Average Error: 3.8 → 1.8
Time: 41.3s
Precision: 64
Internal Precision: 128
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;b - c \le -4.976568559480139 \cdot 10^{+222}:\\ \;\;\;\;\frac{x}{x + e^{(\left(\sqrt{a + t} \cdot z\right) \cdot \left(\frac{1}{t}\right) + \left(\left(-\left(b - c\right)\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{3.0 \cdot t}\right)\right))_* \cdot 2.0} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{(\left(\left(\frac{5.0}{6.0} + a\right) - \frac{\frac{2.0}{3.0}}{t}\right) \cdot \left(c - b\right) + \left(\frac{z}{t} \cdot \sqrt{a + t}\right))_* \cdot 2.0}}\\ \end{array}\]

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 (- b c) < -4.976568559480139e+222

    1. Initial program 7.2

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

    3. Applied div-inv7.2

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

    4. Applied fma-neg4.1

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

    if -4.976568559480139e+222 < (- b c)

    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. Using strategy rm

    3. Applied div-inv3.3

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

    4. Applied prod-diff18.6

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

    5. Simplified16.9

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

    6. Simplified1.5

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left((\left(\left(\frac{5.0}{6.0} + a\right) - \frac{\frac{2.0}{3.0}}{t}\right) \cdot \left(c - b\right) + \left(\sqrt{t + a} \cdot \frac{z}{t}\right))_* + \color{blue}{0}\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b - c \le -4.976568559480139 \cdot 10^{+222}:\\ \;\;\;\;\frac{x}{x + e^{(\left(\sqrt{a + t} \cdot z\right) \cdot \left(\frac{1}{t}\right) + \left(\left(-\left(b - c\right)\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{3.0 \cdot t}\right)\right))_* \cdot 2.0} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{(\left(\left(\frac{5.0}{6.0} + a\right) - \frac{\frac{2.0}{3.0}}{t}\right) \cdot \left(c - b\right) + \left(\frac{z}{t} \cdot \sqrt{a + t}\right))_* \cdot 2.0}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019030 +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)))))))))))