Octave 3.8, jcobi/4

Average Error: 54.2 → 36.9

Time: 37.0s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\beta \le 1.737031389650685423817438758506316217418 \cdot 10^{208}:\\ \;\;\;\;\sqrt{\frac{i}{\left(\sqrt{1} + \mathsf{fma}\left(2, i, \alpha + \beta\right)\right) \cdot \frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}} \cdot \sqrt{\frac{i}{\left(\sqrt{1} + \mathsf{fma}\left(2, i, \alpha + \beta\right)\right) \cdot \frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r84164 = i;
        double r84165 = alpha;
        double r84166 = beta;
        double r84167 = r84165 + r84166;
        double r84168 = r84167 + r84164;
        double r84169 = r84164 * r84168;
        double r84170 = r84166 * r84165;
        double r84171 = r84170 + r84169;
        double r84172 = r84169 * r84171;
        double r84173 = 2.0;
        double r84174 = r84173 * r84164;
        double r84175 = r84167 + r84174;
        double r84176 = r84175 * r84175;
        double r84177 = r84172 / r84176;
        double r84178 = 1.0;
        double r84179 = r84176 - r84178;
        double r84180 = r84177 / r84179;
        return r84180;
}

↓

double f(double alpha, double beta, double i) {
        double r84181 = beta;
        double r84182 = 1.7370313896506854e+208;
        bool r84183 = r84181 <= r84182;
        double r84184 = i;
        double r84185 = 1.0;
        double r84186 = sqrt(r84185);
        double r84187 = 2.0;
        double r84188 = alpha;
        double r84189 = r84188 + r84181;
        double r84190 = fma(r84187, r84184, r84189);
        double r84191 = r84186 + r84190;
        double r84192 = r84189 + r84184;
        double r84193 = r84184 * r84192;
        double r84194 = fma(r84181, r84188, r84193);
        double r84195 = sqrt(r84194);
        double r84196 = r84190 / r84195;
        double r84197 = r84191 * r84196;
        double r84198 = r84184 / r84197;
        double r84199 = r84192 / r84196;
        double r84200 = r84190 - r84186;
        double r84201 = r84199 / r84200;
        double r84202 = r84198 * r84201;
        double r84203 = sqrt(r84202);
        double r84204 = r84203 * r84203;
        double r84205 = 0.0;
        double r84206 = r84183 ? r84204 : r84205;
        return r84206;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if beta < 1.7370313896506854e+208

   1. Initial program 52.9

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

   3. Applied associate-/l* 38.1

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

   4. Simplified 38.1

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

   6. Applied add-sqr-sqrt 38.1

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

   7. Applied difference-of-squares 38.1

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

   8. Applied add-sqr-sqrt 38.1

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

   9. Applied times-frac 38.1

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

   10. Applied times-frac 38.1

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

   11. Applied times-frac 35.9

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

   12. Simplified 35.9

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

   13. Simplified 35.9

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

   15. Applied add-sqr-sqrt 35.9

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

   if 1.7370313896506854e+208 < 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. Taylor expanded around inf 44.4

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

4. Final simplification 36.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.737031389650685423817438758506316217418 \cdot 10^{208}:\\ \;\;\;\;\sqrt{\frac{i}{\left(\sqrt{1} + \mathsf{fma}\left(2, i, \alpha + \beta\right)\right) \cdot \frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}} \cdot \sqrt{\frac{i}{\left(\sqrt{1} + \mathsf{fma}\left(2, i, \alpha + \beta\right)\right) \cdot \frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +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)))