Octave 3.8, jcobi/4

Average Error: 53.7 → 34.7

Time: 3.2m

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 4.11152951168140068 \cdot 10^{143}:\\ \;\;\;\;\frac{\frac{i}{\frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \alpha + \beta\right), \mathsf{fma}\left(2, i, \alpha + \beta\right), -1\right)}}{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}}{\frac{\frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \alpha + \beta\right), \mathsf{fma}\left(2, i, \alpha + \beta\right), -1\right)}}{\frac{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}}{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\frac{6 \cdot i + \left(3 \cdot \beta + 3 \cdot \alpha\right)}{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r609651 = i;
        double r609652 = alpha;
        double r609653 = beta;
        double r609654 = r609652 + r609653;
        double r609655 = r609654 + r609651;
        double r609656 = r609651 * r609655;
        double r609657 = r609653 * r609652;
        double r609658 = r609657 + r609656;
        double r609659 = r609656 * r609658;
        double r609660 = 2.0;
        double r609661 = r609660 * r609651;
        double r609662 = r609654 + r609661;
        double r609663 = r609662 * r609662;
        double r609664 = r609659 / r609663;
        double r609665 = 1.0;
        double r609666 = r609663 - r609665;
        double r609667 = r609664 / r609666;
        return r609667;
}

↓

double f(double alpha, double beta, double i) {
        double r609668 = i;
        double r609669 = 4.111529511681401e+143;
        bool r609670 = r609668 <= r609669;
        double r609671 = 2.0;
        double r609672 = alpha;
        double r609673 = beta;
        double r609674 = r609672 + r609673;
        double r609675 = fma(r609671, r609668, r609674);
        double r609676 = 1.0;
        double r609677 = -r609676;
        double r609678 = fma(r609675, r609675, r609677);
        double r609679 = sqrt(r609678);
        double r609680 = r609674 + r609668;
        double r609681 = r609668 * r609680;
        double r609682 = fma(r609673, r609672, r609681);
        double r609683 = sqrt(r609682);
        double r609684 = r609679 / r609683;
        double r609685 = r609668 / r609684;
        double r609686 = r609683 / r609675;
        double r609687 = r609679 / r609686;
        double r609688 = r609680 / r609675;
        double r609689 = r609687 / r609688;
        double r609690 = r609685 / r609689;
        double r609691 = 6.0;
        double r609692 = r609691 * r609668;
        double r609693 = 3.0;
        double r609694 = r609693 * r609673;
        double r609695 = r609693 * r609672;
        double r609696 = r609694 + r609695;
        double r609697 = r609692 + r609696;
        double r609698 = r609697 / r609688;
        double r609699 = r609668 / r609698;
        double r609700 = r609670 ? r609690 : r609699;
        return r609700;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if i < 4.111529511681401e+143

   1. Initial program 42.1

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

   2. Simplified 15.2

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

   4. Applied *-un-lft-identity 15.2

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

   5. Applied times-frac 15.1

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

   6. Applied associate-/l* 15.1

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

   8. Applied *-un-lft-identity 15.1

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

   9. Applied *-un-lft-identity 15.1

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

   10. Applied times-frac 15.1

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

   11. Applied *-un-lft-identity 15.1

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

   12. Applied add-sqr-sqrt 15.2

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

   13. Applied times-frac 15.2

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

   14. Applied add-sqr-sqrt 15.2

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

   15. Applied times-frac 15.2

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

   16. Applied times-frac 15.2

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

   17. Applied associate-/r* 15.2

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

   if 4.111529511681401e+143 < 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. Simplified 61.2

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

   4. Applied *-un-lft-identity 61.2

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

   5. Applied times-frac 61.2

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

   6. Applied associate-/l* 61.2

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

   7. Taylor expanded around 0 51.9

      \[\leadsto \frac{\frac{i}{1}}{\frac{\color{blue}{6 \cdot i + \left(3 \cdot \beta + 3 \cdot \alpha\right)}}{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 34.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 4.11152951168140068 \cdot 10^{143}:\\ \;\;\;\;\frac{\frac{i}{\frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \alpha + \beta\right), \mathsf{fma}\left(2, i, \alpha + \beta\right), -1\right)}}{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}}{\frac{\frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \alpha + \beta\right), \mathsf{fma}\left(2, i, \alpha + \beta\right), -1\right)}}{\frac{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}}{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\frac{6 \cdot i + \left(3 \cdot \beta + 3 \cdot \alpha\right)}{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (> alpha -1.0) (> beta -1.0) (> i 1.0))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i)))) (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))