Average Error: 2.0 → 1.1
Time: 2.8m
Precision: 64
Internal Precision: 576
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\sqrt[3]{\frac{\frac{{a}^{\left(t - 1.0\right)}}{e^{b}}}{\frac{\frac{y}{x}}{{z}^{y}}}} \cdot \sqrt[3]{\frac{\frac{{a}^{\left(t - 1.0\right)}}{e^{b}}}{\frac{\frac{y}{x}}{{z}^{y}}}}\right) \cdot \sqrt[3]{\frac{{z}^{y}}{y} \cdot \left({a}^{\left(t - 1.0\right)} \cdot \frac{x}{e^{b}}\right)} \le 1.0568766691707109 \cdot 10^{-302}:\\ \;\;\;\;\frac{x}{\frac{e^{b} \cdot y}{{z}^{y} \cdot {a}^{\left(t - 1.0\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\\ \end{array}\]

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 (* (* (cbrt (/ (/ (pow a (- t 1.0)) (exp b)) (/ (/ y x) (pow z y)))) (cbrt (/ (/ (pow a (- t 1.0)) (exp b)) (/ (/ y x) (pow z y))))) (cbrt (* (/ (pow z y) y) (* (pow a (- t 1.0)) (/ x (exp b)))))) < 1.0568766691707109e-302

    1. Initial program 2.7

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

    3. Applied associate-/l*1.8

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

    4. Applied simplify1.1

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

    if 1.0568766691707109e-302 < (* (* (cbrt (/ (/ (pow a (- t 1.0)) (exp b)) (/ (/ y x) (pow z y)))) (cbrt (/ (/ (pow a (- t 1.0)) (exp b)) (/ (/ y x) (pow z y))))) (cbrt (* (/ (pow z y) y) (* (pow a (- t 1.0)) (/ x (exp b))))))

    1. Initial program 1.0

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

Runtime

Time bar (total: 2.8m)Debug logProfile

herbie shell --seed '#(1071979731 1496239409 439705970 2863295848 982327776 189749553)' 
(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))