Average Error: 24.4 → 11.5
Time: 43.6s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.857471580661007580128468637249079462587 \cdot 10^{202}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{\sqrt{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}{\sqrt{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} \cdot \left(\beta + \alpha\right) + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)} + \left(\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1.857471580661007580128468637249079462587 \cdot 10^{202}:\\
\;\;\;\;\frac{\frac{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{\sqrt{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}{\sqrt{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} \cdot \left(\beta + \alpha\right) + 1}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)} + \left(\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r6079090 = alpha;
        double r6079091 = beta;
        double r6079092 = r6079090 + r6079091;
        double r6079093 = r6079091 - r6079090;
        double r6079094 = r6079092 * r6079093;
        double r6079095 = 2.0;
        double r6079096 = i;
        double r6079097 = r6079095 * r6079096;
        double r6079098 = r6079092 + r6079097;
        double r6079099 = r6079094 / r6079098;
        double r6079100 = r6079098 + r6079095;
        double r6079101 = r6079099 / r6079100;
        double r6079102 = 1.0;
        double r6079103 = r6079101 + r6079102;
        double r6079104 = r6079103 / r6079095;
        return r6079104;
}
↓
double f(double alpha, double beta, double i) {
        double r6079105 = alpha;
        double r6079106 = 1.8574715806610076e+202;
        bool r6079107 = r6079105 <= r6079106;
        double r6079108 = beta;
        double r6079109 = r6079108 - r6079105;
        double r6079110 = i;
        double r6079111 = 2.0;
        double r6079112 = r6079110 * r6079111;
        double r6079113 = r6079108 + r6079105;
        double r6079114 = r6079112 + r6079113;
        double r6079115 = r6079109 / r6079114;
        double r6079116 = r6079111 + r6079114;
        double r6079117 = sqrt(r6079116);
        double r6079118 = r6079115 / r6079117;
        double r6079119 = r6079118 / r6079117;
        double r6079120 = r6079119 * r6079113;
        double r6079121 = 1.0;
        double r6079122 = r6079120 + r6079121;
        double r6079123 = r6079122 / r6079111;
        double r6079124 = 8.0;
        double r6079125 = r6079105 * r6079105;
        double r6079126 = r6079105 * r6079125;
        double r6079127 = r6079124 / r6079126;
        double r6079128 = r6079111 / r6079105;
        double r6079129 = 4.0;
        double r6079130 = r6079129 / r6079125;
        double r6079131 = r6079128 - r6079130;
        double r6079132 = r6079127 + r6079131;
        double r6079133 = r6079132 / r6079111;
        double r6079134 = r6079107 ? r6079123 : r6079133;
        return r6079134;
}

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 < 1.8574715806610076e+202

    1. Initial program 19.3

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

    3. Applied *-un-lft-identity19.3

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

    4. Applied *-un-lft-identity19.3

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

    5. Applied times-frac7.7

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

    6. Applied times-frac7.7

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

    7. Simplified7.7

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

    9. Applied add-sqr-sqrt7.7

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

    10. Applied associate-/r*7.7

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

    if 1.8574715806610076e+202 < alpha

    1. Initial program 64.0

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

    3. Applied *-un-lft-identity64.0

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

    4. Applied *-un-lft-identity64.0

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

    5. Applied times-frac50.7

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

    6. Applied times-frac50.8

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

    7. Simplified50.8

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

    8. Taylor expanded around inf 40.8

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}}{2}\]

    9. Simplified40.8

      \[\leadsto \frac{\color{blue}{\frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)} + \left(\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.857471580661007580128468637249079462587 \cdot 10^{202}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{\sqrt{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}{\sqrt{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} \cdot \left(\beta + \alpha\right) + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)} + \left(\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1.0) (> beta -1.0) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))