Octave 3.8, jcobi/4

Average Error: 53.9 → 10.9

Time: 13.4s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 4.0079005132304603 \cdot 10^{132}:\\ \;\;\;\;\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}{1}} \cdot \left(\frac{i}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{i}{4} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r169478 = i;
        double r169479 = alpha;
        double r169480 = beta;
        double r169481 = r169479 + r169480;
        double r169482 = r169481 + r169478;
        double r169483 = r169478 * r169482;
        double r169484 = r169480 * r169479;
        double r169485 = r169484 + r169483;
        double r169486 = r169483 * r169485;
        double r169487 = 2.0;
        double r169488 = r169487 * r169478;
        double r169489 = r169481 + r169488;
        double r169490 = r169489 * r169489;
        double r169491 = r169486 / r169490;
        double r169492 = 1.0;
        double r169493 = r169490 - r169492;
        double r169494 = r169491 / r169493;
        return r169494;
}

↓

double f(double alpha, double beta, double i) {
        double r169495 = i;
        double r169496 = 4.0079005132304603e+132;
        bool r169497 = r169495 <= r169496;
        double r169498 = 1.0;
        double r169499 = alpha;
        double r169500 = beta;
        double r169501 = r169499 + r169500;
        double r169502 = 2.0;
        double r169503 = r169502 * r169495;
        double r169504 = r169501 + r169503;
        double r169505 = 1.0;
        double r169506 = sqrt(r169505);
        double r169507 = r169504 + r169506;
        double r169508 = r169507 / r169498;
        double r169509 = r169498 / r169508;
        double r169510 = r169504 - r169506;
        double r169511 = r169501 + r169495;
        double r169512 = r169495 * r169511;
        double r169513 = fma(r169500, r169499, r169512);
        double r169514 = r169510 / r169513;
        double r169515 = r169495 / r169514;
        double r169516 = fma(r169495, r169502, r169501);
        double r169517 = r169511 / r169516;
        double r169518 = r169517 / r169516;
        double r169519 = r169515 * r169518;
        double r169520 = r169509 * r169519;
        double r169521 = 4.0;
        double r169522 = r169495 / r169521;
        double r169523 = r169522 * r169518;
        double r169524 = log(r169523);
        double r169525 = exp(r169524);
        double r169526 = r169497 ? r169520 : r169525;
        return r169526;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if i < 4.0079005132304603e+132

   1. Initial program 40.6

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

   2. Simplified 37.0

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

   4. Applied associate-/l* 29.2

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

   6. Applied div-inv 29.3

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

   7. Applied times-frac 17.9

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

   8. Applied times-frac 15.0

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

   9. Simplified 14.9

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

   11. Applied *-un-lft-identity 14.9

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

   12. Applied add-sqr-sqrt 14.9

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

   13. Applied difference-of-squares 14.9

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

   14. Applied times-frac 10.5

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

   15. Applied *-un-lft-identity 10.5

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

   16. Applied times-frac 10.6

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

   17. Applied associate-*l* 10.6

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

   if 4.0079005132304603e+132 < 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 63.8

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

   4. Applied associate-/l* 63.8

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

   6. Applied div-inv 63.8

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

   7. Applied times-frac 58.7

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

   8. Applied times-frac 58.5

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

   9. Simplified 58.5

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

   10. Taylor expanded around 0 11.2

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

   12. Applied add-exp-log 16.4

       \[\leadsto \frac{i}{4} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\color{blue}{e^{\log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}}\]

   13. Applied add-exp-log 17.2

       \[\leadsto \frac{i}{4} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\color{blue}{e^{\log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}}}{e^{\log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}\]

   14. Applied add-exp-log 17.3

       \[\leadsto \frac{i}{4} \cdot \frac{\frac{\color{blue}{e^{\log \left(\left(\alpha + \beta\right) + i\right)}}}{e^{\log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}}{e^{\log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}\]

   15. Applied div-exp 17.3

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

   16. Applied div-exp 17.3

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

   17. Applied add-exp-log 17.3

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

   18. Applied add-exp-log 17.1

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

   19. Applied div-exp 17.5

       \[\leadsto \color{blue}{e^{\log i - \log 4}} \cdot e^{\left(\log \left(\left(\alpha + \beta\right) + i\right) - \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)\right) - \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}\]

   20. Applied prod-exp 17.5

       \[\leadsto \color{blue}{e^{\left(\log i - \log 4\right) + \left(\left(\log \left(\left(\alpha + \beta\right) + i\right) - \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)\right) - \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)\right)}}\]

   21. Simplified 11.1

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

4. Final simplification 10.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 4.0079005132304603 \cdot 10^{132}:\\ \;\;\;\;\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}{1}} \cdot \left(\frac{i}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{i}{4} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :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)))