Average Error: 3.9 → 0.9
Time: 3.7m
Precision: 64
Internal Precision: 384
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.9669780720966174 \cdot 10^{+68}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\\ \mathbf{if}\;t \le 3.1727279318285767 \cdot 10^{-45}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \frac{\left(\frac{2.0}{3.0 \cdot t} + \left(\frac{5.0}{6.0} + a\right)\right) \cdot \left(z \cdot \sqrt{t + a} - \left(\left(b - c\right) \cdot t\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{3.0 \cdot t}\right)\right)}{t \cdot \left(\left(a + \frac{5.0}{6.0}\right) + \frac{2.0}{t \cdot 3.0}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\\ \end{array}\]

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 < -2.9669780720966174e+68 or 3.1727279318285767e-45 < t

    1. Initial program 2.6

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

    3. Applied associate-/l*0.3

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

    if -2.9669780720966174e+68 < t < 3.1727279318285767e-45

    1. Initial program 5.3

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

    3. Applied flip--15.6

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

    4. Applied associate-*r/16.1

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

    5. Applied frac-sub17.5

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

    6. Applied simplify1.4

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

Runtime

Time bar (total: 3.7m)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' 
(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)))))))))))