Octave 3.8, jcobi/4

Average Error: 54.6 → 12.8

Time: 50.4s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\beta \le 5.188945744706776056846709445192721671049 \cdot 10^{218}:\\ \;\;\;\;\sqrt[3]{\left(\left(\frac{\frac{i}{\frac{\left(\beta + \alpha\right) + i \cdot 2}{i + \left(\beta + \alpha\right)}}}{\sqrt{1} + \left(\left(\beta + \alpha\right) + i \cdot 2\right)} \cdot \frac{\mathsf{fma}\left(0.25, \beta + \alpha, i \cdot 0.5\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - \sqrt{1}}\right) \cdot \left(\frac{\frac{i}{\frac{\left(\beta + \alpha\right) + i \cdot 2}{i + \left(\beta + \alpha\right)}}}{\sqrt{1} + \left(\left(\beta + \alpha\right) + i \cdot 2\right)} \cdot \frac{\mathsf{fma}\left(0.25, \beta + \alpha, i \cdot 0.5\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - \sqrt{1}}\right)\right) \cdot \left(\frac{\frac{i}{\frac{\left(\beta + \alpha\right) + i \cdot 2}{i + \left(\beta + \alpha\right)}}}{\sqrt{1} + \left(\left(\beta + \alpha\right) + i \cdot 2\right)} \cdot \frac{\mathsf{fma}\left(0.25, \beta + \alpha, i \cdot 0.5\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - \sqrt{1}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - \sqrt{1}} \cdot \frac{\frac{i}{\frac{\left(\beta + \alpha\right) + i \cdot 2}{i + \left(\beta + \alpha\right)}}}{\sqrt{1} + \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r4280208 = i;
        double r4280209 = alpha;
        double r4280210 = beta;
        double r4280211 = r4280209 + r4280210;
        double r4280212 = r4280211 + r4280208;
        double r4280213 = r4280208 * r4280212;
        double r4280214 = r4280210 * r4280209;
        double r4280215 = r4280214 + r4280213;
        double r4280216 = r4280213 * r4280215;
        double r4280217 = 2.0;
        double r4280218 = r4280217 * r4280208;
        double r4280219 = r4280211 + r4280218;
        double r4280220 = r4280219 * r4280219;
        double r4280221 = r4280216 / r4280220;
        double r4280222 = 1.0;
        double r4280223 = r4280220 - r4280222;
        double r4280224 = r4280221 / r4280223;
        return r4280224;
}

↓

double f(double alpha, double beta, double i) {
        double r4280225 = beta;
        double r4280226 = 5.188945744706776e+218;
        bool r4280227 = r4280225 <= r4280226;
        double r4280228 = i;
        double r4280229 = alpha;
        double r4280230 = r4280225 + r4280229;
        double r4280231 = 2.0;
        double r4280232 = r4280228 * r4280231;
        double r4280233 = r4280230 + r4280232;
        double r4280234 = r4280228 + r4280230;
        double r4280235 = r4280233 / r4280234;
        double r4280236 = r4280228 / r4280235;
        double r4280237 = 1.0;
        double r4280238 = sqrt(r4280237);
        double r4280239 = r4280238 + r4280233;
        double r4280240 = r4280236 / r4280239;
        double r4280241 = 0.25;
        double r4280242 = 0.5;
        double r4280243 = r4280228 * r4280242;
        double r4280244 = fma(r4280241, r4280230, r4280243);
        double r4280245 = r4280233 - r4280238;
        double r4280246 = r4280244 / r4280245;
        double r4280247 = r4280240 * r4280246;
        double r4280248 = r4280247 * r4280247;
        double r4280249 = r4280248 * r4280247;
        double r4280250 = cbrt(r4280249);
        double r4280251 = r4280228 / r4280245;
        double r4280252 = r4280251 * r4280240;
        double r4280253 = r4280227 ? r4280250 : r4280252;
        return r4280253;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if beta < 5.188945744706776e+218

   1. Initial program 53.7

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

   3. Applied add-sqr-sqrt 53.7

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

   4. Applied difference-of-squares 53.7

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

   5. Applied times-frac 38.6

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

   6. Applied times-frac 36.2

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

   8. Applied associate-/l* 36.2

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

   9. Taylor expanded around 0 14.4

      \[\leadsto \frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\color{blue}{0.5 \cdot i + \left(0.25 \cdot \beta + 0.25 \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\]

   10. Simplified 14.4

       \[\leadsto \frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\color{blue}{\mathsf{fma}\left(0.25, \alpha + \beta, 0.5 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\]
   11. Using strategy rm

   12. Applied add-cbrt-cube 12.7

       \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\mathsf{fma}\left(0.25, \alpha + \beta, 0.5 \cdot i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\right) \cdot \left(\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\mathsf{fma}\left(0.25, \alpha + \beta, 0.5 \cdot i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\right)\right) \cdot \left(\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\mathsf{fma}\left(0.25, \alpha + \beta, 0.5 \cdot i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\right)}}\]

   if 5.188945744706776e+218 < beta

   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 add-sqr-sqrt 64.0

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

   4. Applied difference-of-squares 64.0

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

   5. Applied times-frac 58.1

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

   6. Applied times-frac 57.7

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

   8. Applied associate-/l* 57.7

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

   9. Taylor expanded around inf 13.8

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

4. Final simplification 12.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 5.188945744706776056846709445192721671049 \cdot 10^{218}:\\ \;\;\;\;\sqrt[3]{\left(\left(\frac{\frac{i}{\frac{\left(\beta + \alpha\right) + i \cdot 2}{i + \left(\beta + \alpha\right)}}}{\sqrt{1} + \left(\left(\beta + \alpha\right) + i \cdot 2\right)} \cdot \frac{\mathsf{fma}\left(0.25, \beta + \alpha, i \cdot 0.5\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - \sqrt{1}}\right) \cdot \left(\frac{\frac{i}{\frac{\left(\beta + \alpha\right) + i \cdot 2}{i + \left(\beta + \alpha\right)}}}{\sqrt{1} + \left(\left(\beta + \alpha\right) + i \cdot 2\right)} \cdot \frac{\mathsf{fma}\left(0.25, \beta + \alpha, i \cdot 0.5\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - \sqrt{1}}\right)\right) \cdot \left(\frac{\frac{i}{\frac{\left(\beta + \alpha\right) + i \cdot 2}{i + \left(\beta + \alpha\right)}}}{\sqrt{1} + \left(\left(\beta + \alpha\right) + i \cdot 2\right)} \cdot \frac{\mathsf{fma}\left(0.25, \beta + \alpha, i \cdot 0.5\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - \sqrt{1}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - \sqrt{1}} \cdot \frac{\frac{i}{\frac{\left(\beta + \alpha\right) + i \cdot 2}{i + \left(\beta + \alpha\right)}}}{\sqrt{1} + \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +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)))