Average Error: 23.8 → 11.6
Time: 1.8m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 6.089121605647622 \cdot 10^{+121}:\\ \;\;\;\;\frac{e^{\log \left(\mathsf{fma}\left(\left(\beta + \alpha\right), \left(\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\right), 1.0\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{1}{\alpha \cdot \alpha}\right), \left(\frac{8.0}{\alpha} - 4.0\right), \left(\frac{2.0}{\alpha}\right)\right)}{2.0}\\ \end{array}\]
\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 6.089121605647622 \cdot 10^{+121}:\\
\;\;\;\;\frac{e^{\log \left(\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)\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 r8704104 = alpha;
        double r8704105 = beta;
        double r8704106 = r8704104 + r8704105;
        double r8704107 = r8704105 - r8704104;
        double r8704108 = r8704106 * r8704107;
        double r8704109 = 2.0;
        double r8704110 = i;
        double r8704111 = r8704109 * r8704110;
        double r8704112 = r8704106 + r8704111;
        double r8704113 = r8704108 / r8704112;
        double r8704114 = 2.0;
        double r8704115 = r8704112 + r8704114;
        double r8704116 = r8704113 / r8704115;
        double r8704117 = 1.0;
        double r8704118 = r8704116 + r8704117;
        double r8704119 = r8704118 / r8704114;
        return r8704119;
}
↓
double f(double alpha, double beta, double i) {
        double r8704120 = alpha;
        double r8704121 = 6.089121605647622e+121;
        bool r8704122 = r8704120 <= r8704121;
        double r8704123 = beta;
        double r8704124 = r8704123 + r8704120;
        double r8704125 = r8704123 - r8704120;
        double r8704126 = 2.0;
        double r8704127 = i;
        double r8704128 = r8704126 * r8704127;
        double r8704129 = r8704124 + r8704128;
        double r8704130 = r8704125 / r8704129;
        double r8704131 = 2.0;
        double r8704132 = r8704131 + r8704129;
        double r8704133 = r8704130 / r8704132;
        double r8704134 = 1.0;
        double r8704135 = fma(r8704124, r8704133, r8704134);
        double r8704136 = log(r8704135);
        double r8704137 = exp(r8704136);
        double r8704138 = r8704137 / r8704131;
        double r8704139 = 1.0;
        double r8704140 = r8704120 * r8704120;
        double r8704141 = r8704139 / r8704140;
        double r8704142 = 8.0;
        double r8704143 = r8704142 / r8704120;
        double r8704144 = 4.0;
        double r8704145 = r8704143 - r8704144;
        double r8704146 = r8704131 / r8704120;
        double r8704147 = fma(r8704141, r8704145, r8704146);
        double r8704148 = r8704147 / r8704131;
        double r8704149 = r8704122 ? r8704138 : r8704148;
        return r8704149;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if alpha < 6.089121605647622e+121

    1. Initial program 14.9

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

    3. Applied *-un-lft-identity14.9

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

    4. Applied *-un-lft-identity14.9

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

    5. Applied times-frac4.0

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

    6. Applied times-frac4.0

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

    7. Applied fma-def4.0

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

    8. Simplified4.0

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

    10. Applied add-exp-log4.0

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

    if 6.089121605647622e+121 < alpha

    1. Initial program 59.8

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

    2. Taylor expanded around -inf 42.2

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

    3. Simplified42.2

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{1}{\alpha \cdot \alpha}\right), \left(\frac{8.0}{\alpha} - 4.0\right), \left(\frac{2.0}{\alpha}\right)\right)}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 6.089121605647622 \cdot 10^{+121}:\\ \;\;\;\;\frac{e^{\log \left(\mathsf{fma}\left(\left(\beta + \alpha\right), \left(\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\right), 1.0\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{1}{\alpha \cdot \alpha}\right), \left(\frac{8.0}{\alpha} - 4.0\right), \left(\frac{2.0}{\alpha}\right)\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 +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))