Average Error: 1.9 → 1.3
Time: 1.3m
Precision: 64
Internal Precision: 576
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -0.7642216222482152 \lor \neg \left(y \le 4.409430372676395 \cdot 10^{-05}\right):\\ \;\;\;\;\frac{\sqrt{{e}^{\left((y \cdot \left(\log z\right) + \left((\left(\log a\right) \cdot \left(t - 1.0\right) + \left(-b\right))_*\right))_*\right)}} \cdot \left(x \cdot \sqrt{{e}^{\left(\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b\right)}}\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{e^{(\left(\log a\right) \cdot t + \left(-b\right))_*} \cdot \left({z}^{y} \cdot {a}^{\left(-1.0\right)}\right)}}\\ \end{array}\]

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 < -0.7642216222482152 or 4.409430372676395e-05 < y

    1. Initial program 0.1

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

    3. Applied *-un-lft-identity0.1

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

    4. Applied exp-prod0.1

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

    5. Simplified0.1

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

    7. Applied add-sqr-sqrt0.1

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

    8. Applied associate-*r*0.1

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

    9. Simplified0.1

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

    if -0.7642216222482152 < y < 4.409430372676395e-05

    1. Initial program 3.6

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

    3. Applied *-un-lft-identity3.6

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

    4. Applied exp-prod3.7

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

    5. Simplified3.7

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

    6. Taylor expanded around inf 3.6

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

    7. Simplified2.2

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

    9. Applied associate-/l*2.4

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -0.7642216222482152 \lor \neg \left(y \le 4.409430372676395 \cdot 10^{-05}\right):\\ \;\;\;\;\frac{\sqrt{{e}^{\left((y \cdot \left(\log z\right) + \left((\left(\log a\right) \cdot \left(t - 1.0\right) + \left(-b\right))_*\right))_*\right)}} \cdot \left(x \cdot \sqrt{{e}^{\left(\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b\right)}}\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{e^{(\left(\log a\right) \cdot t + \left(-b\right))_*} \cdot \left({z}^{y} \cdot {a}^{\left(-1.0\right)}\right)}}\\ \end{array}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018234 +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))