Octave 3.8, jcobi/4

Average Error: 54.0 → 11.7

Time: 1.8m

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 3.464835701411268256074169445036287456604 \cdot 10^{111}:\\ \;\;\;\;\frac{i}{\frac{\sqrt{1} + \left(2 \cdot i + \left(\alpha + \beta\right)\right)}{\log \left(e^{\frac{\left(\alpha + \beta\right) + i}{2 \cdot i + \left(\alpha + \beta\right)}}\right)}} \cdot \frac{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}{2 \cdot i + \left(\alpha + \beta\right)}}{\left(2 \cdot i + \left(\alpha + \beta\right)\right) - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{\frac{i}{\frac{\sqrt{1} + \left(2 \cdot i + \left(\alpha + \beta\right)\right)}{\frac{\left(\alpha + \beta\right) + i}{2 \cdot i + \left(\alpha + \beta\right)}}}} \cdot \sqrt[3]{\frac{i}{\frac{\sqrt{1} + \left(2 \cdot i + \left(\alpha + \beta\right)\right)}{\frac{\left(\alpha + \beta\right) + i}{2 \cdot i + \left(\alpha + \beta\right)}}}}\right) \cdot \sqrt[3]{\frac{i}{\frac{\sqrt{1} + \left(2 \cdot i + \left(\alpha + \beta\right)\right)}{\frac{\left(\alpha + \beta\right) + i}{2 \cdot i + \left(\alpha + \beta\right)}}}}\right) \cdot \frac{\left(\beta \cdot 0.25 + 0.5 \cdot i\right) + 0.25 \cdot \alpha}{\left(2 \cdot i + \left(\alpha + \beta\right)\right) - \sqrt{1}}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r5528043 = i;
        double r5528044 = alpha;
        double r5528045 = beta;
        double r5528046 = r5528044 + r5528045;
        double r5528047 = r5528046 + r5528043;
        double r5528048 = r5528043 * r5528047;
        double r5528049 = r5528045 * r5528044;
        double r5528050 = r5528049 + r5528048;
        double r5528051 = r5528048 * r5528050;
        double r5528052 = 2.0;
        double r5528053 = r5528052 * r5528043;
        double r5528054 = r5528046 + r5528053;
        double r5528055 = r5528054 * r5528054;
        double r5528056 = r5528051 / r5528055;
        double r5528057 = 1.0;
        double r5528058 = r5528055 - r5528057;
        double r5528059 = r5528056 / r5528058;
        return r5528059;
}

↓

double f(double alpha, double beta, double i) {
        double r5528060 = i;
        double r5528061 = 3.464835701411268e+111;
        bool r5528062 = r5528060 <= r5528061;
        double r5528063 = 1.0;
        double r5528064 = sqrt(r5528063);
        double r5528065 = 2.0;
        double r5528066 = r5528065 * r5528060;
        double r5528067 = alpha;
        double r5528068 = beta;
        double r5528069 = r5528067 + r5528068;
        double r5528070 = r5528066 + r5528069;
        double r5528071 = r5528064 + r5528070;
        double r5528072 = r5528069 + r5528060;
        double r5528073 = r5528072 / r5528070;
        double r5528074 = exp(r5528073);
        double r5528075 = log(r5528074);
        double r5528076 = r5528071 / r5528075;
        double r5528077 = r5528060 / r5528076;
        double r5528078 = r5528072 * r5528060;
        double r5528079 = r5528067 * r5528068;
        double r5528080 = r5528078 + r5528079;
        double r5528081 = r5528080 / r5528070;
        double r5528082 = r5528070 - r5528064;
        double r5528083 = r5528081 / r5528082;
        double r5528084 = r5528077 * r5528083;
        double r5528085 = r5528071 / r5528073;
        double r5528086 = r5528060 / r5528085;
        double r5528087 = cbrt(r5528086);
        double r5528088 = r5528087 * r5528087;
        double r5528089 = r5528088 * r5528087;
        double r5528090 = 0.25;
        double r5528091 = r5528068 * r5528090;
        double r5528092 = 0.5;
        double r5528093 = r5528092 * r5528060;
        double r5528094 = r5528091 + r5528093;
        double r5528095 = r5528090 * r5528067;
        double r5528096 = r5528094 + r5528095;
        double r5528097 = r5528096 / r5528082;
        double r5528098 = r5528089 * r5528097;
        double r5528099 = r5528062 ? r5528084 : r5528098;
        return r5528099;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if i < 3.464835701411268e+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. Using strategy rm

   3. Applied add-sqr-sqrt 36.2

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

   4. Applied difference-of-squares 36.2

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

   5. Applied times-frac 14.7

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

   6. Applied times-frac 10.4

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

   8. Applied *-un-lft-identity 10.4

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

   9. Applied times-frac 10.4

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

   10. Applied associate-/l* 10.4

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

   12. Applied add-log-exp 10.4

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

   if 3.464835701411268e+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. 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 54.1

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

   6. Applied times-frac 53.6

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

   8. Applied *-un-lft-identity 53.6

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

   9. Applied times-frac 53.5

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

   10. Applied associate-/l* 53.5

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

   11. Taylor expanded around 0 12.4

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

   13. Applied add-cube-cbrt 12.4

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

4. Final simplification 11.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 3.464835701411268256074169445036287456604 \cdot 10^{111}:\\ \;\;\;\;\frac{i}{\frac{\sqrt{1} + \left(2 \cdot i + \left(\alpha + \beta\right)\right)}{\log \left(e^{\frac{\left(\alpha + \beta\right) + i}{2 \cdot i + \left(\alpha + \beta\right)}}\right)}} \cdot \frac{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}{2 \cdot i + \left(\alpha + \beta\right)}}{\left(2 \cdot i + \left(\alpha + \beta\right)\right) - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{\frac{i}{\frac{\sqrt{1} + \left(2 \cdot i + \left(\alpha + \beta\right)\right)}{\frac{\left(\alpha + \beta\right) + i}{2 \cdot i + \left(\alpha + \beta\right)}}}} \cdot \sqrt[3]{\frac{i}{\frac{\sqrt{1} + \left(2 \cdot i + \left(\alpha + \beta\right)\right)}{\frac{\left(\alpha + \beta\right) + i}{2 \cdot i + \left(\alpha + \beta\right)}}}}\right) \cdot \sqrt[3]{\frac{i}{\frac{\sqrt{1} + \left(2 \cdot i + \left(\alpha + \beta\right)\right)}{\frac{\left(\alpha + \beta\right) + i}{2 \cdot i + \left(\alpha + \beta\right)}}}}\right) \cdot \frac{\left(\beta \cdot 0.25 + 0.5 \cdot i\right) + 0.25 \cdot \alpha}{\left(2 \cdot i + \left(\alpha + \beta\right)\right) - \sqrt{1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(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)))