Average Error: 3.6 → 2.4
Time: 2.1m
Precision: 64
Internal Precision: 128
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.7707342386183387 \cdot 10^{-279}:\\ \;\;\;\;\frac{x}{y \cdot e^{(z \cdot \left(\frac{\sqrt{t + a}}{t}\right) + \left(\left(-\left(b - c\right)\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right))_* \cdot 2.0} + x}\\ \mathbf{elif}\;t \le 3.450515419609681 \cdot 10^{-215}:\\ \;\;\;\;\frac{x}{y \cdot e^{\frac{\left(\left(t \cdot 3.0\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) \cdot \left(\sqrt{t + a} \cdot z\right) - t \cdot \left(\left(\left(a \cdot a - \frac{5.0}{6.0} \cdot \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right) - \left(a - \frac{5.0}{6.0}\right) \cdot 2.0\right) \cdot \left(b - c\right)\right)}{t \cdot \left(\left(t \cdot 3.0\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right)} \cdot 2.0} + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot e^{(z \cdot \left(\frac{\sqrt{t + a}}{t}\right) + \left(\left(-\left(b - c\right)\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right))_* \cdot 2.0} + x}\\ \end{array}\]

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 < -1.7707342386183387e-279 or 3.450515419609681e-215 < t

    1. Initial program 2.9

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

    3. Applied *-un-lft-identity2.9

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

    4. Applied times-frac2.0

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

    5. Applied fma-neg1.3

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

    6. Simplified1.3

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

    if -1.7707342386183387e-279 < t < 3.450515419609681e-215

    1. Initial program 9.3

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

    3. Applied flip-+14.1

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

    4. Applied frac-sub14.1

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

    5. Applied associate-*r/14.1

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

    6. Applied frac-sub11.2

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

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.7707342386183387 \cdot 10^{-279}:\\ \;\;\;\;\frac{x}{y \cdot e^{(z \cdot \left(\frac{\sqrt{t + a}}{t}\right) + \left(\left(-\left(b - c\right)\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right))_* \cdot 2.0} + x}\\ \mathbf{elif}\;t \le 3.450515419609681 \cdot 10^{-215}:\\ \;\;\;\;\frac{x}{y \cdot e^{\frac{\left(\left(t \cdot 3.0\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) \cdot \left(\sqrt{t + a} \cdot z\right) - t \cdot \left(\left(\left(a \cdot a - \frac{5.0}{6.0} \cdot \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right) - \left(a - \frac{5.0}{6.0}\right) \cdot 2.0\right) \cdot \left(b - c\right)\right)}{t \cdot \left(\left(t \cdot 3.0\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right)} \cdot 2.0} + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot e^{(z \cdot \left(\frac{\sqrt{t + a}}{t}\right) + \left(\left(-\left(b - c\right)\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right))_* \cdot 2.0} + x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019090 +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)))))))))))