Average Error: 3.7 → 4.0
Time: 37.3s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -5.869458934952860147042895437611662296973 \cdot 10^{-296} \lor \neg \left(t \le 3.585798647479515118878164820573513844078 \cdot 10^{-110}\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{0.6666666666666666296592325124947819858789}{t}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{t \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \end{array}\]
\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 -5.869458934952860147042895437611662296973 \cdot 10^{-296} \lor \neg \left(t \le 3.585798647479515118878164820573513844078 \cdot 10^{-110}\right):\\
\;\;\;\;\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{0.6666666666666666296592325124947819858789}{t}\right)\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - t \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)}{t \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r84370 = x;
        double r84371 = y;
        double r84372 = 2.0;
        double r84373 = z;
        double r84374 = t;
        double r84375 = a;
        double r84376 = r84374 + r84375;
        double r84377 = sqrt(r84376);
        double r84378 = r84373 * r84377;
        double r84379 = r84378 / r84374;
        double r84380 = b;
        double r84381 = c;
        double r84382 = r84380 - r84381;
        double r84383 = 5.0;
        double r84384 = 6.0;
        double r84385 = r84383 / r84384;
        double r84386 = r84375 + r84385;
        double r84387 = 3.0;
        double r84388 = r84374 * r84387;
        double r84389 = r84372 / r84388;
        double r84390 = r84386 - r84389;
        double r84391 = r84382 * r84390;
        double r84392 = r84379 - r84391;
        double r84393 = r84372 * r84392;
        double r84394 = exp(r84393);
        double r84395 = r84371 * r84394;
        double r84396 = r84370 + r84395;
        double r84397 = r84370 / r84396;
        return r84397;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r84398 = t;
        double r84399 = -5.86945893495286e-296;
        bool r84400 = r84398 <= r84399;
        double r84401 = 3.585798647479515e-110;
        bool r84402 = r84398 <= r84401;
        double r84403 = !r84402;
        bool r84404 = r84400 || r84403;
        double r84405 = x;
        double r84406 = y;
        double r84407 = 2.0;
        double r84408 = z;
        double r84409 = a;
        double r84410 = r84398 + r84409;
        double r84411 = sqrt(r84410);
        double r84412 = r84408 * r84411;
        double r84413 = r84412 / r84398;
        double r84414 = b;
        double r84415 = c;
        double r84416 = r84414 - r84415;
        double r84417 = 5.0;
        double r84418 = 6.0;
        double r84419 = r84417 / r84418;
        double r84420 = r84409 + r84419;
        double r84421 = 0.6666666666666666;
        double r84422 = r84421 / r84398;
        double r84423 = r84420 - r84422;
        double r84424 = r84416 * r84423;
        double r84425 = r84413 - r84424;
        double r84426 = r84407 * r84425;
        double r84427 = exp(r84426);
        double r84428 = r84406 * r84427;
        double r84429 = r84405 + r84428;
        double r84430 = r84405 / r84429;
        double r84431 = r84409 - r84419;
        double r84432 = 3.0;
        double r84433 = r84398 * r84432;
        double r84434 = r84431 * r84433;
        double r84435 = r84412 * r84434;
        double r84436 = r84409 * r84409;
        double r84437 = r84419 * r84419;
        double r84438 = r84436 - r84437;
        double r84439 = r84438 * r84433;
        double r84440 = r84431 * r84407;
        double r84441 = r84439 - r84440;
        double r84442 = r84416 * r84441;
        double r84443 = r84398 * r84442;
        double r84444 = r84435 - r84443;
        double r84445 = r84398 * r84434;
        double r84446 = r84444 / r84445;
        double r84447 = r84407 * r84446;
        double r84448 = exp(r84447);
        double r84449 = r84406 * r84448;
        double r84450 = r84405 + r84449;
        double r84451 = r84405 / r84450;
        double r84452 = r84404 ? r84430 : r84451;
        return r84452;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if t < -5.86945893495286e-296 or 3.585798647479515e-110 < t

    1. Initial program 3.0

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

    2. Taylor expanded around 0 3.0

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \color{blue}{\frac{0.6666666666666666296592325124947819858789}{t}}\right)\right)}}\]

    if -5.86945893495286e-296 < t < 3.585798647479515e-110

    1. Initial program 6.6

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

    3. Applied flip-+10.6

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

    4. Applied frac-sub10.7

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

    5. Applied associate-*r/10.7

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

    6. Applied frac-sub7.8

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

  4. Final simplification4.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.869458934952860147042895437611662296973 \cdot 10^{-296} \lor \neg \left(t \le 3.585798647479515118878164820573513844078 \cdot 10^{-110}\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{0.6666666666666666296592325124947819858789}{t}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{t \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(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)))))))))))