Octave 3.8, jcobi/4

Average Error: 54.1 → 39.1

Time: 25.4s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.054433814820164952872972799641075608789 \cdot 10^{117}:\\ \;\;\;\;\frac{\left(\left(i + \left(\alpha + \beta\right)\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\right) \cdot \frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \alpha + \beta\right), \mathsf{fma}\left(2, i, \alpha + \beta\right), -1\right)} \cdot \mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \alpha + \beta\right), \mathsf{fma}\left(2, i, \alpha + \beta\right), -1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i + \left(\alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \alpha + \beta\right), \mathsf{fma}\left(2, i, \alpha + \beta\right), -1\right)}} \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\right) \cdot 0\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r79826 = i;
        double r79827 = alpha;
        double r79828 = beta;
        double r79829 = r79827 + r79828;
        double r79830 = r79829 + r79826;
        double r79831 = r79826 * r79830;
        double r79832 = r79828 * r79827;
        double r79833 = r79832 + r79831;
        double r79834 = r79831 * r79833;
        double r79835 = 2.0;
        double r79836 = r79835 * r79826;
        double r79837 = r79829 + r79836;
        double r79838 = r79837 * r79837;
        double r79839 = r79834 / r79838;
        double r79840 = 1.0;
        double r79841 = r79838 - r79840;
        double r79842 = r79839 / r79841;
        return r79842;
}

↓

double f(double alpha, double beta, double i) {
        double r79843 = alpha;
        double r79844 = 1.054433814820165e+117;
        bool r79845 = r79843 <= r79844;
        double r79846 = i;
        double r79847 = beta;
        double r79848 = r79843 + r79847;
        double r79849 = r79846 + r79848;
        double r79850 = 2.0;
        double r79851 = fma(r79850, r79846, r79848);
        double r79852 = r79846 / r79851;
        double r79853 = r79849 * r79852;
        double r79854 = r79848 + r79846;
        double r79855 = r79846 * r79854;
        double r79856 = fma(r79847, r79843, r79855);
        double r79857 = 1.0;
        double r79858 = -r79857;
        double r79859 = fma(r79851, r79851, r79858);
        double r79860 = sqrt(r79859);
        double r79861 = r79860 * r79851;
        double r79862 = r79856 / r79861;
        double r79863 = r79853 * r79862;
        double r79864 = r79863 / r79860;
        double r79865 = r79849 / r79860;
        double r79866 = r79865 * r79852;
        double r79867 = 0.0;
        double r79868 = r79866 * r79867;
        double r79869 = r79845 ? r79864 : r79868;
        return r79869;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if alpha < 1.054433814820165e+117

   1. Initial program 51.6

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

   3. Applied add-sqr-sqrt 51.6

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

   4. Applied times-frac 36.1

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

   5. Applied times-frac 36.1

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

   6. Simplified 36.1

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

   7. Simplified 36.1

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

   9. Applied associate-*l/ 36.1

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

   10. Applied associate-*l/ 36.1

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

   if 1.054433814820165e+117 < alpha

   1. Initial program 63.2

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

   3. Applied add-sqr-sqrt 63.2

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

   4. Applied times-frac 53.4

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

   5. Applied times-frac 53.4

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

   6. Simplified 53.4

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

   7. Simplified 53.4

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

   8. Taylor expanded around inf 50.2

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

4. Final simplification 39.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.054433814820164952872972799641075608789 \cdot 10^{117}:\\ \;\;\;\;\frac{\left(\left(i + \left(\alpha + \beta\right)\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\right) \cdot \frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \alpha + \beta\right), \mathsf{fma}\left(2, i, \alpha + \beta\right), -1\right)} \cdot \mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \alpha + \beta\right), \mathsf{fma}\left(2, i, \alpha + \beta\right), -1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i + \left(\alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \alpha + \beta\right), \mathsf{fma}\left(2, i, \alpha + \beta\right), -1\right)}} \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\right) \cdot 0\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +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)))