Average Error: 4.0 → 1.5
Time: 5.8m
Precision: 64
Internal Precision: 576
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot e^{2.0 \cdot \sqrt[3]{{\left(\frac{\left(\left(3.0 \cdot t\right) \cdot z\right) \cdot \left(a - \frac{5.0}{6.0}\right) - \frac{\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(b - c\right) \cdot t\right)}{\sqrt{t + a}} \cdot \left(\left(a + \frac{5.0}{6.0}\right) \cdot \left(3.0 \cdot t\right) - 2.0\right)}{\frac{t}{\sqrt{t + a}} \cdot \left(\left(3.0 \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right)}\right)}^{3}}} \le -0.0:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \sqrt[3]{{\left(\frac{\left(\left(3.0 \cdot t\right) \cdot z\right) \cdot \left(a - \frac{5.0}{6.0}\right) - \frac{\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(b - c\right) \cdot t\right)}{\sqrt{t + a}} \cdot \left(\left(a + \frac{5.0}{6.0}\right) \cdot \left(3.0 \cdot t\right) - 2.0\right)}{\frac{t}{\sqrt{t + a}} \cdot \left(\left(3.0 \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right)}\right)}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot e^{\left(\frac{\sqrt{t + a}}{t} \cdot \left(\left(3.0 \cdot t\right) \cdot \frac{z}{3.0 \cdot t} - \frac{\frac{\left(b - c\right) \cdot t}{\sqrt{t + a}}}{3.0 \cdot t} \cdot \left(\left(a + \frac{5.0}{6.0}\right) \cdot \left(3.0 \cdot t\right) - 2.0\right)\right)\right) \cdot 2.0} + x}\\ \end{array}\]

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 (* y (exp (* 2.0 (cbrt (pow (/ (- (* (- a (/ 5.0 6.0)) (* (* t 3.0) z)) (* (- (* (+ a (/ 5.0 6.0)) (* t 3.0)) 2.0) (/ (* (* (- b c) t) (- a (/ 5.0 6.0))) (sqrt (+ t a))))) (* (/ t (sqrt (+ t a))) (* (- a (/ 5.0 6.0)) (* t 3.0)))) 3))))) < -0.0

    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 associate-/l*3.7

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

    5. Applied flip-+5.3

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

    6. Applied frac-sub5.4

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

    7. Applied associate-*r/5.4

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

    8. Applied frac-sub5.5

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

    9. Applied simplify1.6

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

    11. Applied add-cbrt-cube14.0

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

    12. Applied add-cbrt-cube20.2

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

    13. Applied cbrt-undiv20.2

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

    14. Applied simplify0.1

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

    if -0.0 < (* y (exp (* 2.0 (cbrt (pow (/ (- (* (- a (/ 5.0 6.0)) (* (* t 3.0) z)) (* (- (* (+ a (/ 5.0 6.0)) (* t 3.0)) 2.0) (/ (* (* (- b c) t) (- a (/ 5.0 6.0))) (sqrt (+ t a))))) (* (/ t (sqrt (+ t a))) (* (- a (/ 5.0 6.0)) (* t 3.0)))) 3)))))

    1. Initial program 5.2

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

    3. Applied associate-/l*2.4

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

    5. Applied flip-+3.6

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

    6. Applied frac-sub20.7

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

    7. Applied associate-*r/21.2

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

    8. Applied frac-sub37.9

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

    9. Applied simplify37.0

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

    11. Applied *-un-lft-identity37.0

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

    12. Applied times-frac26.3

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

    13. Applied simplify26.3

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

    14. Applied simplify3.1

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{\sqrt{t + a}}{t} \cdot \color{blue}{\left(\frac{t \cdot 3.0}{1} \cdot \frac{z}{t \cdot 3.0} - \frac{\left(t \cdot 3.0\right) \cdot \left(a + \frac{5.0}{6.0}\right) - 2.0}{1} \cdot \frac{\frac{\left(b - c\right) \cdot t}{\sqrt{t + a}}}{t \cdot 3.0}\right)}\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Applied simplify1.5

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;y \cdot e^{2.0 \cdot \sqrt[3]{{\left(\frac{\left(\left(3.0 \cdot t\right) \cdot z\right) \cdot \left(a - \frac{5.0}{6.0}\right) - \frac{\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(b - c\right) \cdot t\right)}{\sqrt{t + a}} \cdot \left(\left(a + \frac{5.0}{6.0}\right) \cdot \left(3.0 \cdot t\right) - 2.0\right)}{\frac{t}{\sqrt{t + a}} \cdot \left(\left(3.0 \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right)}\right)}^{3}}} \le -0.0:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \sqrt[3]{{\left(\frac{\left(\left(3.0 \cdot t\right) \cdot z\right) \cdot \left(a - \frac{5.0}{6.0}\right) - \frac{\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(b - c\right) \cdot t\right)}{\sqrt{t + a}} \cdot \left(\left(a + \frac{5.0}{6.0}\right) \cdot \left(3.0 \cdot t\right) - 2.0\right)}{\frac{t}{\sqrt{t + a}} \cdot \left(\left(3.0 \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right)}\right)}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot e^{\left(\frac{\sqrt{t + a}}{t} \cdot \left(\left(3.0 \cdot t\right) \cdot \frac{z}{3.0 \cdot t} - \frac{\frac{\left(b - c\right) \cdot t}{\sqrt{t + a}}}{3.0 \cdot t} \cdot \left(\left(a + \frac{5.0}{6.0}\right) \cdot \left(3.0 \cdot t\right) - 2.0\right)\right)\right) \cdot 2.0} + x}\\ \end{array}}\]

Runtime

Time bar (total: 5.8m)Debug logProfile

herbie shell --seed '#(1071948828 1180510430 2986424009 997076509 406109801 420189285)' 
(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)))))))))))