Average Error: 3.6 → 4.0
Time: 2.4m
Precision: 64
Internal Precision: 128
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 5.42921230049757 \cdot 10^{+257}:\\ \;\;\;\;\frac{x}{x + e^{\log \left(e^{\frac{\sqrt{t + a} \cdot z}{t} - \left(b - c\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{t \cdot 3.0}\right)}\right) \cdot 2.0} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + \sqrt{e^{2.0 \cdot \log \left(e^{c \cdot 0.8333333333333334 - \left(b - c\right) \cdot a}\right)}} \cdot \left(y \cdot \sqrt{e^{2.0 \cdot \log \left(e^{c \cdot 0.8333333333333334 - \left(b - c\right) \cdot a}\right)}}\right)}\\ \end{array}\]

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 2 regimes

  2. if t < 5.42921230049757e+257

    1. Initial program 3.6

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

    3. Applied add-log-exp3.6

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

    if 5.42921230049757e+257 < t

    1. Initial program 4.5

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

    3. Applied add-log-exp4.5

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

    4. Taylor expanded around inf 9.7

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

    5. Simplified8.3

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

    7. Applied add-sqr-sqrt8.3

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

    8. Applied associate-*r*8.3

      \[\leadsto \frac{x}{x + \color{blue}{\left(y \cdot \sqrt{e^{2.0 \cdot \log \left(e^{0.8333333333333334 \cdot c - \left(b - c\right) \cdot a}\right)}}\right) \cdot \sqrt{e^{2.0 \cdot \log \left(e^{0.8333333333333334 \cdot c - \left(b - c\right) \cdot a}\right)}}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification4.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 5.42921230049757 \cdot 10^{+257}:\\ \;\;\;\;\frac{x}{x + e^{\log \left(e^{\frac{\sqrt{t + a} \cdot z}{t} - \left(b - c\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{t \cdot 3.0}\right)}\right) \cdot 2.0} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + \sqrt{e^{2.0 \cdot \log \left(e^{c \cdot 0.8333333333333334 - \left(b - c\right) \cdot a}\right)}} \cdot \left(y \cdot \sqrt{e^{2.0 \cdot \log \left(e^{c \cdot 0.8333333333333334 - \left(b - c\right) \cdot a}\right)}}\right)}\\ \end{array}\]

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed 2018336 
(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)))))))))))