Average Error: 28.2 → 2.5
Time: 4.5m
Precision: 64
Internal Precision: 1408
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1486774482200.535:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1.0\right)}}{\left(b + 1\right) - y \cdot \log z}}{y}\\ \mathbf{if}\;t \le 1.324949139524979 \cdot 10^{-102}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{{a}^{\left(t - 1.0\right)}}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1.0\right)}}{\left(b + 1\right) - y \cdot \log z}}{y}\\ \end{array}\]

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 t < -1486774482200.535 or 1.324949139524979e-102 < t

    1. Initial program 38.5

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

    2. Applied simplify34.8

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

    4. Applied associate-*l/29.6

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

    5. Taylor expanded around 0 1.0

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

    if -1486774482200.535 < t < 1.324949139524979e-102

    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. Applied simplify12.5

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

    4. Applied pow-to-exp12.6

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

    5. Applied div-exp6.2

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

Runtime

Time bar (total: 4.5m)Debug log

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