Average Error: 1.9 → 1.4
Time: 1.4m
Precision: 64
Internal Precision: 576
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.31677741273226 \cdot 10^{-174} \lor \neg \left(y \le 0.11857505804325559\right):\\ \;\;\;\;\frac{x \cdot e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{{z}^{y} \cdot e^{(\left(\log a\right) \cdot t + \left(-b\right))_*}}}}{{a}^{1.0}}\\ \end{array}\]

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 < -1.31677741273226e-174 or 0.11857505804325559 < y

    1. Initial program 0.8

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

    if -1.31677741273226e-174 < y < 0.11857505804325559

    1. Initial program 3.8

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

    2. Taylor expanded around inf 3.8

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

    3. Simplified2.5

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

    5. Applied associate-/l*2.4

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

    7. Applied pow-neg2.4

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

    8. Applied un-div-inv2.4

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

    9. Applied associate-*r/2.4

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

    10. Applied associate-/r/2.3

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

    12. Applied associate-/r*2.5

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.31677741273226 \cdot 10^{-174} \lor \neg \left(y \le 0.11857505804325559\right):\\ \;\;\;\;\frac{x \cdot e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{{z}^{y} \cdot e^{(\left(\log a\right) \cdot t + \left(-b\right))_*}}}}{{a}^{1.0}}\\ \end{array}\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018225 +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))