Average Error: 52.5 → 10.7
Time: 7.3m
Precision: 64
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 3.9665626390492635 \cdot 10^{+140}:\\ \;\;\;\;\frac{i}{\frac{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + i \cdot 2}}} \cdot \frac{1}{\left(\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}\right) \cdot \frac{1}{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\frac{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + i \cdot 2}}} \cdot \frac{1}{\frac{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}{\frac{1}{4} \cdot \left(\alpha + \beta\right) + i \cdot \frac{1}{2}}}\\ \end{array}\]
double f(double alpha, double beta, double i) {
        double r46171206 = i;
        double r46171207 = alpha;
        double r46171208 = beta;
        double r46171209 = r46171207 + r46171208;
        double r46171210 = r46171209 + r46171206;
        double r46171211 = r46171206 * r46171210;
        double r46171212 = r46171208 * r46171207;
        double r46171213 = r46171212 + r46171211;
        double r46171214 = r46171211 * r46171213;
        double r46171215 = 2.0;
        double r46171216 = r46171215 * r46171206;
        double r46171217 = r46171209 + r46171216;
        double r46171218 = r46171217 * r46171217;
        double r46171219 = r46171214 / r46171218;
        double r46171220 = 1.0;
        double r46171221 = r46171218 - r46171220;
        double r46171222 = r46171219 / r46171221;
        return r46171222;
}
↓
double f(double alpha, double beta, double i) {
        double r46171223 = i;
        double r46171224 = 3.9665626390492635e+140;
        bool r46171225 = r46171223 <= r46171224;
        double r46171226 = 1.0;
        double r46171227 = sqrt(r46171226);
        double r46171228 = alpha;
        double r46171229 = beta;
        double r46171230 = r46171228 + r46171229;
        double r46171231 = 2.0;
        double r46171232 = r46171223 * r46171231;
        double r46171233 = r46171230 + r46171232;
        double r46171234 = r46171227 + r46171233;
        double r46171235 = r46171230 + r46171223;
        double r46171236 = r46171235 / r46171233;
        double r46171237 = r46171234 / r46171236;
        double r46171238 = r46171223 / r46171237;
        double r46171239 = 1.0;
        double r46171240 = r46171233 - r46171227;
        double r46171241 = r46171235 * r46171223;
        double r46171242 = r46171228 * r46171229;
        double r46171243 = r46171241 + r46171242;
        double r46171244 = r46171243 / r46171233;
        double r46171245 = r46171239 / r46171244;
        double r46171246 = r46171240 * r46171245;
        double r46171247 = r46171239 / r46171246;
        double r46171248 = r46171238 * r46171247;
        double r46171249 = 0.25;
        double r46171250 = r46171249 * r46171230;
        double r46171251 = 0.5;
        double r46171252 = r46171223 * r46171251;
        double r46171253 = r46171250 + r46171252;
        double r46171254 = r46171240 / r46171253;
        double r46171255 = r46171239 / r46171254;
        double r46171256 = r46171238 * r46171255;
        double r46171257 = r46171225 ? r46171248 : r46171256;
        return r46171257;
}


\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}\;i \le 3.9665626390492635 \cdot 10^{+140}:\\
\;\;\;\;\frac{i}{\frac{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + i \cdot 2}}} \cdot \frac{1}{\left(\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}\right) \cdot \frac{1}{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{i}{\frac{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + i \cdot 2}}} \cdot \frac{1}{\frac{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}{\frac{1}{4} \cdot \left(\alpha + \beta\right) + i \cdot \frac{1}{2}}}\\

\end{array}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 3.9665626390492635e+140

    1. Initial program 41.2

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

    3. Applied add-sqr-sqrt41.2

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

    4. Applied difference-of-squares41.2

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

    5. Applied times-frac15.0

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

    6. Applied times-frac10.6

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

    8. Applied *-un-lft-identity10.6

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

    9. Applied associate-/l*10.6

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

    11. Applied *-un-lft-identity10.6

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

    12. Applied times-frac10.5

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

    13. Applied associate-/l*10.5

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

    14. Simplified10.5

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

    16. Applied div-inv10.6

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

    if 3.9665626390492635e+140 < i

    1. Initial program 62.1

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

    3. Applied add-sqr-sqrt62.1

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

    4. Applied difference-of-squares62.1

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

    5. Applied times-frac58.8

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

    6. Applied times-frac58.7

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

    8. Applied *-un-lft-identity58.7

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

    9. Applied associate-/l*58.7

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

    11. Applied *-un-lft-identity58.7

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

    12. Applied times-frac58.7

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

    13. Applied associate-/l*58.7

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

    14. Simplified58.7

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

    15. Taylor expanded around 0 10.8

      \[\leadsto \frac{i}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}}{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}} \cdot \frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}{\color{blue}{\frac{1}{2} \cdot i + \left(\frac{1}{4} \cdot \beta + \frac{1}{4} \cdot \alpha\right)}}}\]

    16. Simplified10.8

      \[\leadsto \frac{i}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}}{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}} \cdot \frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}{\color{blue}{\left(\alpha + \beta\right) \cdot \frac{1}{4} + \frac{1}{2} \cdot i}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 3.9665626390492635 \cdot 10^{+140}:\\ \;\;\;\;\frac{i}{\frac{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + i \cdot 2}}} \cdot \frac{1}{\left(\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}\right) \cdot \frac{1}{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\frac{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + i \cdot 2}}} \cdot \frac{1}{\frac{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}{\frac{1}{4} \cdot \left(\alpha + \beta\right) + i \cdot \frac{1}{2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019101 
(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)))