Average Error: 3.7 → 1.3
Time: 8.8m
Precision: 64
Internal Precision: 128
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.8021617235846273 \cdot 10^{-107}:\\ \;\;\;\;\frac{x}{e^{\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}} - \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{t \cdot 3.0}\right) \cdot \left(b - c\right)\right) \cdot 2.0} \cdot y + x}\\ \mathbf{elif}\;t \le 4.524535961405479 \cdot 10^{-173}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{\frac{\left(\frac{z}{\sqrt[3]{t}} \cdot \sqrt{t + a}\right) \cdot \left(\left(t \cdot 3.0\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) - \left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(b - c\right)\right) \cdot \left(\left(\left(t \cdot 3.0\right) \cdot \left(\frac{5.0}{6.0} + a\right) - 2.0\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right)}{\left(\left(t \cdot 3.0\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)} \cdot 2.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}} - \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{t \cdot 3.0}\right) \cdot \left(b - c\right)\right) \cdot 2.0} \cdot y + x}\\ \end{array}\]

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 < -1.8021617235846273e-107 or 4.524535961405479e-173 < t

    1. Initial program 2.5

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

    3. Applied add-cube-cbrt2.5

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

    4. Applied times-frac0.9

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

    if -1.8021617235846273e-107 < t < 4.524535961405479e-173

    1. Initial program 7.1

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

    3. Applied add-cube-cbrt7.1

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

    4. Applied times-frac7.2

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\color{blue}{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
    5. Using strategy rm

    6. Applied flip-+11.1

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

    7. Applied frac-sub11.1

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

    8. Applied associate-*r/11.1

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

    9. Applied associate-*l/10.9

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

    10. Applied frac-sub6.8

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

    11. Simplified2.4

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.8021617235846273 \cdot 10^{-107}:\\ \;\;\;\;\frac{x}{e^{\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}} - \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{t \cdot 3.0}\right) \cdot \left(b - c\right)\right) \cdot 2.0} \cdot y + x}\\ \mathbf{elif}\;t \le 4.524535961405479 \cdot 10^{-173}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{\frac{\left(\frac{z}{\sqrt[3]{t}} \cdot \sqrt{t + a}\right) \cdot \left(\left(t \cdot 3.0\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) - \left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(b - c\right)\right) \cdot \left(\left(\left(t \cdot 3.0\right) \cdot \left(\frac{5.0}{6.0} + a\right) - 2.0\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right)}{\left(\left(t \cdot 3.0\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)} \cdot 2.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}} - \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{t \cdot 3.0}\right) \cdot \left(b - c\right)\right) \cdot 2.0} \cdot y + x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019091 
(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)))))))))))