
        double r3930122 = n;
        double r3930123 = r3930120 / r3930122;
        double r3930124 = pow(r3930121, r3930123);
        double r3930125 = pow(r3930119, r3930123);
        double r3930126 = r3930124 - r3930125;
        return r3930126;
}

↓

double f(double x, double n) {
        double r3930127 = n;
        double r3930128 = -5.374271620846741e+21;
        bool r3930129 = r3930127 <= r3930128;
        double r3930130 = 1.0;
        double r3930131 = x;
        double r3930132 = r3930130 / r3930131;
        double r3930133 = r3930132 / r3930131;
        double r3930134 = r3930133 / r3930127;
        double r3930135 = -0.5;
        double r3930136 = -0.6666666666666666;
        double r3930137 = r3930127 * r3930127;
        double r3930138 = r3930136 / r3930137;
        double r3930139 = -0.3333333333333333;
        double r3930140 = r3930139 / r3930137;
        double r3930141 = r3930138 + r3930140;
        double r3930142 = log(r3930132);
        double r3930143 = r3930142 / r3930131;
        double r3930144 = r3930141 * r3930143;
        double r3930145 = r3930130 / r3930127;
        double r3930146 = r3930145 / r3930131;
        double r3930147 = r3930144 + r3930146;
        double r3930148 = fma(r3930134, r3930135, r3930147);
        double r3930149 = cbrt(r3930131);
        double r3930150 = pow(r3930149, r3930145);
        double r3930151 = -r3930150;
        double r3930152 = r3930149 * r3930149;
        double r3930153 = pow(r3930152, r3930145);
        double r3930154 = r3930150 * r3930153;
        double r3930155 = fma(r3930151, r3930153, r3930154);
        double r3930156 = r3930148 + r3930155;
        double r3930157 = 2.44541556257196e-310;
        bool r3930158 = r3930127 <= r3930157;
        double r3930159 = r3930130 + r3930131;
        double r3930160 = cbrt(r3930159);
        double r3930161 = r3930160 * r3930160;
        double r3930162 = pow(r3930161, r3930145);
        double r3930163 = pow(r3930160, r3930145);
        double r3930164 = -r3930153;
        double r3930165 = r3930150 * r3930164;
        double r3930166 = fma(r3930162, r3930163, r3930165);
        double r3930167 = r3930166 * r3930166;
        double r3930168 = r3930167 * r3930166;
        double r3930169 = cbrt(r3930168);
        double r3930170 = r3930169 + r3930155;
        double r3930171 = 2.054730747417339e+31;
        bool r3930172 = r3930127 <= r3930171;
        double r3930173 = log1p(r3930131);
        double r3930174 = r3930173 / r3930127;
        double r3930175 = exp(r3930174);
        double r3930176 = pow(r3930131, r3930145);
        double r3930177 = r3930175 - r3930176;
        double r3930178 = r3930172 ? r3930177 : r3930156;
        double r3930179 = r3930158 ? r3930170 : r3930178;
        double r3930180 = r3930129 ? r3930156 : r3930179;
        return r3930180;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if n < -5.374271620846741e+21 or 2.054730747417339e+31 < n

   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 add-cube-cbrt 44.8

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

   4. Applied unpow-prod-down 44.8

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

   5. Applied add-cube-cbrt 44.8

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

   6. Applied unpow-prod-down 44.8

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

   7. Applied prod-diff 44.8

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

   8. Taylor expanded around inf 33.3

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

   9. Simplified 32.7

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

   if -5.374271620846741e+21 < n < 2.44541556257196e-310

   1. Initial program 2.9

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

   3. Applied add-cube-cbrt 2.9

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

   4. Applied unpow-prod-down 2.9

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

   5. Applied add-cube-cbrt 2.9

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

   6. Applied unpow-prod-down 2.9

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

   7. Applied prod-diff 2.9

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

   9. Applied add-cbrt-cube 2.9

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

   if 2.44541556257196e-310 < n < 2.054730747417339e+31

   1. Initial program 29.0

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

   3. Applied add-exp-log 29.0

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

   4. Applied pow-exp 29.0

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

   5. Simplified 8.7

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

4. Final simplification 19.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -5.374271620846741 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\frac{1}{x}}{x}}{n}, \frac{-1}{2}, \left(\frac{\frac{-2}{3}}{n \cdot n} + \frac{\frac{-1}{3}}{n \cdot n}\right) \cdot \frac{\log \left(\frac{1}{x}\right)}{x} + \frac{\frac{1}{n}}{x}\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)\\ \mathbf{elif}\;n \le 2.44541556257196 \cdot 10^{-310}:\\ \;\;\;\;\sqrt[3]{\left(\mathsf{fma}\left({\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\right) \cdot \mathsf{fma}\left({\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\right)\right) \cdot \mathsf{fma}\left({\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\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)\\ \mathbf{elif}\;n \le 2.054730747417339 \cdot 10^{+31}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\frac{1}{x}}{x}}{n}, \frac{-1}{2}, \left(\frac{\frac{-2}{3}}{n \cdot n} + \frac{\frac{-1}{3}}{n \cdot n}\right) \cdot \frac{\log \left(\frac{1}{x}\right)}{x} + \frac{\frac{1}{n}}{x}\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)\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))