Octave 3.8, jcobi/4

Average Error: 52.9 → 10.9

Time: 1.1m

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 2.736160234700738 \cdot 10^{+130}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\sqrt{1.0} + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\left(\beta + \alpha\right) + i}{2 \cdot i + \left(\beta + \alpha\right)}}}} \cdot \left(\frac{i}{\sqrt{\frac{\sqrt{1.0} + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\left(\beta + \alpha\right) + i}{2 \cdot i + \left(\beta + \alpha\right)}}}} \cdot \frac{\frac{\alpha \cdot \beta + i \cdot \left(\left(\beta + \alpha\right) + i\right)}{2 \cdot i + \left(\beta + \alpha\right)}}{\left(2 \cdot i + \left(\beta + \alpha\right)\right) - \sqrt{1.0}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{\frac{1}{2} \cdot i + \left(\alpha \cdot \frac{1}{4} + \frac{1}{4} \cdot \beta\right)}{\left(2 \cdot i + \left(\beta + \alpha\right)\right) - \sqrt{1.0}} \cdot \frac{i}{\frac{\sqrt{1.0} + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\left(\beta + \alpha\right) + i}{2 \cdot i + \left(\beta + \alpha\right)}}}\right)}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r2861775 = i;
        double r2861776 = alpha;
        double r2861777 = beta;
        double r2861778 = r2861776 + r2861777;
        double r2861779 = r2861778 + r2861775;
        double r2861780 = r2861775 * r2861779;
        double r2861781 = r2861777 * r2861776;
        double r2861782 = r2861781 + r2861780;
        double r2861783 = r2861780 * r2861782;
        double r2861784 = 2.0;
        double r2861785 = r2861784 * r2861775;
        double r2861786 = r2861778 + r2861785;
        double r2861787 = r2861786 * r2861786;
        double r2861788 = r2861783 / r2861787;
        double r2861789 = 1.0;
        double r2861790 = r2861787 - r2861789;
        double r2861791 = r2861788 / r2861790;
        return r2861791;
}

↓

double f(double alpha, double beta, double i) {
        double r2861792 = i;
        double r2861793 = 2.736160234700738e+130;
        bool r2861794 = r2861792 <= r2861793;
        double r2861795 = 1.0;
        double r2861796 = 1.0;
        double r2861797 = sqrt(r2861796);
        double r2861798 = 2.0;
        double r2861799 = r2861798 * r2861792;
        double r2861800 = beta;
        double r2861801 = alpha;
        double r2861802 = r2861800 + r2861801;
        double r2861803 = r2861799 + r2861802;
        double r2861804 = r2861797 + r2861803;
        double r2861805 = r2861802 + r2861792;
        double r2861806 = r2861805 / r2861803;
        double r2861807 = r2861804 / r2861806;
        double r2861808 = sqrt(r2861807);
        double r2861809 = r2861795 / r2861808;
        double r2861810 = r2861792 / r2861808;
        double r2861811 = r2861801 * r2861800;
        double r2861812 = r2861792 * r2861805;
        double r2861813 = r2861811 + r2861812;
        double r2861814 = r2861813 / r2861803;
        double r2861815 = r2861803 - r2861797;
        double r2861816 = r2861814 / r2861815;
        double r2861817 = r2861810 * r2861816;
        double r2861818 = r2861809 * r2861817;
        double r2861819 = 0.5;
        double r2861820 = r2861819 * r2861792;
        double r2861821 = 0.25;
        double r2861822 = r2861801 * r2861821;
        double r2861823 = r2861821 * r2861800;
        double r2861824 = r2861822 + r2861823;
        double r2861825 = r2861820 + r2861824;
        double r2861826 = r2861825 / r2861815;
        double r2861827 = r2861792 / r2861807;
        double r2861828 = r2861826 * r2861827;
        double r2861829 = log(r2861828);
        double r2861830 = exp(r2861829);
        double r2861831 = r2861794 ? r2861818 : r2861830;
        return r2861831;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if i < 2.736160234700738e+130

   1. Initial program 40.4

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

   3. Applied add-sqr-sqrt 40.4

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

   4. Applied difference-of-squares 40.4

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

   5. Applied times-frac 15.7

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

   6. Applied times-frac 10.5

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

   8. Applied *-un-lft-identity 10.5

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

   9. Applied times-frac 10.5

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

   10. Applied associate-/l* 10.4

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

   12. Applied add-sqr-sqrt 10.7

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

   13. Applied *-un-lft-identity 10.7

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

   14. Applied times-frac 10.7

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

   15. Applied associate-*l* 10.7

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

   if 2.736160234700738e+130 < i

   1. Initial program 62.1

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

   3. Applied add-sqr-sqrt 62.1

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

   4. Applied difference-of-squares 62.1

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

   5. Applied times-frac 56.4

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

   6. Applied times-frac 56.2

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

   8. Applied *-un-lft-identity 56.2

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

   9. Applied times-frac 56.2

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

   10. Applied associate-/l* 56.2

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

   11. Taylor expanded around 0 11.0

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

   13. Applied add-exp-log 11.0

       \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{i}{1}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}}{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}} \cdot \frac{\frac{1}{2} \cdot i + \left(\frac{1}{4} \cdot \beta + \frac{1}{4} \cdot \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 10.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 2.736160234700738 \cdot 10^{+130}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\sqrt{1.0} + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\left(\beta + \alpha\right) + i}{2 \cdot i + \left(\beta + \alpha\right)}}}} \cdot \left(\frac{i}{\sqrt{\frac{\sqrt{1.0} + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\left(\beta + \alpha\right) + i}{2 \cdot i + \left(\beta + \alpha\right)}}}} \cdot \frac{\frac{\alpha \cdot \beta + i \cdot \left(\left(\beta + \alpha\right) + i\right)}{2 \cdot i + \left(\beta + \alpha\right)}}{\left(2 \cdot i + \left(\beta + \alpha\right)\right) - \sqrt{1.0}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{\frac{1}{2} \cdot i + \left(\alpha \cdot \frac{1}{4} + \frac{1}{4} \cdot \beta\right)}{\left(2 \cdot i + \left(\beta + \alpha\right)\right) - \sqrt{1.0}} \cdot \frac{i}{\frac{\sqrt{1.0} + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\left(\beta + \alpha\right) + i}{2 \cdot i + \left(\beta + \alpha\right)}}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 
(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)))