Average Error: 2.0 → 1.1
Time: 3.2m
Precision: 64
Internal Precision: 384
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;(\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_* \le -446162.13997763954:\\ \;\;\;\;\frac{x \cdot {\left({\left(e^{2 + 1}\right)}^{\left((y \cdot \left(\log z\right) + \left((\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_*\right))_*\right)}\right)}^{\frac{1}{3}}}{y}\\ \mathbf{if}\;(\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_* \le 24344.583430404997:\\ \;\;\;\;\frac{\frac{\frac{{z}^{y}}{{a}^{1.0}}}{\frac{e^{b}}{{a}^{t}}}}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {\left({\left(e^{2 + 1}\right)}^{\left((y \cdot \left(\log z\right) + \left((\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_*\right))_*\right)}\right)}^{\frac{1}{3}}}{y}\\ \end{array}\]

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 (fma (- t 1.0) (log a) (- b)) < -446162.13997763954 or 24344.583430404997 < (fma (- t 1.0) (log a) (- b))

    1. Initial program 0

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

    3. Applied add-cbrt-cube0

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

    4. Applied simplify0

      \[\leadsto \frac{x \cdot \sqrt[3]{\color{blue}{{\left(e^{2 + 1}\right)}^{\left((y \cdot \left(\log z\right) + \left((\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_*\right))_*\right)}}}}{y}\]
    5. Using strategy rm

    6. Applied pow1/30

      \[\leadsto \frac{x \cdot \color{blue}{{\left({\left(e^{2 + 1}\right)}^{\left((y \cdot \left(\log z\right) + \left((\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_*\right))_*\right)}\right)}^{\frac{1}{3}}}}{y}\]

    if -446162.13997763954 < (fma (- t 1.0) (log a) (- b)) < 24344.583430404997

    1. Initial program 6.8

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

    2. Taylor expanded around inf 6.8

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

    3. Applied simplify3.7

      \[\leadsto \color{blue}{\frac{\frac{\frac{{z}^{y}}{{a}^{1.0}}}{\frac{e^{b}}{{a}^{t}}}}{\frac{y}{x}}}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 3.2m)Debug logProfile

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' +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))