Average Error: 2.0 → 0.8
Time: 2.0m
Precision: 64
Internal Precision: 576
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{{z}^{y} \cdot {a}^{\left(-1.0\right)}}{\left(\frac{y}{x} \cdot e^{b}\right) \cdot {a}^{\left(-t\right)}} \le 7.664300061046819 \cdot 10^{+220}:\\ \;\;\;\;\frac{{z}^{y} \cdot {a}^{\left(-1.0\right)}}{\left(\frac{y}{x} \cdot e^{b}\right) \cdot {a}^{\left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {\left(e^{\sqrt[3]{(\left(\log z\right) \cdot y + \left((\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_*\right))_*} \cdot \sqrt[3]{(\left(\log z\right) \cdot y + \left((\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_*\right))_*}}\right)}^{\left(\sqrt[3]{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}\right)}}{y}\\ \end{array}\]

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 2 regimes

  2. if (/ (* (pow z y) (pow a (- 1.0))) (* (* (/ y x) (exp b)) (pow a (- t)))) < 7.664300061046819e+220

    1. Initial program 2.9

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

    2. Taylor expanded around inf 2.9

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

    3. Applied simplify1.0

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

    if 7.664300061046819e+220 < (/ (* (pow z y) (pow a (- 1.0))) (* (* (/ y x) (exp b)) (pow a (- t))))

    1. Initial program 0.3

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

    3. Applied add-cube-cbrt0.4

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

    4. Applied exp-prod0.4

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

    5. Applied simplify0.4

      \[\leadsto \frac{x \cdot {\color{blue}{\left(e^{\sqrt[3]{(\left(\log z\right) \cdot y + \left((\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_*\right))_*} \cdot \sqrt[3]{(\left(\log z\right) \cdot y + \left((\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_*\right))_*}}\right)}}^{\left(\sqrt[3]{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}\right)}}{y}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed 2018166 +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))