Octave 3.8, jcobi/2

Average Error: 24.1 → 12.1

Time: 13.3s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 3.75440885493052108 \cdot 10^{164}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}\right)\right), \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}{2}\\ \mathbf{elif}\;\alpha \le 5.6771218984373863 \cdot 10^{261}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{{\alpha}^{2}}, \frac{8}{\alpha} - 4, \frac{2}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt[3]{\alpha + \beta} \cdot \sqrt[3]{\alpha + \beta}\right) \cdot \frac{\sqrt[3]{\alpha + \beta}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}\right)\right), \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}{2}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r105109 = alpha;
        double r105110 = beta;
        double r105111 = r105109 + r105110;
        double r105112 = r105110 - r105109;
        double r105113 = r105111 * r105112;
        double r105114 = 2.0;
        double r105115 = i;
        double r105116 = r105114 * r105115;
        double r105117 = r105111 + r105116;
        double r105118 = r105113 / r105117;
        double r105119 = r105117 + r105114;
        double r105120 = r105118 / r105119;
        double r105121 = 1.0;
        double r105122 = r105120 + r105121;
        double r105123 = r105122 / r105114;
        return r105123;
}

↓

double f(double alpha, double beta, double i) {
        double r105124 = alpha;
        double r105125 = 3.754408854930521e+164;
        bool r105126 = r105124 <= r105125;
        double r105127 = beta;
        double r105128 = r105124 + r105127;
        double r105129 = 2.0;
        double r105130 = i;
        double r105131 = fma(r105129, r105130, r105128);
        double r105132 = r105131 + r105129;
        double r105133 = r105128 / r105132;
        double r105134 = log1p(r105133);
        double r105135 = expm1(r105134);
        double r105136 = r105127 - r105124;
        double r105137 = r105136 / r105131;
        double r105138 = 1.0;
        double r105139 = fma(r105135, r105137, r105138);
        double r105140 = r105139 / r105129;
        double r105141 = 5.677121898437386e+261;
        bool r105142 = r105124 <= r105141;
        double r105143 = 1.0;
        double r105144 = 2.0;
        double r105145 = pow(r105124, r105144);
        double r105146 = r105143 / r105145;
        double r105147 = 8.0;
        double r105148 = r105147 / r105124;
        double r105149 = 4.0;
        double r105150 = r105148 - r105149;
        double r105151 = r105129 / r105124;
        double r105152 = fma(r105146, r105150, r105151);
        double r105153 = r105152 / r105129;
        double r105154 = cbrt(r105128);
        double r105155 = r105154 * r105154;
        double r105156 = r105154 / r105132;
        double r105157 = r105155 * r105156;
        double r105158 = log1p(r105157);
        double r105159 = expm1(r105158);
        double r105160 = fma(r105159, r105137, r105138);
        double r105161 = r105160 / r105129;
        double r105162 = r105142 ? r105153 : r105161;
        double r105163 = r105126 ? r105140 : r105162;
        return r105163;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if alpha < 3.754408854930521e+164

   1. Initial program 16.6

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

   2. Simplified 5.8

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

   4. Applied expm1-log1p-u 5.8

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}\right)\right)}, \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}{2}\]

   if 3.754408854930521e+164 < alpha < 5.677121898437386e+261

   1. Initial program 64.0

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

   2. Simplified 45.2

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

   4. Applied expm1-log1p-u 45.2

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}\right)\right)}, \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}{2}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 45.6

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}\right)\right), \frac{\beta - \alpha}{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}}, 1\right)}{2}\]

   7. Applied associate-/r* 45.6

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}\right)\right), \color{blue}{\frac{\frac{\beta - \alpha}{\sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}}{\sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}}, 1\right)}{2}\]

   8. Taylor expanded around inf 40.5

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

   9. Simplified 40.5

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{{\alpha}^{2}}, \frac{8}{\alpha} - 4, \frac{2}{\alpha}\right)}}{2}\]

   if 5.677121898437386e+261 < alpha

   1. Initial program 64.0

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

   2. Simplified 54.4

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

   4. Applied expm1-log1p-u 54.4

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}\right)\right)}, \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}{2}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 54.4

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + \beta}{\color{blue}{1 \cdot \left(\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2\right)}}\right)\right), \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}{2}\]

   7. Applied add-cube-cbrt 56.1

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{\left(\sqrt[3]{\alpha + \beta} \cdot \sqrt[3]{\alpha + \beta}\right) \cdot \sqrt[3]{\alpha + \beta}}}{1 \cdot \left(\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2\right)}\right)\right), \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}{2}\]

   8. Applied times-frac 55.9

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\sqrt[3]{\alpha + \beta} \cdot \sqrt[3]{\alpha + \beta}}{1} \cdot \frac{\sqrt[3]{\alpha + \beta}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}\right)\right), \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}{2}\]

   9. Simplified 55.9

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(\sqrt[3]{\alpha + \beta} \cdot \sqrt[3]{\alpha + \beta}\right)} \cdot \frac{\sqrt[3]{\alpha + \beta}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}\right)\right), \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}{2}\]
3. Recombined 3 regimes into one program.

4. Final simplification 12.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3.75440885493052108 \cdot 10^{164}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}\right)\right), \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}{2}\\ \mathbf{elif}\;\alpha \le 5.6771218984373863 \cdot 10^{261}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{{\alpha}^{2}}, \frac{8}{\alpha} - 4, \frac{2}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt[3]{\alpha + \beta} \cdot \sqrt[3]{\alpha + \beta}\right) \cdot \frac{\sqrt[3]{\alpha + \beta}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}\right)\right), \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2)) 1) 2))