Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2

Average Error: 4.0 → 5.1

Time: 9.2s

Precision: 64

Math

\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.0159840576688351 \cdot 10^{48}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.83333333333333337 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \le 4.062741574579635 \cdot 10^{-103}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{z \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - \frac{t}{\sqrt{t + a}} \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{\frac{t}{\sqrt{t + a}} \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - {\left({\left(\frac{2}{t \cdot 3}\right)}^{3}\right)}^{\frac{1}{3}}\right)\right)}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r78107 = x;
        double r78108 = y;
        double r78109 = 2.0;
        double r78110 = z;
        double r78111 = t;
        double r78112 = a;
        double r78113 = r78111 + r78112;
        double r78114 = sqrt(r78113);
        double r78115 = r78110 * r78114;
        double r78116 = r78115 / r78111;
        double r78117 = b;
        double r78118 = c;
        double r78119 = r78117 - r78118;
        double r78120 = 5.0;
        double r78121 = 6.0;
        double r78122 = r78120 / r78121;
        double r78123 = r78112 + r78122;
        double r78124 = 3.0;
        double r78125 = r78111 * r78124;
        double r78126 = r78109 / r78125;
        double r78127 = r78123 - r78126;
        double r78128 = r78119 * r78127;
        double r78129 = r78116 - r78128;
        double r78130 = r78109 * r78129;
        double r78131 = exp(r78130);
        double r78132 = r78108 * r78131;
        double r78133 = r78107 + r78132;
        double r78134 = r78107 / r78133;
        return r78134;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r78135 = t;
        double r78136 = -2.015984057668835e+48;
        bool r78137 = r78135 <= r78136;
        double r78138 = x;
        double r78139 = y;
        double r78140 = 2.0;
        double r78141 = a;
        double r78142 = c;
        double r78143 = r78141 * r78142;
        double r78144 = 0.8333333333333334;
        double r78145 = r78144 * r78142;
        double r78146 = r78143 + r78145;
        double r78147 = b;
        double r78148 = r78141 * r78147;
        double r78149 = r78146 - r78148;
        double r78150 = r78140 * r78149;
        double r78151 = exp(r78150);
        double r78152 = r78139 * r78151;
        double r78153 = r78138 + r78152;
        double r78154 = r78138 / r78153;
        double r78155 = 4.062741574579635e-103;
        bool r78156 = r78135 <= r78155;
        double r78157 = z;
        double r78158 = 5.0;
        double r78159 = 6.0;
        double r78160 = r78158 / r78159;
        double r78161 = r78141 - r78160;
        double r78162 = 3.0;
        double r78163 = r78135 * r78162;
        double r78164 = r78161 * r78163;
        double r78165 = r78157 * r78164;
        double r78166 = r78135 + r78141;
        double r78167 = sqrt(r78166);
        double r78168 = r78135 / r78167;
        double r78169 = r78147 - r78142;
        double r78170 = r78141 * r78141;
        double r78171 = r78160 * r78160;
        double r78172 = r78170 - r78171;
        double r78173 = r78172 * r78163;
        double r78174 = r78161 * r78140;
        double r78175 = r78173 - r78174;
        double r78176 = r78169 * r78175;
        double r78177 = r78168 * r78176;
        double r78178 = r78165 - r78177;
        double r78179 = r78168 * r78164;
        double r78180 = r78178 / r78179;
        double r78181 = r78140 * r78180;
        double r78182 = exp(r78181);
        double r78183 = r78139 * r78182;
        double r78184 = r78138 + r78183;
        double r78185 = r78138 / r78184;
        double r78186 = r78157 / r78168;
        double r78187 = r78141 + r78160;
        double r78188 = r78140 / r78163;
        double r78189 = 3.0;
        double r78190 = pow(r78188, r78189);
        double r78191 = 0.3333333333333333;
        double r78192 = pow(r78190, r78191);
        double r78193 = r78187 - r78192;
        double r78194 = r78169 * r78193;
        double r78195 = r78186 - r78194;
        double r78196 = r78140 * r78195;
        double r78197 = exp(r78196);
        double r78198 = r78139 * r78197;
        double r78199 = r78138 + r78198;
        double r78200 = r78138 / r78199;
        double r78201 = r78156 ? r78185 : r78200;
        double r78202 = r78137 ? r78154 : r78201;
        return r78202;
}

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

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if t < -2.015984057668835e+48

   1. Initial program 5.3

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

   2. Taylor expanded around inf 7.8

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(a \cdot c + 0.83333333333333337 \cdot c\right) - a \cdot b\right)}}}\]

   if -2.015984057668835e+48 < t < 4.062741574579635e-103

   1. Initial program 6.4

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
   2. Using strategy rm

   3. Applied associate-/l* 7.6

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\frac{z}{\frac{t}{\sqrt{t + a}}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
   4. Using strategy rm

   5. Applied flip-+ 10.2

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\color{blue}{\frac{a \cdot a - \frac{5}{6} \cdot \frac{5}{6}}{a - \frac{5}{6}}} - \frac{2}{t \cdot 3}\right)\right)}}\]

   6. Applied frac-sub 10.6

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \color{blue}{\frac{\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2}{\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)}}\right)}}\]

   7. Applied associate-*r/ 10.6

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \color{blue}{\frac{\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)}{\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)}}\right)}}\]

   8. Applied frac-sub 11.4

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\frac{z \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - \frac{t}{\sqrt{t + a}} \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{\frac{t}{\sqrt{t + a}} \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}}\]

   if 4.062741574579635e-103 < t

   1. Initial program 2.1

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
   2. Using strategy rm

   3. Applied associate-/l* 0.3

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\frac{z}{\frac{t}{\sqrt{t + a}}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
   4. Using strategy rm

   5. Applied add-cbrt-cube 0.3

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot \color{blue}{\sqrt[3]{\left(3 \cdot 3\right) \cdot 3}}}\right)\right)}}\]

   6. Applied add-cbrt-cube 0.3

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{\color{blue}{\sqrt[3]{\left(t \cdot t\right) \cdot t}} \cdot \sqrt[3]{\left(3 \cdot 3\right) \cdot 3}}\right)\right)}}\]

   7. Applied cbrt-unprod 0.3

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{\color{blue}{\sqrt[3]{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\left(3 \cdot 3\right) \cdot 3\right)}}}\right)\right)}}\]

   8. Applied add-cbrt-cube 0.3

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{\color{blue}{\sqrt[3]{\left(2 \cdot 2\right) \cdot 2}}}{\sqrt[3]{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\left(3 \cdot 3\right) \cdot 3\right)}}\right)\right)}}\]

   9. Applied cbrt-undiv 0.3

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \color{blue}{\sqrt[3]{\frac{\left(2 \cdot 2\right) \cdot 2}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\left(3 \cdot 3\right) \cdot 3\right)}}}\right)\right)}}\]

   10. Simplified 0.3

       \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \sqrt[3]{\color{blue}{{\left(\frac{2}{t \cdot 3}\right)}^{3}}}\right)\right)}}\]
   11. Using strategy rm

   12. Applied pow1/3 0.3

       \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \color{blue}{{\left({\left(\frac{2}{t \cdot 3}\right)}^{3}\right)}^{\frac{1}{3}}}\right)\right)}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 5.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.0159840576688351 \cdot 10^{48}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.83333333333333337 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \le 4.062741574579635 \cdot 10^{-103}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{z \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - \frac{t}{\sqrt{t + a}} \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{\frac{t}{\sqrt{t + a}} \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - {\left({\left(\frac{2}{t \cdot 3}\right)}^{3}\right)}^{\frac{1}{3}}\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3)))))))))))