Octave 3.8, jcobi/4

Average Error: 52.3 → 10.8

Time: 48.8s

Precision: 64

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

Math

\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]

↓

\[\begin{array}{l} \mathbf{if}\;i \le 6.711396437353354 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\frac{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}{\frac{\alpha \cdot \beta + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, \frac{1}{2}, \frac{1}{4} \cdot \left(\alpha + \beta\right)\right) \cdot \frac{i \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + i \cdot 2}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r5137173 = i;
        double r5137174 = alpha;
        double r5137175 = beta;
        double r5137176 = r5137174 + r5137175;
        double r5137177 = r5137176 + r5137173;
        double r5137178 = r5137173 * r5137177;
        double r5137179 = r5137175 * r5137174;
        double r5137180 = r5137179 + r5137178;
        double r5137181 = r5137178 * r5137180;
        double r5137182 = 2.0;
        double r5137183 = r5137182 * r5137173;
        double r5137184 = r5137176 + r5137183;
        double r5137185 = r5137184 * r5137184;
        double r5137186 = r5137181 / r5137185;
        double r5137187 = 1.0;
        double r5137188 = r5137185 - r5137187;
        double r5137189 = r5137186 / r5137188;
        return r5137189;
}

↓

double f(double alpha, double beta, double i) {
        double r5137190 = i;
        double r5137191 = 6.711396437353354e+139;
        bool r5137192 = r5137190 <= r5137191;
        double r5137193 = alpha;
        double r5137194 = beta;
        double r5137195 = r5137193 + r5137194;
        double r5137196 = r5137195 + r5137190;
        double r5137197 = r5137190 * r5137196;
        double r5137198 = 2.0;
        double r5137199 = r5137190 * r5137198;
        double r5137200 = r5137195 + r5137199;
        double r5137201 = r5137197 / r5137200;
        double r5137202 = 1.0;
        double r5137203 = sqrt(r5137202);
        double r5137204 = r5137203 + r5137200;
        double r5137205 = r5137201 / r5137204;
        double r5137206 = r5137200 - r5137203;
        double r5137207 = r5137193 * r5137194;
        double r5137208 = r5137207 + r5137197;
        double r5137209 = r5137208 / r5137200;
        double r5137210 = r5137206 / r5137209;
        double r5137211 = r5137205 / r5137210;
        double r5137212 = 0.5;
        double r5137213 = 0.25;
        double r5137214 = r5137213 * r5137195;
        double r5137215 = fma(r5137190, r5137212, r5137214);
        double r5137216 = r5137196 / r5137200;
        double r5137217 = r5137190 * r5137216;
        double r5137218 = r5137217 / r5137204;
        double r5137219 = r5137215 * r5137218;
        double r5137220 = r5137219 / r5137206;
        double r5137221 = r5137192 ? r5137211 : r5137220;
        return r5137221;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if i < 6.711396437353354e+139

   1. Initial program 40.6

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 40.6

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \color{blue}{\sqrt{1.0} \cdot \sqrt{1.0}}}\]

   4. Applied difference-of-squares 40.6

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}\right)}}\]

   5. Applied times-frac 15.9

      \[\leadsto \frac{\color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}\right)}\]

   6. Applied times-frac 11.4

      \[\leadsto \color{blue}{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}}\]
   7. Using strategy rm

   8. Applied associate-*r/ 11.4

      \[\leadsto \color{blue}{\frac{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}}\]
   9. Using strategy rm

   10. Applied associate-/l* 11.4

       \[\leadsto \color{blue}{\frac{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}}\]

   if 6.711396437353354e+139 < i

   1. Initial program 62.1

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 62.1

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \color{blue}{\sqrt{1.0} \cdot \sqrt{1.0}}}\]

   4. Applied difference-of-squares 62.1

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}\right)}}\]

   5. Applied times-frac 58.8

      \[\leadsto \frac{\color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}\right)}\]

   6. Applied times-frac 58.6

      \[\leadsto \color{blue}{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}}\]
   7. Using strategy rm

   8. Applied associate-*r/ 58.6

      \[\leadsto \color{blue}{\frac{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}}\]
   9. Using strategy rm

   10. Applied *-un-lft-identity 58.6

       \[\leadsto \frac{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}\]

   11. Applied times-frac 58.6

       \[\leadsto \frac{\frac{\color{blue}{\frac{i}{1} \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}\]

   12. Simplified 58.6

       \[\leadsto \frac{\frac{\color{blue}{i} \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}\]

   13. Taylor expanded around 0 10.3

       \[\leadsto \frac{\frac{i \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \color{blue}{\left(\frac{1}{2} \cdot i + \left(\frac{1}{4} \cdot \beta + \frac{1}{4} \cdot \alpha\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}\]

   14. Simplified 10.3

       \[\leadsto \frac{\frac{i \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \color{blue}{\mathsf{fma}\left(i, \frac{1}{2}, \frac{1}{4} \cdot \left(\alpha + \beta\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 10.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 6.711396437353354 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\frac{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}{\frac{\alpha \cdot \beta + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, \frac{1}{2}, \frac{1}{4} \cdot \left(\alpha + \beta\right)\right) \cdot \frac{i \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + i \cdot 2}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :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))) 1.0)))