Octave 3.8, jcobi/2

Average Error: 24.4 → 12.4

Time: 23.7s

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.0834670219855457169593522556106251749 \cdot 10^{150} \lor \neg \left(\alpha \le 2.831345554250201259889587794621430415771 \cdot 10^{263}\right):\\ \;\;\;\;\frac{\sqrt[3]{{\left(\left(\alpha + \beta\right) \cdot \frac{{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right)}^{1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r103973 = alpha;
        double r103974 = beta;
        double r103975 = r103973 + r103974;
        double r103976 = r103974 - r103973;
        double r103977 = r103975 * r103976;
        double r103978 = 2.0;
        double r103979 = i;
        double r103980 = r103978 * r103979;
        double r103981 = r103975 + r103980;
        double r103982 = r103977 / r103981;
        double r103983 = r103981 + r103978;
        double r103984 = r103982 / r103983;
        double r103985 = 1.0;
        double r103986 = r103984 + r103985;
        double r103987 = r103986 / r103978;
        return r103987;
}

↓

double f(double alpha, double beta, double i) {
        double r103988 = alpha;
        double r103989 = 1.0834670219855457e+150;
        bool r103990 = r103988 <= r103989;
        double r103991 = 2.831345554250201e+263;
        bool r103992 = r103988 <= r103991;
        double r103993 = !r103992;
        bool r103994 = r103990 || r103993;
        double r103995 = beta;
        double r103996 = r103988 + r103995;
        double r103997 = r103995 - r103988;
        double r103998 = 2.0;
        double r103999 = i;
        double r104000 = r103998 * r103999;
        double r104001 = r103996 + r104000;
        double r104002 = r103997 / r104001;
        double r104003 = 1.0;
        double r104004 = pow(r104002, r104003);
        double r104005 = r104001 + r103998;
        double r104006 = r104004 / r104005;
        double r104007 = r103996 * r104006;
        double r104008 = 1.0;
        double r104009 = r104007 + r104008;
        double r104010 = 3.0;
        double r104011 = pow(r104009, r104010);
        double r104012 = cbrt(r104011);
        double r104013 = r104012 / r103998;
        double r104014 = r104003 / r103988;
        double r104015 = r103998 * r104014;
        double r104016 = 8.0;
        double r104017 = pow(r103988, r104010);
        double r104018 = r104003 / r104017;
        double r104019 = r104016 * r104018;
        double r104020 = r104015 + r104019;
        double r104021 = 4.0;
        double r104022 = 2.0;
        double r104023 = pow(r103988, r104022);
        double r104024 = r104003 / r104023;
        double r104025 = r104021 * r104024;
        double r104026 = r104020 - r104025;
        double r104027 = r104026 / r103998;
        double r104028 = r103994 ? r104013 : r104027;
        return r104028;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if alpha < 1.0834670219855457e+150

   1. Initial program 16.0

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

   3. Applied *-un-lft-identity 16.0

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

   4. Applied *-un-lft-identity 16.0

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

   5. Applied times-frac 5.4

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

   6. Applied times-frac 5.4

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

   7. Simplified 5.4

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

   9. Applied pow1 5.4

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

   11. Applied add-cbrt-cube 5.4

       \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\left(\alpha + \beta\right) \cdot \frac{{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right)}^{1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right) \cdot \left(\left(\alpha + \beta\right) \cdot \frac{{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right)}^{1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right)\right) \cdot \left(\left(\alpha + \beta\right) \cdot \frac{{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right)}^{1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right)}}}{2}\]

   12. Simplified 5.4

       \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\left(\alpha + \beta\right) \cdot \frac{{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right)}^{1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right)}^{3}}}}{2}\]

   if 1.0834670219855457e+150 < alpha < 2.831345554250201e+263

   1. Initial program 63.5

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

   2. Taylor expanded around inf 41.3

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}}{2}\]

   if 2.831345554250201e+263 < alpha

   1. Initial program 64.0

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

   3. Applied *-un-lft-identity 64.0

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

   4. Applied *-un-lft-identity 64.0

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

   5. Applied times-frac 54.2

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

   6. Applied times-frac 54.4

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

   7. Simplified 54.4

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

   9. Applied add-sqr-sqrt 55.0

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

   10. Applied associate-*l* 55.0

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

4. Final simplification 12.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.0834670219855457169593522556106251749 \cdot 10^{150} \lor \neg \left(\alpha \le 2.831345554250201259889587794621430415771 \cdot 10^{263}\right):\\ \;\;\;\;\frac{\sqrt[3]{{\left(\left(\alpha + \beta\right) \cdot \frac{{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right)}^{1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2)) 1) 2))