Octave 3.8, jcobi/4

Average Error: 54.1 → 36.7

Time: 31.6s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\beta \le 1.440748976813595748532594826853542348803 \cdot 10^{203}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}}}{\sqrt{\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}}} \cdot \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\frac{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}}}{\sqrt{\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}} \cdot \mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r194522 = i;
        double r194523 = alpha;
        double r194524 = beta;
        double r194525 = r194523 + r194524;
        double r194526 = r194525 + r194522;
        double r194527 = r194522 * r194526;
        double r194528 = r194524 * r194523;
        double r194529 = r194528 + r194527;
        double r194530 = r194527 * r194529;
        double r194531 = 2.0;
        double r194532 = r194531 * r194522;
        double r194533 = r194525 + r194532;
        double r194534 = r194533 * r194533;
        double r194535 = r194530 / r194534;
        double r194536 = 1.0;
        double r194537 = r194534 - r194536;
        double r194538 = r194535 / r194537;
        return r194538;
}

↓

double f(double alpha, double beta, double i) {
        double r194539 = beta;
        double r194540 = 1.4407489768135957e+203;
        bool r194541 = r194539 <= r194540;
        double r194542 = 1.0;
        double r194543 = 2.0;
        double r194544 = i;
        double r194545 = alpha;
        double r194546 = r194545 + r194539;
        double r194547 = fma(r194543, r194544, r194546);
        double r194548 = 1.0;
        double r194549 = sqrt(r194548);
        double r194550 = r194547 + r194549;
        double r194551 = sqrt(r194550);
        double r194552 = r194546 + r194544;
        double r194553 = r194544 * r194552;
        double r194554 = fma(r194539, r194545, r194553);
        double r194555 = r194547 - r194549;
        double r194556 = r194554 / r194555;
        double r194557 = sqrt(r194556);
        double r194558 = r194551 / r194557;
        double r194559 = r194542 / r194558;
        double r194560 = r194558 * r194547;
        double r194561 = r194553 / r194560;
        double r194562 = r194561 / r194547;
        double r194563 = r194559 * r194562;
        double r194564 = 0.0;
        double r194565 = r194543 * r194544;
        double r194566 = r194546 + r194565;
        double r194567 = r194566 * r194566;
        double r194568 = r194564 / r194567;
        double r194569 = r194567 - r194548;
        double r194570 = r194568 / r194569;
        double r194571 = r194541 ? r194563 : r194570;
        return r194571;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if beta < 1.4407489768135957e+203

   1. Initial program 52.8

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

   3. Applied times-frac 37.6

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

   4. Applied associate-/l* 37.6

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

   5. Simplified 37.6

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

   7. Applied add-sqr-sqrt 37.6

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

   8. Applied difference-of-squares 37.6

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

   9. Applied associate-/l* 36.1

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

   11. Applied add-sqr-sqrt 36.3

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

   12. Applied add-sqr-sqrt 36.1

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

   13. Applied times-frac 36.1

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

   14. Applied associate-*l* 36.1

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

   16. Applied *-un-lft-identity 36.1

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

   17. Applied times-frac 35.6

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

   18. Simplified 35.6

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

   if 1.4407489768135957e+203 < 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. Taylor expanded around 0 45.5

      \[\leadsto \frac{\frac{\color{blue}{0}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
3. Recombined 2 regimes into one program.

4. Final simplification 36.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.440748976813595748532594826853542348803 \cdot 10^{203}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}}}{\sqrt{\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}}} \cdot \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\frac{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}}}{\sqrt{\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}} \cdot \mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :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)))