Octave 3.8, jcobi/3

Average Error: 3.5 → 2.5

Time: 15.5s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 9.8095473062592694 \cdot 10^{111}:\\ \;\;\;\;\frac{\frac{\frac{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, 1, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - \frac{1}{\alpha}\right) + \frac{2}{{\alpha}^{2}}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, 1, 1\right)}\\ \end{array}\]

C

double f(double alpha, double beta) {
        double r143195 = alpha;
        double r143196 = beta;
        double r143197 = r143195 + r143196;
        double r143198 = r143196 * r143195;
        double r143199 = r143197 + r143198;
        double r143200 = 1.0;
        double r143201 = r143199 + r143200;
        double r143202 = 2.0;
        double r143203 = r143202 * r143200;
        double r143204 = r143197 + r143203;
        double r143205 = r143201 / r143204;
        double r143206 = r143205 / r143204;
        double r143207 = r143204 + r143200;
        double r143208 = r143206 / r143207;
        return r143208;
}

↓

double f(double alpha, double beta) {
        double r143209 = alpha;
        double r143210 = 9.80954730625927e+111;
        bool r143211 = r143209 <= r143210;
        double r143212 = 1.0;
        double r143213 = beta;
        double r143214 = r143209 + r143213;
        double r143215 = fma(r143209, r143213, r143214);
        double r143216 = r143212 + r143215;
        double r143217 = 2.0;
        double r143218 = fma(r143212, r143217, r143214);
        double r143219 = r143216 / r143218;
        double r143220 = r143219 / r143218;
        double r143221 = fma(r143217, r143212, r143212);
        double r143222 = r143214 + r143221;
        double r143223 = r143220 / r143222;
        double r143224 = 1.0;
        double r143225 = r143212 / r143209;
        double r143226 = r143224 - r143225;
        double r143227 = 2.0;
        double r143228 = pow(r143209, r143227);
        double r143229 = r143217 / r143228;
        double r143230 = r143226 + r143229;
        double r143231 = r143230 / r143218;
        double r143232 = r143231 / r143222;
        double r143233 = r143211 ? r143223 : r143232;
        return r143233;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if alpha < 9.80954730625927e+111

   1. Initial program 0.8

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

   2. Simplified 0.8

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

   4. Applied *-un-lft-identity 0.8

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

   5. Applied *-un-lft-identity 0.8

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

   6. Applied *-un-lft-identity 0.8

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

   7. Applied times-frac 0.8

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

   8. Applied times-frac 0.8

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

   9. Simplified 0.8

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

   if 9.80954730625927e+111 < alpha

   1. Initial program 13.8

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

   2. Simplified 13.8

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

   4. Applied *-un-lft-identity 13.8

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

   5. Applied *-un-lft-identity 13.8

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

   6. Applied *-un-lft-identity 13.8

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

   7. Applied times-frac 13.8

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

   8. Applied times-frac 13.8

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

   9. Simplified 13.8

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

   10. Taylor expanded around inf 8.9

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

   11. Simplified 8.9

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

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 9.8095473062592694 \cdot 10^{111}:\\ \;\;\;\;\frac{\frac{\frac{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, 1, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - \frac{1}{\alpha}\right) + \frac{2}{{\alpha}^{2}}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, 1, 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1)))