Octave 3.8, jcobi/4

Average Error: 54.4 → 38.1

Time: 1.5m

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 4.817585748747987305707799364642926876965 \cdot 10^{97}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\alpha + \left(\beta + 2 \cdot i\right)}}}{\frac{\sqrt{\left(\left(\beta + 2 \cdot i\right) - \sqrt{1}\right) + \alpha}}{i}} \cdot \left(\frac{\sqrt{i \cdot \left(\beta + \left(i + \alpha\right)\right) + \beta \cdot \alpha}}{\sqrt{\left(2 \cdot i + \left(\beta + \alpha\right)\right) - \sqrt{1}}} \cdot \frac{\sqrt{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right) + \beta \cdot \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}}{\frac{\left(\beta + 2 \cdot i\right) + \left(\alpha + \sqrt{1}\right)}{\frac{\beta + \left(i + \alpha\right)}{2 \cdot i + \left(\beta + \alpha\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot 0.5 + \left(\beta + \alpha\right) \cdot 0.25}{\frac{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right) - 1}{i \cdot \frac{i + \left(\beta + \alpha\right)}{\alpha + \left(\beta + 2 \cdot i\right)}}}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r248046 = i;
        double r248047 = alpha;
        double r248048 = beta;
        double r248049 = r248047 + r248048;
        double r248050 = r248049 + r248046;
        double r248051 = r248046 * r248050;
        double r248052 = r248048 * r248047;
        double r248053 = r248052 + r248051;
        double r248054 = r248051 * r248053;
        double r248055 = 2.0;
        double r248056 = r248055 * r248046;
        double r248057 = r248049 + r248056;
        double r248058 = r248057 * r248057;
        double r248059 = r248054 / r248058;
        double r248060 = 1.0;
        double r248061 = r248058 - r248060;
        double r248062 = r248059 / r248061;
        return r248062;
}

↓

double f(double alpha, double beta, double i) {
        double r248063 = i;
        double r248064 = 4.817585748747987e+97;
        bool r248065 = r248063 <= r248064;
        double r248066 = 1.0;
        double r248067 = alpha;
        double r248068 = beta;
        double r248069 = 2.0;
        double r248070 = r248069 * r248063;
        double r248071 = r248068 + r248070;
        double r248072 = r248067 + r248071;
        double r248073 = r248066 / r248072;
        double r248074 = sqrt(r248073);
        double r248075 = 1.0;
        double r248076 = sqrt(r248075);
        double r248077 = r248071 - r248076;
        double r248078 = r248077 + r248067;
        double r248079 = sqrt(r248078);
        double r248080 = r248079 / r248063;
        double r248081 = r248074 / r248080;
        double r248082 = r248063 + r248067;
        double r248083 = r248068 + r248082;
        double r248084 = r248063 * r248083;
        double r248085 = r248068 * r248067;
        double r248086 = r248084 + r248085;
        double r248087 = sqrt(r248086);
        double r248088 = r248068 + r248067;
        double r248089 = r248070 + r248088;
        double r248090 = r248089 - r248076;
        double r248091 = sqrt(r248090);
        double r248092 = r248087 / r248091;
        double r248093 = r248086 / r248089;
        double r248094 = sqrt(r248093);
        double r248095 = r248067 + r248076;
        double r248096 = r248071 + r248095;
        double r248097 = r248083 / r248089;
        double r248098 = r248096 / r248097;
        double r248099 = r248094 / r248098;
        double r248100 = r248092 * r248099;
        double r248101 = r248081 * r248100;
        double r248102 = 0.5;
        double r248103 = r248063 * r248102;
        double r248104 = 0.25;
        double r248105 = r248088 * r248104;
        double r248106 = r248103 + r248105;
        double r248107 = r248072 * r248072;
        double r248108 = r248107 - r248075;
        double r248109 = r248063 + r248088;
        double r248110 = r248109 / r248072;
        double r248111 = r248063 * r248110;
        double r248112 = r248108 / r248111;
        double r248113 = r248106 / r248112;
        double r248114 = r248065 ? r248101 : r248113;
        return r248114;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if i < 4.817585748747987e+97

   1. Initial program 32.8

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

   2. Simplified 13.7

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

   4. Applied add-sqr-sqrt 13.7

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

   5. Applied difference-of-squares 13.7

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

   6. Applied times-frac 11.0

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

   7. Applied add-sqr-sqrt 11.3

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

   8. Applied times-frac 9.4

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

   9. Simplified 9.4

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

   10. Simplified 9.4

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

   12. Applied *-un-lft-identity 9.4

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

   13. Applied add-sqr-sqrt 9.4

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

   14. Applied times-frac 9.3

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

   15. Applied div-inv 9.3

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

   16. Applied sqrt-prod 9.4

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

   17. Applied times-frac 9.5

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

   18. Applied associate-*r* 9.4

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

   19. Simplified 9.4

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

   if 4.817585748747987e+97 < i

   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 52.6

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

   3. Taylor expanded around 0 50.9

      \[\leadsto \frac{\color{blue}{0.5 \cdot i + \left(0.25 \cdot \beta + 0.25 \cdot \alpha\right)}}{\frac{\left(\left(2 \cdot i + \beta\right) + \alpha\right) \cdot \left(\left(2 \cdot i + \beta\right) + \alpha\right) - 1}{\frac{\left(\alpha + \beta\right) + i}{\left(2 \cdot i + \beta\right) + \alpha} \cdot i}}\]

   4. Simplified 50.9

      \[\leadsto \frac{\color{blue}{0.25 \cdot \left(\alpha + \beta\right) + 0.5 \cdot i}}{\frac{\left(\left(2 \cdot i + \beta\right) + \alpha\right) \cdot \left(\left(2 \cdot i + \beta\right) + \alpha\right) - 1}{\frac{\left(\alpha + \beta\right) + i}{\left(2 \cdot i + \beta\right) + \alpha} \cdot i}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 38.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 4.817585748747987305707799364642926876965 \cdot 10^{97}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\alpha + \left(\beta + 2 \cdot i\right)}}}{\frac{\sqrt{\left(\left(\beta + 2 \cdot i\right) - \sqrt{1}\right) + \alpha}}{i}} \cdot \left(\frac{\sqrt{i \cdot \left(\beta + \left(i + \alpha\right)\right) + \beta \cdot \alpha}}{\sqrt{\left(2 \cdot i + \left(\beta + \alpha\right)\right) - \sqrt{1}}} \cdot \frac{\sqrt{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right) + \beta \cdot \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}}{\frac{\left(\beta + 2 \cdot i\right) + \left(\alpha + \sqrt{1}\right)}{\frac{\beta + \left(i + \alpha\right)}{2 \cdot i + \left(\beta + \alpha\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot 0.5 + \left(\beta + \alpha\right) \cdot 0.25}{\frac{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right) - 1}{i \cdot \frac{i + \left(\beta + \alpha\right)}{\alpha + \left(\beta + 2 \cdot i\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (> alpha -1.0) (> beta -1.0) (> i 1.0))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i)))) (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))