Octave 3.8, jcobi/4

Average Error: 54.0 → 38.5

Time: 10.1s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\beta \le 1.7873990429441248 \cdot 10^{144}:\\ \;\;\;\;\frac{\frac{i}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right), \mathsf{fma}\left(i, 2, \alpha + \beta\right), -1\right)} \cdot \frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\mathsf{fma}\left(2, \alpha \cdot \beta, \mathsf{fma}\left(4, i \cdot \beta, {\beta}^{2}\right)\right)} \cdot \frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\frac{2}{i}}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r113920 = i;
        double r113921 = alpha;
        double r113922 = beta;
        double r113923 = r113921 + r113922;
        double r113924 = r113923 + r113920;
        double r113925 = r113920 * r113924;
        double r113926 = r113922 * r113921;
        double r113927 = r113926 + r113925;
        double r113928 = r113925 * r113927;
        double r113929 = 2.0;
        double r113930 = r113929 * r113920;
        double r113931 = r113923 + r113930;
        double r113932 = r113931 * r113931;
        double r113933 = r113928 / r113932;
        double r113934 = 1.0;
        double r113935 = r113932 - r113934;
        double r113936 = r113933 / r113935;
        return r113936;
}

↓

double f(double alpha, double beta, double i) {
        double r113937 = beta;
        double r113938 = 1.7873990429441248e+144;
        bool r113939 = r113937 <= r113938;
        double r113940 = i;
        double r113941 = 2.0;
        double r113942 = alpha;
        double r113943 = r113942 + r113937;
        double r113944 = fma(r113940, r113941, r113943);
        double r113945 = 1.0;
        double r113946 = -r113945;
        double r113947 = fma(r113944, r113944, r113946);
        double r113948 = r113940 / r113947;
        double r113949 = r113943 + r113940;
        double r113950 = r113949 / r113944;
        double r113951 = r113948 * r113950;
        double r113952 = r113940 * r113949;
        double r113953 = fma(r113937, r113942, r113952);
        double r113954 = r113944 / r113953;
        double r113955 = r113951 / r113954;
        double r113956 = 2.0;
        double r113957 = r113942 * r113937;
        double r113958 = 4.0;
        double r113959 = r113940 * r113937;
        double r113960 = pow(r113937, r113956);
        double r113961 = fma(r113958, r113959, r113960);
        double r113962 = fma(r113956, r113957, r113961);
        double r113963 = r113940 / r113962;
        double r113964 = r113963 * r113950;
        double r113965 = r113941 / r113940;
        double r113966 = r113964 / r113965;
        double r113967 = r113939 ? r113955 : r113966;
        return r113967;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if beta < 1.7873990429441248e+144

   1. Initial program 51.8

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

   2. Simplified 51.3

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

   4. Applied *-un-lft-identity 51.3

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

   5. Applied times-frac 46.7

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

   6. Applied associate-/r* 46.7

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

   7. Simplified 36.2

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

   if 1.7873990429441248e+144 < 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. Simplified 58.7

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

   4. Applied *-un-lft-identity 58.7

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

   5. Applied times-frac 58.7

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

   6. Applied associate-/r* 58.7

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

   7. Simplified 57.1

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

   8. Taylor expanded around inf 49.6

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

   9. Taylor expanded around inf 49.6

      \[\leadsto \frac{\frac{i}{\color{blue}{2 \cdot \left(\alpha \cdot \beta\right) + \left(4 \cdot \left(i \cdot \beta\right) + {\beta}^{2}\right)}} \cdot \frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\frac{2}{i}}\]

   10. Simplified 49.6

       \[\leadsto \frac{\frac{i}{\color{blue}{\mathsf{fma}\left(2, \alpha \cdot \beta, \mathsf{fma}\left(4, i \cdot \beta, {\beta}^{2}\right)\right)}} \cdot \frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\frac{2}{i}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 38.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.7873990429441248 \cdot 10^{144}:\\ \;\;\;\;\frac{\frac{i}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right), \mathsf{fma}\left(i, 2, \alpha + \beta\right), -1\right)} \cdot \frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\mathsf{fma}\left(2, \alpha \cdot \beta, \mathsf{fma}\left(4, i \cdot \beta, {\beta}^{2}\right)\right)} \cdot \frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\frac{2}{i}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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)))