Octave 3.8, jcobi/4

Average Error: 54.1 → 36.7

Time: 32.0s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\beta \le 1.440748976813595748532594826853542348803 \cdot 10^{203}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}}}{\sqrt{\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}}} \cdot \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\frac{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}}}{\sqrt{\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}} \cdot \mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r193951 = i;
        double r193952 = alpha;
        double r193953 = beta;
        double r193954 = r193952 + r193953;
        double r193955 = r193954 + r193951;
        double r193956 = r193951 * r193955;
        double r193957 = r193953 * r193952;
        double r193958 = r193957 + r193956;
        double r193959 = r193956 * r193958;
        double r193960 = 2.0;
        double r193961 = r193960 * r193951;
        double r193962 = r193954 + r193961;
        double r193963 = r193962 * r193962;
        double r193964 = r193959 / r193963;
        double r193965 = 1.0;
        double r193966 = r193963 - r193965;
        double r193967 = r193964 / r193966;
        return r193967;
}

↓

double f(double alpha, double beta, double i) {
        double r193968 = beta;
        double r193969 = 1.4407489768135957e+203;
        bool r193970 = r193968 <= r193969;
        double r193971 = 1.0;
        double r193972 = 2.0;
        double r193973 = i;
        double r193974 = alpha;
        double r193975 = r193974 + r193968;
        double r193976 = fma(r193972, r193973, r193975);
        double r193977 = 1.0;
        double r193978 = sqrt(r193977);
        double r193979 = r193976 + r193978;
        double r193980 = sqrt(r193979);
        double r193981 = r193975 + r193973;
        double r193982 = r193973 * r193981;
        double r193983 = fma(r193968, r193974, r193982);
        double r193984 = r193976 - r193978;
        double r193985 = r193983 / r193984;
        double r193986 = sqrt(r193985);
        double r193987 = r193980 / r193986;
        double r193988 = r193971 / r193987;
        double r193989 = r193987 * r193976;
        double r193990 = r193982 / r193989;
        double r193991 = r193990 / r193976;
        double r193992 = r193988 * r193991;
        double r193993 = 0.0;
        double r193994 = r193970 ? r193992 : r193993;
        return r193994;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if beta < 1.4407489768135957e+203

   1. Initial program 52.8

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

   3. Applied times-frac 37.6

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

   4. Applied associate-/l* 37.6

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

   5. Simplified 37.6

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

   7. Applied add-sqr-sqrt 37.6

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

   8. Applied difference-of-squares 37.6

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

   9. Applied associate-/l* 36.1

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

   11. Applied add-sqr-sqrt 36.3

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

   12. Applied add-sqr-sqrt 36.1

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

   13. Applied times-frac 36.1

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

   14. Applied associate-*l* 36.1

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

   16. Applied *-un-lft-identity 36.1

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

   17. Applied times-frac 35.6

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

   18. Simplified 35.6

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

   if 1.4407489768135957e+203 < beta

   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. Taylor expanded around inf 45.5

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

4. Final simplification 36.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.440748976813595748532594826853542348803 \cdot 10^{203}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}}}{\sqrt{\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}}} \cdot \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\frac{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}}}{\sqrt{\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}} \cdot \mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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)))