Octave 3.8, jcobi/4

Average Error: 52.2 → 10.8

Time: 29.2s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 3.7191341897452675 \cdot 10^{+123}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) - i \cdot 2}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1.0}} \cdot \frac{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\alpha + \beta\right) - i \cdot 2} \cdot \frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log \left(e^{\frac{i \cdot \frac{-1}{8}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1.0}}}\right) \cdot \frac{\left(\alpha + \beta\right) - i \cdot 2}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1.0}}\right)\right)\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r4227165 = i;
        double r4227166 = alpha;
        double r4227167 = beta;
        double r4227168 = r4227166 + r4227167;
        double r4227169 = r4227168 + r4227165;
        double r4227170 = r4227165 * r4227169;
        double r4227171 = r4227167 * r4227166;
        double r4227172 = r4227171 + r4227170;
        double r4227173 = r4227170 * r4227172;
        double r4227174 = 2.0;
        double r4227175 = r4227174 * r4227165;
        double r4227176 = r4227168 + r4227175;
        double r4227177 = r4227176 * r4227176;
        double r4227178 = r4227173 / r4227177;
        double r4227179 = 1.0;
        double r4227180 = r4227177 - r4227179;
        double r4227181 = r4227178 / r4227180;
        return r4227181;
}

↓

double f(double alpha, double beta, double i) {
        double r4227182 = i;
        double r4227183 = 3.7191341897452675e+123;
        bool r4227184 = r4227182 <= r4227183;
        double r4227185 = alpha;
        double r4227186 = beta;
        double r4227187 = r4227185 + r4227186;
        double r4227188 = 2.0;
        double r4227189 = r4227182 * r4227188;
        double r4227190 = r4227187 - r4227189;
        double r4227191 = fma(r4227188, r4227182, r4227187);
        double r4227192 = 1.0;
        double r4227193 = sqrt(r4227192);
        double r4227194 = r4227191 - r4227193;
        double r4227195 = r4227190 / r4227194;
        double r4227196 = r4227187 + r4227182;
        double r4227197 = r4227182 * r4227196;
        double r4227198 = fma(r4227186, r4227185, r4227197);
        double r4227199 = r4227198 / r4227190;
        double r4227200 = r4227197 / r4227191;
        double r4227201 = r4227199 * r4227200;
        double r4227202 = r4227201 / r4227191;
        double r4227203 = r4227191 + r4227193;
        double r4227204 = r4227202 / r4227203;
        double r4227205 = r4227195 * r4227204;
        double r4227206 = -0.125;
        double r4227207 = r4227182 * r4227206;
        double r4227208 = r4227207 / r4227203;
        double r4227209 = exp(r4227208);
        double r4227210 = log(r4227209);
        double r4227211 = r4227210 * r4227195;
        double r4227212 = /* ERROR: no posit support in C */;
        double r4227213 = /* ERROR: no posit support in C */;
        double r4227214 = r4227184 ? r4227205 : r4227213;
        return r4227214;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if i < 3.7191341897452675e+123

   1. Initial program 37.3

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

   3. Applied add-sqr-sqrt 37.3

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

   4. Applied difference-of-squares 37.3

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

   5. Applied flip-+ 37.3

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

   6. Applied associate-*r/ 38.5

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

   7. Applied associate-/r/ 38.5

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

   8. Applied times-frac 38.5

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

   9. Simplified 9.1

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

   10. Simplified 9.1

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

   if 3.7191341897452675e+123 < 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 flip-+ 62.1

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

   6. Applied associate-*r/ 62.1

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

   7. Applied associate-/r/ 62.1

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

   8. Applied times-frac 62.1

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

   9. Simplified 54.6

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

   10. Simplified 54.6

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

   11. Taylor expanded around 0 12.7

       \[\leadsto \frac{\color{blue}{\frac{-1}{8} \cdot i}}{\sqrt{1.0} + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\left(\alpha + \beta\right) - 2 \cdot i}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1.0}}\]
   12. Using strategy rm

   13. Applied insert-posit16 12.4

       \[\leadsto \color{blue}{\left(\left(\frac{\frac{-1}{8} \cdot i}{\sqrt{1.0} + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\left(\alpha + \beta\right) - 2 \cdot i}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1.0}}\right)\right)}\]
   14. Using strategy rm

   15. Applied add-log-exp 11.9

       \[\leadsto \left(\left(\color{blue}{\log \left(e^{\frac{\frac{-1}{8} \cdot i}{\sqrt{1.0} + \mathsf{fma}\left(2, i, \alpha + \beta\right)}}\right)} \cdot \frac{\left(\alpha + \beta\right) - 2 \cdot i}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1.0}}\right)\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 10.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 3.7191341897452675 \cdot 10^{+123}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) - i \cdot 2}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1.0}} \cdot \frac{\frac{\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\alpha + \beta\right) - i \cdot 2} \cdot \frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log \left(e^{\frac{i \cdot \frac{-1}{8}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1.0}}}\right) \cdot \frac{\left(\alpha + \beta\right) - i \cdot 2}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1.0}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +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)))