Average Error: 3.6 → 1.6
Time: 40.1s
Precision: 64
Internal Precision: 320
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{a + t}}{t} - \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{t \cdot 3.0}\right) \cdot \left(b - c\right)\right)} + x} \le 1.0004162554991944:\\ \;\;\;\;\frac{x}{(y \cdot \left(e^{(\left(2.0 \cdot \left(c - b\right)\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{\frac{2.0}{t}}{3.0}\right) + \left(\left(2.0 \cdot \frac{z}{t}\right) \cdot \sqrt{a + t}\right))_*}\right) + x)_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{(y \cdot \left(e^{\left(2.0 \cdot a\right) \cdot \left(c - b\right) - 1.6666666666666667 \cdot b}\right) + x)_*}\\ \end{array}\]

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

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. Initial simplification0.4

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

    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. Initial simplification29.7

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

    3. Taylor expanded around -inf 27.9

      \[\leadsto \frac{x}{(y \cdot \left(e^{\color{blue}{2.0 \cdot \left(a \cdot c\right) - \left(2.0 \cdot \left(a \cdot b\right) + 1.6666666666666667 \cdot b\right)}}\right) + x)_*}\]

    4. Simplified24.2

      \[\leadsto \frac{x}{(y \cdot \left(e^{\color{blue}{\left(2.0 \cdot a\right) \cdot \left(c - b\right) - 1.6666666666666667 \cdot b}}\right) + x)_*}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{a + t}}{t} - \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{t \cdot 3.0}\right) \cdot \left(b - c\right)\right)} + x} \le 1.0004162554991944:\\ \;\;\;\;\frac{x}{(y \cdot \left(e^{(\left(2.0 \cdot \left(c - b\right)\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{\frac{2.0}{t}}{3.0}\right) + \left(\left(2.0 \cdot \frac{z}{t}\right) \cdot \sqrt{a + t}\right))_*}\right) + x)_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{(y \cdot \left(e^{\left(2.0 \cdot a\right) \cdot \left(c - b\right) - 1.6666666666666667 \cdot b}\right) + x)_*}\\ \end{array}\]

Runtime

Time bar (total: 40.1s)Debug logProfile

herbie shell --seed 2018225 +o rules:numerics
(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)))))))))))