Average Error: 54.6 → 11.4
Time: 49.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 7.435536344855911057541128638393389862856 \cdot 10^{121}:\\ \;\;\;\;\left(\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1}} \cdot \sqrt{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}\right) \cdot \frac{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}{\sqrt{1} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}{\sqrt{1} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)} \cdot \frac{\mathsf{fma}\left(0.25, \alpha + \beta, 0.5 \cdot i\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1}}\right)}\\ \end{array}\]
\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 7.435536344855911057541128638393389862856 \cdot 10^{121}:\\
\;\;\;\;\left(\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1}} \cdot \sqrt{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}\right) \cdot \frac{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}{\sqrt{1} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}\\

\mathbf{else}:\\
\;\;\;\;e^{\log \left(\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}{\sqrt{1} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)} \cdot \frac{\mathsf{fma}\left(0.25, \alpha + \beta, 0.5 \cdot i\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1}}\right)}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r4305152 = i;
        double r4305153 = alpha;
        double r4305154 = beta;
        double r4305155 = r4305153 + r4305154;
        double r4305156 = r4305155 + r4305152;
        double r4305157 = r4305152 * r4305156;
        double r4305158 = r4305154 * r4305153;
        double r4305159 = r4305158 + r4305157;
        double r4305160 = r4305157 * r4305159;
        double r4305161 = 2.0;
        double r4305162 = r4305161 * r4305152;
        double r4305163 = r4305155 + r4305162;
        double r4305164 = r4305163 * r4305163;
        double r4305165 = r4305160 / r4305164;
        double r4305166 = 1.0;
        double r4305167 = r4305164 - r4305166;
        double r4305168 = r4305165 / r4305167;
        return r4305168;
}
↓
double f(double alpha, double beta, double i) {
        double r4305169 = i;
        double r4305170 = 7.435536344855911e+121;
        bool r4305171 = r4305169 <= r4305170;
        double r4305172 = alpha;
        double r4305173 = beta;
        double r4305174 = r4305172 + r4305173;
        double r4305175 = r4305174 + r4305169;
        double r4305176 = r4305175 * r4305169;
        double r4305177 = r4305172 * r4305173;
        double r4305178 = r4305176 + r4305177;
        double r4305179 = sqrt(r4305178);
        double r4305180 = 2.0;
        double r4305181 = r4305169 * r4305180;
        double r4305182 = r4305174 + r4305181;
        double r4305183 = r4305179 / r4305182;
        double r4305184 = 1.0;
        double r4305185 = sqrt(r4305184);
        double r4305186 = r4305182 - r4305185;
        double r4305187 = r4305183 / r4305186;
        double r4305188 = r4305187 * r4305179;
        double r4305189 = r4305182 / r4305175;
        double r4305190 = r4305169 / r4305189;
        double r4305191 = r4305185 + r4305182;
        double r4305192 = r4305190 / r4305191;
        double r4305193 = r4305188 * r4305192;
        double r4305194 = 0.25;
        double r4305195 = 0.5;
        double r4305196 = r4305195 * r4305169;
        double r4305197 = fma(r4305194, r4305174, r4305196);
        double r4305198 = r4305197 / r4305186;
        double r4305199 = r4305192 * r4305198;
        double r4305200 = log(r4305199);
        double r4305201 = exp(r4305200);
        double r4305202 = r4305171 ? r4305193 : r4305201;
        return r4305202;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 7.435536344855911e+121

    1. Initial program 39.9

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

    3. Applied add-sqr-sqrt39.9

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

    4. Applied difference-of-squares39.9

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

    5. Applied times-frac15.4

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

    6. Applied times-frac10.3

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

    8. Applied associate-/l*10.3

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

    10. Applied *-un-lft-identity10.3

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

    11. Applied *-un-lft-identity10.3

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

    12. Applied add-sqr-sqrt10.3

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

    13. Applied times-frac10.3

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

    14. Applied times-frac10.3

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

    if 7.435536344855911e+121 < 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. Using strategy rm

    3. Applied add-sqr-sqrt64.0

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

    4. Applied difference-of-squares64.0

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

    5. Applied times-frac56.4

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

    6. Applied times-frac56.1

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

    8. Applied associate-/l*56.1

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

    9. Taylor expanded around 0 12.0

      \[\leadsto \frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\color{blue}{0.5 \cdot i + \left(0.25 \cdot \beta + 0.25 \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\]

    10. Simplified12.0

      \[\leadsto \frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\color{blue}{\mathsf{fma}\left(0.25, \alpha + \beta, 0.5 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\]
    11. Using strategy rm

    12. Applied add-exp-log12.0

      \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\mathsf{fma}\left(0.25, \alpha + \beta, 0.5 \cdot i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 7.435536344855911057541128638393389862856 \cdot 10^{121}:\\ \;\;\;\;\left(\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1}} \cdot \sqrt{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}\right) \cdot \frac{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}{\sqrt{1} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}{\sqrt{1} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)} \cdot \frac{\mathsf{fma}\left(0.25, \alpha + \beta, 0.5 \cdot i\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (> alpha -1.0) (> beta -1.0) (> i 1.0))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i)))) (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))