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

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

↓

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

C

double f(double alpha, double beta, double i) {
        double r20778016 = i;
        double r20778017 = alpha;
        double r20778018 = beta;
        double r20778019 = r20778017 + r20778018;
        double r20778020 = r20778019 + r20778016;
        double r20778021 = r20778016 * r20778020;
        double r20778022 = r20778018 * r20778017;
        double r20778023 = r20778022 + r20778021;
        double r20778024 = r20778021 * r20778023;
        double r20778025 = 2.0;
        double r20778026 = r20778025 * r20778016;
        double r20778027 = r20778019 + r20778026;
        double r20778028 = r20778027 * r20778027;
        double r20778029 = r20778024 / r20778028;
        double r20778030 = 1.0;
        double r20778031 = r20778028 - r20778030;
        double r20778032 = r20778029 / r20778031;
        return r20778032;
}

↓

double f(double alpha, double beta, double i) {
        double r20778033 = beta;
        double r20778034 = 8.636484859471028e+55;
        bool r20778035 = r20778033 <= r20778034;
        double r20778036 = alpha;
        double r20778037 = r20778036 + r20778033;
        double r20778038 = 0.25;
        double r20778039 = i;
        double r20778040 = 0.5;
        double r20778041 = r20778039 * r20778040;
        double r20778042 = fma(r20778037, r20778038, r20778041);
        double r20778043 = 1.0;
        double r20778044 = sqrt(r20778043);
        double r20778045 = 2.0;
        double r20778046 = fma(r20778045, r20778039, r20778037);
        double r20778047 = r20778044 + r20778046;
        double r20778048 = r20778042 / r20778047;
        double r20778049 = r20778039 + r20778037;
        double r20778050 = r20778039 / r20778046;
        double r20778051 = r20778049 * r20778050;
        double r20778052 = r20778046 - r20778044;
        double r20778053 = r20778051 / r20778052;
        double r20778054 = exp(r20778053);
        double r20778055 = log(r20778054);
        double r20778056 = r20778048 * r20778055;
        double r20778057 = r20778039 / r20778047;
        double r20778058 = r20778057 * r20778053;
        double r20778059 = r20778035 ? r20778056 : r20778058;
        return r20778059;
}

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

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

   4. Applied add-sqr-sqrt 49.5

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

   5. Applied difference-of-squares 49.5

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

   6. Applied times-frac 34.8

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

   7. Applied times-frac 33.5

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

   9. Applied *-un-lft-identity 33.5

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

   10. Applied times-frac 33.4

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

   11. Taylor expanded around 0 10.5

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

   12. Simplified 10.5

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

   14. Applied add-log-exp 9.7

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

   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

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

   4. Applied add-sqr-sqrt 60.1

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

   5. Applied difference-of-squares 60.1

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

   6. Applied times-frac 49.0

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

   7. Applied times-frac 44.5

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

   9. Applied *-un-lft-identity 44.5

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

   10. Applied times-frac 44.5

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

   11. Taylor expanded around -inf 26.6

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

4. Final simplification 14.4

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

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)))