Octave 3.8, jcobi/4

Average Error: 53.9 → 36.7

Time: 29.1s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.537706138612880776985619096873344020409 \cdot 10^{194}:\\ \;\;\;\;\frac{i}{\left(\sqrt{1} + \mathsf{fma}\left(2, i, \alpha + \beta\right)\right) \cdot \frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r108196 = i;
        double r108197 = alpha;
        double r108198 = beta;
        double r108199 = r108197 + r108198;
        double r108200 = r108199 + r108196;
        double r108201 = r108196 * r108200;
        double r108202 = r108198 * r108197;
        double r108203 = r108202 + r108201;
        double r108204 = r108201 * r108203;
        double r108205 = 2.0;
        double r108206 = r108205 * r108196;
        double r108207 = r108199 + r108206;
        double r108208 = r108207 * r108207;
        double r108209 = r108204 / r108208;
        double r108210 = 1.0;
        double r108211 = r108208 - r108210;
        double r108212 = r108209 / r108211;
        return r108212;
}

↓

double f(double alpha, double beta, double i) {
        double r108213 = alpha;
        double r108214 = 1.5377061386128808e+194;
        bool r108215 = r108213 <= r108214;
        double r108216 = i;
        double r108217 = 1.0;
        double r108218 = sqrt(r108217);
        double r108219 = 2.0;
        double r108220 = beta;
        double r108221 = r108213 + r108220;
        double r108222 = fma(r108219, r108216, r108221);
        double r108223 = r108218 + r108222;
        double r108224 = r108221 + r108216;
        double r108225 = r108216 * r108224;
        double r108226 = fma(r108220, r108213, r108225);
        double r108227 = sqrt(r108226);
        double r108228 = r108222 / r108227;
        double r108229 = r108223 * r108228;
        double r108230 = r108216 / r108229;
        double r108231 = r108224 / r108228;
        double r108232 = r108222 - r108218;
        double r108233 = r108231 / r108232;
        double r108234 = r108230 * r108233;
        double r108235 = 0.0;
        double r108236 = r108215 ? r108234 : r108235;
        return r108236;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if alpha < 1.5377061386128808e+194

   1. Initial program 52.5

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

   3. Applied associate-/l* 37.9

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

   4. Simplified 37.9

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

   6. Applied add-sqr-sqrt 37.9

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

   7. Applied difference-of-squares 37.9

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

   8. Applied add-sqr-sqrt 37.9

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

   9. Applied times-frac 37.9

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

   10. Applied times-frac 37.9

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

   11. Applied times-frac 35.6

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

   12. Simplified 35.6

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

   13. Simplified 35.6

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

   if 1.5377061386128808e+194 < alpha

   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. Using strategy rm

   3. Applied associate-/l* 57.4

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

   4. Simplified 57.4

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

   5. Taylor expanded around inf 44.8

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

4. Final simplification 36.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.537706138612880776985619096873344020409 \cdot 10^{194}:\\ \;\;\;\;\frac{i}{\left(\sqrt{1} + \mathsf{fma}\left(2, i, \alpha + \beta\right)\right) \cdot \frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +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)))