Average Error: 3.6 → 1.3
Time: 2.8m
Precision: 64
Internal Precision: 576
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot e^{2.0 \cdot \frac{\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(3.0 \cdot t\right) \cdot z\right) - \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) \cdot \left(3.0 \cdot t\right) - 2.0\right)\right) \cdot \frac{\left(b - c\right) \cdot t}{\sqrt{a + t}}}{\left(\left(3.0 \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) \cdot \frac{t}{\sqrt{a + t}}}} \le 4.0617163594434 \cdot 10^{-310}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \frac{\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(3.0 \cdot t\right) \cdot z\right) - \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) \cdot \left(3.0 \cdot t\right) - 2.0\right)\right) \cdot \frac{\left(b - c\right) \cdot t}{\sqrt{a + t}}}{\left(\left(3.0 \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) \cdot \frac{t}{\sqrt{a + t}}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{\left(\frac{z}{\frac{t}{\sqrt{a + t}}} - \left(b - c\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{3.0 \cdot t}\right)\right) \cdot 2.0} \cdot y + x}\\ \end{array}\]

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (* y (exp (* 2.0 (/ (- (* (* z (* 3.0 t)) (- a (/ 5.0 6.0))) (* (/ (* (- b c) t) (sqrt (+ t a))) (* (- a (/ 5.0 6.0)) (- (* (* 3.0 t) (+ a (/ 5.0 6.0))) 2.0)))) (* (/ t (sqrt (+ t a))) (* (- a (/ 5.0 6.0)) (* t 3.0))))))) < 4.0617163594434e-310

    1. Initial program 3.1

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

    3. Applied associate-/l*3.8

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

    5. Applied flip-+4.9

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

    6. Applied frac-sub4.9

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

    7. Applied associate-*r/4.9

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

    8. Applied frac-sub4.5

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

    9. Simplified0.4

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

    if 4.0617163594434e-310 < (* y (exp (* 2.0 (/ (- (* (* z (* 3.0 t)) (- a (/ 5.0 6.0))) (* (/ (* (- b c) t) (sqrt (+ t a))) (* (- a (/ 5.0 6.0)) (- (* (* 3.0 t) (+ a (/ 5.0 6.0))) 2.0)))) (* (/ t (sqrt (+ t a))) (* (- a (/ 5.0 6.0)) (* t 3.0)))))))

    1. Initial program 4.3

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

    3. Applied associate-/l*2.4

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot e^{2.0 \cdot \frac{\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(3.0 \cdot t\right) \cdot z\right) - \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) \cdot \left(3.0 \cdot t\right) - 2.0\right)\right) \cdot \frac{\left(b - c\right) \cdot t}{\sqrt{a + t}}}{\left(\left(3.0 \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) \cdot \frac{t}{\sqrt{a + t}}}} \le 4.0617163594434 \cdot 10^{-310}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \frac{\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(3.0 \cdot t\right) \cdot z\right) - \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) \cdot \left(3.0 \cdot t\right) - 2.0\right)\right) \cdot \frac{\left(b - c\right) \cdot t}{\sqrt{a + t}}}{\left(\left(3.0 \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) \cdot \frac{t}{\sqrt{a + t}}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{\left(\frac{z}{\frac{t}{\sqrt{a + t}}} - \left(b - c\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{3.0 \cdot t}\right)\right) \cdot 2.0} \cdot y + x}\\ \end{array}\]

Runtime

Time bar (total: 2.8m)Debug logProfile

herbie shell --seed 2018215 
(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)))))))))))