Octave 3.8, jcobi/4

Average Error: 54.0 → 11.6

Time: 46.3s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 3.365899413275378359645560424080333100681 \cdot 10^{111}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(\beta, \alpha, \left(\left(\alpha + \beta\right) + i\right) \cdot i\right)}}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(\beta, \alpha, \left(\left(\alpha + \beta\right) + i\right) \cdot i\right)}}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{i}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \alpha, \mathsf{fma}\left(0.5, i, \beta \cdot 0.25\right)\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}} \cdot \left(\left(\sqrt[3]{\frac{\frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{i}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}} \cdot \sqrt[3]{\frac{\frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{i}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}\right) \cdot \sqrt[3]{\frac{\frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{i}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}\right)\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r4060125 = i;
        double r4060126 = alpha;
        double r4060127 = beta;
        double r4060128 = r4060126 + r4060127;
        double r4060129 = r4060128 + r4060125;
        double r4060130 = r4060125 * r4060129;
        double r4060131 = r4060127 * r4060126;
        double r4060132 = r4060131 + r4060130;
        double r4060133 = r4060130 * r4060132;
        double r4060134 = 2.0;
        double r4060135 = r4060134 * r4060125;
        double r4060136 = r4060128 + r4060135;
        double r4060137 = r4060136 * r4060136;
        double r4060138 = r4060133 / r4060137;
        double r4060139 = 1.0;
        double r4060140 = r4060137 - r4060139;
        double r4060141 = r4060138 / r4060140;
        return r4060141;
}

↓

double f(double alpha, double beta, double i) {
        double r4060142 = i;
        double r4060143 = 3.3658994132753784e+111;
        bool r4060144 = r4060142 <= r4060143;
        double r4060145 = beta;
        double r4060146 = alpha;
        double r4060147 = r4060146 + r4060145;
        double r4060148 = r4060147 + r4060142;
        double r4060149 = r4060148 * r4060142;
        double r4060150 = fma(r4060145, r4060146, r4060149);
        double r4060151 = sqrt(r4060150);
        double r4060152 = 2.0;
        double r4060153 = fma(r4060152, r4060142, r4060147);
        double r4060154 = r4060153 / r4060151;
        double r4060155 = r4060151 / r4060154;
        double r4060156 = 1.0;
        double r4060157 = sqrt(r4060156);
        double r4060158 = r4060153 + r4060157;
        double r4060159 = r4060155 / r4060158;
        double r4060160 = r4060153 / r4060142;
        double r4060161 = r4060148 / r4060160;
        double r4060162 = r4060153 - r4060157;
        double r4060163 = r4060161 / r4060162;
        double r4060164 = r4060159 * r4060163;
        double r4060165 = 0.25;
        double r4060166 = 0.5;
        double r4060167 = r4060145 * r4060165;
        double r4060168 = fma(r4060166, r4060142, r4060167);
        double r4060169 = fma(r4060165, r4060146, r4060168);
        double r4060170 = r4060169 / r4060158;
        double r4060171 = cbrt(r4060163);
        double r4060172 = r4060171 * r4060171;
        double r4060173 = r4060172 * r4060171;
        double r4060174 = r4060170 * r4060173;
        double r4060175 = r4060144 ? r4060164 : r4060174;
        return r4060175;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if i < 3.3658994132753784e+111

   1. Initial program 36.2

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

   2. Simplified 36.2

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

   4. Applied add-sqr-sqrt 36.2

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

   5. Applied difference-of-squares 36.2

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

   6. Applied times-frac 14.7

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

   7. Applied times-frac 10.4

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

   9. Applied associate-/l* 10.4

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

   11. Applied add-sqr-sqrt 10.4

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

   12. Applied associate-/l* 10.4

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

   if 3.3658994132753784e+111 < i

   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. Simplified 64.0

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

   4. Applied add-sqr-sqrt 64.0

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

   5. Applied difference-of-squares 64.0

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

   6. Applied times-frac 54.1

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

   7. Applied times-frac 53.6

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

   9. Applied associate-/l* 53.5

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

   10. Taylor expanded around 0 12.4

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

   11. Simplified 12.4

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

   13. Applied add-cube-cbrt 12.4

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

4. Final simplification 11.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 3.365899413275378359645560424080333100681 \cdot 10^{111}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(\beta, \alpha, \left(\left(\alpha + \beta\right) + i\right) \cdot i\right)}}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(\beta, \alpha, \left(\left(\alpha + \beta\right) + i\right) \cdot i\right)}}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{i}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \alpha, \mathsf{fma}\left(0.5, i, \beta \cdot 0.25\right)\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}} \cdot \left(\left(\sqrt[3]{\frac{\frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{i}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}} \cdot \sqrt[3]{\frac{\frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{i}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}\right) \cdot \sqrt[3]{\frac{\frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{i}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +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)))