Octave 3.8, jcobi/4

Average Error: 54.2 → 10.5

Time: 41.9s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 3.615708469216375512155137715842033763303 \cdot 10^{121}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(i, \left(\alpha + i\right) + \beta, \alpha \cdot \beta\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}} \cdot \left(\frac{1}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}} \cdot \frac{\left(\alpha + i\right) + \beta}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{i}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot 0.5\right) \cdot \frac{\frac{\frac{\left(\alpha + i\right) + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{i}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r168743 = i;
        double r168744 = alpha;
        double r168745 = beta;
        double r168746 = r168744 + r168745;
        double r168747 = r168746 + r168743;
        double r168748 = r168743 * r168747;
        double r168749 = r168745 * r168744;
        double r168750 = r168749 + r168748;
        double r168751 = r168748 * r168750;
        double r168752 = 2.0;
        double r168753 = r168752 * r168743;
        double r168754 = r168746 + r168753;
        double r168755 = r168754 * r168754;
        double r168756 = r168751 / r168755;
        double r168757 = 1.0;
        double r168758 = r168755 - r168757;
        double r168759 = r168756 / r168758;
        return r168759;
}

↓

double f(double alpha, double beta, double i) {
        double r168760 = i;
        double r168761 = 3.6157084692163755e+121;
        bool r168762 = r168760 <= r168761;
        double r168763 = alpha;
        double r168764 = r168763 + r168760;
        double r168765 = beta;
        double r168766 = r168764 + r168765;
        double r168767 = r168763 * r168765;
        double r168768 = fma(r168760, r168766, r168767);
        double r168769 = 2.0;
        double r168770 = r168763 + r168765;
        double r168771 = fma(r168769, r168760, r168770);
        double r168772 = r168768 / r168771;
        double r168773 = 1.0;
        double r168774 = sqrt(r168773);
        double r168775 = r168771 + r168774;
        double r168776 = r168772 / r168775;
        double r168777 = 1.0;
        double r168778 = r168771 - r168774;
        double r168779 = r168777 / r168778;
        double r168780 = r168771 / r168760;
        double r168781 = r168766 / r168780;
        double r168782 = r168779 * r168781;
        double r168783 = r168776 * r168782;
        double r168784 = 0.5;
        double r168785 = r168760 * r168784;
        double r168786 = r168766 / r168778;
        double r168787 = r168786 / r168780;
        double r168788 = r168787 / r168775;
        double r168789 = r168785 * r168788;
        double r168790 = r168762 ? r168783 : r168789;
        return r168790;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if i < 3.6157084692163755e+121

   1. Initial program 39.2

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

   2. Simplified 39.2

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

   4. Applied add-sqr-sqrt 39.2

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

   5. Applied difference-of-squares 39.2

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

   6. Applied times-frac 14.9

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

   7. Applied times-frac 9.7

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

   8. Simplified 9.7

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

   9. Simplified 11.7

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

   11. Applied *-un-lft-identity 11.7

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

   12. Applied times-frac 9.7

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

   13. Simplified 9.7

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

   if 3.6157084692163755e+121 < 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

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

   4. Applied add-sqr-sqrt 64.0

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

   5. Applied difference-of-squares 64.0

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

   6. Applied times-frac 56.6

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

   7. Applied times-frac 56.1

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

   8. Simplified 56.1

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

   9. Simplified 56.1

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

   10. Taylor expanded around inf 11.1

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

   12. Applied div-inv 11.2

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

   13. Applied associate-*l* 11.2

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

   14. Simplified 11.0

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

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 3.615708469216375512155137715842033763303 \cdot 10^{121}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(i, \left(\alpha + i\right) + \beta, \alpha \cdot \beta\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}} \cdot \left(\frac{1}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}} \cdot \frac{\left(\alpha + i\right) + \beta}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{i}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot 0.5\right) \cdot \frac{\frac{\frac{\left(\alpha + i\right) + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{i}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +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)))