Octave 3.8, jcobi/4

Average Error: 52.4 → 11.2

Time: 26.4s

Precision: 64

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

Math

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

↓

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

C

double f(double alpha, double beta, double i) {
        double r3610534 = i;
        double r3610535 = alpha;
        double r3610536 = beta;
        double r3610537 = r3610535 + r3610536;
        double r3610538 = r3610537 + r3610534;
        double r3610539 = r3610534 * r3610538;
        double r3610540 = r3610536 * r3610535;
        double r3610541 = r3610540 + r3610539;
        double r3610542 = r3610539 * r3610541;
        double r3610543 = 2.0;
        double r3610544 = r3610543 * r3610534;
        double r3610545 = r3610537 + r3610544;
        double r3610546 = r3610545 * r3610545;
        double r3610547 = r3610542 / r3610546;
        double r3610548 = 1.0;
        double r3610549 = r3610546 - r3610548;
        double r3610550 = r3610547 / r3610549;
        return r3610550;
}

↓

double f(double alpha, double beta, double i) {
        double r3610551 = i;
        double r3610552 = 2.495505368754722e+120;
        bool r3610553 = r3610551 <= r3610552;
        double r3610554 = alpha;
        double r3610555 = beta;
        double r3610556 = r3610554 + r3610555;
        double r3610557 = r3610556 + r3610551;
        double r3610558 = r3610554 * r3610555;
        double r3610559 = fma(r3610557, r3610551, r3610558);
        double r3610560 = 2.0;
        double r3610561 = fma(r3610560, r3610551, r3610556);
        double r3610562 = r3610559 / r3610561;
        double r3610563 = 1.0;
        double r3610564 = sqrt(r3610563);
        double r3610565 = r3610564 + r3610561;
        double r3610566 = r3610562 / r3610565;
        double r3610567 = r3610551 / r3610561;
        double r3610568 = r3610567 * r3610557;
        double r3610569 = r3610561 - r3610564;
        double r3610570 = r3610568 / r3610569;
        double r3610571 = r3610566 * r3610570;
        double r3610572 = 0.25;
        double r3610573 = 0.5;
        double r3610574 = r3610551 * r3610573;
        double r3610575 = fma(r3610572, r3610556, r3610574);
        double r3610576 = r3610575 / r3610565;
        double r3610577 = /* ERROR: no posit support in C */;
        double r3610578 = /* ERROR: no posit support in C */;
        double r3610579 = r3610576 * r3610578;
        double r3610580 = r3610553 ? r3610571 : r3610579;
        return r3610580;
}

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. Simplified 37.3

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

   4. Applied add-sqr-sqrt 37.3

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

   5. Applied difference-of-squares 37.3

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

   6. Applied times-frac 14.7

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

   7. Applied times-frac 10.0

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

   9. Applied *-un-lft-identity 10.0

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

   10. Applied times-frac 10.0

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

   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. Simplified 62.1

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

   4. Applied add-sqr-sqrt 62.1

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

   5. Applied difference-of-squares 62.1

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

   6. Applied times-frac 54.9

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

   7. Applied times-frac 54.6

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

   9. Applied *-un-lft-identity 54.6

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

   10. Applied times-frac 54.6

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

   11. Taylor expanded around 0 11.8

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

   12. Simplified 11.8

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

   14. Applied insert-posit16 12.0

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

4. Final simplification 11.2

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

Reproduce

herbie shell --seed 2019163 +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)))