Average Error: 2.1 → 3.0
Time: 1.4m
Precision: 64
Internal Precision: 576
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le 4.370672124418482 \cdot 10^{-65} \lor \neg \left(z \le 4.34260666699038 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{x \cdot e^{\left(\left(t - 1.0\right) \cdot \log a + y \cdot \log z\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{y}{x \cdot \left({z}^{y} \cdot e^{(t \cdot \left(\log a\right) + \left(-b\right))_*}\right)}}}{{a}^{1.0}}\\ \end{array}\]

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 z < 4.370672124418482e-65 or 4.34260666699038e-11 < z

    1. Initial program 2.1

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

    2. Taylor expanded around 0 2.1

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

    3. Simplified2.1

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

    if 4.370672124418482e-65 < z < 4.34260666699038e-11

    1. Initial program 1.8

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

    2. Taylor expanded around inf 1.7

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

    3. Simplified12.3

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

    5. Applied clear-num12.3

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

    7. Applied pow-neg12.3

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

    8. Applied associate-*l/12.3

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

    9. Applied associate-*l/12.3

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

    10. Applied associate-*r/12.3

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

    11. Applied associate-/r/12.2

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

    12. Applied associate-/r*12.3

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

  4. Final simplification3.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 4.370672124418482 \cdot 10^{-65} \lor \neg \left(z \le 4.34260666699038 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{x \cdot e^{\left(\left(t - 1.0\right) \cdot \log a + y \cdot \log z\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{y}{x \cdot \left({z}^{y} \cdot e^{(t \cdot \left(\log a\right) + \left(-b\right))_*}\right)}}}{{a}^{1.0}}\\ \end{array}\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018250 +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))