Average Error: 3.8 → 3.7
Time: 1.4m
Precision: 64
Internal Precision: 128
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.7864061371423284 \cdot 10^{+23}:\\ \;\;\;\;\frac{x}{y \cdot e^{2.0 \cdot \left(c \cdot 0.8333333333333334 - \left(b - c\right) \cdot a\right)} + x}\\ \mathbf{elif}\;t \le 1.3928975876369032 \cdot 10^{-06}:\\ \;\;\;\;\frac{x}{y \cdot e^{\frac{\left(\left(t \cdot 3.0\right) \cdot z\right) \cdot \left(a - \frac{5.0}{6.0}\right) - \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right) - 2.0\right)\right) \cdot \frac{t \cdot \left(b - c\right)}{\sqrt{t + a}}}{\left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)\right) \cdot \frac{t}{\sqrt{t + a}}} \cdot 2.0} + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{\left(\frac{\sqrt{t + a} \cdot z}{t} - \left(\left(a + \frac{5.0}{6.0}\right) - \log \left(e^{\frac{2.0}{t \cdot 3.0}}\right)\right) \cdot \left(b - c\right)\right) \cdot 2.0} \cdot y + x}\\ \end{array}\]

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 < -3.7864061371423284e+23

    1. Initial program 2.8

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

    3. Applied associate-/l*0.3

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

    4. Taylor expanded around -inf 8.4

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

    5. Simplified4.1

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

    if -3.7864061371423284e+23 < t < 1.3928975876369032e-06

    1. Initial program 5.1

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

    3. Applied associate-/l*6.4

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

    5. Applied flip-+9.1

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

    6. Applied frac-sub9.2

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

    7. Applied associate-*r/9.2

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

    8. Applied frac-sub9.5

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

    9. Simplified4.6

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

    if 1.3928975876369032e-06 < t

    1. Initial program 2.5

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

    3. Applied add-log-exp2.6

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \color{blue}{\log \left(e^{\frac{2.0}{t \cdot 3.0}}\right)}\right)\right)}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification3.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.7864061371423284 \cdot 10^{+23}:\\ \;\;\;\;\frac{x}{y \cdot e^{2.0 \cdot \left(c \cdot 0.8333333333333334 - \left(b - c\right) \cdot a\right)} + x}\\ \mathbf{elif}\;t \le 1.3928975876369032 \cdot 10^{-06}:\\ \;\;\;\;\frac{x}{y \cdot e^{\frac{\left(\left(t \cdot 3.0\right) \cdot z\right) \cdot \left(a - \frac{5.0}{6.0}\right) - \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right) - 2.0\right)\right) \cdot \frac{t \cdot \left(b - c\right)}{\sqrt{t + a}}}{\left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)\right) \cdot \frac{t}{\sqrt{t + a}}} \cdot 2.0} + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{\left(\frac{\sqrt{t + a} \cdot z}{t} - \left(\left(a + \frac{5.0}{6.0}\right) - \log \left(e^{\frac{2.0}{t \cdot 3.0}}\right)\right) \cdot \left(b - c\right)\right) \cdot 2.0} \cdot y + x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019021 
(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)))))))))))