Average Error: 52.7 → 35.7
Time: 38.1s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 1.447602491634548 \cdot 10^{+178}:\\ \;\;\;\;\frac{\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right) + \alpha \cdot \beta}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{1.0} + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) - \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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}\;\beta \le 1.447602491634548 \cdot 10^{+178}:\\
\;\;\;\;\frac{\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right) + \alpha \cdot \beta}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{1.0} + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) - \sqrt{1.0}}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r4054278 = i;
        double r4054279 = alpha;
        double r4054280 = beta;
        double r4054281 = r4054279 + r4054280;
        double r4054282 = r4054281 + r4054278;
        double r4054283 = r4054278 * r4054282;
        double r4054284 = r4054280 * r4054279;
        double r4054285 = r4054284 + r4054283;
        double r4054286 = r4054283 * r4054285;
        double r4054287 = 2.0;
        double r4054288 = r4054287 * r4054278;
        double r4054289 = r4054281 + r4054288;
        double r4054290 = r4054289 * r4054289;
        double r4054291 = r4054286 / r4054290;
        double r4054292 = 1.0;
        double r4054293 = r4054290 - r4054292;
        double r4054294 = r4054291 / r4054293;
        return r4054294;
}
↓
double f(double alpha, double beta, double i) {
        double r4054295 = beta;
        double r4054296 = 1.447602491634548e+178;
        bool r4054297 = r4054295 <= r4054296;
        double r4054298 = i;
        double r4054299 = alpha;
        double r4054300 = r4054295 + r4054299;
        double r4054301 = r4054298 + r4054300;
        double r4054302 = r4054298 * r4054301;
        double r4054303 = r4054299 * r4054295;
        double r4054304 = r4054302 + r4054303;
        double r4054305 = 2.0;
        double r4054306 = r4054305 * r4054298;
        double r4054307 = r4054300 + r4054306;
        double r4054308 = r4054304 / r4054307;
        double r4054309 = r4054302 / r4054307;
        double r4054310 = r4054308 * r4054309;
        double r4054311 = 1.0;
        double r4054312 = sqrt(r4054311);
        double r4054313 = r4054312 + r4054307;
        double r4054314 = r4054310 / r4054313;
        double r4054315 = r4054307 - r4054312;
        double r4054316 = r4054314 / r4054315;
        double r4054317 = 0.0;
        double r4054318 = r4054297 ? r4054316 : r4054317;
        return r4054318;
}

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 beta < 1.447602491634548e+178

    1. Initial program 51.1

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

    3. Applied add-sqr-sqrt51.1

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

    4. Applied difference-of-squares51.1

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

    5. Applied times-frac35.8

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

    6. Applied times-frac34.0

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

    8. Applied associate-*r/34.0

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

    10. Applied associate-*l/34.0

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

    if 1.447602491634548e+178 < beta

    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 45.9

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

  4. Final simplification35.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.447602491634548 \cdot 10^{+178}:\\ \;\;\;\;\frac{\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right) + \alpha \cdot \beta}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{1.0} + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) - \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +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)))