Average Error: 3.6 → 1.7
Time: 1.7m
Precision: 64
Internal Precision: 320
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{a + t}}{t} - \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{t \cdot 3.0}\right) \cdot \left(b - c\right)\right)} + x} \le 1.0004162554991944:\\ \;\;\;\;\frac{x}{y \cdot e^{\left(\frac{z \cdot \sqrt{a + t}}{t} - \left(\left(\frac{5.0}{6.0} + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \left(b - c\right)\right) \cdot 2.0} + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{2.0 \cdot \left(0.8333333333333334 \cdot c - \left(b - c\right) \cdot a\right)} \cdot y + x}\\ \end{array}\]

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 2 regimes

  2. if (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))) < 1.0004162554991944

    1. Initial program 0.7

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

    2. Taylor expanded around inf 0.7

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

    if 1.0004162554991944 < (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))))))

    1. Initial program 62.2

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

    2. Taylor expanded around -inf 26.6

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

    3. Simplified22.9

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

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{a + t}}{t} - \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{t \cdot 3.0}\right) \cdot \left(b - c\right)\right)} + x} \le 1.0004162554991944:\\ \;\;\;\;\frac{x}{y \cdot e^{\left(\frac{z \cdot \sqrt{a + t}}{t} - \left(\left(\frac{5.0}{6.0} + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \left(b - c\right)\right) \cdot 2.0} + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{2.0 \cdot \left(0.8333333333333334 \cdot c - \left(b - c\right) \cdot a\right)} \cdot y + x}\\ \end{array}\]

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2018225 
(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)))))))))))