Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2

Average Error: 3.9 → 5.1

Time: 22.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.58864618275117732889084966831500711282 \cdot 10^{-55} \lor \neg \left(t \le 2592927404032904812036096\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\sqrt[3]{t} \cdot \log \left(e^{\sqrt[3]{t}}\right)} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right) \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r59079 = x;
        double r59080 = y;
        double r59081 = 2.0;
        double r59082 = z;
        double r59083 = t;
        double r59084 = a;
        double r59085 = r59083 + r59084;
        double r59086 = sqrt(r59085);
        double r59087 = r59082 * r59086;
        double r59088 = r59087 / r59083;
        double r59089 = b;
        double r59090 = c;
        double r59091 = r59089 - r59090;
        double r59092 = 5.0;
        double r59093 = 6.0;
        double r59094 = r59092 / r59093;
        double r59095 = r59084 + r59094;
        double r59096 = 3.0;
        double r59097 = r59083 * r59096;
        double r59098 = r59081 / r59097;
        double r59099 = r59095 - r59098;
        double r59100 = r59091 * r59099;
        double r59101 = r59088 - r59100;
        double r59102 = r59081 * r59101;
        double r59103 = exp(r59102);
        double r59104 = r59080 * r59103;
        double r59105 = r59079 + r59104;
        double r59106 = r59079 / r59105;
        return r59106;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r59107 = t;
        double r59108 = -1.5886461827511773e-55;
        bool r59109 = r59107 <= r59108;
        double r59110 = 2.592927404032905e+24;
        bool r59111 = r59107 <= r59110;
        double r59112 = !r59111;
        bool r59113 = r59109 || r59112;
        double r59114 = x;
        double r59115 = y;
        double r59116 = 2.0;
        double r59117 = z;
        double r59118 = cbrt(r59107);
        double r59119 = exp(r59118);
        double r59120 = log(r59119);
        double r59121 = r59118 * r59120;
        double r59122 = r59117 / r59121;
        double r59123 = a;
        double r59124 = r59107 + r59123;
        double r59125 = sqrt(r59124);
        double r59126 = r59125 / r59118;
        double r59127 = r59122 * r59126;
        double r59128 = b;
        double r59129 = c;
        double r59130 = r59128 - r59129;
        double r59131 = 5.0;
        double r59132 = 6.0;
        double r59133 = r59131 / r59132;
        double r59134 = r59123 + r59133;
        double r59135 = 3.0;
        double r59136 = r59107 * r59135;
        double r59137 = r59116 / r59136;
        double r59138 = r59134 - r59137;
        double r59139 = r59130 * r59138;
        double r59140 = r59127 - r59139;
        double r59141 = r59116 * r59140;
        double r59142 = exp(r59141);
        double r59143 = r59115 * r59142;
        double r59144 = r59114 + r59143;
        double r59145 = r59114 / r59144;
        double r59146 = r59117 * r59126;
        double r59147 = r59123 - r59133;
        double r59148 = r59147 * r59136;
        double r59149 = r59146 * r59148;
        double r59150 = r59123 * r59123;
        double r59151 = r59133 * r59133;
        double r59152 = r59150 - r59151;
        double r59153 = r59152 * r59136;
        double r59154 = r59147 * r59116;
        double r59155 = r59153 - r59154;
        double r59156 = r59130 * r59155;
        double r59157 = r59118 * r59118;
        double r59158 = r59156 * r59157;
        double r59159 = r59149 - r59158;
        double r59160 = r59157 * r59148;
        double r59161 = r59159 / r59160;
        double r59162 = r59116 * r59161;
        double r59163 = exp(r59162);
        double r59164 = r59115 * r59163;
        double r59165 = r59114 + r59164;
        double r59166 = r59114 / r59165;
        double r59167 = r59113 ? r59145 : r59166;
        return r59167;
}

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 2 regimes

2. if t < -1.5886461827511773e-55 or 2.592927404032905e+24 < t

   1. Initial program 2.9

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

   3. Applied add-cube-cbrt 2.9

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

   4. Applied times-frac 0.4

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

   6. Applied add-log-exp 4.8

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

   if -1.5886461827511773e-55 < t < 2.592927404032905e+24

   1. Initial program 5.0

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

   3. Applied add-cube-cbrt 5.0

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

   4. Applied times-frac 5.2

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

   6. Applied add-log-exp 26.7

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

   8. Applied flip-+ 28.0

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

   9. Applied frac-sub 28.0

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

   10. Applied associate-*r/ 28.0

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

   11. Applied associate-*l/ 28.2

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

   12. Applied frac-sub 39.8

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

   13. Simplified 25.6

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

   14. Simplified 5.4

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

4. Final simplification 5.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.58864618275117732889084966831500711282 \cdot 10^{-55} \lor \neg \left(t \le 2592927404032904812036096\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\sqrt[3]{t} \cdot \log \left(e^{\sqrt[3]{t}}\right)} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right) \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(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)))))))))))