Average Error: 1.9 → 1.3
Time: 1.6m
Precision: 64
Internal Precision: 576
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}} \le -6.6585205798721 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{\frac{y}{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}\\ \mathbf{elif}\;\frac{y}{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}} \le 2.4517247083719316 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{x}{y}}{{a}^{1.0}} + \left(\log a \cdot \frac{t}{y} + \log z\right) \cdot \frac{x}{{a}^{1.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}\\ \end{array}\]

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (/ y (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) < -6.6585205798721e-40 or 2.4517247083719316e+87 < (/ y (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b)))

    1. Initial program 0.9

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

    3. Applied associate-/l*0.3

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

    if -6.6585205798721e-40 < (/ y (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) < 2.4517247083719316e+87

    1. Initial program 9.4

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

    2. Taylor expanded around 0 12.2

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

    3. Simplified8.5

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}} \le -6.6585205798721 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{\frac{y}{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}\\ \mathbf{elif}\;\frac{y}{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}} \le 2.4517247083719316 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{x}{y}}{{a}^{1.0}} + \left(\log a \cdot \frac{t}{y} + \log z\right) \cdot \frac{x}{{a}^{1.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}\\ \end{array}\]

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018225 
(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))