Octave 3.8, jcobi/4

Average Error: 52.3 → 10.5

Time: 28.6s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 6.263709625600619 \cdot 10^{+146}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) - i \cdot 2}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1.0}} \cdot \frac{\frac{\frac{1}{\frac{\left(\alpha + \beta\right) - i \cdot 2}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}} \cdot \frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log \left(e^{\frac{i \cdot \frac{-1}{8}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1.0}}}\right) \cdot \frac{\left(\alpha + \beta\right) - i \cdot 2}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1.0}}\right)\right)\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r3846640 = i;
        double r3846641 = alpha;
        double r3846642 = beta;
        double r3846643 = r3846641 + r3846642;
        double r3846644 = r3846643 + r3846640;
        double r3846645 = r3846640 * r3846644;
        double r3846646 = r3846642 * r3846641;
        double r3846647 = r3846646 + r3846645;
        double r3846648 = r3846645 * r3846647;
        double r3846649 = 2.0;
        double r3846650 = r3846649 * r3846640;
        double r3846651 = r3846643 + r3846650;
        double r3846652 = r3846651 * r3846651;
        double r3846653 = r3846648 / r3846652;
        double r3846654 = 1.0;
        double r3846655 = r3846652 - r3846654;
        double r3846656 = r3846653 / r3846655;
        return r3846656;
}

↓

double f(double alpha, double beta, double i) {
        double r3846657 = i;
        double r3846658 = 6.263709625600619e+146;
        bool r3846659 = r3846657 <= r3846658;
        double r3846660 = alpha;
        double r3846661 = beta;
        double r3846662 = r3846660 + r3846661;
        double r3846663 = 2.0;
        double r3846664 = r3846657 * r3846663;
        double r3846665 = r3846662 - r3846664;
        double r3846666 = fma(r3846663, r3846657, r3846662);
        double r3846667 = 1.0;
        double r3846668 = sqrt(r3846667);
        double r3846669 = r3846666 - r3846668;
        double r3846670 = r3846665 / r3846669;
        double r3846671 = 1.0;
        double r3846672 = r3846662 + r3846657;
        double r3846673 = r3846657 * r3846672;
        double r3846674 = fma(r3846661, r3846660, r3846673);
        double r3846675 = r3846665 / r3846674;
        double r3846676 = r3846671 / r3846675;
        double r3846677 = r3846673 / r3846666;
        double r3846678 = r3846676 * r3846677;
        double r3846679 = r3846678 / r3846666;
        double r3846680 = r3846666 + r3846668;
        double r3846681 = r3846679 / r3846680;
        double r3846682 = r3846670 * r3846681;
        double r3846683 = -0.125;
        double r3846684 = r3846657 * r3846683;
        double r3846685 = r3846684 / r3846680;
        double r3846686 = exp(r3846685);
        double r3846687 = log(r3846686);
        double r3846688 = r3846687 * r3846670;
        double r3846689 = /* ERROR: no posit support in C */;
        double r3846690 = /* ERROR: no posit support in C */;
        double r3846691 = r3846659 ? r3846682 : r3846690;
        return r3846691;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if i < 6.263709625600619e+146

   1. Initial program 41.9

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

   3. Applied add-sqr-sqrt 41.9

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

   4. Applied difference-of-squares 41.9

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

   5. Applied flip-+ 41.9

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

   6. Applied associate-*r/ 42.8

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

   7. Applied associate-/r/ 42.9

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

   8. Applied times-frac 42.9

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

   9. Simplified 10.4

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

   10. Simplified 10.4

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

   12. Applied clear-num 10.5

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

   if 6.263709625600619e+146 < i

   1. Initial program 62.1

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

   3. Applied add-sqr-sqrt 62.1

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

   4. Applied difference-of-squares 62.1

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

   5. Applied flip-+ 62.1

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

   6. Applied associate-*r/ 62.1

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

   7. Applied associate-/r/ 62.1

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

   8. Applied times-frac 62.1

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

   9. Simplified 60.1

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

   10. Simplified 60.1

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

   11. Taylor expanded around 0 11.2

       \[\leadsto \frac{\color{blue}{\frac{-1}{8} \cdot i}}{\sqrt{1.0} + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\left(\alpha + \beta\right) - 2 \cdot i}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1.0}}\]
   12. Using strategy rm

   13. Applied insert-posit16 10.8

       \[\leadsto \color{blue}{\left(\left(\frac{\frac{-1}{8} \cdot i}{\sqrt{1.0} + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\left(\alpha + \beta\right) - 2 \cdot i}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1.0}}\right)\right)}\]
   14. Using strategy rm

   15. Applied add-log-exp 10.6

       \[\leadsto \left(\left(\color{blue}{\log \left(e^{\frac{\frac{-1}{8} \cdot i}{\sqrt{1.0} + \mathsf{fma}\left(2, i, \alpha + \beta\right)}}\right)} \cdot \frac{\left(\alpha + \beta\right) - 2 \cdot i}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1.0}}\right)\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 6.263709625600619 \cdot 10^{+146}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) - i \cdot 2}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1.0}} \cdot \frac{\frac{\frac{1}{\frac{\left(\alpha + \beta\right) - i \cdot 2}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}} \cdot \frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log \left(e^{\frac{i \cdot \frac{-1}{8}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1.0}}}\right) \cdot \frac{\left(\alpha + \beta\right) - i \cdot 2}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1.0}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +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)))