Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2

Average Error: 3.9 → 4.1

Time: 9.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.097264455817546718684511622823663412103 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\sqrt[3]{t} \cdot \log \left(e^{\sqrt[3]{t}}\right)} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \mathbf{elif}\;t \le 1.207880239692675796069901468393483526346 \cdot 10^{-104}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - {\left({\left(\frac{2}{t \cdot 3}\right)}^{3}\right)}^{\frac{1}{3}}\right)\right)}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r76224 = x;
        double r76225 = y;
        double r76226 = 2.0;
        double r76227 = z;
        double r76228 = t;
        double r76229 = a;
        double r76230 = r76228 + r76229;
        double r76231 = sqrt(r76230);
        double r76232 = r76227 * r76231;
        double r76233 = r76232 / r76228;
        double r76234 = b;
        double r76235 = c;
        double r76236 = r76234 - r76235;
        double r76237 = 5.0;
        double r76238 = 6.0;
        double r76239 = r76237 / r76238;
        double r76240 = r76229 + r76239;
        double r76241 = 3.0;
        double r76242 = r76228 * r76241;
        double r76243 = r76226 / r76242;
        double r76244 = r76240 - r76243;
        double r76245 = r76236 * r76244;
        double r76246 = r76233 - r76245;
        double r76247 = r76226 * r76246;
        double r76248 = exp(r76247);
        double r76249 = r76225 * r76248;
        double r76250 = r76224 + r76249;
        double r76251 = r76224 / r76250;
        return r76251;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r76252 = t;
        double r76253 = -2.0972644558175467e-14;
        bool r76254 = r76252 <= r76253;
        double r76255 = x;
        double r76256 = y;
        double r76257 = 2.0;
        double r76258 = z;
        double r76259 = cbrt(r76252);
        double r76260 = exp(r76259);
        double r76261 = log(r76260);
        double r76262 = r76259 * r76261;
        double r76263 = r76258 / r76262;
        double r76264 = a;
        double r76265 = r76252 + r76264;
        double r76266 = sqrt(r76265);
        double r76267 = r76266 / r76259;
        double r76268 = r76263 * r76267;
        double r76269 = b;
        double r76270 = c;
        double r76271 = r76269 - r76270;
        double r76272 = 5.0;
        double r76273 = 6.0;
        double r76274 = r76272 / r76273;
        double r76275 = r76264 + r76274;
        double r76276 = 3.0;
        double r76277 = r76252 * r76276;
        double r76278 = r76257 / r76277;
        double r76279 = r76275 - r76278;
        double r76280 = r76271 * r76279;
        double r76281 = r76268 - r76280;
        double r76282 = r76257 * r76281;
        double r76283 = exp(r76282);
        double r76284 = r76256 * r76283;
        double r76285 = r76255 + r76284;
        double r76286 = r76255 / r76285;
        double r76287 = 1.2078802396926758e-104;
        bool r76288 = r76252 <= r76287;
        double r76289 = r76258 * r76267;
        double r76290 = r76264 - r76274;
        double r76291 = r76290 * r76277;
        double r76292 = r76289 * r76291;
        double r76293 = r76259 * r76259;
        double r76294 = r76264 * r76264;
        double r76295 = r76274 * r76274;
        double r76296 = r76294 - r76295;
        double r76297 = r76296 * r76277;
        double r76298 = r76290 * r76257;
        double r76299 = r76297 - r76298;
        double r76300 = r76271 * r76299;
        double r76301 = r76293 * r76300;
        double r76302 = r76292 - r76301;
        double r76303 = r76293 * r76291;
        double r76304 = r76302 / r76303;
        double r76305 = r76257 * r76304;
        double r76306 = exp(r76305);
        double r76307 = r76256 * r76306;
        double r76308 = r76255 + r76307;
        double r76309 = r76255 / r76308;
        double r76310 = r76258 * r76266;
        double r76311 = r76310 / r76252;
        double r76312 = 3.0;
        double r76313 = pow(r76278, r76312);
        double r76314 = 0.3333333333333333;
        double r76315 = pow(r76313, r76314);
        double r76316 = r76275 - r76315;
        double r76317 = r76271 * r76316;
        double r76318 = r76311 - r76317;
        double r76319 = r76257 * r76318;
        double r76320 = exp(r76319);
        double r76321 = r76256 * r76320;
        double r76322 = r76255 + r76321;
        double r76323 = r76255 / r76322;
        double r76324 = r76288 ? r76309 : r76323;
        double r76325 = r76254 ? r76286 : r76324;
        return r76325;
}

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 < -2.0972644558175467e-14

   1. Initial program 4.1

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

   3. Applied add-cube-cbrt 4.1

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

   4. Applied times-frac 0.6

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

   6. Applied add-log-exp 3.9

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

   if -2.0972644558175467e-14 < t < 1.2078802396926758e-104

   1. Initial program 6.5

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

   3. Applied add-cube-cbrt 6.5

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

   4. Applied times-frac 6.7

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

   6. Applied flip-+ 10.0

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

   7. Applied frac-sub 10.1

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

   8. Applied associate-*r/ 10.1

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

   9. Applied associate-*l/ 9.9

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

   10. Applied frac-sub 7.0

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

   if 1.2078802396926758e-104 < t

   1. Initial program 2.3

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

   3. Applied add-cbrt-cube 2.3

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

   4. Applied add-cbrt-cube 2.3

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

   5. Applied cbrt-unprod 2.3

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

   6. Applied add-cbrt-cube 2.3

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

   7. Applied cbrt-undiv 2.3

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

   8. Simplified 2.3

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

   10. Applied pow1/3 2.3

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

4. Final simplification 4.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.097264455817546718684511622823663412103 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\sqrt[3]{t} \cdot \log \left(e^{\sqrt[3]{t}}\right)} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \mathbf{elif}\;t \le 1.207880239692675796069901468393483526346 \cdot 10^{-104}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - {\left({\left(\frac{2}{t \cdot 3}\right)}^{3}\right)}^{\frac{1}{3}}\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(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)))))))))))