Octave 3.8, jcobi/4

Average Error: 54.1 → 10.9

Time: 32.6s

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]

Math

\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]

↓

\[\begin{array}{l} \mathbf{if}\;i \le 2.31369047250441019949697212907331269344 \cdot 10^{144}:\\ \;\;\;\;\frac{\sqrt{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}{\frac{\sqrt{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}}{\left(\alpha + \beta\right) + 2 \cdot i}}} \cdot \frac{i}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}{i \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}} \cdot \frac{0.25 \cdot \alpha + \left(0.5 \cdot i + 0.25 \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r84871 = i;
        double r84872 = alpha;
        double r84873 = beta;
        double r84874 = r84872 + r84873;
        double r84875 = r84874 + r84871;
        double r84876 = r84871 * r84875;
        double r84877 = r84873 * r84872;
        double r84878 = r84877 + r84876;
        double r84879 = r84876 * r84878;
        double r84880 = 2.0;
        double r84881 = r84880 * r84871;
        double r84882 = r84874 + r84881;
        double r84883 = r84882 * r84882;
        double r84884 = r84879 / r84883;
        double r84885 = 1.0;
        double r84886 = r84883 - r84885;
        double r84887 = r84884 / r84886;
        return r84887;
}

↓

double f(double alpha, double beta, double i) {
        double r84888 = i;
        double r84889 = 2.3136904725044102e+144;
        bool r84890 = r84888 <= r84889;
        double r84891 = beta;
        double r84892 = alpha;
        double r84893 = r84891 * r84892;
        double r84894 = r84892 + r84891;
        double r84895 = r84894 + r84888;
        double r84896 = r84888 * r84895;
        double r84897 = r84893 + r84896;
        double r84898 = sqrt(r84897);
        double r84899 = 2.0;
        double r84900 = r84899 * r84888;
        double r84901 = r84894 + r84900;
        double r84902 = 1.0;
        double r84903 = sqrt(r84902);
        double r84904 = r84901 - r84903;
        double r84905 = r84898 / r84901;
        double r84906 = r84904 / r84905;
        double r84907 = r84898 / r84906;
        double r84908 = r84901 + r84903;
        double r84909 = r84895 / r84901;
        double r84910 = r84908 / r84909;
        double r84911 = r84888 / r84910;
        double r84912 = r84907 * r84911;
        double r84913 = 1.0;
        double r84914 = r84888 * r84909;
        double r84915 = r84908 / r84914;
        double r84916 = r84913 / r84915;
        double r84917 = 0.25;
        double r84918 = r84917 * r84892;
        double r84919 = 0.5;
        double r84920 = r84919 * r84888;
        double r84921 = r84917 * r84891;
        double r84922 = r84920 + r84921;
        double r84923 = r84918 + r84922;
        double r84924 = r84923 / r84904;
        double r84925 = r84916 * r84924;
        double r84926 = r84890 ? r84912 : r84925;
        return r84926;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if i < 2.3136904725044102e+144

   1. Initial program 42.8

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 42.8

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \color{blue}{\sqrt{1} \cdot \sqrt{1}}}\]

   4. Applied difference-of-squares 42.8

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}\right)}}\]

   5. Applied times-frac 15.1

      \[\leadsto \frac{\color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}\right)}\]

   6. Applied times-frac 10.6

      \[\leadsto \color{blue}{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 10.6

      \[\leadsto \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\]

   9. Applied times-frac 10.6

      \[\leadsto \frac{\color{blue}{\frac{i}{1} \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\]

   10. Applied associate-/l* 10.6

       \[\leadsto \color{blue}{\frac{\frac{i}{1}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\]
   11. Using strategy rm

   12. Applied *-un-lft-identity 10.6

       \[\leadsto \frac{\frac{i}{1}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\]

   13. Applied add-sqr-sqrt 10.6

       \[\leadsto \frac{\frac{i}{1}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}} \cdot \frac{\frac{\color{blue}{\sqrt{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)} \cdot \sqrt{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}}}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\]

   14. Applied times-frac 10.6

       \[\leadsto \frac{\frac{i}{1}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}} \cdot \frac{\color{blue}{\frac{\sqrt{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}}{1} \cdot \frac{\sqrt{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\]

   15. Applied associate-/l* 10.6

       \[\leadsto \frac{\frac{i}{1}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}} \cdot \color{blue}{\frac{\frac{\sqrt{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}}{1}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}{\frac{\sqrt{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}}{\left(\alpha + \beta\right) + 2 \cdot i}}}}\]

   if 2.3136904725044102e+144 < i

   1. Initial program 64.0

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 64.0

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \color{blue}{\sqrt{1} \cdot \sqrt{1}}}\]

   4. Applied difference-of-squares 64.0

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}\right)}}\]

   5. Applied times-frac 61.8

      \[\leadsto \frac{\color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}\right)}\]

   6. Applied times-frac 61.6

      \[\leadsto \color{blue}{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 61.6

      \[\leadsto \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\]

   9. Applied times-frac 61.6

      \[\leadsto \frac{\color{blue}{\frac{i}{1} \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\]

   10. Applied associate-/l* 61.6

       \[\leadsto \color{blue}{\frac{\frac{i}{1}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\]

   11. Taylor expanded around 0 11.2

       \[\leadsto \frac{\frac{i}{1}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}} \cdot \frac{\color{blue}{0.25 \cdot \alpha + \left(0.5 \cdot i + 0.25 \cdot \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\]
   12. Using strategy rm

   13. Applied clear-num 11.2

       \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\frac{i}{1}}}} \cdot \frac{0.25 \cdot \alpha + \left(0.5 \cdot i + 0.25 \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\]

   14. Simplified 11.1

       \[\leadsto \frac{1}{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}{i \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}}} \cdot \frac{0.25 \cdot \alpha + \left(0.5 \cdot i + 0.25 \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 10.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 2.31369047250441019949697212907331269344 \cdot 10^{144}:\\ \;\;\;\;\frac{\sqrt{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}{\frac{\sqrt{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}}{\left(\alpha + \beta\right) + 2 \cdot i}}} \cdot \frac{i}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}{i \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}} \cdot \frac{0.25 \cdot \alpha + \left(0.5 \cdot i + 0.25 \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 1))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1)))