Octave 3.8, jcobi/2

Average Error: 23.2 → 12.0

Time: 33.5s

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]

Math

\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.6141649384551372 \cdot 10^{+41}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0\right) \cdot \left(\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0\right)\right)}}{2.0}\\ \mathbf{elif}\;\alpha \le 5.9556047485876425 \cdot 10^{+87}:\\ \;\;\;\;\frac{\left(\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{2.0}{\alpha}}{2.0}\\ \mathbf{elif}\;\alpha \le 1.9640182087305322 \cdot 10^{+237}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} \cdot \left(\left(\beta + \alpha\right) \cdot \left(\sqrt[3]{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} \cdot \sqrt[3]{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}\right)\right) + 1.0}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{2.0}{\alpha}}{2.0}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r6509939 = alpha;
        double r6509940 = beta;
        double r6509941 = r6509939 + r6509940;
        double r6509942 = r6509940 - r6509939;
        double r6509943 = r6509941 * r6509942;
        double r6509944 = 2.0;
        double r6509945 = i;
        double r6509946 = r6509944 * r6509945;
        double r6509947 = r6509941 + r6509946;
        double r6509948 = r6509943 / r6509947;
        double r6509949 = 2.0;
        double r6509950 = r6509947 + r6509949;
        double r6509951 = r6509948 / r6509950;
        double r6509952 = 1.0;
        double r6509953 = r6509951 + r6509952;
        double r6509954 = r6509953 / r6509949;
        return r6509954;
}

↓

double f(double alpha, double beta, double i) {
        double r6509955 = alpha;
        double r6509956 = 2.6141649384551372e+41;
        bool r6509957 = r6509955 <= r6509956;
        double r6509958 = beta;
        double r6509959 = r6509958 + r6509955;
        double r6509960 = r6509958 - r6509955;
        double r6509961 = i;
        double r6509962 = 2.0;
        double r6509963 = r6509961 * r6509962;
        double r6509964 = r6509963 + r6509959;
        double r6509965 = r6509960 / r6509964;
        double r6509966 = 2.0;
        double r6509967 = r6509966 + r6509964;
        double r6509968 = r6509965 / r6509967;
        double r6509969 = r6509959 * r6509968;
        double r6509970 = 1.0;
        double r6509971 = r6509969 + r6509970;
        double r6509972 = r6509971 * r6509971;
        double r6509973 = r6509971 * r6509972;
        double r6509974 = cbrt(r6509973);
        double r6509975 = r6509974 / r6509966;
        double r6509976 = 5.9556047485876425e+87;
        bool r6509977 = r6509955 <= r6509976;
        double r6509978 = 8.0;
        double r6509979 = r6509955 * r6509955;
        double r6509980 = r6509955 * r6509979;
        double r6509981 = r6509978 / r6509980;
        double r6509982 = 4.0;
        double r6509983 = r6509982 / r6509979;
        double r6509984 = r6509981 - r6509983;
        double r6509985 = r6509966 / r6509955;
        double r6509986 = r6509984 + r6509985;
        double r6509987 = r6509986 / r6509966;
        double r6509988 = 1.9640182087305322e+237;
        bool r6509989 = r6509955 <= r6509988;
        double r6509990 = cbrt(r6509968);
        double r6509991 = r6509990 * r6509990;
        double r6509992 = r6509959 * r6509991;
        double r6509993 = r6509990 * r6509992;
        double r6509994 = r6509993 + r6509970;
        double r6509995 = r6509994 / r6509966;
        double r6509996 = r6509989 ? r6509995 : r6509987;
        double r6509997 = r6509977 ? r6509987 : r6509996;
        double r6509998 = r6509957 ? r6509975 : r6509997;
        return r6509998;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if alpha < 2.6141649384551372e+41

   1. Initial program 11.3

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 11.3

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)}} + 1.0}{2.0}\]

   4. Applied *-un-lft-identity 11.3

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

   5. Applied times-frac 1.2

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

   6. Applied times-frac 1.2

      \[\leadsto \frac{\color{blue}{\frac{\frac{\alpha + \beta}{1}}{1} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

   7. Simplified 1.2

      \[\leadsto \frac{\color{blue}{\left(\alpha + \beta\right)} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
   8. Using strategy rm

   9. Applied add-cbrt-cube 1.2

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right) \cdot \left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right)\right) \cdot \left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right)}}}{2.0}\]

   if 2.6141649384551372e+41 < alpha < 5.9556047485876425e+87 or 1.9640182087305322e+237 < alpha

   1. Initial program 53.0

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 53.0

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)}} + 1.0}{2.0}\]

   4. Applied *-un-lft-identity 53.0

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

   5. Applied times-frac 42.1

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

   6. Applied times-frac 42.1

      \[\leadsto \frac{\color{blue}{\frac{\frac{\alpha + \beta}{1}}{1} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

   7. Simplified 42.1

      \[\leadsto \frac{\color{blue}{\left(\alpha + \beta\right)} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

   8. Taylor expanded around inf 40.8

      \[\leadsto \frac{\color{blue}{\left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right) - 4.0 \cdot \frac{1}{{\alpha}^{2}}}}{2.0}\]

   9. Simplified 40.8

      \[\leadsto \frac{\color{blue}{\left(\frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{2.0}{\alpha}}}{2.0}\]

   if 5.9556047485876425e+87 < alpha < 1.9640182087305322e+237

   1. Initial program 53.0

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 53.0

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)}} + 1.0}{2.0}\]

   4. Applied *-un-lft-identity 53.0

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

   5. Applied times-frac 37.6

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

   6. Applied times-frac 37.5

      \[\leadsto \frac{\color{blue}{\frac{\frac{\alpha + \beta}{1}}{1} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

   7. Simplified 37.5

      \[\leadsto \frac{\color{blue}{\left(\alpha + \beta\right)} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 37.5

      \[\leadsto \frac{\left(\alpha + \beta\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \sqrt[3]{\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right) \cdot \sqrt[3]{\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right)} + 1.0}{2.0}\]

   10. Applied associate-*r* 37.5

       \[\leadsto \frac{\color{blue}{\left(\left(\alpha + \beta\right) \cdot \left(\sqrt[3]{\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \sqrt[3]{\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right)\right) \cdot \sqrt[3]{\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}} + 1.0}{2.0}\]
3. Recombined 3 regimes into one program.

4. Final simplification 12.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.6141649384551372 \cdot 10^{+41}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0\right) \cdot \left(\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0\right)\right)}}{2.0}\\ \mathbf{elif}\;\alpha \le 5.9556047485876425 \cdot 10^{+87}:\\ \;\;\;\;\frac{\left(\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{2.0}{\alpha}}{2.0}\\ \mathbf{elif}\;\alpha \le 1.9640182087305322 \cdot 10^{+237}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} \cdot \left(\left(\beta + \alpha\right) \cdot \left(\sqrt[3]{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} \cdot \sqrt[3]{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}\right)\right) + 1.0}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{2.0}{\alpha}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))