Octave 3.8, jcobi/4

Average Error: 54.2 → 47.4

Time: 25.2s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\beta \le 6.5582632000498063 \cdot 10^{98}:\\ \;\;\;\;\frac{-i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\frac{\frac{\left(-{\left(\left(\alpha + \beta\right) + 2 \cdot i\right)}^{3}\right) + \mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot 1}{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}} \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{\frac{i}{{\left(\left(\alpha + \beta\right) + 2 \cdot i\right)}^{3} + \mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \left(-1\right)}}\right)}^{\left(\frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\right)}\right)\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r123401 = i;
        double r123402 = alpha;
        double r123403 = beta;
        double r123404 = r123402 + r123403;
        double r123405 = r123404 + r123401;
        double r123406 = r123401 * r123405;
        double r123407 = r123403 * r123402;
        double r123408 = r123407 + r123406;
        double r123409 = r123406 * r123408;
        double r123410 = 2.0;
        double r123411 = r123410 * r123401;
        double r123412 = r123404 + r123411;
        double r123413 = r123412 * r123412;
        double r123414 = r123409 / r123413;
        double r123415 = 1.0;
        double r123416 = r123413 - r123415;
        double r123417 = r123414 / r123416;
        return r123417;
}

↓

double f(double alpha, double beta, double i) {
        double r123418 = beta;
        double r123419 = 6.558263200049806e+98;
        bool r123420 = r123418 <= r123419;
        double r123421 = i;
        double r123422 = alpha;
        double r123423 = r123422 + r123418;
        double r123424 = r123423 + r123421;
        double r123425 = r123421 * r123424;
        double r123426 = -r123425;
        double r123427 = 2.0;
        double r123428 = r123427 * r123421;
        double r123429 = r123423 + r123428;
        double r123430 = 3.0;
        double r123431 = pow(r123429, r123430);
        double r123432 = -r123431;
        double r123433 = fma(r123421, r123427, r123423);
        double r123434 = 1.0;
        double r123435 = r123433 * r123434;
        double r123436 = r123432 + r123435;
        double r123437 = fma(r123418, r123422, r123425);
        double r123438 = sqrt(r123437);
        double r123439 = r123436 / r123438;
        double r123440 = r123439 * r123433;
        double r123441 = r123440 / r123438;
        double r123442 = r123426 / r123441;
        double r123443 = -r123434;
        double r123444 = r123433 * r123443;
        double r123445 = r123431 + r123444;
        double r123446 = r123421 / r123445;
        double r123447 = exp(r123446);
        double r123448 = r123433 / r123437;
        double r123449 = r123424 / r123448;
        double r123450 = pow(r123447, r123449);
        double r123451 = log(r123450);
        double r123452 = r123420 ? r123442 : r123451;
        return r123452;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if beta < 6.558263200049806e+98

   1. Initial program 51.5

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

   2. Simplified 50.5

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

   4. Applied add-sqr-sqrt 50.5

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

   5. Applied times-frac 45.9

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

   6. Simplified 46.0

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

   8. Applied frac-2neg 46.0

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

   9. Simplified 46.3

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

   if 6.558263200049806e+98 < beta

   1. Initial program 63.2

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

   2. Simplified 59.2

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

   4. Applied *-un-lft-identity 59.2

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

   5. Applied times-frac 58.9

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

   6. Applied times-frac 58.9

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

   7. Simplified 58.9

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

   9. Applied add-log-exp 59.2

      \[\leadsto \color{blue}{\log \left(e^{\frac{i \cdot 1}{{\left(\left(\alpha + \beta\right) + 2 \cdot i\right)}^{3} + \mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \left(-1\right)} \cdot \frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}\right)}\]

   10. Simplified 51.1

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

4. Final simplification 47.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 6.5582632000498063 \cdot 10^{98}:\\ \;\;\;\;\frac{-i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\frac{\frac{\left(-{\left(\left(\alpha + \beta\right) + 2 \cdot i\right)}^{3}\right) + \mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot 1}{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}} \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{\frac{i}{{\left(\left(\alpha + \beta\right) + 2 \cdot i\right)}^{3} + \mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \left(-1\right)}}\right)}^{\left(\frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +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)))