Average Error: 3.8 → 8.7
Time: 1.4m
Precision: 64
Internal Precision: 576
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -7.80300254621126 \cdot 10^{-238}:\\ \;\;\;\;\frac{x}{x + e^{2.0 \cdot \left(\frac{\sqrt{t + a} \cdot z}{t} - \left(b \cdot 0.8333333333333334 + \left(c - b\right) \cdot \frac{0.6666666666666666}{t}\right)\right)} \cdot y}\\ \mathbf{elif}\;t \le 2.209876178238086 \cdot 10^{+97}:\\ \;\;\;\;\frac{x}{y \cdot e^{\frac{\left(\left(a - \frac{5.0}{6.0}\right) \cdot t\right) \cdot \left(\sqrt{t + a} \cdot z\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5.0}{6.0} \cdot \frac{5.0}{6.0}\right) \cdot t - \left(a - \frac{5.0}{6.0}\right) \cdot 0.6666666666666666\right)\right)}{t \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot t\right)} \cdot 2.0} + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{\left(c \cdot 0.8333333333333334 - a \cdot \left(b - c\right)\right) \cdot 2.0} \cdot y + x}\\ \end{array}\]

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 3 regimes

  2. if t < -7.80300254621126e-238

    1. Initial program 4.1

      \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]

    2. Taylor expanded around inf 4.1

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \color{blue}{\frac{0.6666666666666666}{t}}\right)\right)}}\]

    3. Taylor expanded around 0 10.2

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \color{blue}{\left(\left(0.6666666666666666 \cdot \frac{c}{t} + 0.8333333333333334 \cdot b\right) - 0.6666666666666666 \cdot \frac{b}{t}\right)}\right)}}\]

    4. Simplified9.2

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \color{blue}{\left(0.8333333333333334 \cdot b + \left(c - b\right) \cdot \frac{0.6666666666666666}{t}\right)}\right)}}\]

    if -7.80300254621126e-238 < t < 2.209876178238086e+97

    1. Initial program 4.2

      \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]

    2. Taylor expanded around inf 4.2

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \color{blue}{\frac{0.6666666666666666}{t}}\right)\right)}}\]
    3. Using strategy rm

    4. Applied flip-+6.1

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\color{blue}{\frac{a \cdot a - \frac{5.0}{6.0} \cdot \frac{5.0}{6.0}}{a - \frac{5.0}{6.0}}} - \frac{0.6666666666666666}{t}\right)\right)}}\]

    5. Applied frac-sub6.8

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \color{blue}{\frac{\left(a \cdot a - \frac{5.0}{6.0} \cdot \frac{5.0}{6.0}\right) \cdot t - \left(a - \frac{5.0}{6.0}\right) \cdot 0.6666666666666666}{\left(a - \frac{5.0}{6.0}\right) \cdot t}}\right)}}\]

    6. Applied associate-*r/6.9

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \color{blue}{\frac{\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5.0}{6.0} \cdot \frac{5.0}{6.0}\right) \cdot t - \left(a - \frac{5.0}{6.0}\right) \cdot 0.6666666666666666\right)}{\left(a - \frac{5.0}{6.0}\right) \cdot t}}\right)}}\]

    7. Applied frac-sub6.9

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \color{blue}{\frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot t\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5.0}{6.0} \cdot \frac{5.0}{6.0}\right) \cdot t - \left(a - \frac{5.0}{6.0}\right) \cdot 0.6666666666666666\right)\right)}{t \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot t\right)}}}}\]

    if 2.209876178238086e+97 < t

    1. Initial program 3.0

      \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]

    2. Taylor expanded around inf 3.0

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \color{blue}{\frac{0.6666666666666666}{t}}\right)\right)}}\]

    3. Taylor expanded around inf 12.1

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \color{blue}{\left(\left(a \cdot c + 0.8333333333333334 \cdot c\right) - a \cdot b\right)}}}\]

    4. Simplified11.1

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \color{blue}{\left(0.8333333333333334 \cdot c - \left(b - c\right) \cdot a\right)}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -7.80300254621126 \cdot 10^{-238}:\\ \;\;\;\;\frac{x}{x + e^{2.0 \cdot \left(\frac{\sqrt{t + a} \cdot z}{t} - \left(b \cdot 0.8333333333333334 + \left(c - b\right) \cdot \frac{0.6666666666666666}{t}\right)\right)} \cdot y}\\ \mathbf{elif}\;t \le 2.209876178238086 \cdot 10^{+97}:\\ \;\;\;\;\frac{x}{y \cdot e^{\frac{\left(\left(a - \frac{5.0}{6.0}\right) \cdot t\right) \cdot \left(\sqrt{t + a} \cdot z\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5.0}{6.0} \cdot \frac{5.0}{6.0}\right) \cdot t - \left(a - \frac{5.0}{6.0}\right) \cdot 0.6666666666666666\right)\right)}{t \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot t\right)} \cdot 2.0} + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{\left(c \cdot 0.8333333333333334 - a \cdot \left(b - c\right)\right) \cdot 2.0} \cdot y + x}\\ \end{array}\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018255 
(FPCore (x y z t a b c)
  :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)))))))))))