Average Error: 3.7 → 1.8
Time: 5.8m
Precision: 64
Internal Precision: 384
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -7.642850582532975 \cdot 10^{-06}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\\ \mathbf{if}\;t \le 8.450570625098171 \cdot 10^{-106}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \frac{\left(\frac{2.0}{3.0 \cdot t} + \left(\frac{5.0}{6.0} + a\right)\right) \cdot \left(z \cdot \sqrt{t + a} - \left(\left(b - c\right) \cdot t\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{3.0 \cdot t}\right)\right)}{t \cdot \left(\left(a + \frac{5.0}{6.0}\right) + \frac{2.0}{t \cdot 3.0}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\\ \end{array}\]

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 3 regimes

  2. if t < -7.642850582532975e-06

    1. Initial program 3.5

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

    if -7.642850582532975e-06 < t < 8.450570625098171e-106

    1. Initial program 6.0

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

    3. Applied flip--18.8

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

    4. Applied associate-*r/19.2

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

    5. Applied frac-sub19.9

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

    6. Applied simplify1.1

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

    if 8.450570625098171e-106 < t

    1. Initial program 2.1

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

Runtime

Time bar (total: 5.8m)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit
(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)))))))))))