Octave 3.8, jcobi/4

Average Error: 52.5 → 14.4

Time: 3.5m

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\beta \le 8.636484859471028 \cdot 10^{+55}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\alpha + \beta\right), \frac{1}{4}, \left(i \cdot \frac{1}{2}\right)\right)}{\sqrt{1.0} + \mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)} \cdot \log \left(e^{\frac{\left(i + \left(\alpha + \beta\right)\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)}}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right) - \sqrt{1.0}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\sqrt{1.0} + \mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)} \cdot \frac{\left(i + \left(\alpha + \beta\right)\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)}}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right) - \sqrt{1.0}}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r18698895 = i;
        double r18698896 = alpha;
        double r18698897 = beta;
        double r18698898 = r18698896 + r18698897;
        double r18698899 = r18698898 + r18698895;
        double r18698900 = r18698895 * r18698899;
        double r18698901 = r18698897 * r18698896;
        double r18698902 = r18698901 + r18698900;
        double r18698903 = r18698900 * r18698902;
        double r18698904 = 2.0;
        double r18698905 = r18698904 * r18698895;
        double r18698906 = r18698898 + r18698905;
        double r18698907 = r18698906 * r18698906;
        double r18698908 = r18698903 / r18698907;
        double r18698909 = 1.0;
        double r18698910 = r18698907 - r18698909;
        double r18698911 = r18698908 / r18698910;
        return r18698911;
}

↓

double f(double alpha, double beta, double i) {
        double r18698912 = beta;
        double r18698913 = 8.636484859471028e+55;
        bool r18698914 = r18698912 <= r18698913;
        double r18698915 = alpha;
        double r18698916 = r18698915 + r18698912;
        double r18698917 = 0.25;
        double r18698918 = i;
        double r18698919 = 0.5;
        double r18698920 = r18698918 * r18698919;
        double r18698921 = fma(r18698916, r18698917, r18698920);
        double r18698922 = 1.0;
        double r18698923 = sqrt(r18698922);
        double r18698924 = 2.0;
        double r18698925 = fma(r18698924, r18698918, r18698916);
        double r18698926 = r18698923 + r18698925;
        double r18698927 = r18698921 / r18698926;
        double r18698928 = r18698918 + r18698916;
        double r18698929 = r18698918 / r18698925;
        double r18698930 = r18698928 * r18698929;
        double r18698931 = r18698925 - r18698923;
        double r18698932 = r18698930 / r18698931;
        double r18698933 = exp(r18698932);
        double r18698934 = log(r18698933);
        double r18698935 = r18698927 * r18698934;
        double r18698936 = r18698918 / r18698926;
        double r18698937 = r18698936 * r18698932;
        double r18698938 = r18698914 ? r18698935 : r18698937;
        return r18698938;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if beta < 8.636484859471028e+55

   1. Initial program 49.5

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

   2. Simplified 49.5

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

   4. Applied add-sqr-sqrt 49.5

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

   5. Applied difference-of-squares 49.5

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

   6. Applied times-frac 34.8

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

   7. Applied times-frac 33.5

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

   9. Applied *-un-lft-identity 33.5

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(\left(\alpha + \beta\right) + i\right), i, \left(\beta \cdot \alpha\right)\right)}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)}}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right) + \sqrt{1.0}} \cdot \frac{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i}{\color{blue}{1 \cdot \mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)}}}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right) - \sqrt{1.0}}\]

   10. Applied times-frac 33.4

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(\left(\alpha + \beta\right) + i\right), i, \left(\beta \cdot \alpha\right)\right)}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)}}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right) + \sqrt{1.0}} \cdot \frac{\color{blue}{\frac{\left(\alpha + \beta\right) + i}{1} \cdot \frac{i}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)}}}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right) - \sqrt{1.0}}\]

   11. Taylor expanded around 0 10.5

       \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot i + \left(\frac{1}{4} \cdot \beta + \frac{1}{4} \cdot \alpha\right)}}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right) + \sqrt{1.0}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{1} \cdot \frac{i}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)}}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right) - \sqrt{1.0}}\]

   12. Simplified 10.5

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\alpha + \beta\right), \frac{1}{4}, \left(\frac{1}{2} \cdot i\right)\right)}}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right) + \sqrt{1.0}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{1} \cdot \frac{i}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)}}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right) - \sqrt{1.0}}\]
   13. Using strategy rm

   14. Applied add-log-exp 9.7

       \[\leadsto \frac{\mathsf{fma}\left(\left(\alpha + \beta\right), \frac{1}{4}, \left(\frac{1}{2} \cdot i\right)\right)}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right) + \sqrt{1.0}} \cdot \color{blue}{\log \left(e^{\frac{\frac{\left(\alpha + \beta\right) + i}{1} \cdot \frac{i}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)}}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right) - \sqrt{1.0}}}\right)}\]

   if 8.636484859471028e+55 < beta

   1. Initial program 60.1

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

   2. Simplified 60.1

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

   4. Applied add-sqr-sqrt 60.1

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

   5. Applied difference-of-squares 60.1

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

   6. Applied times-frac 49.0

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

   7. Applied times-frac 44.5

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

   9. Applied *-un-lft-identity 44.5

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(\left(\alpha + \beta\right) + i\right), i, \left(\beta \cdot \alpha\right)\right)}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)}}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right) + \sqrt{1.0}} \cdot \frac{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i}{\color{blue}{1 \cdot \mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)}}}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right) - \sqrt{1.0}}\]

   10. Applied times-frac 44.5

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(\left(\alpha + \beta\right) + i\right), i, \left(\beta \cdot \alpha\right)\right)}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)}}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right) + \sqrt{1.0}} \cdot \frac{\color{blue}{\frac{\left(\alpha + \beta\right) + i}{1} \cdot \frac{i}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)}}}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right) - \sqrt{1.0}}\]

   11. Taylor expanded around -inf 26.6

       \[\leadsto \frac{\color{blue}{i}}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right) + \sqrt{1.0}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{1} \cdot \frac{i}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)}}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right) - \sqrt{1.0}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 14.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 8.636484859471028 \cdot 10^{+55}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\alpha + \beta\right), \frac{1}{4}, \left(i \cdot \frac{1}{2}\right)\right)}{\sqrt{1.0} + \mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)} \cdot \log \left(e^{\frac{\left(i + \left(\alpha + \beta\right)\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)}}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right) - \sqrt{1.0}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\sqrt{1.0} + \mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)} \cdot \frac{\left(i + \left(\alpha + \beta\right)\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)}}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right) - \sqrt{1.0}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 +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)))