Average Error: 3.8 → 0.6
Time: 4.6m
Precision: 64
Internal Precision: 384
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}} \le 1.0:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \sqrt[3]{\frac{\left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(z \cdot t\right) \cdot 3.0\right) - \frac{\left(b - c\right) \cdot t}{\frac{\sqrt{t + a}}{a - \frac{5.0}{6.0}}} \cdot \left(\left(t \cdot 3.0\right) \cdot \left(a + \frac{5.0}{6.0}\right) - 2.0\right)\right) \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(z \cdot t\right) \cdot 3.0\right) - \frac{\left(b - c\right) \cdot t}{\frac{\sqrt{t + a}}{a - \frac{5.0}{6.0}}} \cdot \left(\left(t \cdot 3.0\right) \cdot \left(a + \frac{5.0}{6.0}\right) - 2.0\right)\right)}{\frac{{\left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)\right)}^{3} \cdot \frac{\frac{\left(t \cdot t\right) \cdot t}{\sqrt{t + a}}}{\sqrt{t + a} \cdot \sqrt{t + a}}}{\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(z \cdot t\right) \cdot 3.0\right) - \frac{\left(b - c\right) \cdot t}{\frac{\sqrt{t + a}}{a - \frac{5.0}{6.0}}} \cdot \left(\left(t \cdot 3.0\right) \cdot \left(a + \frac{5.0}{6.0}\right) - 2.0\right)}}}}}\\ \end{array}\]

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 (/ x (+ x (* y (exp (* 2.0 (- (/ z (/ t (sqrt (+ t a)))) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))) < 1.0

    1. Initial program 1.2

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

    3. Applied associate-/l*0.1

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

    if 1.0 < (/ x (+ x (* y (exp (* 2.0 (- (/ z (/ t (sqrt (+ t a)))) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))))))

    1. Initial program 53.9

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

    3. Applied associate-/l*61.2

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

    5. Applied flip-+59.2

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

    6. Applied frac-sub59.2

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

    7. Applied associate-*r/59.2

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

    8. Applied frac-sub31.9

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

    9. Applied simplify11.4

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

    11. Applied add-cbrt-cube12.9

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

    12. Applied add-cbrt-cube12.9

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

    13. Applied cbrt-unprod12.9

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

    14. Applied add-cbrt-cube14.8

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

    15. Applied cbrt-undiv14.8

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

    16. Applied simplify10.7

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

Runtime

Time bar (total: 4.6m)Debug logProfile

herbie shell --seed '#(1070609872 3456127585 2380521889 2328837196 1765472538 734540918)' 
(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)))))))))))