Octave 3.8, jcobi/4

Average Error: 52.4 → 11.1

Time: 40.1s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 2.495505368754722 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}}{\sqrt{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}} \cdot \frac{\sqrt{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}}{\sqrt{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}\right) \cdot \frac{\frac{1}{4} \cdot \left(\alpha + \beta\right) + \frac{1}{2} \cdot i}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r4192729 = i;
        double r4192730 = alpha;
        double r4192731 = beta;
        double r4192732 = r4192730 + r4192731;
        double r4192733 = r4192732 + r4192729;
        double r4192734 = r4192729 * r4192733;
        double r4192735 = r4192731 * r4192730;
        double r4192736 = r4192735 + r4192734;
        double r4192737 = r4192734 * r4192736;
        double r4192738 = 2.0;
        double r4192739 = r4192738 * r4192729;
        double r4192740 = r4192732 + r4192739;
        double r4192741 = r4192740 * r4192740;
        double r4192742 = r4192737 / r4192741;
        double r4192743 = 1.0;
        double r4192744 = r4192741 - r4192743;
        double r4192745 = r4192742 / r4192744;
        return r4192745;
}

↓

double f(double alpha, double beta, double i) {
        double r4192746 = i;
        double r4192747 = 2.495505368754722e+120;
        bool r4192748 = r4192746 <= r4192747;
        double r4192749 = alpha;
        double r4192750 = beta;
        double r4192751 = r4192749 + r4192750;
        double r4192752 = 2.0;
        double r4192753 = r4192746 * r4192752;
        double r4192754 = r4192751 + r4192753;
        double r4192755 = r4192751 + r4192746;
        double r4192756 = r4192754 / r4192755;
        double r4192757 = r4192746 / r4192756;
        double r4192758 = 1.0;
        double r4192759 = sqrt(r4192758);
        double r4192760 = r4192759 + r4192754;
        double r4192761 = r4192757 / r4192760;
        double r4192762 = r4192755 * r4192746;
        double r4192763 = r4192749 * r4192750;
        double r4192764 = r4192762 + r4192763;
        double r4192765 = r4192764 / r4192754;
        double r4192766 = r4192761 * r4192765;
        double r4192767 = r4192754 - r4192759;
        double r4192768 = r4192766 / r4192767;
        double r4192769 = sqrt(r4192757);
        double r4192770 = sqrt(r4192760);
        double r4192771 = r4192769 / r4192770;
        double r4192772 = r4192771 * r4192771;
        double r4192773 = 0.25;
        double r4192774 = r4192773 * r4192751;
        double r4192775 = 0.5;
        double r4192776 = r4192775 * r4192746;
        double r4192777 = r4192774 + r4192776;
        double r4192778 = r4192777 / r4192767;
        double r4192779 = r4192772 * r4192778;
        double r4192780 = r4192748 ? r4192768 : r4192779;
        return r4192780;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if i < 2.495505368754722e+120

   1. Initial program 37.3

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

   3. Applied add-sqr-sqrt 37.3

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

   4. Applied difference-of-squares 37.3

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

   5. Applied times-frac 14.7

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

   6. Applied times-frac 10.0

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

   8. Applied associate-/l* 10.0

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

   10. Applied associate-*r/ 9.9

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

   if 2.495505368754722e+120 < 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 54.9

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

   6. Applied times-frac 54.6

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

   8. Applied associate-/l* 54.6

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

   9. Taylor expanded around 0 11.8

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

   10. Simplified 11.8

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

   12. Applied add-sqr-sqrt 12.3

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

   13. Applied add-sqr-sqrt 11.8

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

   14. Applied times-frac 11.8

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

4. Final simplification 11.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 2.495505368754722 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}}{\sqrt{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}} \cdot \frac{\sqrt{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}}{\sqrt{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}\right) \cdot \frac{\frac{1}{4} \cdot \left(\alpha + \beta\right) + \frac{1}{2} \cdot i}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(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)))