Octave 3.8, jcobi/4

Average Error: 54.0 → 11.6

Time: 1.2m

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 3.365899413275378359645560424080333100681 \cdot 10^{111}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(\beta, \alpha, \left(\left(\alpha + \beta\right) + i\right) \cdot i\right)}}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(\beta, \alpha, \left(\left(\alpha + \beta\right) + i\right) \cdot i\right)}}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{i}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \alpha, \mathsf{fma}\left(0.5, i, \beta \cdot 0.25\right)\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}} \cdot \left(\left(\sqrt[3]{\frac{\frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{i}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}} \cdot \sqrt[3]{\frac{\frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{i}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}\right) \cdot \sqrt[3]{\frac{\frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{i}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}\right)\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r4207583 = i;
        double r4207584 = alpha;
        double r4207585 = beta;
        double r4207586 = r4207584 + r4207585;
        double r4207587 = r4207586 + r4207583;
        double r4207588 = r4207583 * r4207587;
        double r4207589 = r4207585 * r4207584;
        double r4207590 = r4207589 + r4207588;
        double r4207591 = r4207588 * r4207590;
        double r4207592 = 2.0;
        double r4207593 = r4207592 * r4207583;
        double r4207594 = r4207586 + r4207593;
        double r4207595 = r4207594 * r4207594;
        double r4207596 = r4207591 / r4207595;
        double r4207597 = 1.0;
        double r4207598 = r4207595 - r4207597;
        double r4207599 = r4207596 / r4207598;
        return r4207599;
}

↓

double f(double alpha, double beta, double i) {
        double r4207600 = i;
        double r4207601 = 3.3658994132753784e+111;
        bool r4207602 = r4207600 <= r4207601;
        double r4207603 = beta;
        double r4207604 = alpha;
        double r4207605 = r4207604 + r4207603;
        double r4207606 = r4207605 + r4207600;
        double r4207607 = r4207606 * r4207600;
        double r4207608 = fma(r4207603, r4207604, r4207607);
        double r4207609 = sqrt(r4207608);
        double r4207610 = 2.0;
        double r4207611 = fma(r4207610, r4207600, r4207605);
        double r4207612 = r4207611 / r4207609;
        double r4207613 = r4207609 / r4207612;
        double r4207614 = 1.0;
        double r4207615 = sqrt(r4207614);
        double r4207616 = r4207611 + r4207615;
        double r4207617 = r4207613 / r4207616;
        double r4207618 = r4207611 / r4207600;
        double r4207619 = r4207606 / r4207618;
        double r4207620 = r4207611 - r4207615;
        double r4207621 = r4207619 / r4207620;
        double r4207622 = r4207617 * r4207621;
        double r4207623 = 0.25;
        double r4207624 = 0.5;
        double r4207625 = r4207603 * r4207623;
        double r4207626 = fma(r4207624, r4207600, r4207625);
        double r4207627 = fma(r4207623, r4207604, r4207626);
        double r4207628 = r4207627 / r4207616;
        double r4207629 = cbrt(r4207621);
        double r4207630 = r4207629 * r4207629;
        double r4207631 = r4207630 * r4207629;
        double r4207632 = r4207628 * r4207631;
        double r4207633 = r4207602 ? r4207622 : r4207632;
        return r4207633;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if i < 3.3658994132753784e+111

   1. Initial program 36.2

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

   2. Simplified 36.2

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

   4. Applied add-sqr-sqrt 36.2

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

   5. Applied difference-of-squares 36.2

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

   6. Applied times-frac 14.7

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

   7. Applied times-frac 10.4

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

   9. Applied associate-/l* 10.4

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

   11. Applied add-sqr-sqrt 10.4

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

   12. Applied associate-/l* 10.4

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

   if 3.3658994132753784e+111 < 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. Simplified 64.0

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

   4. Applied add-sqr-sqrt 64.0

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

   5. Applied difference-of-squares 64.0

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

   6. Applied times-frac 54.1

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

   7. Applied times-frac 53.6

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

   9. Applied associate-/l* 53.5

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

   10. Taylor expanded around 0 12.4

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

   11. Simplified 12.4

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

   13. Applied add-cube-cbrt 12.4

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

4. Final simplification 11.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 3.365899413275378359645560424080333100681 \cdot 10^{111}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(\beta, \alpha, \left(\left(\alpha + \beta\right) + i\right) \cdot i\right)}}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(\beta, \alpha, \left(\left(\alpha + \beta\right) + i\right) \cdot i\right)}}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{i}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \alpha, \mathsf{fma}\left(0.5, i, \beta \cdot 0.25\right)\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}} \cdot \left(\left(\sqrt[3]{\frac{\frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{i}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}} \cdot \sqrt[3]{\frac{\frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{i}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}\right) \cdot \sqrt[3]{\frac{\frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{i}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +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)))