\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
Test:
Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2
Bits:
128 bits
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
Time: 30.2 s
Input Error: 27.0
Output Error: 1.6
Log:
Profile: 🕒
\(\begin{cases} \frac{x}{x + e^{\left(2.0 \cdot c\right) \cdot 0.8333333333333333 + \left(2.0 \cdot 1.0\right) \cdot \left(a \cdot \left(c - b\right)\right)} \cdot y} & \text{when } t \le -1.4045077231774965 \cdot 10^{-84} \\ \frac{x}{x + y \cdot e^{2.0 \cdot \frac{\left(\left(3.0 \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) \cdot \left(z \cdot \sqrt{a + t}\right) - \left(\left(\left(b - c\right) \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) \cdot \left(\left(3.0 \cdot t\right) \cdot \left(a + \frac{5.0}{6.0}\right) - 2.0\right)}{\left(a - \frac{5.0}{6.0}\right) \cdot \left({t}^2 \cdot 3.0\right)}}} & \text{when } t \le 5.203540092866018 \cdot 10^{+102} \\ \frac{x}{x + {\left(e^{2.0}\right)}^{\left(\frac{z}{t} \cdot \sqrt{a + t} - \left(b \cdot 0.8333333333333334 - \frac{0.6666666666666666}{t} \cdot \left(b - c\right)\right)\right)} \cdot y} & \text{when } t \le 1.0313555018265321 \cdot 10^{+169} \\ \frac{x}{x + e^{\left(2.0 \cdot c\right) \cdot 0.8333333333333333 + \left(2.0 \cdot 1.0\right) \cdot \left(a \cdot \left(c - b\right)\right)} \cdot y} & \text{otherwise} \end{cases}\)

    if t < -1.4045077231774965e-84 or 1.0313555018265321e+169 < t

    1. Started with
      \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
      43.7
    2. Using strategy rm
      43.7
    3. Applied flip-+ to get
      \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\color{red}{\left(a + \frac{5.0}{6.0}\right)} - \frac{2.0}{t \cdot 3.0}\right)\right)}} \leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\color{blue}{\frac{{a}^2 - {\left(\frac{5.0}{6.0}\right)}^2}{a - \frac{5.0}{6.0}}} - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
      43.7
    4. Applied frac-sub to get
      \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \color{red}{\left(\frac{{a}^2 - {\left(\frac{5.0}{6.0}\right)}^2}{a - \frac{5.0}{6.0}} - \frac{2.0}{t \cdot 3.0}\right)}\right)}} \leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \color{blue}{\frac{\left({a}^2 - {\left(\frac{5.0}{6.0}\right)}^2\right) \cdot \left(t \cdot 3.0\right) - \left(a - \frac{5.0}{6.0}\right) \cdot 2.0}{\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)}}\right)}}\]
      51.0
    5. Applied associate-*r/ to get
      \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \color{red}{\left(b - c\right) \cdot \frac{\left({a}^2 - {\left(\frac{5.0}{6.0}\right)}^2\right) \cdot \left(t \cdot 3.0\right) - \left(a - \frac{5.0}{6.0}\right) \cdot 2.0}{\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)}}\right)}} \leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \color{blue}{\frac{\left(b - c\right) \cdot \left(\left({a}^2 - {\left(\frac{5.0}{6.0}\right)}^2\right) \cdot \left(t \cdot 3.0\right) - \left(a - \frac{5.0}{6.0}\right) \cdot 2.0\right)}{\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)}}\right)}}\]
      51.1
    6. Applied frac-sub to get
      \[\frac{x}{x + y \cdot e^{2.0 \cdot \color{red}{\left(\frac{z \cdot \sqrt{t + a}}{t} - \frac{\left(b - c\right) \cdot \left(\left({a}^2 - {\left(\frac{5.0}{6.0}\right)}^2\right) \cdot \left(t \cdot 3.0\right) - \left(a - \frac{5.0}{6.0}\right) \cdot 2.0\right)}{\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)}\right)}}} \leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \color{blue}{\frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left({a}^2 - {\left(\frac{5.0}{6.0}\right)}^2\right) \cdot \left(t \cdot 3.0\right) - \left(a - \frac{5.0}{6.0}\right) \cdot 2.0\right)\right)}{t \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)\right)}}}}\]
      58.8
    7. Applied simplify to get
      \[\frac{x}{x + y \cdot e^{2.0 \cdot \frac{\color{red}{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left({a}^2 - {\left(\frac{5.0}{6.0}\right)}^2\right) \cdot \left(t \cdot 3.0\right) - \left(a - \frac{5.0}{6.0}\right) \cdot 2.0\right)\right)}}{t \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)\right)}}} \leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \frac{\color{blue}{\left(\left(3.0 \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) \cdot \left(z \cdot \sqrt{a + t}\right) - \left(\left(\left(b - c\right) \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) \cdot \left(\left(3.0 \cdot t\right) \cdot \left(a + \frac{5.0}{6.0}\right) - 2.0\right)}}{t \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)\right)}}}\]
      58.8
    8. Applied simplify to get
      \[\frac{x}{x + y \cdot e^{2.0 \cdot \frac{\left(\left(3.0 \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) \cdot \left(z \cdot \sqrt{a + t}\right) - \left(\left(\left(b - c\right) \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) \cdot \left(\left(3.0 \cdot t\right) \cdot \left(a + \frac{5.0}{6.0}\right) - 2.0\right)}{\color{red}{t \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)\right)}}}} \leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \frac{\left(\left(3.0 \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) \cdot \left(z \cdot \sqrt{a + t}\right) - \left(\left(\left(b - c\right) \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) \cdot \left(\left(3.0 \cdot t\right) \cdot \left(a + \frac{5.0}{6.0}\right) - 2.0\right)}{\color{blue}{\left(a - \frac{5.0}{6.0}\right) \cdot \left({t}^2 \cdot 3.0\right)}}}}\]
      58.8
    9. Applied taylor to get
      \[\frac{x}{x + y \cdot e^{2.0 \cdot \frac{\left(\left(3.0 \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) \cdot \left(z \cdot \sqrt{a + t}\right) - \left(\left(\left(b - c\right) \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) \cdot \left(\left(3.0 \cdot t\right) \cdot \left(a + \frac{5.0}{6.0}\right) - 2.0\right)}{\left(a - \frac{5.0}{6.0}\right) \cdot \left({t}^2 \cdot 3.0\right)}}} \leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\left(0.8333333333333333 \cdot c + 1.0 \cdot \left(c \cdot a\right)\right) - 1.0 \cdot \left(b \cdot a\right)\right)}}\]
      1.2
    10. Taylor expanded around inf to get
      \[\frac{x}{x + y \cdot e^{2.0 \cdot \color{red}{\left(\left(0.8333333333333333 \cdot c + 1.0 \cdot \left(c \cdot a\right)\right) - 1.0 \cdot \left(b \cdot a\right)\right)}}} \leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \color{blue}{\left(\left(0.8333333333333333 \cdot c + 1.0 \cdot \left(c \cdot a\right)\right) - 1.0 \cdot \left(b \cdot a\right)\right)}}}\]
      1.2
    11. Applied simplify to get
      \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\left(0.8333333333333333 \cdot c + 1.0 \cdot \left(c \cdot a\right)\right) - 1.0 \cdot \left(b \cdot a\right)\right)}} \leadsto \frac{x}{x + e^{\left(2.0 \cdot c\right) \cdot 0.8333333333333333 + \left(2.0 \cdot 1.0\right) \cdot \left(a \cdot \left(c - b\right)\right)} \cdot y}\]
      0.0

    12. Applied final simplification

    if -1.4045077231774965e-84 < t < 5.203540092866018e+102

    1. Started with
      \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
      4.1
    2. Using strategy rm
      4.1
    3. Applied flip-+ to get
      \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\color{red}{\left(a + \frac{5.0}{6.0}\right)} - \frac{2.0}{t \cdot 3.0}\right)\right)}} \leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\color{blue}{\frac{{a}^2 - {\left(\frac{5.0}{6.0}\right)}^2}{a - \frac{5.0}{6.0}}} - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
      6.1
    4. Applied frac-sub to get
      \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \color{red}{\left(\frac{{a}^2 - {\left(\frac{5.0}{6.0}\right)}^2}{a - \frac{5.0}{6.0}} - \frac{2.0}{t \cdot 3.0}\right)}\right)}} \leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \color{blue}{\frac{\left({a}^2 - {\left(\frac{5.0}{6.0}\right)}^2\right) \cdot \left(t \cdot 3.0\right) - \left(a - \frac{5.0}{6.0}\right) \cdot 2.0}{\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)}}\right)}}\]
      6.9
    5. Applied associate-*r/ to get
      \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \color{red}{\left(b - c\right) \cdot \frac{\left({a}^2 - {\left(\frac{5.0}{6.0}\right)}^2\right) \cdot \left(t \cdot 3.0\right) - \left(a - \frac{5.0}{6.0}\right) \cdot 2.0}{\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)}}\right)}} \leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \color{blue}{\frac{\left(b - c\right) \cdot \left(\left({a}^2 - {\left(\frac{5.0}{6.0}\right)}^2\right) \cdot \left(t \cdot 3.0\right) - \left(a - \frac{5.0}{6.0}\right) \cdot 2.0\right)}{\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)}}\right)}}\]
      6.9
    6. Applied frac-sub to get
      \[\frac{x}{x + y \cdot e^{2.0 \cdot \color{red}{\left(\frac{z \cdot \sqrt{t + a}}{t} - \frac{\left(b - c\right) \cdot \left(\left({a}^2 - {\left(\frac{5.0}{6.0}\right)}^2\right) \cdot \left(t \cdot 3.0\right) - \left(a - \frac{5.0}{6.0}\right) \cdot 2.0\right)}{\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)}\right)}}} \leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \color{blue}{\frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left({a}^2 - {\left(\frac{5.0}{6.0}\right)}^2\right) \cdot \left(t \cdot 3.0\right) - \left(a - \frac{5.0}{6.0}\right) \cdot 2.0\right)\right)}{t \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)\right)}}}}\]
      7.3
    7. Applied simplify to get
      \[\frac{x}{x + y \cdot e^{2.0 \cdot \frac{\color{red}{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left({a}^2 - {\left(\frac{5.0}{6.0}\right)}^2\right) \cdot \left(t \cdot 3.0\right) - \left(a - \frac{5.0}{6.0}\right) \cdot 2.0\right)\right)}}{t \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)\right)}}} \leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \frac{\color{blue}{\left(\left(3.0 \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) \cdot \left(z \cdot \sqrt{a + t}\right) - \left(\left(\left(b - c\right) \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) \cdot \left(\left(3.0 \cdot t\right) \cdot \left(a + \frac{5.0}{6.0}\right) - 2.0\right)}}{t \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)\right)}}}\]
      4.2
    8. Applied simplify to get
      \[\frac{x}{x + y \cdot e^{2.0 \cdot \frac{\left(\left(3.0 \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) \cdot \left(z \cdot \sqrt{a + t}\right) - \left(\left(\left(b - c\right) \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) \cdot \left(\left(3.0 \cdot t\right) \cdot \left(a + \frac{5.0}{6.0}\right) - 2.0\right)}{\color{red}{t \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)\right)}}}} \leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \frac{\left(\left(3.0 \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) \cdot \left(z \cdot \sqrt{a + t}\right) - \left(\left(\left(b - c\right) \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) \cdot \left(\left(3.0 \cdot t\right) \cdot \left(a + \frac{5.0}{6.0}\right) - 2.0\right)}{\color{blue}{\left(a - \frac{5.0}{6.0}\right) \cdot \left({t}^2 \cdot 3.0\right)}}}}\]
      4.2

    if 5.203540092866018e+102 < t < 1.0313555018265321e+169

    1. Started with
      \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
      19.8
    2. Applied taylor to get
      \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}} \leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(\left(0.8333333333333334 \cdot b + 0.6666666666666666 \cdot \frac{c}{t}\right) - 0.6666666666666666 \cdot \frac{b}{t}\right)\right)}}\]
      0.4
    3. Taylor expanded around 0 to get
      \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \color{red}{\left(\left(0.8333333333333334 \cdot b + 0.6666666666666666 \cdot \frac{c}{t}\right) - 0.6666666666666666 \cdot \frac{b}{t}\right)}\right)}} \leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \color{blue}{\left(\left(0.8333333333333334 \cdot b + 0.6666666666666666 \cdot \frac{c}{t}\right) - 0.6666666666666666 \cdot \frac{b}{t}\right)}\right)}}\]
      0.4
    4. Applied simplify to get
      \[\color{red}{\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(\left(0.8333333333333334 \cdot b + 0.6666666666666666 \cdot \frac{c}{t}\right) - 0.6666666666666666 \cdot \frac{b}{t}\right)\right)}}} \leadsto \color{blue}{\frac{x}{x + {\left(e^{2.0}\right)}^{\left(\frac{z}{t} \cdot \sqrt{a + t} - \left(b \cdot 0.8333333333333334 - \frac{0.6666666666666666}{t} \cdot \left(b - c\right)\right)\right)} \cdot y}}\]
      0.0

  1. Removed slow pow expressions

Original test:


(lambda ((x default) (y default) (z default) (t default) (a default) (b default) (c default))
  #:name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))