Octave 3.8, jcobi/2

Average Error: 23.8 → 7.4

Time: 1.6m

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{2 \cdot i + \left(\beta + \alpha\right)}}{2 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} \le -0.9999999999999998889776975374843459576368:\\ \;\;\;\;\frac{\left(\frac{8}{\left(\alpha \cdot \alpha\right) \cdot \alpha} - \frac{4}{\alpha \cdot \alpha}\right) + \frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log \left(e^{1 \cdot \left(1 \cdot 1\right)} \cdot e^{\left(\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}\right) \cdot \left(\left(\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}\right) \cdot \left(\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}\right)\right)}\right)}{\frac{1 \cdot 1 - \left(\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}\right) \cdot \left(\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}\right)}{\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)} + 1} \cdot 1 + \left(\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}\right) \cdot \left(\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}\right)}}{2}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r5017073 = alpha;
        double r5017074 = beta;
        double r5017075 = r5017073 + r5017074;
        double r5017076 = r5017074 - r5017073;
        double r5017077 = r5017075 * r5017076;
        double r5017078 = 2.0;
        double r5017079 = i;
        double r5017080 = r5017078 * r5017079;
        double r5017081 = r5017075 + r5017080;
        double r5017082 = r5017077 / r5017081;
        double r5017083 = r5017081 + r5017078;
        double r5017084 = r5017082 / r5017083;
        double r5017085 = 1.0;
        double r5017086 = r5017084 + r5017085;
        double r5017087 = r5017086 / r5017078;
        return r5017087;
}

↓

double f(double alpha, double beta, double i) {
        double r5017088 = beta;
        double r5017089 = alpha;
        double r5017090 = r5017088 + r5017089;
        double r5017091 = r5017088 - r5017089;
        double r5017092 = r5017090 * r5017091;
        double r5017093 = 2.0;
        double r5017094 = i;
        double r5017095 = r5017093 * r5017094;
        double r5017096 = r5017095 + r5017090;
        double r5017097 = r5017092 / r5017096;
        double r5017098 = r5017093 + r5017096;
        double r5017099 = r5017097 / r5017098;
        double r5017100 = -0.9999999999999999;
        bool r5017101 = r5017099 <= r5017100;
        double r5017102 = 8.0;
        double r5017103 = r5017089 * r5017089;
        double r5017104 = r5017103 * r5017089;
        double r5017105 = r5017102 / r5017104;
        double r5017106 = 4.0;
        double r5017107 = r5017106 / r5017103;
        double r5017108 = r5017105 - r5017107;
        double r5017109 = r5017093 / r5017089;
        double r5017110 = r5017108 + r5017109;
        double r5017111 = r5017110 / r5017093;
        double r5017112 = 1.0;
        double r5017113 = r5017112 * r5017112;
        double r5017114 = r5017112 * r5017113;
        double r5017115 = exp(r5017114);
        double r5017116 = r5017093 + r5017090;
        double r5017117 = r5017116 + r5017095;
        double r5017118 = r5017090 / r5017117;
        double r5017119 = r5017091 / r5017096;
        double r5017120 = r5017118 * r5017119;
        double r5017121 = r5017120 * r5017120;
        double r5017122 = r5017120 * r5017121;
        double r5017123 = exp(r5017122);
        double r5017124 = r5017115 * r5017123;
        double r5017125 = log(r5017124);
        double r5017126 = r5017113 - r5017121;
        double r5017127 = r5017120 + r5017112;
        double r5017128 = r5017126 / r5017127;
        double r5017129 = r5017128 * r5017112;
        double r5017130 = r5017129 + r5017121;
        double r5017131 = r5017125 / r5017130;
        double r5017132 = r5017131 / r5017093;
        double r5017133 = r5017101 ? r5017111 : r5017132;
        return r5017133;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) < -0.9999999999999999

   1. Initial program 63.3

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

   2. Taylor expanded around inf 32.7

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}}{2}\]

   3. Simplified 32.7

      \[\leadsto \frac{\color{blue}{\left(\frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4}{\alpha \cdot \alpha}\right) + \frac{2}{\alpha}}}{2}\]

   if -0.9999999999999999 < (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0))

   1. Initial program 12.8

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

   3. Applied flip3-+ 12.8

      \[\leadsto \frac{\color{blue}{\frac{{\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right)}^{3} + {1}^{3}}{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + \left(1 \cdot 1 - \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot 1\right)}}}{2}\]

   4. Simplified 12.8

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

   5. Simplified 0.3

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

   7. Applied add-log-exp 0.3

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

   8. Applied add-log-exp 0.3

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

   9. Applied sum-log 0.3

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

   11. Applied flip-- 0.3

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

4. Final simplification 7.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{2 \cdot i + \left(\beta + \alpha\right)}}{2 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} \le -0.9999999999999998889776975374843459576368:\\ \;\;\;\;\frac{\left(\frac{8}{\left(\alpha \cdot \alpha\right) \cdot \alpha} - \frac{4}{\alpha \cdot \alpha}\right) + \frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log \left(e^{1 \cdot \left(1 \cdot 1\right)} \cdot e^{\left(\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}\right) \cdot \left(\left(\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}\right) \cdot \left(\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}\right)\right)}\right)}{\frac{1 \cdot 1 - \left(\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}\right) \cdot \left(\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}\right)}{\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)} + 1} \cdot 1 + \left(\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}\right) \cdot \left(\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1.0) (> beta -1.0) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))