Average Error: 9.7 → 1.9
Time: 6.6m
Precision: 64
Internal Precision: 1408
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -3.5047403910759882 \cdot 10^{-133}:\\ \;\;\;\;\frac{x}{y} \cdot e^{\log a \cdot \left(t - 1.0\right) - \left(b - \log z \cdot y\right)}\\ \mathbf{if}\;y \le 0.05204455380571942:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1.0\right)}}{\frac{e^{b}}{{z}^{y}}}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot e^{\log a \cdot \left(t - 1.0\right) - \left(b - \log z \cdot y\right)}\\ \end{array}\]

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 2 regimes

  2. if y < -3.5047403910759882e-133 or 0.05204455380571942 < y

    1. Initial program 0.6

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

    2. Applied simplify24.4

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

    4. Applied pow-to-exp24.4

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

    5. Applied div-exp14.9

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

    6. Applied pow-to-exp15.3

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

    7. Applied div-exp1.3

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

    if -3.5047403910759882e-133 < y < 0.05204455380571942

    1. Initial program 20.2

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

    2. Applied simplify15.9

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

    4. Applied associate-*l/2.6

      \[\leadsto \color{blue}{\frac{x \cdot \frac{{a}^{\left(t - 1.0\right)}}{\frac{e^{b}}{{z}^{y}}}}{y}}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 6.6m)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit +o reduce:binary-search
(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))