Octave 3.8, jcobi/4

Average Error: 52.8 → 35.4

Time: 3.7m

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 4.0145625771446497 \cdot 10^{+217}:\\ \;\;\;\;\sqrt{\frac{\frac{\mathsf{fma}\left(\left(i + \left(\beta + \alpha\right)\right), i, \left(\beta \cdot \alpha\right)\right)}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}}{\sqrt{1.0} + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}} \cdot \left(\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right) - \sqrt{1.0}} \cdot \sqrt{\frac{\frac{\sqrt{\mathsf{fma}\left(\left(i + \left(\beta + \alpha\right)\right), i, \left(\beta \cdot \alpha\right)\right)}}{\sqrt{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}} \cdot \frac{\sqrt{\mathsf{fma}\left(\left(i + \left(\beta + \alpha\right)\right), i, \left(\beta \cdot \alpha\right)\right)}}{\sqrt{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}}}{\sqrt{1.0} + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r26829860 = i;
        double r26829861 = alpha;
        double r26829862 = beta;
        double r26829863 = r26829861 + r26829862;
        double r26829864 = r26829863 + r26829860;
        double r26829865 = r26829860 * r26829864;
        double r26829866 = r26829862 * r26829861;
        double r26829867 = r26829866 + r26829865;
        double r26829868 = r26829865 * r26829867;
        double r26829869 = 2.0;
        double r26829870 = r26829869 * r26829860;
        double r26829871 = r26829863 + r26829870;
        double r26829872 = r26829871 * r26829871;
        double r26829873 = r26829868 / r26829872;
        double r26829874 = 1.0;
        double r26829875 = r26829872 - r26829874;
        double r26829876 = r26829873 / r26829875;
        return r26829876;
}

↓

double f(double alpha, double beta, double i) {
        double r26829877 = alpha;
        double r26829878 = 4.0145625771446497e+217;
        bool r26829879 = r26829877 <= r26829878;
        double r26829880 = i;
        double r26829881 = beta;
        double r26829882 = r26829881 + r26829877;
        double r26829883 = r26829880 + r26829882;
        double r26829884 = r26829881 * r26829877;
        double r26829885 = fma(r26829883, r26829880, r26829884);
        double r26829886 = 2.0;
        double r26829887 = fma(r26829886, r26829880, r26829882);
        double r26829888 = r26829885 / r26829887;
        double r26829889 = 1.0;
        double r26829890 = sqrt(r26829889);
        double r26829891 = r26829890 + r26829887;
        double r26829892 = r26829888 / r26829891;
        double r26829893 = sqrt(r26829892);
        double r26829894 = r26829880 * r26829883;
        double r26829895 = r26829894 / r26829887;
        double r26829896 = r26829887 - r26829890;
        double r26829897 = r26829895 / r26829896;
        double r26829898 = sqrt(r26829885);
        double r26829899 = sqrt(r26829887);
        double r26829900 = r26829898 / r26829899;
        double r26829901 = r26829900 * r26829900;
        double r26829902 = r26829901 / r26829891;
        double r26829903 = sqrt(r26829902);
        double r26829904 = r26829897 * r26829903;
        double r26829905 = r26829893 * r26829904;
        double r26829906 = 0.0;
        double r26829907 = r26829879 ? r26829905 : r26829906;
        return r26829907;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if alpha < 4.0145625771446497e+217

   1. Initial program 51.7

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

   2. Simplified 51.7

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

   4. Applied add-sqr-sqrt 51.7

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

   5. Applied difference-of-squares 51.7

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

   6. Applied times-frac 36.8

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

   7. Applied times-frac 34.4

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

   9. Applied add-sqr-sqrt 34.4

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

   10. Applied associate-*l* 34.4

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

   12. Applied add-sqr-sqrt 34.5

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

   13. Applied add-sqr-sqrt 34.5

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

   14. Applied times-frac 34.5

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

   if 4.0145625771446497e+217 < alpha

   1. Initial program 62.6

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

   2. Simplified 62.6

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

   4. Applied add-sqr-sqrt 62.6

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

   5. Applied difference-of-squares 62.6

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

   6. Applied times-frac 56.2

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

   7. Applied times-frac 55.7

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

   8. Taylor expanded around inf 43.2

      \[\leadsto \color{blue}{0}\]
3. Recombined 2 regimes into one program.

4. Final simplification 35.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 4.0145625771446497 \cdot 10^{+217}:\\ \;\;\;\;\sqrt{\frac{\frac{\mathsf{fma}\left(\left(i + \left(\beta + \alpha\right)\right), i, \left(\beta \cdot \alpha\right)\right)}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}}{\sqrt{1.0} + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}} \cdot \left(\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right) - \sqrt{1.0}} \cdot \sqrt{\frac{\frac{\sqrt{\mathsf{fma}\left(\left(i + \left(\beta + \alpha\right)\right), i, \left(\beta \cdot \alpha\right)\right)}}{\sqrt{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}} \cdot \frac{\sqrt{\mathsf{fma}\left(\left(i + \left(\beta + \alpha\right)\right), i, \left(\beta \cdot \alpha\right)\right)}}{\sqrt{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}}}{\sqrt{1.0} + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +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)))