Average Error: 3.5 → 1.6
Time: 2.5m
Precision: 64
Internal Precision: 384
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 8.880244365518877 \cdot 10^{-162}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \frac{\left(\left(a + \frac{5.0}{6.0}\right) + \frac{\frac{2.0}{t}}{3.0}\right) \cdot \left(z - \frac{\left(b - c\right) \cdot t}{\sqrt{a + t}} \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{\frac{2.0}{t}}{3.0}\right)\right)}{\frac{t}{\sqrt{t + a}} \cdot \left(\left(a + \frac{5.0}{6.0}\right) + \frac{2.0}{t \cdot 3.0}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\\ \end{array}\]

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 < 8.880244365518877e-162

    1. Initial program 5.6

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

    3. Applied associate-/l*6.9

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

    5. Applied flip--17.8

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

    6. Applied associate-*r/18.1

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

    7. Applied frac-sub22.2

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

    8. Applied simplify3.2

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

    if 8.880244365518877e-162 < t

    1. Initial program 2.3

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

    3. Applied associate-/l*0.7

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

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' 
(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)))))))))))