2nthrt (problem 3.4.6)

Average Error: 29.4 → 22.1

Time: 55.3s

Precision: 64

• could not determine a ground truth for program body

  1. x = 8.245083282969932e+191
  2. n = 8.666561730544872e-18

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -23787117.0056510865688323974609375:\\ \;\;\;\;{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + \left(\frac{1}{{n}^{2}} \cdot \left(\frac{\log x \cdot \frac{1}{3}}{x} + \frac{\frac{-2}{3}}{\frac{x}{-\log x}}\right) + \left(\frac{\frac{1}{n}}{x} - \frac{0.5}{\mathsf{log1p}\left(\mathsf{expm1}\left({x}^{2} \cdot n\right)\right)}\right)\right)\\ \mathbf{elif}\;n \le 96674705739.3252410888671875:\\ \;\;\;\;\mathsf{fma}\left({\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right)} \cdot \sqrt{\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right)} - \frac{0.5}{{x}^{2} \cdot n}\\ \end{array}\]

C

double f(double x, double n) {
        double r122935 = x;
        double r122936 = 1.0;
        double r122937 = r122935 + r122936;
        double r122938 = n;
        double r122939 = r122936 / r122938;
        double r122940 = pow(r122937, r122939);
        double r122941 = pow(r122935, r122939);
        double r122942 = r122940 - r122941;
        return r122942;
}

↓

double f(double x, double n) {
        double r122943 = n;
        double r122944 = -23787117.005651087;
        bool r122945 = r122943 <= r122944;
        double r122946 = x;
        double r122947 = cbrt(r122946);
        double r122948 = r122947 * r122947;
        double r122949 = 1.0;
        double r122950 = r122949 / r122943;
        double r122951 = pow(r122948, r122950);
        double r122952 = pow(r122947, r122950);
        double r122953 = -r122952;
        double r122954 = r122953 + r122952;
        double r122955 = r122951 * r122954;
        double r122956 = 2.0;
        double r122957 = pow(r122943, r122956);
        double r122958 = r122949 / r122957;
        double r122959 = log(r122946);
        double r122960 = 0.3333333333333333;
        double r122961 = r122959 * r122960;
        double r122962 = r122961 / r122946;
        double r122963 = -0.6666666666666666;
        double r122964 = -r122959;
        double r122965 = r122946 / r122964;
        double r122966 = r122963 / r122965;
        double r122967 = r122962 + r122966;
        double r122968 = r122958 * r122967;
        double r122969 = r122950 / r122946;
        double r122970 = 0.5;
        double r122971 = pow(r122946, r122956);
        double r122972 = r122971 * r122943;
        double r122973 = expm1(r122972);
        double r122974 = log1p(r122973);
        double r122975 = r122970 / r122974;
        double r122976 = r122969 - r122975;
        double r122977 = r122968 + r122976;
        double r122978 = r122955 + r122977;
        double r122979 = 96674705739.32524;
        bool r122980 = r122943 <= r122979;
        double r122981 = r122946 + r122949;
        double r122982 = cbrt(r122981);
        double r122983 = r122982 * r122982;
        double r122984 = pow(r122983, r122950);
        double r122985 = pow(r122982, r122950);
        double r122986 = r122952 * r122951;
        double r122987 = -r122986;
        double r122988 = fma(r122984, r122985, r122987);
        double r122989 = r122988 + r122955;
        double r122990 = r122949 / r122946;
        double r122991 = 1.0;
        double r122992 = r122991 / r122943;
        double r122993 = r122964 / r122957;
        double r122994 = r122992 - r122993;
        double r122995 = r122990 * r122994;
        double r122996 = sqrt(r122995);
        double r122997 = r122996 * r122996;
        double r122998 = r122970 / r122972;
        double r122999 = r122997 - r122998;
        double r123000 = r122980 ? r122989 : r122999;
        double r123001 = r122945 ? r122978 : r123000;
        return r123001;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if n < -23787117.005651087

   1. Initial program 45.0

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

   3. Applied add-cube-cbrt 45.0

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

   4. Applied unpow-prod-down 45.0

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

   5. Applied add-cube-cbrt 45.0

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

   6. Applied unpow-prod-down 45.0

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

   7. Applied prod-diff 45.0

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

   8. Simplified 45.0

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

   9. Taylor expanded around inf 33.2

      \[\leadsto \color{blue}{\left(\left(1 \cdot \frac{\log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right)}{x \cdot {n}^{2}} + \left(1 \cdot \frac{1}{x \cdot n} + 1 \cdot \frac{\log \left({\left(\frac{1}{x}\right)}^{\frac{-2}{3}}\right)}{x \cdot {n}^{2}}\right)\right) - 0.5 \cdot \frac{1}{{x}^{2} \cdot n}\right)} + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\]

   10. Simplified 32.6

       \[\leadsto \color{blue}{\left(\frac{1}{{n}^{2}} \cdot \left(\frac{\left(-\log x\right) \cdot \frac{-1}{3}}{x} + \frac{\frac{-2}{3}}{\frac{x}{-\log x}}\right) + \left(\frac{\frac{1}{n}}{x} - \frac{0.5}{{x}^{2} \cdot n}\right)\right)} + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\]
   11. Using strategy rm

   12. Applied log1p-expm1-u 32.5

       \[\leadsto \left(\frac{1}{{n}^{2}} \cdot \left(\frac{\left(-\log x\right) \cdot \frac{-1}{3}}{x} + \frac{\frac{-2}{3}}{\frac{x}{-\log x}}\right) + \left(\frac{\frac{1}{n}}{x} - \frac{0.5}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({x}^{2} \cdot n\right)\right)}}\right)\right) + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\]

   if -23787117.005651087 < n < 96674705739.32524

   1. Initial program 8.8

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

   3. Applied add-cube-cbrt 8.8

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

   4. Applied unpow-prod-down 8.8

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

   5. Applied add-cube-cbrt 8.9

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

   6. Applied unpow-prod-down 8.8

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

   7. Applied prod-diff 8.8

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

   8. Simplified 8.9

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

   if 96674705739.32524 < n

   1. Initial program 44.8

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

   2. Taylor expanded around inf 32.1

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

   3. Simplified 31.5

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}}\]
   4. Using strategy rm

   5. Applied add-sqr-sqrt 31.5

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right)} \cdot \sqrt{\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right)}} - \frac{0.5}{{x}^{2} \cdot n}\]
3. Recombined 3 regimes into one program.

4. Final simplification 22.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -23787117.0056510865688323974609375:\\ \;\;\;\;{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + \left(\frac{1}{{n}^{2}} \cdot \left(\frac{\log x \cdot \frac{1}{3}}{x} + \frac{\frac{-2}{3}}{\frac{x}{-\log x}}\right) + \left(\frac{\frac{1}{n}}{x} - \frac{0.5}{\mathsf{log1p}\left(\mathsf{expm1}\left({x}^{2} \cdot n\right)\right)}\right)\right)\\ \mathbf{elif}\;n \le 96674705739.3252410888671875:\\ \;\;\;\;\mathsf{fma}\left({\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right)} \cdot \sqrt{\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right)} - \frac{0.5}{{x}^{2} \cdot n}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))