Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2

Average Error: 2.0 → 3.6

Time: 21.7s

Precision: 64

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

Math

\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]

↓

\[\begin{array}{l} \mathbf{if}\;a \le 6.704663585819556267416471937206933320954 \cdot 10^{-216}:\\ \;\;\;\;\frac{x}{\left(\sqrt[3]{y} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{y}}} \cdot \frac{\frac{{a}^{\left(-1\right)}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}{\sqrt[3]{y}}\\ \mathbf{elif}\;a \le 1.299885454868772367621751764068226642771 \cdot 10^{-207}:\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{\frac{e^{b}}{{a}^{\left(t - 1\right)}}}}{y}\\ \mathbf{elif}\;a \le 5.576242448142597018860216064040225357932 \cdot 10^{-92}:\\ \;\;\;\;\frac{x}{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{y}\right)} \cdot \frac{\frac{{a}^{\left(-1\right)}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}{\sqrt[3]{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{{a}^{1}}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}{y}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r106099 = x;
        double r106100 = y;
        double r106101 = z;
        double r106102 = log(r106101);
        double r106103 = r106100 * r106102;
        double r106104 = t;
        double r106105 = 1.0;
        double r106106 = r106104 - r106105;
        double r106107 = a;
        double r106108 = log(r106107);
        double r106109 = r106106 * r106108;
        double r106110 = r106103 + r106109;
        double r106111 = b;
        double r106112 = r106110 - r106111;
        double r106113 = exp(r106112);
        double r106114 = r106099 * r106113;
        double r106115 = r106114 / r106100;
        return r106115;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r106116 = a;
        double r106117 = 6.704663585819556e-216;
        bool r106118 = r106116 <= r106117;
        double r106119 = x;
        double r106120 = y;
        double r106121 = cbrt(r106120);
        double r106122 = cbrt(r106121);
        double r106123 = r106122 * r106122;
        double r106124 = r106121 * r106123;
        double r106125 = r106124 * r106122;
        double r106126 = r106119 / r106125;
        double r106127 = 1.0;
        double r106128 = -r106127;
        double r106129 = pow(r106116, r106128);
        double r106130 = z;
        double r106131 = log(r106130);
        double r106132 = -r106131;
        double r106133 = log(r106116);
        double r106134 = -r106133;
        double r106135 = t;
        double r106136 = b;
        double r106137 = fma(r106134, r106135, r106136);
        double r106138 = fma(r106120, r106132, r106137);
        double r106139 = exp(r106138);
        double r106140 = r106129 / r106139;
        double r106141 = r106140 / r106121;
        double r106142 = r106126 * r106141;
        double r106143 = 1.2998854548687724e-207;
        bool r106144 = r106116 <= r106143;
        double r106145 = pow(r106130, r106120);
        double r106146 = exp(r106136);
        double r106147 = r106135 - r106127;
        double r106148 = pow(r106116, r106147);
        double r106149 = r106146 / r106148;
        double r106150 = r106145 / r106149;
        double r106151 = r106150 / r106120;
        double r106152 = r106119 * r106151;
        double r106153 = 5.576242448142597e-92;
        bool r106154 = r106116 <= r106153;
        double r106155 = r106121 * r106121;
        double r106156 = cbrt(r106155);
        double r106157 = r106122 * r106121;
        double r106158 = r106156 * r106157;
        double r106159 = r106119 / r106158;
        double r106160 = r106159 * r106141;
        double r106161 = pow(r106116, r106127);
        double r106162 = r106119 / r106161;
        double r106163 = r106162 / r106139;
        double r106164 = r106163 / r106120;
        double r106165 = r106154 ? r106160 : r106164;
        double r106166 = r106144 ? r106152 : r106165;
        double r106167 = r106118 ? r106142 : r106166;
        return r106167;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Derivation

1. Split input into 4 regimes

2. if a < 6.704663585819556e-216

   1. Initial program 0.7

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 0.7

      \[\leadsto \frac{x \cdot e^{\color{blue}{1 \cdot \left(\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b\right)}}}{y}\]

   4. Applied exp-prod 0.8

      \[\leadsto \frac{x \cdot \color{blue}{{\left(e^{1}\right)}^{\left(\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b\right)}}}{y}\]

   5. Simplified 0.8

      \[\leadsto \frac{x \cdot {\color{blue}{e}}^{\left(\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b\right)}}{y}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt 0.8

      \[\leadsto \frac{x \cdot {e}^{\left(\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b\right)}}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}\]

   8. Applied times-frac 7.0

      \[\leadsto \color{blue}{\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{{e}^{\left(\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b\right)}}{\sqrt[3]{y}}}\]

   9. Simplified 7.0

      \[\leadsto \frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \color{blue}{\frac{{e}^{\left(\mathsf{fma}\left(y, \log z, \left(t - 1\right) \cdot \log a\right) - b\right)}}{\sqrt[3]{y}}}\]

   10. Taylor expanded around inf 7.1

       \[\leadsto \frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\color{blue}{e^{1 \cdot \log \left(\frac{1}{a}\right) - \left(y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)\right)}}}{\sqrt[3]{y}}\]

   11. Simplified 6.8

       \[\leadsto \frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\color{blue}{\frac{{a}^{\left(-1\right)}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{\sqrt[3]{y}}\]
   12. Using strategy rm

   13. Applied add-cube-cbrt 6.8

       \[\leadsto \frac{x}{\sqrt[3]{y} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}\right)}} \cdot \frac{\frac{{a}^{\left(-1\right)}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}{\sqrt[3]{y}}\]

   14. Applied associate-*r* 6.8

       \[\leadsto \frac{x}{\color{blue}{\left(\sqrt[3]{y} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{y}}}} \cdot \frac{\frac{{a}^{\left(-1\right)}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}{\sqrt[3]{y}}\]

   if 6.704663585819556e-216 < a < 1.2998854548687724e-207

   1. Initial program 0.8

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 0.8

      \[\leadsto \frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{\color{blue}{1 \cdot y}}\]

   4. Applied times-frac 5.6

      \[\leadsto \color{blue}{\frac{x}{1} \cdot \frac{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}}\]

   5. Simplified 5.6

      \[\leadsto \color{blue}{x} \cdot \frac{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]

   6. Simplified 22.0

      \[\leadsto x \cdot \color{blue}{\frac{\frac{{z}^{y}}{\frac{e^{b}}{{a}^{\left(t - 1\right)}}}}{y}}\]

   if 1.2998854548687724e-207 < a < 5.576242448142597e-92

   1. Initial program 0.9

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 0.9

      \[\leadsto \frac{x \cdot e^{\color{blue}{1 \cdot \left(\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b\right)}}}{y}\]

   4. Applied exp-prod 0.9

      \[\leadsto \frac{x \cdot \color{blue}{{\left(e^{1}\right)}^{\left(\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b\right)}}}{y}\]

   5. Simplified 0.9

      \[\leadsto \frac{x \cdot {\color{blue}{e}}^{\left(\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b\right)}}{y}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt 0.9

      \[\leadsto \frac{x \cdot {e}^{\left(\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b\right)}}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}\]

   8. Applied times-frac 4.4

      \[\leadsto \color{blue}{\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{{e}^{\left(\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b\right)}}{\sqrt[3]{y}}}\]

   9. Simplified 4.4

      \[\leadsto \frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \color{blue}{\frac{{e}^{\left(\mathsf{fma}\left(y, \log z, \left(t - 1\right) \cdot \log a\right) - b\right)}}{\sqrt[3]{y}}}\]

   10. Taylor expanded around inf 4.3

       \[\leadsto \frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\color{blue}{e^{1 \cdot \log \left(\frac{1}{a}\right) - \left(y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)\right)}}}{\sqrt[3]{y}}\]

   11. Simplified 3.7

       \[\leadsto \frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\color{blue}{\frac{{a}^{\left(-1\right)}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}}{\sqrt[3]{y}}\]
   12. Using strategy rm

   13. Applied add-cube-cbrt 3.7

       \[\leadsto \frac{x}{\sqrt[3]{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}} \cdot \sqrt[3]{y}} \cdot \frac{\frac{{a}^{\left(-1\right)}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}{\sqrt[3]{y}}\]

   14. Applied cbrt-prod 3.7

       \[\leadsto \frac{x}{\color{blue}{\left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)} \cdot \sqrt[3]{y}} \cdot \frac{\frac{{a}^{\left(-1\right)}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}{\sqrt[3]{y}}\]

   15. Applied associate-*l* 3.7

       \[\leadsto \frac{x}{\color{blue}{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{y}\right)}} \cdot \frac{\frac{{a}^{\left(-1\right)}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}{\sqrt[3]{y}}\]

   if 5.576242448142597e-92 < a

   1. Initial program 2.6

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 2.6

      \[\leadsto \frac{x \cdot e^{\color{blue}{1 \cdot \left(\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b\right)}}}{y}\]

   4. Applied exp-prod 2.7

      \[\leadsto \frac{x \cdot \color{blue}{{\left(e^{1}\right)}^{\left(\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b\right)}}}{y}\]

   5. Simplified 2.7

      \[\leadsto \frac{x \cdot {\color{blue}{e}}^{\left(\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b\right)}}{y}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt 2.7

      \[\leadsto \frac{x \cdot {e}^{\left(\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b\right)}}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}\]

   8. Applied times-frac 5.5

      \[\leadsto \color{blue}{\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{{e}^{\left(\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b\right)}}{\sqrt[3]{y}}}\]

   9. Simplified 5.5

      \[\leadsto \frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \color{blue}{\frac{{e}^{\left(\mathsf{fma}\left(y, \log z, \left(t - 1\right) \cdot \log a\right) - b\right)}}{\sqrt[3]{y}}}\]

   10. Taylor expanded around inf 2.7

       \[\leadsto \color{blue}{\frac{x \cdot e^{1 \cdot \log \left(\frac{1}{a}\right) - \left(y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)\right)}}{y}}\]

   11. Simplified 2.6

       \[\leadsto \color{blue}{\frac{\frac{\frac{x}{{a}^{1}}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}{y}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 3.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le 6.704663585819556267416471937206933320954 \cdot 10^{-216}:\\ \;\;\;\;\frac{x}{\left(\sqrt[3]{y} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{y}}} \cdot \frac{\frac{{a}^{\left(-1\right)}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}{\sqrt[3]{y}}\\ \mathbf{elif}\;a \le 1.299885454868772367621751764068226642771 \cdot 10^{-207}:\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{\frac{e^{b}}{{a}^{\left(t - 1\right)}}}}{y}\\ \mathbf{elif}\;a \le 5.576242448142597018860216064040225357932 \cdot 10^{-92}:\\ \;\;\;\;\frac{x}{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{y}\right)} \cdot \frac{\frac{{a}^{\left(-1\right)}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}{\sqrt[3]{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{{a}^{1}}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1) (log a))) b))) y))