Octave 3.8, jcobi/3

Average Error: 3.6 → 2.5

Time: 25.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.838436704407160387237702454048743370255 \cdot 10^{213}:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt{1}}}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right) + 1}} \cdot \frac{\frac{\frac{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(\sqrt{0.5}, \beta, \mathsf{fma}\left(0.75, \alpha \cdot \sqrt{0.5}, 1 \cdot \sqrt{0.5}\right)\right) - 0.125 \cdot \frac{\beta}{\sqrt{0.5}}}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right) + 1}\\ \end{array}\]

C

double f(double alpha, double beta) {
        double r180651 = alpha;
        double r180652 = beta;
        double r180653 = r180651 + r180652;
        double r180654 = r180652 * r180651;
        double r180655 = r180653 + r180654;
        double r180656 = 1.0;
        double r180657 = r180655 + r180656;
        double r180658 = 2.0;
        double r180659 = r180658 * r180656;
        double r180660 = r180653 + r180659;
        double r180661 = r180657 / r180660;
        double r180662 = r180661 / r180660;
        double r180663 = r180660 + r180656;
        double r180664 = r180662 / r180663;
        return r180664;
}

↓

double f(double alpha, double beta) {
        double r180665 = alpha;
        double r180666 = 1.8384367044071604e+213;
        bool r180667 = r180665 <= r180666;
        double r180668 = 1.0;
        double r180669 = sqrt(r180668);
        double r180670 = r180668 / r180669;
        double r180671 = 1.0;
        double r180672 = 2.0;
        double r180673 = beta;
        double r180674 = r180665 + r180673;
        double r180675 = fma(r180671, r180672, r180674);
        double r180676 = sqrt(r180675);
        double r180677 = r180670 / r180676;
        double r180678 = r180675 + r180671;
        double r180679 = sqrt(r180678);
        double r180680 = r180677 / r180679;
        double r180681 = fma(r180665, r180673, r180674);
        double r180682 = r180671 + r180681;
        double r180683 = r180682 / r180676;
        double r180684 = r180683 / r180675;
        double r180685 = r180684 / r180679;
        double r180686 = r180680 * r180685;
        double r180687 = 0.5;
        double r180688 = sqrt(r180687);
        double r180689 = 0.75;
        double r180690 = r180665 * r180688;
        double r180691 = r180671 * r180688;
        double r180692 = fma(r180689, r180690, r180691);
        double r180693 = fma(r180688, r180673, r180692);
        double r180694 = 0.125;
        double r180695 = r180673 / r180688;
        double r180696 = r180694 * r180695;
        double r180697 = r180693 - r180696;
        double r180698 = r180697 / r180676;
        double r180699 = r180698 / r180675;
        double r180700 = r180699 / r180678;
        double r180701 = r180667 ? r180686 : r180700;
        return r180701;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if alpha < 1.8384367044071604e+213

   1. Initial program 2.0

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

   2. Simplified 2.0

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

   4. Applied add-sqr-sqrt 2.5

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

   5. Applied associate-/r* 2.1

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

   7. Applied add-sqr-sqrt 2.6

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

   8. Applied add-sqr-sqrt 2.7

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

   9. Applied *-un-lft-identity 2.7

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

   10. Applied *-un-lft-identity 2.7

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

   11. Applied sqrt-prod 2.7

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

   12. Applied *-un-lft-identity 2.7

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

   13. Applied times-frac 2.7

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

   14. Applied times-frac 2.7

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

   15. Applied times-frac 2.7

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

   16. Applied times-frac 2.3

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

   17. Simplified 2.3

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

   18. Simplified 2.2

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

   if 1.8384367044071604e+213 < alpha

   1. Initial program 18.3

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

   2. Simplified 18.3

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

   4. Applied add-sqr-sqrt 18.3

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

   5. Applied associate-/r* 18.3

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

   6. Taylor expanded around 0 5.4

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\sqrt{0.5} \cdot \beta + \left(0.75 \cdot \left(\alpha \cdot \sqrt{0.5}\right) + 1 \cdot \sqrt{0.5}\right)\right) - 0.125 \cdot \frac{\beta}{\sqrt{0.5}}}}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right) + 1}\]

   7. Simplified 5.4

      \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(\sqrt{0.5}, \beta, \mathsf{fma}\left(0.75, \alpha \cdot \sqrt{0.5}, 1 \cdot \sqrt{0.5}\right)\right) - 0.125 \cdot \frac{\beta}{\sqrt{0.5}}}}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right) + 1}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.838436704407160387237702454048743370255 \cdot 10^{213}:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt{1}}}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right) + 1}} \cdot \frac{\frac{\frac{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(\sqrt{0.5}, \beta, \mathsf{fma}\left(0.75, \alpha \cdot \sqrt{0.5}, 1 \cdot \sqrt{0.5}\right)\right) - 0.125 \cdot \frac{\beta}{\sqrt{0.5}}}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1)))