Octave 3.8, jcobi/4

Average Error: 54.1 → 11.5

Time: 37.7s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 1.243617314926794442030271342965650187278 \cdot 10^{103}:\\ \;\;\;\;\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{0.25 \cdot \alpha + \left(0.5 \cdot i + 0.25 \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\right)}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r155045 = i;
        double r155046 = alpha;
        double r155047 = beta;
        double r155048 = r155046 + r155047;
        double r155049 = r155048 + r155045;
        double r155050 = r155045 * r155049;
        double r155051 = r155047 * r155046;
        double r155052 = r155051 + r155050;
        double r155053 = r155050 * r155052;
        double r155054 = 2.0;
        double r155055 = r155054 * r155045;
        double r155056 = r155048 + r155055;
        double r155057 = r155056 * r155056;
        double r155058 = r155053 / r155057;
        double r155059 = 1.0;
        double r155060 = r155057 - r155059;
        double r155061 = r155058 / r155060;
        return r155061;
}

↓

double f(double alpha, double beta, double i) {
        double r155062 = i;
        double r155063 = 1.2436173149267944e+103;
        bool r155064 = r155062 <= r155063;
        double r155065 = alpha;
        double r155066 = beta;
        double r155067 = r155065 + r155066;
        double r155068 = 2.0;
        double r155069 = r155068 * r155062;
        double r155070 = r155067 + r155069;
        double r155071 = r155067 + r155062;
        double r155072 = r155070 / r155071;
        double r155073 = r155062 / r155072;
        double r155074 = 1.0;
        double r155075 = sqrt(r155074);
        double r155076 = r155070 + r155075;
        double r155077 = r155073 / r155076;
        double r155078 = r155066 * r155065;
        double r155079 = r155062 * r155071;
        double r155080 = r155078 + r155079;
        double r155081 = r155080 / r155070;
        double r155082 = r155070 - r155075;
        double r155083 = r155081 / r155082;
        double r155084 = r155077 * r155083;
        double r155085 = 0.25;
        double r155086 = r155085 * r155065;
        double r155087 = 0.5;
        double r155088 = r155087 * r155062;
        double r155089 = r155085 * r155066;
        double r155090 = r155088 + r155089;
        double r155091 = r155086 + r155090;
        double r155092 = r155091 / r155082;
        double r155093 = r155077 * r155092;
        double r155094 = log(r155093);
        double r155095 = exp(r155094);
        double r155096 = r155064 ? r155084 : r155095;
        return r155096;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if i < 1.2436173149267944e+103

   1. Initial program 33.8

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

   3. Applied add-sqr-sqrt 33.8

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

   4. Applied difference-of-squares 33.8

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

   5. Applied times-frac 13.7

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

   6. Applied times-frac 9.0

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

   8. Applied associate-/l* 8.9

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

   if 1.2436173149267944e+103 < 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. Using strategy rm

   3. Applied add-sqr-sqrt 64.0

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

   4. Applied difference-of-squares 64.0

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

   5. Applied times-frac 52.8

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

   6. Applied times-frac 52.2

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

   8. Applied associate-/l* 52.2

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

   9. Taylor expanded around 0 12.8

      \[\leadsto \frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\color{blue}{0.25 \cdot \alpha + \left(0.5 \cdot i + 0.25 \cdot \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\]
   10. Using strategy rm

   11. Applied add-exp-log 17.9

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

   12. Applied add-exp-log 16.9

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

   13. Applied div-exp 16.9

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

   14. Applied add-exp-log 18.0

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

   15. Applied add-exp-log 18.4

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

   16. Applied add-exp-log 18.8

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

   17. Applied div-exp 18.8

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

   18. Applied add-exp-log 18.9

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

   19. Applied div-exp 18.9

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

   20. Applied div-exp 18.9

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

   21. Applied prod-exp 18.9

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

   22. Simplified 12.8

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

4. Final simplification 11.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 1.243617314926794442030271342965650187278 \cdot 10^{103}:\\ \;\;\;\;\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{0.25 \cdot \alpha + \left(0.5 \cdot i + 0.25 \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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)))