Average Error: 3.7 → 1.8
Time: 3.0m
Precision: 64
Internal Precision: 384
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.4525767485989455 \cdot 10^{+63}:\\ \;\;\;\;\frac{x}{(y \cdot \left({\left(e^{2.0}\right)}^{\left((0.8333333333333334 \cdot c + \left((\left(\sqrt{t + a}\right) \cdot \left(\frac{z}{t}\right) + \left(\left(-a\right) \cdot \left(b - c\right)\right))_*\right))_*\right)}\right) + x)_*}\\ \mathbf{if}\;t \le 9.909816380988951 \cdot 10^{-135}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \frac{\left(\frac{2.0}{3.0 \cdot t} + \left(\frac{5.0}{6.0} + a\right)\right) \cdot \left(z \cdot \sqrt{t + a} - \left(\left(b - c\right) \cdot t\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{3.0 \cdot t}\right)\right)}{(t \cdot \left(a + \frac{5.0}{6.0}\right) + \left(\frac{2.0}{3.0}\right))_*}}}\\ \mathbf{if}\;t \le +\infty:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot (\left(z \cdot \sqrt{t + a}\right) \cdot \left(\frac{1}{t}\right) + \left(-\left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot (\left(z \cdot \sqrt{t + a}\right) \cdot \left(\frac{1}{t}\right) + \left(-\left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right))_*}}\\ \end{array}\]

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 < -1.4525767485989455e+63

    1. Initial program 3.3

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

    2. Applied simplify0.4

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

    3. Taylor expanded around inf 6.6

      \[\leadsto \frac{x}{(y \cdot \left({\left(e^{2.0}\right)}^{\left(\frac{z}{t} \cdot \sqrt{t + a} - \color{blue}{\left(b \cdot a - \left(c \cdot a + 0.8333333333333334 \cdot c\right)\right)}\right)}\right) + x)_*}\]

    4. Applied simplify0.2

      \[\leadsto \color{blue}{\frac{x}{(y \cdot \left({\left(e^{2.0}\right)}^{\left((0.8333333333333334 \cdot c + \left((\left(\sqrt{t + a}\right) \cdot \left(\frac{z}{t}\right) + \left(\left(-a\right) \cdot \left(b - c\right)\right))_*\right))_*\right)}\right) + x)_*}}\]

    if -1.4525767485989455e+63 < t < 9.909816380988951e-135

    1. Initial program 5.9

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

    3. Applied flip--18.0

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

    4. Applied associate-*r/18.5

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

    5. Applied frac-sub19.8

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

    6. Applied simplify1.5

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

    7. Applied simplify1.5

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

    if 9.909816380988951e-135 < 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 div-inv2.3

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\color{blue}{\left(z \cdot \sqrt{t + a}\right) \cdot \frac{1}{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]

    4. Applied fma-neg2.1

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \color{blue}{(\left(z \cdot \sqrt{t + a}\right) \cdot \left(\frac{1}{t}\right) + \left(-\left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right))_*}}}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 3.0m)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' +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)))))))))))