Octave 3.8, jcobi/2

Average Error: 24.1 → 11.0

Time: 23.0s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.63898582180695576 \cdot 10^{156}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}, \frac{\frac{1}{\frac{\frac{\mathsf{fma}\left(2, i, \beta\right) + \alpha}{\alpha + \beta}}{\beta - \alpha}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}, 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 7.99999999999999911 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r143931 = alpha;
        double r143932 = beta;
        double r143933 = r143931 + r143932;
        double r143934 = r143932 - r143931;
        double r143935 = r143933 * r143934;
        double r143936 = 2.0;
        double r143937 = i;
        double r143938 = r143936 * r143937;
        double r143939 = r143933 + r143938;
        double r143940 = r143935 / r143939;
        double r143941 = r143939 + r143936;
        double r143942 = r143940 / r143941;
        double r143943 = 1.0;
        double r143944 = r143942 + r143943;
        double r143945 = r143944 / r143936;
        return r143945;
}

↓

double f(double alpha, double beta, double i) {
        double r143946 = alpha;
        double r143947 = 1.6389858218069558e+156;
        bool r143948 = r143946 <= r143947;
        double r143949 = 1.0;
        double r143950 = beta;
        double r143951 = r143946 + r143950;
        double r143952 = 2.0;
        double r143953 = i;
        double r143954 = r143952 * r143953;
        double r143955 = r143951 + r143954;
        double r143956 = r143955 + r143952;
        double r143957 = sqrt(r143956);
        double r143958 = r143949 / r143957;
        double r143959 = fma(r143952, r143953, r143950);
        double r143960 = r143959 + r143946;
        double r143961 = r143960 / r143951;
        double r143962 = r143950 - r143946;
        double r143963 = r143961 / r143962;
        double r143964 = r143949 / r143963;
        double r143965 = r143964 / r143957;
        double r143966 = 1.0;
        double r143967 = fma(r143958, r143965, r143966);
        double r143968 = r143967 / r143952;
        double r143969 = r143949 / r143946;
        double r143970 = 7.999999999999999;
        double r143971 = 3.0;
        double r143972 = pow(r143946, r143971);
        double r143973 = r143949 / r143972;
        double r143974 = r143970 * r143973;
        double r143975 = 4.0;
        double r143976 = 2.0;
        double r143977 = pow(r143946, r143976);
        double r143978 = r143949 / r143977;
        double r143979 = r143975 * r143978;
        double r143980 = r143974 - r143979;
        double r143981 = fma(r143952, r143969, r143980);
        double r143982 = r143981 / r143952;
        double r143983 = r143948 ? r143968 : r143982;
        return r143983;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if alpha < 1.6389858218069558e+156

   1. Initial program 16.1

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

   3. Applied clear-num 16.1

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

   4. Simplified 5.2

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

   6. Applied add-sqr-sqrt 5.2

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

   7. Applied *-un-lft-identity 5.2

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

   8. Applied *-un-lft-identity 5.2

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

   9. Applied times-frac 5.2

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

   10. Applied *-un-lft-identity 5.2

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

   11. Applied times-frac 5.2

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

   12. Applied times-frac 5.2

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

   13. Applied fma-def 5.2

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

   if 1.6389858218069558e+156 < 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 clear-num 64.0

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

   4. Simplified 47.6

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

   6. Applied add-cube-cbrt 48.2

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

   7. Applied *-un-lft-identity 48.2

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

   8. Applied add-cube-cbrt 47.8

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

   9. Applied *-un-lft-identity 47.8

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

   10. Applied times-frac 47.9

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

   11. Applied times-frac 47.9

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

   12. Applied *-un-lft-identity 47.9

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

   13. Applied times-frac 47.8

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

   14. Applied times-frac 47.8

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

   15. Applied fma-def 47.8

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\frac{1}{\frac{\frac{1}{\sqrt[3]{\alpha + \beta} \cdot \sqrt[3]{\alpha + \beta}}}{1}}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}, \frac{\frac{1}{\frac{\frac{\mathsf{fma}\left(2, i, \beta\right) + \alpha}{\sqrt[3]{\alpha + \beta}}}{\beta - \alpha}}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}, 1\right)}}{2}\]
   16. Using strategy rm

   17. Applied add-cube-cbrt 48.0

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

   18. Applied cbrt-prod 48.1

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

   19. Taylor expanded around inf 39.8

       \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{\alpha} + 7.99999999999999911 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}}{2}\]

   20. Simplified 39.8

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(2, \frac{1}{\alpha}, 7.99999999999999911 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}}{2}\]
3. Recombined 2 regimes into one program.

4. Final simplification 11.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.63898582180695576 \cdot 10^{156}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}, \frac{\frac{1}{\frac{\frac{\mathsf{fma}\left(2, i, \beta\right) + \alpha}{\alpha + \beta}}{\beta - \alpha}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}, 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 7.99999999999999911 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(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))