Average Error: 23.4 → 12.3
Time: 2.9m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.778527467618903 \cdot 10^{+23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\beta + \alpha\right), \left(\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\right), 1.0\right)}{2.0}\\ \mathbf{elif}\;\alpha \le 7.90075049213558 \cdot 10^{+53}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{1}{\alpha \cdot \alpha}\right), \left(\frac{8.0}{\alpha} - 4.0\right), \left(\frac{2.0}{\alpha}\right)\right)}{2.0}\\ \mathbf{elif}\;\alpha \le 4.755359132529547 \cdot 10^{+94}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{\beta + \alpha}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right), \left(\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right), 1.0\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{1}{\alpha \cdot \alpha}\right), \left(\frac{8.0}{\alpha} - 4.0\right), \left(\frac{2.0}{\alpha}\right)\right)}{2.0}\\ \end{array}\]
\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 2.778527467618903 \cdot 10^{+23}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\left(\beta + \alpha\right), \left(\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\right), 1.0\right)}{2.0}\\

\mathbf{elif}\;\alpha \le 7.90075049213558 \cdot 10^{+53}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{1}{\alpha \cdot \alpha}\right), \left(\frac{8.0}{\alpha} - 4.0\right), \left(\frac{2.0}{\alpha}\right)\right)}{2.0}\\

\mathbf{elif}\;\alpha \le 4.755359132529547 \cdot 10^{+94}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{\beta + \alpha}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right), \left(\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right), 1.0\right)}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{1}{\alpha \cdot \alpha}\right), \left(\frac{8.0}{\alpha} - 4.0\right), \left(\frac{2.0}{\alpha}\right)\right)}{2.0}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r9806035 = alpha;
        double r9806036 = beta;
        double r9806037 = r9806035 + r9806036;
        double r9806038 = r9806036 - r9806035;
        double r9806039 = r9806037 * r9806038;
        double r9806040 = 2.0;
        double r9806041 = i;
        double r9806042 = r9806040 * r9806041;
        double r9806043 = r9806037 + r9806042;
        double r9806044 = r9806039 / r9806043;
        double r9806045 = 2.0;
        double r9806046 = r9806043 + r9806045;
        double r9806047 = r9806044 / r9806046;
        double r9806048 = 1.0;
        double r9806049 = r9806047 + r9806048;
        double r9806050 = r9806049 / r9806045;
        return r9806050;
}
↓
double f(double alpha, double beta, double i) {
        double r9806051 = alpha;
        double r9806052 = 2.778527467618903e+23;
        bool r9806053 = r9806051 <= r9806052;
        double r9806054 = beta;
        double r9806055 = r9806054 + r9806051;
        double r9806056 = r9806054 - r9806051;
        double r9806057 = 2.0;
        double r9806058 = i;
        double r9806059 = r9806057 * r9806058;
        double r9806060 = r9806055 + r9806059;
        double r9806061 = r9806056 / r9806060;
        double r9806062 = 2.0;
        double r9806063 = r9806062 + r9806060;
        double r9806064 = r9806061 / r9806063;
        double r9806065 = 1.0;
        double r9806066 = fma(r9806055, r9806064, r9806065);
        double r9806067 = r9806066 / r9806062;
        double r9806068 = 7.90075049213558e+53;
        bool r9806069 = r9806051 <= r9806068;
        double r9806070 = 1.0;
        double r9806071 = r9806051 * r9806051;
        double r9806072 = r9806070 / r9806071;
        double r9806073 = 8.0;
        double r9806074 = r9806073 / r9806051;
        double r9806075 = 4.0;
        double r9806076 = r9806074 - r9806075;
        double r9806077 = r9806062 / r9806051;
        double r9806078 = fma(r9806072, r9806076, r9806077);
        double r9806079 = r9806078 / r9806062;
        double r9806080 = 4.755359132529547e+94;
        bool r9806081 = r9806051 <= r9806080;
        double r9806082 = sqrt(r9806063);
        double r9806083 = r9806055 / r9806082;
        double r9806084 = r9806061 / r9806082;
        double r9806085 = fma(r9806083, r9806084, r9806065);
        double r9806086 = r9806085 / r9806062;
        double r9806087 = r9806081 ? r9806086 : r9806079;
        double r9806088 = r9806069 ? r9806079 : r9806087;
        double r9806089 = r9806053 ? r9806067 : r9806088;
        return r9806089;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 3 regimes

  2. if alpha < 2.778527467618903e+23

    1. Initial program 11.2

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

    3. Applied *-un-lft-identity11.2

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

    4. Applied *-un-lft-identity11.2

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

    5. Applied times-frac0.4

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

    6. Applied times-frac0.4

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

    7. Applied fma-def0.4

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

    8. Simplified0.4

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

    if 2.778527467618903e+23 < alpha < 7.90075049213558e+53 or 4.755359132529547e+94 < alpha

    1. Initial program 53.7

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

    2. Taylor expanded around inf 42.0

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

    3. Simplified42.0

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{1}{\alpha \cdot \alpha}\right), \left(\frac{8.0}{\alpha} - 4.0\right), \left(\frac{2.0}{\alpha}\right)\right)}}{2.0}\]

    if 7.90075049213558e+53 < alpha < 4.755359132529547e+94

    1. Initial program 37.6

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

    3. Applied add-sqr-sqrt37.6

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

    4. Applied *-un-lft-identity37.6

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

    5. Applied times-frac26.6

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

    6. Applied times-frac26.6

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

    7. Applied fma-def26.5

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{\frac{\alpha + \beta}{1}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right), \left(\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right), 1.0\right)}}{2.0}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification12.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.778527467618903 \cdot 10^{+23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\beta + \alpha\right), \left(\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\right), 1.0\right)}{2.0}\\ \mathbf{elif}\;\alpha \le 7.90075049213558 \cdot 10^{+53}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{1}{\alpha \cdot \alpha}\right), \left(\frac{8.0}{\alpha} - 4.0\right), \left(\frac{2.0}{\alpha}\right)\right)}{2.0}\\ \mathbf{elif}\;\alpha \le 4.755359132529547 \cdot 10^{+94}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{\beta + \alpha}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right), \left(\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right), 1.0\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{1}{\alpha \cdot \alpha}\right), \left(\frac{8.0}{\alpha} - 4.0\right), \left(\frac{2.0}{\alpha}\right)\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(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))