Average Error: 3.6 → 2.5
Time: 32.6s
Precision: 64
Internal precision: 384
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
⬇
\[\begin{array}{l} \mathbf{if}\;t \le 2.237233773871134 \cdot 10^{-299}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\\ \mathbf{if}\;t \le 2.375652348111882 \cdot 10^{-175}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \frac{\left(\left(3.0 \cdot t\right) \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot z\right) - \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(3.0 \cdot t\right) \cdot \left(\frac{5.0}{6.0} + a\right) - 2.0\right)\right) \cdot \left(\left(b - c\right) \cdot t\right)}{t \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\\ \end{array}\]

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

Derivation

  1. Split input into 2 regimes.

  2. if t < 2.237233773871134e-299 or 2.375652348111882e-175 < t

    1. Initial program 3.2

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

    3. Applied associate-/l* 2.0

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

    if 2.237233773871134e-299 < t < 2.375652348111882e-175

    1. Initial program 6.8

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

    3. Applied flip-+ 11.4

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

    4. Applied frac-sub 11.4

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

    5. Applied associate-*r/ 11.4

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

    6. Applied frac-sub 9.9

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

    7. Applied simplify 6.4

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \frac{\color{blue}{\left(\left(3.0 \cdot t\right) \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot z\right) - \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(3.0 \cdot t\right) \cdot \left(\frac{5.0}{6.0} + a\right) - 2.0\right)\right) \cdot \left(\left(b - c\right) \cdot t\right)}}{t \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)\right)}}}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 32.6s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1064524629 4159152179 2999149171 575749698 4006532819 692958815)'
(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)))))))))))