Average Error: 23.5 → 11.0
Time: 1.9m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 7.05557034354749 \cdot 10^{+237}:\\ \;\;\;\;\frac{e^{\log \left(\mathsf{fma}\left(\left(\beta + \alpha\right), \left(\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\right), 1.0\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{1}{\alpha \cdot \alpha}\right), \left(\frac{8.0}{\alpha} - 4.0\right), \left(\frac{2.0}{\alpha}\right)\right)}{2.0}\\ \end{array}\]
\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 7.05557034354749 \cdot 10^{+237}:\\
\;\;\;\;\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 r8779100 = alpha;
        double r8779101 = beta;
        double r8779102 = r8779100 + r8779101;
        double r8779103 = r8779101 - r8779100;
        double r8779104 = r8779102 * r8779103;
        double r8779105 = 2.0;
        double r8779106 = i;
        double r8779107 = r8779105 * r8779106;
        double r8779108 = r8779102 + r8779107;
        double r8779109 = r8779104 / r8779108;
        double r8779110 = 2.0;
        double r8779111 = r8779108 + r8779110;
        double r8779112 = r8779109 / r8779111;
        double r8779113 = 1.0;
        double r8779114 = r8779112 + r8779113;
        double r8779115 = r8779114 / r8779110;
        return r8779115;
}
↓
double f(double alpha, double beta, double i) {
        double r8779116 = alpha;
        double r8779117 = 7.05557034354749e+237;
        bool r8779118 = r8779116 <= r8779117;
        double r8779119 = beta;
        double r8779120 = r8779119 + r8779116;
        double r8779121 = r8779119 - r8779116;
        double r8779122 = 2.0;
        double r8779123 = i;
        double r8779124 = r8779122 * r8779123;
        double r8779125 = r8779120 + r8779124;
        double r8779126 = r8779121 / r8779125;
        double r8779127 = 2.0;
        double r8779128 = r8779127 + r8779125;
        double r8779129 = r8779126 / r8779128;
        double r8779130 = 1.0;
        double r8779131 = fma(r8779120, r8779129, r8779130);
        double r8779132 = log(r8779131);
        double r8779133 = exp(r8779132);
        double r8779134 = r8779133 / r8779127;
        double r8779135 = 1.0;
        double r8779136 = r8779116 * r8779116;
        double r8779137 = r8779135 / r8779136;
        double r8779138 = 8.0;
        double r8779139 = r8779138 / r8779116;
        double r8779140 = 4.0;
        double r8779141 = r8779139 - r8779140;
        double r8779142 = r8779127 / r8779116;
        double r8779143 = fma(r8779137, r8779141, r8779142);
        double r8779144 = r8779143 / r8779127;
        double r8779145 = r8779118 ? r8779134 : r8779144;
        return r8779145;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if alpha < 7.05557034354749e+237

    1. Initial program 20.3

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

    3. Applied *-un-lft-identity20.3

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

    4. Applied *-un-lft-identity20.3

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

    5. Applied times-frac8.7

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

    6. Applied times-frac8.6

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

    7. Applied fma-def8.7

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

    8. Simplified8.7

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

    10. Applied add-exp-log8.6

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

    if 7.05557034354749e+237 < alpha

    1. Initial program 63.2

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

    2. Taylor expanded around inf 40.5

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

    3. Simplified40.5

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

  4. Final simplification11.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 7.05557034354749 \cdot 10^{+237}:\\ \;\;\;\;\frac{e^{\log \left(\mathsf{fma}\left(\left(\beta + \alpha\right), \left(\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\right), 1.0\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{1}{\alpha \cdot \alpha}\right), \left(\frac{8.0}{\alpha} - 4.0\right), \left(\frac{2.0}{\alpha}\right)\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 +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))