Average Error: 1.9 → 1.5
Time: 1.6m
Precision: 64
Internal Precision: 576
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -5161390.384021355:\\ \;\;\;\;\frac{x}{\frac{y}{\sqrt[3]{e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b} \cdot \left(e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b} \cdot e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b}\right)}}}\\ \mathbf{elif}\;y \le 1.7681062047212044 \cdot 10^{-280}:\\ \;\;\;\;\frac{x}{\frac{y}{\left(\sqrt{{a}^{\left(-1.0\right)} \cdot {z}^{y}} \cdot \sqrt{{a}^{\left(-1.0\right)} \cdot {z}^{y}}\right) \cdot e^{(t \cdot \left(\log a\right) + \left(-b\right))_*}}}\\ \mathbf{elif}\;y \le 6.486193561690206 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{{z}^{y} \cdot e^{(t \cdot \left(\log a\right) + \left(-b\right))_*}}}}{{a}^{1.0}}\\ \mathbf{elif}\;y \le 2.4383762230412337 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{\frac{y}{\left(\sqrt{{a}^{\left(-1.0\right)} \cdot {z}^{y}} \cdot \sqrt{{a}^{\left(-1.0\right)} \cdot {z}^{y}}\right) \cdot e^{(t \cdot \left(\log a\right) + \left(-b\right))_*}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{\sqrt[3]{e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b} \cdot \left(e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b} \cdot e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b}\right)}}}\\ \end{array}\]

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

Derivation

  1. Split input into 3 regimes

  2. if y < -5161390.384021355 or 2.4383762230412337e-32 < y

    1. Initial program 0.1

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
    2. Using strategy rm

    3. Applied associate-/l*0.1

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}}\]
    4. Using strategy rm

    5. Applied add-cbrt-cube0.7

      \[\leadsto \frac{x}{\frac{y}{\color{blue}{\sqrt[3]{\left(e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b} \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}\right) \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}}}\]

    if -5161390.384021355 < y < 1.7681062047212044e-280 or 6.486193561690206e-50 < y < 2.4383762230412337e-32

    1. Initial program 3.7

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
    2. Using strategy rm

    3. Applied associate-/l*3.1

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}}\]
    4. Using strategy rm

    5. Applied add-cbrt-cube11.8

      \[\leadsto \frac{x}{\frac{y}{\color{blue}{\sqrt[3]{\left(e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b} \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}\right) \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}}}\]

    6. Taylor expanded around inf 3.1

      \[\leadsto \frac{x}{\frac{y}{\color{blue}{e^{1.0 \cdot \log \left(\frac{1}{a}\right) - \left(\log \left(\frac{1}{z}\right) \cdot y + \left(b + t \cdot \log \left(\frac{1}{a}\right)\right)\right)}}}}\]

    7. Simplified2.1

      \[\leadsto \frac{x}{\frac{y}{\color{blue}{\left({a}^{\left(-1.0\right)} \cdot {z}^{y}\right) \cdot e^{(t \cdot \left(\log a\right) + \left(-b\right))_*}}}}\]
    8. Using strategy rm

    9. Applied add-sqr-sqrt2.2

      \[\leadsto \frac{x}{\frac{y}{\color{blue}{\left(\sqrt{{a}^{\left(-1.0\right)} \cdot {z}^{y}} \cdot \sqrt{{a}^{\left(-1.0\right)} \cdot {z}^{y}}\right)} \cdot e^{(t \cdot \left(\log a\right) + \left(-b\right))_*}}}\]

    if 1.7681062047212044e-280 < y < 6.486193561690206e-50

    1. Initial program 3.9

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
    2. Using strategy rm

    3. Applied associate-/l*4.1

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}}\]
    4. Using strategy rm

    5. Applied add-cbrt-cube12.9

      \[\leadsto \frac{x}{\frac{y}{\color{blue}{\sqrt[3]{\left(e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b} \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}\right) \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}}}\]

    6. Taylor expanded around inf 4.1

      \[\leadsto \frac{x}{\frac{y}{\color{blue}{e^{1.0 \cdot \log \left(\frac{1}{a}\right) - \left(\log \left(\frac{1}{z}\right) \cdot y + \left(b + t \cdot \log \left(\frac{1}{a}\right)\right)\right)}}}}\]

    7. Simplified2.7

      \[\leadsto \frac{x}{\frac{y}{\color{blue}{\left({a}^{\left(-1.0\right)} \cdot {z}^{y}\right) \cdot e^{(t \cdot \left(\log a\right) + \left(-b\right))_*}}}}\]
    8. Using strategy rm

    9. Applied pow-neg2.7

      \[\leadsto \frac{x}{\frac{y}{\left(\color{blue}{\frac{1}{{a}^{1.0}}} \cdot {z}^{y}\right) \cdot e^{(t \cdot \left(\log a\right) + \left(-b\right))_*}}}\]

    10. Applied associate-*l/2.7

      \[\leadsto \frac{x}{\frac{y}{\color{blue}{\frac{1 \cdot {z}^{y}}{{a}^{1.0}}} \cdot e^{(t \cdot \left(\log a\right) + \left(-b\right))_*}}}\]

    11. Applied associate-*l/2.7

      \[\leadsto \frac{x}{\frac{y}{\color{blue}{\frac{\left(1 \cdot {z}^{y}\right) \cdot e^{(t \cdot \left(\log a\right) + \left(-b\right))_*}}{{a}^{1.0}}}}}\]

    12. Applied associate-/r/2.7

      \[\leadsto \frac{x}{\color{blue}{\frac{y}{\left(1 \cdot {z}^{y}\right) \cdot e^{(t \cdot \left(\log a\right) + \left(-b\right))_*}} \cdot {a}^{1.0}}}\]

    13. Applied associate-/r*2.6

      \[\leadsto \color{blue}{\frac{\frac{x}{\frac{y}{\left(1 \cdot {z}^{y}\right) \cdot e^{(t \cdot \left(\log a\right) + \left(-b\right))_*}}}}{{a}^{1.0}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -5161390.384021355:\\ \;\;\;\;\frac{x}{\frac{y}{\sqrt[3]{e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b} \cdot \left(e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b} \cdot e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b}\right)}}}\\ \mathbf{elif}\;y \le 1.7681062047212044 \cdot 10^{-280}:\\ \;\;\;\;\frac{x}{\frac{y}{\left(\sqrt{{a}^{\left(-1.0\right)} \cdot {z}^{y}} \cdot \sqrt{{a}^{\left(-1.0\right)} \cdot {z}^{y}}\right) \cdot e^{(t \cdot \left(\log a\right) + \left(-b\right))_*}}}\\ \mathbf{elif}\;y \le 6.486193561690206 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{{z}^{y} \cdot e^{(t \cdot \left(\log a\right) + \left(-b\right))_*}}}}{{a}^{1.0}}\\ \mathbf{elif}\;y \le 2.4383762230412337 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{\frac{y}{\left(\sqrt{{a}^{\left(-1.0\right)} \cdot {z}^{y}} \cdot \sqrt{{a}^{\left(-1.0\right)} \cdot {z}^{y}}\right) \cdot e^{(t \cdot \left(\log a\right) + \left(-b\right))_*}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{\sqrt[3]{e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b} \cdot \left(e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b} \cdot e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b}\right)}}}\\ \end{array}\]

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018255 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2"
  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))