Average Error: 52.5 → 35.4
Time: 1.1m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.5516067629565482 \cdot 10^{+212}:\\ \;\;\;\;\frac{i}{\frac{\sqrt{1.0} + \left(2 \cdot i + \left(\alpha + \beta\right)\right)}{\frac{i + \left(\alpha + \beta\right)}{2 \cdot i + \left(\alpha + \beta\right)}}} \cdot \left(\frac{\sqrt{\frac{\left(i + \left(\alpha + \beta\right)\right) \cdot i + \alpha \cdot \beta}{2 \cdot i + \left(\alpha + \beta\right)}}}{\sqrt{\left(2 \cdot i + \left(\alpha + \beta\right)\right) - \sqrt{1.0}}} \cdot \frac{\sqrt{\frac{\left(i + \left(\alpha + \beta\right)\right) \cdot i + \alpha \cdot \beta}{2 \cdot i + \left(\alpha + \beta\right)}}}{\sqrt{\left(2 \cdot i + \left(\alpha + \beta\right)\right) - \sqrt{1.0}}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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 2.5516067629565482 \cdot 10^{+212}:\\
\;\;\;\;\frac{i}{\frac{\sqrt{1.0} + \left(2 \cdot i + \left(\alpha + \beta\right)\right)}{\frac{i + \left(\alpha + \beta\right)}{2 \cdot i + \left(\alpha + \beta\right)}}} \cdot \left(\frac{\sqrt{\frac{\left(i + \left(\alpha + \beta\right)\right) \cdot i + \alpha \cdot \beta}{2 \cdot i + \left(\alpha + \beta\right)}}}{\sqrt{\left(2 \cdot i + \left(\alpha + \beta\right)\right) - \sqrt{1.0}}} \cdot \frac{\sqrt{\frac{\left(i + \left(\alpha + \beta\right)\right) \cdot i + \alpha \cdot \beta}{2 \cdot i + \left(\alpha + \beta\right)}}}{\sqrt{\left(2 \cdot i + \left(\alpha + \beta\right)\right) - \sqrt{1.0}}}\right)\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r3788201 = i;
        double r3788202 = alpha;
        double r3788203 = beta;
        double r3788204 = r3788202 + r3788203;
        double r3788205 = r3788204 + r3788201;
        double r3788206 = r3788201 * r3788205;
        double r3788207 = r3788203 * r3788202;
        double r3788208 = r3788207 + r3788206;
        double r3788209 = r3788206 * r3788208;
        double r3788210 = 2.0;
        double r3788211 = r3788210 * r3788201;
        double r3788212 = r3788204 + r3788211;
        double r3788213 = r3788212 * r3788212;
        double r3788214 = r3788209 / r3788213;
        double r3788215 = 1.0;
        double r3788216 = r3788213 - r3788215;
        double r3788217 = r3788214 / r3788216;
        return r3788217;
}
↓
double f(double alpha, double beta, double i) {
        double r3788218 = alpha;
        double r3788219 = 2.5516067629565482e+212;
        bool r3788220 = r3788218 <= r3788219;
        double r3788221 = i;
        double r3788222 = 1.0;
        double r3788223 = sqrt(r3788222);
        double r3788224 = 2.0;
        double r3788225 = r3788224 * r3788221;
        double r3788226 = beta;
        double r3788227 = r3788218 + r3788226;
        double r3788228 = r3788225 + r3788227;
        double r3788229 = r3788223 + r3788228;
        double r3788230 = r3788221 + r3788227;
        double r3788231 = r3788230 / r3788228;
        double r3788232 = r3788229 / r3788231;
        double r3788233 = r3788221 / r3788232;
        double r3788234 = r3788230 * r3788221;
        double r3788235 = r3788218 * r3788226;
        double r3788236 = r3788234 + r3788235;
        double r3788237 = r3788236 / r3788228;
        double r3788238 = sqrt(r3788237);
        double r3788239 = r3788228 - r3788223;
        double r3788240 = sqrt(r3788239);
        double r3788241 = r3788238 / r3788240;
        double r3788242 = r3788241 * r3788241;
        double r3788243 = r3788233 * r3788242;
        double r3788244 = 0.0;
        double r3788245 = r3788220 ? r3788243 : r3788244;
        return r3788245;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if alpha < 2.5516067629565482e+212

    1. Initial program 51.4

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

    3. Applied add-sqr-sqrt51.4

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

    4. Applied difference-of-squares51.4

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

    5. Applied times-frac37.2

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

    6. Applied times-frac34.6

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

    8. Applied add-sqr-sqrt34.7

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

    9. Applied add-sqr-sqrt34.6

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

    10. Applied times-frac34.6

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

    12. Applied *-un-lft-identity34.6

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

    13. Applied times-frac34.6

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

    14. Applied associate-/l*34.6

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

    if 2.5516067629565482e+212 < 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. Taylor expanded around inf 43.0

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

  4. Final simplification35.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.5516067629565482 \cdot 10^{+212}:\\ \;\;\;\;\frac{i}{\frac{\sqrt{1.0} + \left(2 \cdot i + \left(\alpha + \beta\right)\right)}{\frac{i + \left(\alpha + \beta\right)}{2 \cdot i + \left(\alpha + \beta\right)}}} \cdot \left(\frac{\sqrt{\frac{\left(i + \left(\alpha + \beta\right)\right) \cdot i + \alpha \cdot \beta}{2 \cdot i + \left(\alpha + \beta\right)}}}{\sqrt{\left(2 \cdot i + \left(\alpha + \beta\right)\right) - \sqrt{1.0}}} \cdot \frac{\sqrt{\frac{\left(i + \left(\alpha + \beta\right)\right) \cdot i + \alpha \cdot \beta}{2 \cdot i + \left(\alpha + \beta\right)}}}{\sqrt{\left(2 \cdot i + \left(\alpha + \beta\right)\right) - \sqrt{1.0}}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 +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)))