Average Error: 3.8 → 1.7
Time: 3.0m
Precision: 64
Internal Precision: 320
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -8.340062577609172 \cdot 10^{+218}:\\ \;\;\;\;\frac{x}{e^{2.0 \cdot \left((\left(\frac{\sqrt{t + a}}{t}\right) \cdot z + \left(\left(c - b\right) \cdot \left(\frac{5.0}{6.0} + a\right)\right))_* - \frac{2.0}{t \cdot 3.0} \cdot \left(-\left(b - c\right)\right)\right)} \cdot y + x}\\ \mathbf{if}\;z \le 7.67082581723443 \cdot 10^{+78}:\\ \;\;\;\;\frac{x}{x + e^{2.0 \cdot (\left(\sqrt{t + a} \cdot z\right) \cdot \left(\frac{1}{t}\right) + \left(\left(c - b\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{\frac{2.0}{t}}{3.0}\right)\right))_*} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{2.0 \cdot (\left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{t \cdot 3.0}\right) \cdot \left(c - b\right) + \left(\frac{z}{t} \cdot \sqrt{t + a}\right))_*} \cdot y + x}\\ \end{array}\]

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 z < -8.340062577609172e+218

    1. Initial program 12.8

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

    3. Applied sub-neg12.8

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

    4. Applied distribute-lft-in13.2

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

    5. Applied associate--r+13.2

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

    6. Applied simplify6.9

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

    if -8.340062577609172e+218 < z < 7.67082581723443e+78

    1. Initial program 1.6

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

    3. Applied div-inv1.6

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

    4. Applied fma-neg0.7

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

    5. Applied simplify0.7

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

    if 7.67082581723443e+78 < z

    1. Initial program 8.9

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

    3. Applied add-cube-cbrt8.9

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

    4. Applied prod-diff22.8

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

    5. Applied simplify18.2

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

    6. Applied simplify3.8

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left((\left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{3.0 \cdot t}\right) \cdot \left(c - b\right) + \left(\sqrt{t + a} \cdot \frac{z}{t}\right))_* + \color{blue}{0}\right)}}\]
  3. Recombined 3 regimes into one program.

  4. Applied simplify1.7

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;z \le -8.340062577609172 \cdot 10^{+218}:\\ \;\;\;\;\frac{x}{e^{2.0 \cdot \left((\left(\frac{\sqrt{t + a}}{t}\right) \cdot z + \left(\left(c - b\right) \cdot \left(\frac{5.0}{6.0} + a\right)\right))_* - \frac{2.0}{t \cdot 3.0} \cdot \left(-\left(b - c\right)\right)\right)} \cdot y + x}\\ \mathbf{if}\;z \le 7.67082581723443 \cdot 10^{+78}:\\ \;\;\;\;\frac{x}{x + e^{2.0 \cdot (\left(\sqrt{t + a} \cdot z\right) \cdot \left(\frac{1}{t}\right) + \left(\left(c - b\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{\frac{2.0}{t}}{3.0}\right)\right))_*} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{2.0 \cdot (\left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{t \cdot 3.0}\right) \cdot \left(c - b\right) + \left(\frac{z}{t} \cdot \sqrt{t + a}\right))_*} \cdot y + x}\\ \end{array}}\]

Runtime

Time bar (total: 3.0m)Debug logProfile

herbie shell --seed '#(1072936661 1621281212 3440817831 3219514234 460296804 1258167384)' +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)))))))))))