Octave 3.8, jcobi/4

Average Error: 54.6 → 12.8

Time: 50.0s

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\beta \le 5.188945744706776056846709445192721671049 \cdot 10^{218}:\\ \;\;\;\;\frac{\left(\beta + \alpha\right) \cdot 0.25 + 0.5 \cdot i}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - \sqrt{1}} \cdot \sqrt[3]{\frac{\frac{i}{\frac{\left(\beta + \alpha\right) + i \cdot 2}{i + \left(\beta + \alpha\right)}}}{\sqrt{1} + \left(\left(\beta + \alpha\right) + i \cdot 2\right)} \cdot \left(\frac{\frac{i}{\frac{\left(\beta + \alpha\right) + i \cdot 2}{i + \left(\beta + \alpha\right)}}}{\sqrt{1} + \left(\left(\beta + \alpha\right) + i \cdot 2\right)} \cdot \frac{\frac{i}{\frac{\left(\beta + \alpha\right) + i \cdot 2}{i + \left(\beta + \alpha\right)}}}{\sqrt{1} + \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\left(\beta + \alpha\right) + i \cdot 2}{i + \left(\beta + \alpha\right)}}}{\sqrt{1} + \left(\left(\beta + \alpha\right) + i \cdot 2\right)} \cdot \frac{i}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - \sqrt{1}}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r5365660 = i;
        double r5365661 = alpha;
        double r5365662 = beta;
        double r5365663 = r5365661 + r5365662;
        double r5365664 = r5365663 + r5365660;
        double r5365665 = r5365660 * r5365664;
        double r5365666 = r5365662 * r5365661;
        double r5365667 = r5365666 + r5365665;
        double r5365668 = r5365665 * r5365667;
        double r5365669 = 2.0;
        double r5365670 = r5365669 * r5365660;
        double r5365671 = r5365663 + r5365670;
        double r5365672 = r5365671 * r5365671;
        double r5365673 = r5365668 / r5365672;
        double r5365674 = 1.0;
        double r5365675 = r5365672 - r5365674;
        double r5365676 = r5365673 / r5365675;
        return r5365676;
}

↓

double f(double alpha, double beta, double i) {
        double r5365677 = beta;
        double r5365678 = 5.188945744706776e+218;
        bool r5365679 = r5365677 <= r5365678;
        double r5365680 = alpha;
        double r5365681 = r5365677 + r5365680;
        double r5365682 = 0.25;
        double r5365683 = r5365681 * r5365682;
        double r5365684 = 0.5;
        double r5365685 = i;
        double r5365686 = r5365684 * r5365685;
        double r5365687 = r5365683 + r5365686;
        double r5365688 = 2.0;
        double r5365689 = r5365685 * r5365688;
        double r5365690 = r5365681 + r5365689;
        double r5365691 = 1.0;
        double r5365692 = sqrt(r5365691);
        double r5365693 = r5365690 - r5365692;
        double r5365694 = r5365687 / r5365693;
        double r5365695 = r5365685 + r5365681;
        double r5365696 = r5365690 / r5365695;
        double r5365697 = r5365685 / r5365696;
        double r5365698 = r5365692 + r5365690;
        double r5365699 = r5365697 / r5365698;
        double r5365700 = r5365699 * r5365699;
        double r5365701 = r5365699 * r5365700;
        double r5365702 = cbrt(r5365701);
        double r5365703 = r5365694 * r5365702;
        double r5365704 = r5365685 / r5365693;
        double r5365705 = r5365699 * r5365704;
        double r5365706 = r5365679 ? r5365703 : r5365705;
        return r5365706;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if beta < 5.188945744706776e+218

   1. Initial program 53.7

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

   3. Applied add-sqr-sqrt 53.7

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

   4. Applied difference-of-squares 53.7

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

   5. Applied times-frac 38.6

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

   6. Applied times-frac 36.2

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

   8. Applied associate-/l* 36.2

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

   9. Taylor expanded around 0 14.4

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

   10. Simplified 14.4

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

   12. Applied add-cbrt-cube 12.6

       \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}\right) \cdot \frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}}} \cdot \frac{0.25 \cdot \left(\alpha + \beta\right) + i \cdot 0.5}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\]

   if 5.188945744706776e+218 < beta

   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-sqrt 64.0

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

   4. Applied difference-of-squares 64.0

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

   5. Applied times-frac 58.1

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

   6. Applied times-frac 57.7

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

   8. Applied associate-/l* 57.7

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

   9. Taylor expanded around inf 13.8

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

4. Final simplification 12.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 5.188945744706776056846709445192721671049 \cdot 10^{218}:\\ \;\;\;\;\frac{\left(\beta + \alpha\right) \cdot 0.25 + 0.5 \cdot i}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - \sqrt{1}} \cdot \sqrt[3]{\frac{\frac{i}{\frac{\left(\beta + \alpha\right) + i \cdot 2}{i + \left(\beta + \alpha\right)}}}{\sqrt{1} + \left(\left(\beta + \alpha\right) + i \cdot 2\right)} \cdot \left(\frac{\frac{i}{\frac{\left(\beta + \alpha\right) + i \cdot 2}{i + \left(\beta + \alpha\right)}}}{\sqrt{1} + \left(\left(\beta + \alpha\right) + i \cdot 2\right)} \cdot \frac{\frac{i}{\frac{\left(\beta + \alpha\right) + i \cdot 2}{i + \left(\beta + \alpha\right)}}}{\sqrt{1} + \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\left(\beta + \alpha\right) + i \cdot 2}{i + \left(\beta + \alpha\right)}}}{\sqrt{1} + \left(\left(\beta + \alpha\right) + i \cdot 2\right)} \cdot \frac{i}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - \sqrt{1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(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)))