Average Error: 52.8 → 35.4
Time: 26.6s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 4.0145625771446497 \cdot 10^{+217}:\\ \;\;\;\;\sqrt{\frac{\frac{\mathsf{fma}\left(\left(i + \left(\beta + \alpha\right)\right), i, \left(\beta \cdot \alpha\right)\right)}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}}{\sqrt{1.0} + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}} \cdot \left(\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right) - \sqrt{1.0}} \cdot \sqrt{\frac{\frac{\sqrt{\mathsf{fma}\left(\left(i + \left(\beta + \alpha\right)\right), i, \left(\beta \cdot \alpha\right)\right)}}{\sqrt{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}} \cdot \frac{\sqrt{\mathsf{fma}\left(\left(i + \left(\beta + \alpha\right)\right), i, \left(\beta \cdot \alpha\right)\right)}}{\sqrt{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}}}{\sqrt{1.0} + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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) - 1.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 4.0145625771446497 \cdot 10^{+217}:\\
\;\;\;\;\sqrt{\frac{\frac{\mathsf{fma}\left(\left(i + \left(\beta + \alpha\right)\right), i, \left(\beta \cdot \alpha\right)\right)}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}}{\sqrt{1.0} + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}} \cdot \left(\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right) - \sqrt{1.0}} \cdot \sqrt{\frac{\frac{\sqrt{\mathsf{fma}\left(\left(i + \left(\beta + \alpha\right)\right), i, \left(\beta \cdot \alpha\right)\right)}}{\sqrt{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}} \cdot \frac{\sqrt{\mathsf{fma}\left(\left(i + \left(\beta + \alpha\right)\right), i, \left(\beta \cdot \alpha\right)\right)}}{\sqrt{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}}}{\sqrt{1.0} + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}}\right)\\

\mathbf{else}:\\
\;\;\;\;0\\

\end{array}
double f(double alpha, double beta, double i) {
        double r2147107 = i;
        double r2147108 = alpha;
        double r2147109 = beta;
        double r2147110 = r2147108 + r2147109;
        double r2147111 = r2147110 + r2147107;
        double r2147112 = r2147107 * r2147111;
        double r2147113 = r2147109 * r2147108;
        double r2147114 = r2147113 + r2147112;
        double r2147115 = r2147112 * r2147114;
        double r2147116 = 2.0;
        double r2147117 = r2147116 * r2147107;
        double r2147118 = r2147110 + r2147117;
        double r2147119 = r2147118 * r2147118;
        double r2147120 = r2147115 / r2147119;
        double r2147121 = 1.0;
        double r2147122 = r2147119 - r2147121;
        double r2147123 = r2147120 / r2147122;
        return r2147123;
}
↓
double f(double alpha, double beta, double i) {
        double r2147124 = alpha;
        double r2147125 = 4.0145625771446497e+217;
        bool r2147126 = r2147124 <= r2147125;
        double r2147127 = i;
        double r2147128 = beta;
        double r2147129 = r2147128 + r2147124;
        double r2147130 = r2147127 + r2147129;
        double r2147131 = r2147128 * r2147124;
        double r2147132 = fma(r2147130, r2147127, r2147131);
        double r2147133 = 2.0;
        double r2147134 = fma(r2147133, r2147127, r2147129);
        double r2147135 = r2147132 / r2147134;
        double r2147136 = 1.0;
        double r2147137 = sqrt(r2147136);
        double r2147138 = r2147137 + r2147134;
        double r2147139 = r2147135 / r2147138;
        double r2147140 = sqrt(r2147139);
        double r2147141 = r2147127 * r2147130;
        double r2147142 = r2147141 / r2147134;
        double r2147143 = r2147134 - r2147137;
        double r2147144 = r2147142 / r2147143;
        double r2147145 = sqrt(r2147132);
        double r2147146 = sqrt(r2147134);
        double r2147147 = r2147145 / r2147146;
        double r2147148 = r2147147 * r2147147;
        double r2147149 = r2147148 / r2147138;
        double r2147150 = sqrt(r2147149);
        double r2147151 = r2147144 * r2147150;
        double r2147152 = r2147140 * r2147151;
        double r2147153 = 0.0;
        double r2147154 = r2147126 ? r2147152 : r2147153;
        return r2147154;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if alpha < 4.0145625771446497e+217

    1. Initial program 51.7

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

    2. Simplified51.7

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

    4. Applied add-sqr-sqrt51.7

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

    5. Applied difference-of-squares51.7

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

    6. Applied times-frac36.8

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

    7. Applied times-frac34.4

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

    9. Applied add-sqr-sqrt34.4

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

    10. Applied associate-*l*34.4

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

    12. Applied add-sqr-sqrt34.5

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

    13. Applied add-sqr-sqrt34.5

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

    14. Applied times-frac34.5

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

    if 4.0145625771446497e+217 < alpha

    1. Initial program 62.6

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

    2. Simplified62.6

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

    3. Taylor expanded around -inf 43.2

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification35.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 4.0145625771446497 \cdot 10^{+217}:\\ \;\;\;\;\sqrt{\frac{\frac{\mathsf{fma}\left(\left(i + \left(\beta + \alpha\right)\right), i, \left(\beta \cdot \alpha\right)\right)}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}}{\sqrt{1.0} + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}} \cdot \left(\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right) - \sqrt{1.0}} \cdot \sqrt{\frac{\frac{\sqrt{\mathsf{fma}\left(\left(i + \left(\beta + \alpha\right)\right), i, \left(\beta \cdot \alpha\right)\right)}}{\sqrt{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}} \cdot \frac{\sqrt{\mathsf{fma}\left(\left(i + \left(\beta + \alpha\right)\right), i, \left(\beta \cdot \alpha\right)\right)}}{\sqrt{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}}}{\sqrt{1.0} + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :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))) 1.0)))