Average Error: 4.1 → 1.6
Time: 2.1m
Precision: 64
Internal Precision: 384
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -4.090604796168849 \cdot 10^{-224}:\\ \;\;\;\;\frac{x}{(y \cdot \left(e^{2.0 \cdot (\left(\frac{z}{t}\right) \cdot \left(\sqrt{t + a}\right) + \left(-\left(\left(\frac{5.0}{6.0} + a\right) - \frac{\frac{2.0}{t}}{3.0}\right) \cdot \left(b - c\right)\right))_*}\right) + x)_*}\\ \mathbf{if}\;t \le 1.5573875622572847 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{(y \cdot \left({\left(e^{2.0}\right)}^{\left(\frac{\left(\left(\frac{5.0}{6.0} - a\right) \cdot z\right) \cdot \left(3.0 \cdot \sqrt{t + a}\right) - \left(\left(t \cdot \left(b - c\right)\right) \cdot \left(\frac{5.0}{6.0} - a\right)\right) \cdot (3.0 \cdot \left(\frac{5.0}{6.0}\right) + \left(a \cdot 3.0 - \frac{2.0}{t}\right))_*}{t \cdot \left(\left(\frac{5.0}{6.0} - a\right) \cdot 3.0\right)}\right)}\right) + x)_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{(y \cdot \left(e^{2.0 \cdot (\left(\frac{z}{t}\right) \cdot \left(\sqrt{t + a}\right) + \left(-\left(\left(\frac{5.0}{6.0} + a\right) - \frac{\frac{2.0}{t}}{3.0}\right) \cdot \left(b - c\right)\right))_*}\right) + x)_*}\\ \end{array}\]

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 t < -4.090604796168849e-224

    1. Initial program 4.5

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

    2. Applied simplify3.7

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

    4. Applied fma-neg2.7

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

    6. Applied pow-exp2.7

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

    if -4.090604796168849e-224 < t < 1.5573875622572847e-101

    1. Initial program 7.2

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

    2. Applied simplify8.4

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

    4. Applied flip-+11.8

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

    5. Applied frac-sub17.7

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

    6. Applied associate-*l/17.7

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

    7. Applied associate-*l/16.6

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

    8. Applied frac-sub14.2

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

    9. Applied simplify3.9

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

    if 1.5573875622572847e-101 < t

    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 simplify0.4

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

    4. Applied fma-neg0.2

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

    6. Applied pow-exp0.2

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

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +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)))))))))))