Average Error: 3.9 → 2.7
Time: 3.7m
Precision: 64
Internal Precision: 320
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.049758174600674 \cdot 10^{+29} \lor \neg \left(t \le 1.0790710796675099 \cdot 10^{-256}\right):\\ \;\;\;\;\frac{x}{y \cdot e^{\left(\frac{\sqrt{t + a} \cdot z}{t} - \left(b - c\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{t \cdot 3.0}\right)\right) \cdot 2.0} + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot e^{\left(\left(\sqrt{t + a} \cdot z - \left(0.6666666666666666 \cdot \left(c - b\right) + \left(b \cdot t\right) \cdot 0.8333333333333333\right)\right) \cdot \frac{1}{t}\right) \cdot 2.0} + x}\\ \end{array}\]

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 2 regimes

  2. if t < -3.049758174600674e+29 or 1.0790710796675099e-256 < t

    1. Initial program 2.9

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

    if -3.049758174600674e+29 < t < 1.0790710796675099e-256

    1. Initial program 7.0

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

    3. Applied flip-+10.0

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

    4. Applied frac-sub10.1

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

    5. Applied associate-*r/10.2

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

    6. Applied frac-sub8.2

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

    8. Applied *-un-lft-identity8.2

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \frac{\color{blue}{1 \cdot \left(\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5.0}{6.0} \cdot \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right) - \left(a - \frac{5.0}{6.0}\right) \cdot 2.0\right)\right)\right)}}{t \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)\right)}}}\]

    9. Applied times-frac7.7

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \color{blue}{\left(\frac{1}{t} \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5.0}{6.0} \cdot \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right) - \left(a - \frac{5.0}{6.0}\right) \cdot 2.0\right)\right)}{\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)}\right)}}}\]

    10. Simplified3.6

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{1}{t} \cdot \color{blue}{\left(\frac{\sqrt{t + a} \cdot z}{1} - \frac{\left(b - c\right) \cdot t}{\left(3.0 \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)} \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(3.0 \cdot t\right) \cdot \left(\frac{5.0}{6.0} + a\right) - 2.0\right)\right)\right)}\right)}}\]

    11. Taylor expanded around 0 2.1

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{1}{t} \cdot \left(\frac{\sqrt{t + a} \cdot z}{1} - \color{blue}{\left(\left(0.8333333333333333 \cdot \left(t \cdot b\right) + 0.6666666666666666 \cdot c\right) - 0.6666666666666666 \cdot b\right)}\right)\right)}}\]

    12. Simplified2.1

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{1}{t} \cdot \left(\frac{\sqrt{t + a} \cdot z}{1} - \color{blue}{\left(\left(b \cdot t\right) \cdot 0.8333333333333333 + 0.6666666666666666 \cdot \left(c - b\right)\right)}\right)\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.049758174600674 \cdot 10^{+29} \lor \neg \left(t \le 1.0790710796675099 \cdot 10^{-256}\right):\\ \;\;\;\;\frac{x}{y \cdot e^{\left(\frac{\sqrt{t + a} \cdot z}{t} - \left(b - c\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{t \cdot 3.0}\right)\right) \cdot 2.0} + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot e^{\left(\left(\sqrt{t + a} \cdot z - \left(0.6666666666666666 \cdot \left(c - b\right) + \left(b \cdot t\right) \cdot 0.8333333333333333\right)\right) \cdot \frac{1}{t}\right) \cdot 2.0} + x}\\ \end{array}\]

Runtime

Time bar (total: 3.7m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes3.92.70.13.731.4%
herbie shell --seed 2018274 
(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)))))))))))