Octave 3.8, jcobi/4

Average Error: 53.4 → 10.5

Time: 29.4s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 2.481854927162700235317587220016731075366 \cdot 10^{91}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(i, \alpha + \left(i + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + \sqrt{1}} \cdot \left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \left(\alpha + \left(i + \beta\right)\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{\mathsf{fma}\left(i, 0.5, \mathsf{fma}\left(\frac{1}{i}, 0.125, \left(-\sqrt{1}\right) \cdot 0.25\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)} \cdot \left(\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) - \sqrt{1}} \cdot \frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r111374 = i;
        double r111375 = alpha;
        double r111376 = beta;
        double r111377 = r111375 + r111376;
        double r111378 = r111377 + r111374;
        double r111379 = r111374 * r111378;
        double r111380 = r111376 * r111375;
        double r111381 = r111380 + r111379;
        double r111382 = r111379 * r111381;
        double r111383 = 2.0;
        double r111384 = r111383 * r111374;
        double r111385 = r111377 + r111384;
        double r111386 = r111385 * r111385;
        double r111387 = r111382 / r111386;
        double r111388 = 1.0;
        double r111389 = r111386 - r111388;
        double r111390 = r111387 / r111389;
        return r111390;
}

↓

double f(double alpha, double beta, double i) {
        double r111391 = i;
        double r111392 = 2.4818549271627002e+91;
        bool r111393 = r111391 <= r111392;
        double r111394 = alpha;
        double r111395 = beta;
        double r111396 = r111391 + r111395;
        double r111397 = r111394 + r111396;
        double r111398 = r111394 * r111395;
        double r111399 = fma(r111391, r111397, r111398);
        double r111400 = 2.0;
        double r111401 = r111394 + r111395;
        double r111402 = fma(r111391, r111400, r111401);
        double r111403 = 1.0;
        double r111404 = sqrt(r111403);
        double r111405 = r111402 + r111404;
        double r111406 = r111399 / r111405;
        double r111407 = r111391 / r111402;
        double r111408 = r111407 * r111397;
        double r111409 = r111406 * r111408;
        double r111410 = r111409 / r111402;
        double r111411 = r111402 - r111404;
        double r111412 = r111410 / r111411;
        double r111413 = 0.5;
        double r111414 = r111403 / r111391;
        double r111415 = 0.125;
        double r111416 = -r111404;
        double r111417 = 0.25;
        double r111418 = r111416 * r111417;
        double r111419 = fma(r111414, r111415, r111418);
        double r111420 = fma(r111391, r111413, r111419);
        double r111421 = r111420 / r111402;
        double r111422 = log(r111421);
        double r111423 = exp(r111422);
        double r111424 = r111401 + r111391;
        double r111425 = r111424 / r111411;
        double r111426 = r111425 * r111407;
        double r111427 = r111423 * r111426;
        double r111428 = r111393 ? r111412 : r111427;
        return r111428;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if i < 2.4818549271627002e+91

   1. Initial program 29.0

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

   2. Simplified 29.0

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

   4. Applied add-sqr-sqrt 29.0

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

   5. Applied difference-of-squares 29.0

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

   6. Applied times-frac 12.1

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

   7. Applied times-frac 8.0

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

   8. Simplified 8.0

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

   9. Simplified 8.0

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

   11. Applied *-un-lft-identity 8.0

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

   12. Applied *-un-lft-identity 8.0

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

   13. Applied times-frac 8.0

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

   14. Applied associate-*l* 8.0

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

   15. Simplified 8.0

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

   if 2.4818549271627002e+91 < 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 64.0

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

   4. Applied add-sqr-sqrt 64.0

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

   5. Applied difference-of-squares 64.0

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

   6. Applied times-frac 51.5

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

   7. Applied times-frac 50.4

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

   8. Simplified 50.4

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

   9. Simplified 50.4

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

   10. Taylor expanded around inf 11.6

       \[\leadsto \frac{\color{blue}{\left(0.5 \cdot i + 0.125 \cdot \frac{{\left(\sqrt{1}\right)}^{2}}{i}\right) - 0.25 \cdot \sqrt{1}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \left(\frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) - \sqrt{1}} \cdot \frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)\]

   11. Simplified 11.6

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(i, 0.5, \mathsf{fma}\left(\frac{1}{i}, 0.125, -0.25 \cdot \sqrt{1}\right)\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \left(\frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) - \sqrt{1}} \cdot \frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)\]
   12. Using strategy rm

   13. Applied add-exp-log 16.5

       \[\leadsto \frac{\mathsf{fma}\left(i, 0.5, \mathsf{fma}\left(\frac{1}{i}, 0.125, -0.25 \cdot \sqrt{1}\right)\right)}{\color{blue}{e^{\log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}} \cdot \left(\frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) - \sqrt{1}} \cdot \frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)\]

   14. Applied add-exp-log 15.6

       \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(i, 0.5, \mathsf{fma}\left(\frac{1}{i}, 0.125, -0.25 \cdot \sqrt{1}\right)\right)\right)}}}{e^{\log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}} \cdot \left(\frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) - \sqrt{1}} \cdot \frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)\]

   15. Applied div-exp 15.6

       \[\leadsto \color{blue}{e^{\log \left(\mathsf{fma}\left(i, 0.5, \mathsf{fma}\left(\frac{1}{i}, 0.125, -0.25 \cdot \sqrt{1}\right)\right)\right) - \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}} \cdot \left(\frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) - \sqrt{1}} \cdot \frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)\]

   16. Simplified 11.6

       \[\leadsto e^{\color{blue}{\log \left(\frac{\mathsf{fma}\left(i, 0.5, \mathsf{fma}\left(\frac{1}{i}, 0.125, -0.25 \cdot \sqrt{1}\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)}} \cdot \left(\frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) - \sqrt{1}} \cdot \frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 2.481854927162700235317587220016731075366 \cdot 10^{91}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(i, \alpha + \left(i + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + \sqrt{1}} \cdot \left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \left(\alpha + \left(i + \beta\right)\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{\mathsf{fma}\left(i, 0.5, \mathsf{fma}\left(\frac{1}{i}, 0.125, \left(-\sqrt{1}\right) \cdot 0.25\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)} \cdot \left(\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) - \sqrt{1}} \cdot \frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +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)))