Average Error: 1.9 → 1.2
Time: 59.9s
Precision: 64
Internal Precision: 576
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.1390913778276833 \cdot 10^{-44} \lor \neg \left(y \le 7.277186123736713 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{x}{\frac{y}{e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left({z}^{y} \cdot e^{(t \cdot \left(\log a\right) + \left(-b\right))_*}\right)}{{a}^{1.0}}}{y}\\ \end{array}\]

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 y < -1.1390913778276833e-44 or 7.277186123736713e-11 < y

    1. Initial program 0.2

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

    3. Applied associate-/l*0.2

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

    if -1.1390913778276833e-44 < y < 7.277186123736713e-11

    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. Simplified2.4

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

    5. Applied pow-neg2.4

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

    6. Applied associate-*l/2.4

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

    7. Applied associate-*l/2.4

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

    8. Applied associate-*r/2.4

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.1390913778276833 \cdot 10^{-44} \lor \neg \left(y \le 7.277186123736713 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{x}{\frac{y}{e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left({z}^{y} \cdot e^{(t \cdot \left(\log a\right) + \left(-b\right))_*}\right)}{{a}^{1.0}}}{y}\\ \end{array}\]

Runtime

Time bar (total: 59.9s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes1.81.20.01.833.1%
herbie shell --seed 2018296 +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))