Octave 3.8, jcobi/4

Average Error: 52.3 → 35.3

Time: 5.2m

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.2994741600183278 \cdot 10^{+189}:\\ \;\;\;\;\sqrt{\frac{\frac{i + \left(\alpha + \beta\right)}{\left(\alpha + \beta\right) + i \cdot 2} \cdot i}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}} \cdot \left(\frac{\sqrt{\beta \cdot \alpha + i \cdot \left(i + \left(\alpha + \beta\right)\right)}}{\frac{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}{\frac{\sqrt{\beta \cdot \alpha + i \cdot \left(i + \left(\alpha + \beta\right)\right)}}{\left(\alpha + \beta\right) + i \cdot 2}}} \cdot \sqrt{\frac{\frac{i + \left(\alpha + \beta\right)}{\left(\alpha + \beta\right) + i \cdot 2} \cdot i}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r10169374 = i;
        double r10169375 = alpha;
        double r10169376 = beta;
        double r10169377 = r10169375 + r10169376;
        double r10169378 = r10169377 + r10169374;
        double r10169379 = r10169374 * r10169378;
        double r10169380 = r10169376 * r10169375;
        double r10169381 = r10169380 + r10169379;
        double r10169382 = r10169379 * r10169381;
        double r10169383 = 2.0;
        double r10169384 = r10169383 * r10169374;
        double r10169385 = r10169377 + r10169384;
        double r10169386 = r10169385 * r10169385;
        double r10169387 = r10169382 / r10169386;
        double r10169388 = 1.0;
        double r10169389 = r10169386 - r10169388;
        double r10169390 = r10169387 / r10169389;
        return r10169390;
}

↓

double f(double alpha, double beta, double i) {
        double r10169391 = alpha;
        double r10169392 = 1.2994741600183278e+189;
        bool r10169393 = r10169391 <= r10169392;
        double r10169394 = i;
        double r10169395 = beta;
        double r10169396 = r10169391 + r10169395;
        double r10169397 = r10169394 + r10169396;
        double r10169398 = 2.0;
        double r10169399 = r10169394 * r10169398;
        double r10169400 = r10169396 + r10169399;
        double r10169401 = r10169397 / r10169400;
        double r10169402 = r10169401 * r10169394;
        double r10169403 = 1.0;
        double r10169404 = sqrt(r10169403);
        double r10169405 = r10169404 + r10169400;
        double r10169406 = r10169402 / r10169405;
        double r10169407 = sqrt(r10169406);
        double r10169408 = r10169395 * r10169391;
        double r10169409 = r10169394 * r10169397;
        double r10169410 = r10169408 + r10169409;
        double r10169411 = sqrt(r10169410);
        double r10169412 = r10169400 - r10169404;
        double r10169413 = r10169411 / r10169400;
        double r10169414 = r10169412 / r10169413;
        double r10169415 = r10169411 / r10169414;
        double r10169416 = r10169415 * r10169407;
        double r10169417 = r10169407 * r10169416;
        double r10169418 = 0.0;
        double r10169419 = r10169393 ? r10169417 : r10169418;
        return r10169419;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if alpha < 1.2994741600183278e+189

   1. Initial program 50.9

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

   3. Applied add-sqr-sqrt 50.9

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

   4. Applied difference-of-squares 50.9

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

   5. Applied times-frac 35.9

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

   6. Applied times-frac 33.7

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

   8. Applied *-un-lft-identity 33.7

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

   9. Applied add-sqr-sqrt 33.7

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

   10. Applied times-frac 33.7

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

   11. Applied associate-/l* 33.7

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

   13. Applied *-un-lft-identity 33.7

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

   14. Applied times-frac 33.7

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

   15. Simplified 33.7

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

   17. Applied add-sqr-sqrt 33.7

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

   18. Applied associate-*l* 33.7

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

   if 1.2994741600183278e+189 < alpha

   1. Initial program 62.6

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

   2. Taylor expanded around -inf 46.8

      \[\leadsto \color{blue}{0}\]
3. Recombined 2 regimes into one program.

4. Final simplification 35.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.2994741600183278 \cdot 10^{+189}:\\ \;\;\;\;\sqrt{\frac{\frac{i + \left(\alpha + \beta\right)}{\left(\alpha + \beta\right) + i \cdot 2} \cdot i}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}} \cdot \left(\frac{\sqrt{\beta \cdot \alpha + i \cdot \left(i + \left(\alpha + \beta\right)\right)}}{\frac{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}{\frac{\sqrt{\beta \cdot \alpha + i \cdot \left(i + \left(\alpha + \beta\right)\right)}}{\left(\alpha + \beta\right) + i \cdot 2}}} \cdot \sqrt{\frac{\frac{i + \left(\alpha + \beta\right)}{\left(\alpha + \beta\right) + i \cdot 2} \cdot i}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(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)))