Average Error: 54.1 → 36.4
Time: 21.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 1.362422161808940266728771203266420138041 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\left(\alpha + \beta\right) + i}}}{\sqrt{1} + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \left(\frac{\sqrt{\mathsf{fma}\left(i, \left(\beta + i\right) + \alpha, \beta \cdot \alpha\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}} \cdot \frac{\sqrt{\mathsf{fma}\left(i, \left(\beta + i\right) + \alpha, \beta \cdot \alpha\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\\ \end{array}\]
\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}
↓
\begin{array}{l}
\mathbf{if}\;i \le 1.362422161808940266728771203266420138041 \cdot 10^{154}:\\
\;\;\;\;\frac{\frac{i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\left(\alpha + \beta\right) + i}}}{\sqrt{1} + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \left(\frac{\sqrt{\mathsf{fma}\left(i, \left(\beta + i\right) + \alpha, \beta \cdot \alpha\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}} \cdot \frac{\sqrt{\mathsf{fma}\left(i, \left(\beta + i\right) + \alpha, \beta \cdot \alpha\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{0}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r83130 = i;
        double r83131 = alpha;
        double r83132 = beta;
        double r83133 = r83131 + r83132;
        double r83134 = r83133 + r83130;
        double r83135 = r83130 * r83134;
        double r83136 = r83132 * r83131;
        double r83137 = r83136 + r83135;
        double r83138 = r83135 * r83137;
        double r83139 = 2.0;
        double r83140 = r83139 * r83130;
        double r83141 = r83133 + r83140;
        double r83142 = r83141 * r83141;
        double r83143 = r83138 / r83142;
        double r83144 = 1.0;
        double r83145 = r83142 - r83144;
        double r83146 = r83143 / r83145;
        return r83146;
}
↓
double f(double alpha, double beta, double i) {
        double r83147 = i;
        double r83148 = 1.3624221618089403e+154;
        bool r83149 = r83147 <= r83148;
        double r83150 = 2.0;
        double r83151 = alpha;
        double r83152 = beta;
        double r83153 = r83151 + r83152;
        double r83154 = fma(r83150, r83147, r83153);
        double r83155 = r83153 + r83147;
        double r83156 = r83154 / r83155;
        double r83157 = r83147 / r83156;
        double r83158 = 1.0;
        double r83159 = sqrt(r83158);
        double r83160 = r83159 + r83154;
        double r83161 = r83157 / r83160;
        double r83162 = r83152 + r83147;
        double r83163 = r83162 + r83151;
        double r83164 = r83152 * r83151;
        double r83165 = fma(r83147, r83163, r83164);
        double r83166 = sqrt(r83165);
        double r83167 = r83154 - r83159;
        double r83168 = r83166 / r83167;
        double r83169 = r83166 / r83154;
        double r83170 = r83168 * r83169;
        double r83171 = r83161 * r83170;
        double r83172 = 0.0;
        double r83173 = r83150 * r83147;
        double r83174 = r83153 + r83173;
        double r83175 = r83174 * r83174;
        double r83176 = r83172 / r83175;
        double r83177 = r83175 - r83158;
        double r83178 = r83176 / r83177;
        double r83179 = r83149 ? r83171 : r83178;
        return r83179;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 1.3624221618089403e+154

    1. Initial program 44.4

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

    3. Applied add-sqr-sqrt44.4

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

    4. Applied difference-of-squares44.4

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

    5. Applied times-frac16.1

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

    6. Applied times-frac11.8

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

    7. Simplified11.8

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

    8. Simplified16.1

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

    10. Applied add-sqr-sqrt16.1

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

    11. Applied times-frac11.8

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

    13. Applied associate-/l*11.8

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

    if 1.3624221618089403e+154 < i

    1. Initial program 64.0

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

    2. Taylor expanded around 0 61.9

      \[\leadsto \frac{\frac{\color{blue}{0}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification36.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 1.362422161808940266728771203266420138041 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\left(\alpha + \beta\right) + i}}}{\sqrt{1} + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \left(\frac{\sqrt{\mathsf{fma}\left(i, \left(\beta + i\right) + \alpha, \beta \cdot \alpha\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}} \cdot \frac{\sqrt{\mathsf{fma}\left(i, \left(\beta + i\right) + \alpha, \beta \cdot \alpha\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 1))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1)))