Octave 3.8, jcobi/4

Average Error: 54.2 → 11.1

Time: 13.7s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 4.82276421781825579 \cdot 10^{120}:\\ \;\;\;\;\sqrt{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}} \cdot \sqrt{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{1}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}} \cdot \left(\frac{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}} \cdot \frac{0.25 \cdot \alpha + \left(0.5 \cdot i + 0.25 \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\right)\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r183077 = i;
        double r183078 = alpha;
        double r183079 = beta;
        double r183080 = r183078 + r183079;
        double r183081 = r183080 + r183077;
        double r183082 = r183077 * r183081;
        double r183083 = r183079 * r183078;
        double r183084 = r183083 + r183082;
        double r183085 = r183082 * r183084;
        double r183086 = 2.0;
        double r183087 = r183086 * r183077;
        double r183088 = r183080 + r183087;
        double r183089 = r183088 * r183088;
        double r183090 = r183085 / r183089;
        double r183091 = 1.0;
        double r183092 = r183089 - r183091;
        double r183093 = r183090 / r183092;
        return r183093;
}

↓

double f(double alpha, double beta, double i) {
        double r183094 = i;
        double r183095 = 4.822764217818256e+120;
        bool r183096 = r183094 <= r183095;
        double r183097 = alpha;
        double r183098 = beta;
        double r183099 = r183097 + r183098;
        double r183100 = r183099 + r183094;
        double r183101 = r183094 * r183100;
        double r183102 = 2.0;
        double r183103 = r183102 * r183094;
        double r183104 = r183099 + r183103;
        double r183105 = r183101 / r183104;
        double r183106 = 1.0;
        double r183107 = sqrt(r183106);
        double r183108 = r183104 + r183107;
        double r183109 = r183105 / r183108;
        double r183110 = r183098 * r183097;
        double r183111 = r183110 + r183101;
        double r183112 = r183111 / r183104;
        double r183113 = r183104 - r183107;
        double r183114 = r183112 / r183113;
        double r183115 = r183109 * r183114;
        double r183116 = sqrt(r183115);
        double r183117 = r183116 * r183116;
        double r183118 = 1.0;
        double r183119 = r183094 / r183118;
        double r183120 = sqrt(r183108);
        double r183121 = r183119 / r183120;
        double r183122 = r183100 / r183104;
        double r183123 = r183122 / r183120;
        double r183124 = 0.25;
        double r183125 = r183124 * r183097;
        double r183126 = 0.5;
        double r183127 = r183126 * r183094;
        double r183128 = r183124 * r183098;
        double r183129 = r183127 + r183128;
        double r183130 = r183125 + r183129;
        double r183131 = r183130 / r183113;
        double r183132 = r183123 * r183131;
        double r183133 = r183121 * r183132;
        double r183134 = r183096 ? r183117 : r183133;
        return r183134;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if i < 4.822764217818256e+120

   1. Initial program 38.5

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

   3. Applied add-sqr-sqrt 38.5

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

   4. Applied difference-of-squares 38.5

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

   5. Applied times-frac 14.5

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

   6. Applied times-frac 9.4

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

   8. Applied add-sqr-sqrt 9.4

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

   if 4.822764217818256e+120 < 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. 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 56.1

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

   6. Applied times-frac 55.8

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

   8. Applied add-sqr-sqrt 55.8

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

   9. Applied *-un-lft-identity 55.8

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

   10. Applied times-frac 55.8

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

   11. Applied times-frac 55.8

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

   12. Applied associate-*l* 55.8

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

   13. Taylor expanded around 0 12.2

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

4. Final simplification 11.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 4.82276421781825579 \cdot 10^{120}:\\ \;\;\;\;\sqrt{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}} \cdot \sqrt{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{1}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}} \cdot \left(\frac{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}} \cdot \frac{0.25 \cdot \alpha + \left(0.5 \cdot i + 0.25 \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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)))