Average Error: 3.5 → 1.9
Time: 1.6m
Precision: 64
Internal Precision: 384
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le 6.61690143712692 \cdot 10^{-224}:\\ \;\;\;\;\frac{x}{(y \cdot \left({\left(e^{2.0}\right)}^{\left((\left(\frac{z}{t}\right) \cdot \left(\sqrt{t + a}\right) + \left(-\left(\left(\frac{5.0}{6.0} + a\right) - \frac{\frac{2.0}{t}}{3.0}\right) \cdot \left(b - c\right)\right))_*\right)}\right) + x)_*}\\ \mathbf{if}\;a \le 3.315442387610316 \cdot 10^{+55}:\\ \;\;\;\;\frac{x}{(y \cdot \left({\left(e^{2.0}\right)}^{\left(\frac{\left(\left(3.0 \cdot z\right) \cdot \sqrt{t + a}\right) \cdot (\left(a - \frac{5.0}{6.0}\right) \cdot a + \left(\frac{5.0}{6.0} \cdot \frac{5.0}{6.0}\right))_* - (3.0 \cdot \left((\left(\frac{5.0}{6.0}\right) \cdot \left(\frac{5.0}{6.0} \cdot \frac{5.0}{6.0}\right) + \left({a}^{3}\right))_*\right) + \left(\frac{(\left(a - \frac{5.0}{6.0}\right) \cdot a + \left(\frac{5.0}{6.0} \cdot \frac{5.0}{6.0}\right))_*}{\frac{t}{-2.0}}\right))_* \cdot \left(t \cdot \left(b - c\right)\right)}{\left(t \cdot 3.0\right) \cdot (\left(\frac{5.0}{6.0}\right) \cdot \left(\frac{5.0}{6.0} - a\right) + \left(a \cdot a\right))_*}\right)}\right) + x)_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{(y \cdot \left({\left(e^{2.0}\right)}^{\left((\left(\frac{z}{t}\right) \cdot \left(\sqrt{t + a}\right) + \left(-\left(\left(\frac{5.0}{6.0} + a\right) - \frac{\frac{2.0}{t}}{3.0}\right) \cdot \left(b - c\right)\right))_*\right)}\right) + x)_*}\\ \end{array}\]

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 2 regimes

  2. if a < 6.61690143712692e-224 or 3.315442387610316e+55 < a

    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 simplify2.6

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

    4. Applied fma-neg1.6

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

    if 6.61690143712692e-224 < a < 3.315442387610316e+55

    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. Applied simplify3.0

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

    4. Applied flip3-+3.0

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

    5. Applied frac-sub3.2

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

    6. Applied associate-*l/3.4

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

    7. Applied associate-*l/3.3

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

    8. Applied frac-sub2.9

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

    9. Applied simplify2.4

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

    10. Applied simplify2.5

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

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +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)))))))))))