Octave 3.8, jcobi/4

Average Error: 52.4 → 31.9

Time: 27.3s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.8246756895617496 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(i + \left(\beta + \alpha\right), i, \beta \cdot \alpha\right)}{\sqrt{1.0} + \mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \frac{\frac{i + \left(\beta + \alpha\right)}{\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) - \sqrt{1.0}}{i}}}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \left(\left(i + \frac{i \cdot 1.0}{\alpha \cdot \alpha}\right) - \frac{\sqrt{1.0} \cdot i}{\alpha}\right)\right) \cdot \frac{\frac{i + \left(\beta + \alpha\right)}{\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) - \sqrt{1.0}}{i}}}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r1773422 = i;
        double r1773423 = alpha;
        double r1773424 = beta;
        double r1773425 = r1773423 + r1773424;
        double r1773426 = r1773425 + r1773422;
        double r1773427 = r1773422 * r1773426;
        double r1773428 = r1773424 * r1773423;
        double r1773429 = r1773428 + r1773427;
        double r1773430 = r1773427 * r1773429;
        double r1773431 = 2.0;
        double r1773432 = r1773431 * r1773422;
        double r1773433 = r1773425 + r1773432;
        double r1773434 = r1773433 * r1773433;
        double r1773435 = r1773430 / r1773434;
        double r1773436 = 1.0;
        double r1773437 = r1773434 - r1773436;
        double r1773438 = r1773435 / r1773437;
        return r1773438;
}

↓

double f(double alpha, double beta, double i) {
        double r1773439 = alpha;
        double r1773440 = 1.8246756895617496e+128;
        bool r1773441 = r1773439 <= r1773440;
        double r1773442 = i;
        double r1773443 = beta;
        double r1773444 = r1773443 + r1773439;
        double r1773445 = r1773442 + r1773444;
        double r1773446 = r1773443 * r1773439;
        double r1773447 = fma(r1773445, r1773442, r1773446);
        double r1773448 = 1.0;
        double r1773449 = sqrt(r1773448);
        double r1773450 = 2.0;
        double r1773451 = fma(r1773450, r1773442, r1773444);
        double r1773452 = r1773449 + r1773451;
        double r1773453 = r1773447 / r1773452;
        double r1773454 = r1773453 / r1773451;
        double r1773455 = r1773451 - r1773449;
        double r1773456 = r1773455 / r1773442;
        double r1773457 = r1773445 / r1773456;
        double r1773458 = r1773457 / r1773451;
        double r1773459 = r1773454 * r1773458;
        double r1773460 = 1.0;
        double r1773461 = r1773460 / r1773451;
        double r1773462 = r1773442 * r1773448;
        double r1773463 = r1773439 * r1773439;
        double r1773464 = r1773462 / r1773463;
        double r1773465 = r1773442 + r1773464;
        double r1773466 = r1773449 * r1773442;
        double r1773467 = r1773466 / r1773439;
        double r1773468 = r1773465 - r1773467;
        double r1773469 = r1773461 * r1773468;
        double r1773470 = r1773469 * r1773458;
        double r1773471 = r1773441 ? r1773459 : r1773470;
        return r1773471;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if alpha < 1.8246756895617496e+128

   1. Initial program 50.0

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

   2. Simplified 50.0

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

   4. Applied add-sqr-sqrt 50.0

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

   5. Applied difference-of-squares 50.0

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

   6. Applied times-frac 35.5

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

   7. Applied times-frac 34.0

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

   9. Applied associate-/l* 34.0

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

   if 1.8246756895617496e+128 < alpha

   1. Initial program 62.4

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

   2. Simplified 62.4

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

   4. Applied add-sqr-sqrt 62.4

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

   5. Applied difference-of-squares 62.4

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

   6. Applied times-frac 53.9

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

   7. Applied times-frac 48.6

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

   9. Applied associate-/l* 48.6

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

   11. Applied div-inv 48.6

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

   12. Taylor expanded around inf 23.1

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

   13. Simplified 23.1

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

4. Final simplification 31.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.8246756895617496 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(i + \left(\beta + \alpha\right), i, \beta \cdot \alpha\right)}{\sqrt{1.0} + \mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \frac{\frac{i + \left(\beta + \alpha\right)}{\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) - \sqrt{1.0}}{i}}}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \left(\left(i + \frac{i \cdot 1.0}{\alpha \cdot \alpha}\right) - \frac{\sqrt{1.0} \cdot i}{\alpha}\right)\right) \cdot \frac{\frac{i + \left(\beta + \alpha\right)}{\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right) - \sqrt{1.0}}{i}}}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(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)))