Octave 3.8, jcobi/4

Average Error: 53.4 → 10.5

Time: 29.2s

Precision: 64

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

Math

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

↓

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

C

double f(double alpha, double beta, double i) {
        double r116451 = i;
        double r116452 = alpha;
        double r116453 = beta;
        double r116454 = r116452 + r116453;
        double r116455 = r116454 + r116451;
        double r116456 = r116451 * r116455;
        double r116457 = r116453 * r116452;
        double r116458 = r116457 + r116456;
        double r116459 = r116456 * r116458;
        double r116460 = 2.0;
        double r116461 = r116460 * r116451;
        double r116462 = r116454 + r116461;
        double r116463 = r116462 * r116462;
        double r116464 = r116459 / r116463;
        double r116465 = 1.0;
        double r116466 = r116463 - r116465;
        double r116467 = r116464 / r116466;
        return r116467;
}

↓

double f(double alpha, double beta, double i) {
        double r116468 = i;
        double r116469 = 2.4818549271627002e+91;
        bool r116470 = r116468 <= r116469;
        double r116471 = beta;
        double r116472 = r116471 + r116468;
        double r116473 = alpha;
        double r116474 = r116472 + r116473;
        double r116475 = r116473 * r116471;
        double r116476 = fma(r116468, r116474, r116475);
        double r116477 = 1.0;
        double r116478 = sqrt(r116477);
        double r116479 = 2.0;
        double r116480 = r116471 + r116473;
        double r116481 = fma(r116468, r116479, r116480);
        double r116482 = r116478 + r116481;
        double r116483 = r116476 / r116482;
        double r116484 = r116468 / r116481;
        double r116485 = r116484 * r116474;
        double r116486 = r116483 * r116485;
        double r116487 = r116486 / r116481;
        double r116488 = r116481 - r116478;
        double r116489 = r116487 / r116488;
        double r116490 = r116480 + r116468;
        double r116491 = r116490 / r116488;
        double r116492 = r116491 * r116484;
        double r116493 = 0.5;
        double r116494 = r116477 / r116468;
        double r116495 = 0.125;
        double r116496 = 0.25;
        double r116497 = -r116496;
        double r116498 = r116497 * r116478;
        double r116499 = fma(r116494, r116495, r116498);
        double r116500 = fma(r116468, r116493, r116499);
        double r116501 = r116500 / r116481;
        double r116502 = log(r116501);
        double r116503 = exp(r116502);
        double r116504 = r116492 * r116503;
        double r116505 = r116470 ? r116489 : r116504;
        return r116505;
}

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

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

   12. Applied associate-*l* 8.0

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

   13. Simplified 8.0

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

   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

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

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)))