Average Error: 3.7 → 8.0
Time: 2.7m
Precision: 64
Internal Precision: 384
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le 5.470743572104813 \cdot 10^{+24}:\\ \;\;\;\;\frac{x}{(y \cdot \left({\left(e^{2.0}\right)}^{\left(\frac{\left(\left(3.0 \cdot z\right) \cdot \sqrt{t + a}\right) \cdot (\left(a - \frac{5.0}{6.0}\right) \cdot a + \left(\frac{5.0}{6.0} \cdot \frac{5.0}{6.0}\right))_* - (3.0 \cdot \left((\left(\frac{5.0}{6.0}\right) \cdot \left(\frac{5.0}{6.0} \cdot \frac{5.0}{6.0}\right) + \left({a}^{3}\right))_*\right) + \left(\frac{(\left(a - \frac{5.0}{6.0}\right) \cdot a + \left(\frac{5.0}{6.0} \cdot \frac{5.0}{6.0}\right))_*}{\frac{t}{-2.0}}\right))_* \cdot \left(t \cdot \left(b - c\right)\right)}{\left(t \cdot 3.0\right) \cdot (\left(\frac{5.0}{6.0}\right) \cdot \left(\frac{5.0}{6.0} - a\right) + \left(a \cdot a\right))_*}\right)}\right) + x)_*}\\ \mathbf{if}\;a \le 1.6676422061464537 \cdot 10^{+107}:\\ \;\;\;\;\frac{1}{\frac{(y \cdot \left({\left(e^{2.0}\right)}^{\left((\left(\sqrt{t + a}\right) \cdot \left(\frac{z}{t}\right) + \left(\frac{0.6666666666666666}{t} \cdot \left(b - c\right)\right))_* - 0.8333333333333334 \cdot b\right)}\right) + x)_*}{x}}\\ \mathbf{if}\;a \le 4.860120670957077 \cdot 10^{+136}:\\ \;\;\;\;\frac{x}{(y \cdot \left({\left(e^{2.0}\right)}^{\left((a \cdot \left(c - b\right) + \left(0.8333333333333334 \cdot c\right))_*\right)}\right) + x)_*}\\ \mathbf{if}\;a \le 3.849783140460225 \cdot 10^{+168}:\\ \;\;\;\;\frac{1}{\frac{(y \cdot \left({\left(e^{2.0}\right)}^{\left((\left(\sqrt{t + a}\right) \cdot \left(\frac{z}{t}\right) + \left(\frac{0.6666666666666666}{t} \cdot \left(b - c\right)\right))_* - 0.8333333333333334 \cdot b\right)}\right) + x)_*}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{(y \cdot \left({\left(e^{2.0}\right)}^{\left((a \cdot \left(c - b\right) + \left(0.8333333333333334 \cdot c\right))_*\right)}\right) + x)_*}\\ \end{array}\]

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 3 regimes

  2. if a < 5.470743572104813e+24

    1. Initial program 2.4

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

    2. Applied simplify2.1

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

    4. Applied flip3-+3.8

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

    5. Applied frac-sub3.8

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

    6. Applied associate-*l/3.8

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

    7. Applied associate-*l/4.0

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

    8. Applied frac-sub5.8

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

    9. Applied simplify5.7

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

    10. Applied simplify5.7

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

    if 5.470743572104813e+24 < a < 1.6676422061464537e+107 or 4.860120670957077e+136 < a < 3.849783140460225e+168

    1. Initial program 3.8

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

    2. Applied simplify3.3

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

    3. Taylor expanded around 0 14.8

      \[\leadsto \frac{x}{(y \cdot \left({\left(e^{2.0}\right)}^{\left(\frac{z}{t} \cdot \sqrt{t + a} - \color{blue}{\left(\left(0.8333333333333334 \cdot b + 0.6666666666666666 \cdot \frac{c}{t}\right) - 0.6666666666666666 \cdot \frac{b}{t}\right)}\right)}\right) + x)_*}\]

    4. Applied simplify13.0

      \[\leadsto \color{blue}{\frac{x}{(y \cdot \left({\left(e^{2.0}\right)}^{\left((\left(\sqrt{t + a}\right) \cdot \left(\frac{z}{t}\right) + \left(\frac{0.6666666666666666}{t} \cdot \left(b - c\right)\right))_* - 0.8333333333333334 \cdot b\right)}\right) + x)_*}}\]
    5. Using strategy rm

    6. Applied clear-num13.0

      \[\leadsto \color{blue}{\frac{1}{\frac{(y \cdot \left({\left(e^{2.0}\right)}^{\left((\left(\sqrt{t + a}\right) \cdot \left(\frac{z}{t}\right) + \left(\frac{0.6666666666666666}{t} \cdot \left(b - c\right)\right))_* - 0.8333333333333334 \cdot b\right)}\right) + x)_*}{x}}}\]

    if 1.6676422061464537e+107 < a < 4.860120670957077e+136 or 3.849783140460225e+168 < a

    1. Initial program 6.6

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

    2. Applied simplify5.1

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

    3. Taylor expanded around inf 14.5

      \[\leadsto \frac{x}{(y \cdot \left({\left(e^{2.0}\right)}^{\color{blue}{\left(\left(c \cdot a + 0.8333333333333334 \cdot c\right) - b \cdot a\right)}}\right) + x)_*}\]

    4. Applied simplify10.2

      \[\leadsto \color{blue}{\frac{x}{(y \cdot \left({\left(e^{2.0}\right)}^{\left((a \cdot \left(c - b\right) + \left(0.8333333333333334 \cdot c\right))_*\right)}\right) + x)_*}}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' +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)))))))))))