Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A

Average Error: 2.0 → 3.6

Time: 22.0s

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 r524104 = x;
        double r524105 = y;
        double r524106 = z;
        double r524107 = log(r524106);
        double r524108 = r524105 * r524107;
        double r524109 = t;
        double r524110 = 1.0;
        double r524111 = r524109 - r524110;
        double r524112 = a;
        double r524113 = log(r524112);
        double r524114 = r524111 * r524113;
        double r524115 = r524108 + r524114;
        double r524116 = b;
        double r524117 = r524115 - r524116;
        double r524118 = exp(r524117);
        double r524119 = r524104 * r524118;
        double r524120 = r524119 / r524105;
        return r524120;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r524121 = a;
        double r524122 = 6.704663585819556e-216;
        bool r524123 = r524121 <= r524122;
        double r524124 = x;
        double r524125 = y;
        double r524126 = cbrt(r524125);
        double r524127 = cbrt(r524126);
        double r524128 = r524127 * r524127;
        double r524129 = r524126 * r524128;
        double r524130 = r524129 * r524127;
        double r524131 = r524124 / r524130;
        double r524132 = 1.0;
        double r524133 = -r524132;
        double r524134 = pow(r524121, r524133);
        double r524135 = z;
        double r524136 = log(r524135);
        double r524137 = -r524136;
        double r524138 = log(r524121);
        double r524139 = -r524138;
        double r524140 = t;
        double r524141 = b;
        double r524142 = fma(r524139, r524140, r524141);
        double r524143 = fma(r524125, r524137, r524142);
        double r524144 = exp(r524143);
        double r524145 = r524134 / r524144;
        double r524146 = r524145 / r524126;
        double r524147 = r524131 * r524146;
        double r524148 = 1.2998854548687724e-207;
        bool r524149 = r524121 <= r524148;
        double r524150 = pow(r524135, r524125);
        double r524151 = exp(r524141);
        double r524152 = r524140 - r524132;
        double r524153 = pow(r524121, r524152);
        double r524154 = r524151 / r524153;
        double r524155 = r524150 / r524154;
        double r524156 = r524155 / r524125;
        double r524157 = r524124 * r524156;
        double r524158 = 5.576242448142597e-92;
        bool r524159 = r524121 <= r524158;
        double r524160 = r524126 * r524126;
        double r524161 = cbrt(r524160);
        double r524162 = r524127 * r524126;
        double r524163 = r524161 * r524162;
        double r524164 = r524124 / r524163;
        double r524165 = r524164 * r524146;
        double r524166 = pow(r524121, r524132);
        double r524167 = r524124 / r524166;
        double r524168 = r524167 / r524144;
        double r524169 = r524168 / r524125;
        double r524170 = r524159 ? r524165 : r524169;
        double r524171 = r524149 ? r524157 : r524170;
        double r524172 = r524123 ? r524147 : r524171;
        return r524172;
}

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

Target

Original	2.0
Target	11.0
Herbie	3.6

\[\begin{array}{l} \mathbf{if}\;t \lt -0.8845848504127471478852839936735108494759:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{elif}\;t \lt 852031.228837407310493290424346923828125:\\ \;\;\;\;\frac{\frac{x}{y} \cdot {a}^{\left(t - 1\right)}}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \end{array}\]

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, A"
  :precision binary64

  :herbie-target
  (if (< t -0.8845848504127471) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z)))) (if (< t 852031.2288374073) (/ (* (/ x y) (pow a (- t 1))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z))))))

  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1) (log a))) b))) y))