Octave 3.8, jcobi/4

Average Error: 51.9 → 10.3

Time: 59.0s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 1.2331841345710615 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)} \cdot \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right) + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + i \cdot 2}}{\sqrt{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}} \cdot \frac{i}{\sqrt{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}\right) \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, i, \left(\left(\alpha + \beta\right) \cdot \frac{1}{4}\right)\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r4455666 = i;
        double r4455667 = alpha;
        double r4455668 = beta;
        double r4455669 = r4455667 + r4455668;
        double r4455670 = r4455669 + r4455666;
        double r4455671 = r4455666 * r4455670;
        double r4455672 = r4455668 * r4455667;
        double r4455673 = r4455672 + r4455671;
        double r4455674 = r4455671 * r4455673;
        double r4455675 = 2.0;
        double r4455676 = r4455675 * r4455666;
        double r4455677 = r4455669 + r4455676;
        double r4455678 = r4455677 * r4455677;
        double r4455679 = r4455674 / r4455678;
        double r4455680 = 1.0;
        double r4455681 = r4455678 - r4455680;
        double r4455682 = r4455679 / r4455681;
        return r4455682;
}

↓

double f(double alpha, double beta, double i) {
        double r4455683 = i;
        double r4455684 = 1.2331841345710615e+151;
        bool r4455685 = r4455683 <= r4455684;
        double r4455686 = alpha;
        double r4455687 = beta;
        double r4455688 = r4455686 + r4455687;
        double r4455689 = r4455688 + r4455683;
        double r4455690 = r4455683 * r4455689;
        double r4455691 = 2.0;
        double r4455692 = r4455683 * r4455691;
        double r4455693 = r4455688 + r4455692;
        double r4455694 = r4455690 / r4455693;
        double r4455695 = 1.0;
        double r4455696 = sqrt(r4455695);
        double r4455697 = r4455696 + r4455693;
        double r4455698 = r4455694 / r4455697;
        double r4455699 = r4455686 * r4455687;
        double r4455700 = r4455690 + r4455699;
        double r4455701 = r4455700 / r4455693;
        double r4455702 = r4455693 - r4455696;
        double r4455703 = r4455701 / r4455702;
        double r4455704 = r4455698 * r4455703;
        double r4455705 = r4455689 / r4455693;
        double r4455706 = sqrt(r4455697);
        double r4455707 = r4455705 / r4455706;
        double r4455708 = r4455683 / r4455706;
        double r4455709 = r4455707 * r4455708;
        double r4455710 = 0.5;
        double r4455711 = 0.25;
        double r4455712 = r4455688 * r4455711;
        double r4455713 = fma(r4455710, r4455683, r4455712);
        double r4455714 = r4455713 / r4455702;
        double r4455715 = r4455709 * r4455714;
        double r4455716 = r4455685 ? r4455704 : r4455715;
        return r4455716;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if i < 1.2331841345710615e+151

   1. Initial program 41.1

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

   3. Applied add-sqr-sqrt 41.1

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

   4. Applied difference-of-squares 41.1

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

   5. Applied times-frac 14.6

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

   6. Applied times-frac 10.3

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

   if 1.2331841345710615e+151 < 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 61.4

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

   6. Applied times-frac 61.3

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

   8. Applied add-sqr-sqrt 61.3

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

   9. Applied *-un-lft-identity 61.3

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

   10. Applied times-frac 61.3

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

   11. Applied times-frac 61.3

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

   12. Taylor expanded around 0 10.2

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

   13. Simplified 10.2

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

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 1.2331841345710615 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)} \cdot \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right) + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + i \cdot 2}}{\sqrt{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}} \cdot \frac{i}{\sqrt{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}\right) \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, i, \left(\left(\alpha + \beta\right) \cdot \frac{1}{4}\right)\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 +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)))