Average Error: 24.1 → 1.6
Time: 1.7m
Precision: 64
Internal precision: 128
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
⬇
\[\begin{array}{l} \mathbf{if}\;y \le -6.408792807834615 \cdot 10^{+52}:\\ \;\;\;\;\frac{\frac{{z}^{y} \cdot x}{y \cdot a}}{\left(1 + b\right) - t \cdot \log a}\\ \mathbf{if}\;y \le -69.525409230544:\\ \;\;\;\;\frac{\frac{x}{y} + \left(x \cdot \log z + \frac{1}{2} \cdot \left(y \cdot \left(x \cdot {\left(\log z\right)}^2\right)\right)\right)}{e^{b - \log a \cdot \left(t - 1.0\right)}}\\ \mathbf{if}\;y \le 1.4591773572605082 \cdot 10^{-06}:\\ \;\;\;\;\frac{{z}^{y} \cdot x}{e^{b - \log a \cdot \left(t - 1.0\right)} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{z}^{y} \cdot x}{y \cdot a}}{\left(1 + b\right) - t \cdot \log a}\\ \end{array}\]

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 3 regimes.

  2. if y < -6.408792807834615e+52 or 1.4591773572605082e-06 < y

    1. Initial program 36.4

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

    2. Applied simplify 27.1

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

    3. Applied taylor 1.6

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

    4. Taylor expanded around 0 1.6

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

    5. Applied simplify 0.1

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

    if -6.408792807834615e+52 < y < -69.525409230544

    1. Initial program 23.4

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

    2. Applied simplify 45.3

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

    4. Applied pow-to-exp 45.3

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

    5. Applied div-exp 41.9

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

    6. Applied taylor 5.8

      \[\leadsto \frac{\frac{x}{y} + \left(x \cdot \log z + \frac{1}{2} \cdot \left(y \cdot \left(x \cdot {\left(\log z\right)}^2\right)\right)\right)}{e^{b - \log a \cdot \left(t - 1.0\right)}}\]

    7. Taylor expanded around 0 5.8

      \[\leadsto \frac{\color{blue}{\frac{x}{y} + \left(x \cdot \log z + \frac{1}{2} \cdot \left(y \cdot \left(x \cdot {\left(\log z\right)}^2\right)\right)\right)}}{e^{b - \log a \cdot \left(t - 1.0\right)}}\]

    if -69.525409230544 < y < 1.4591773572605082e-06

    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. Applied simplify 18.4

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

    4. Applied associate-*r/ 18.4

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

    5. Applied associate-/l/ 9.5

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

    7. Applied pow-to-exp 10.8

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

    8. Applied div-exp 3.8

      \[\leadsto \frac{{z}^{y} \cdot x}{\color{blue}{e^{b - \log a \cdot \left(t - 1.0\right)}} \cdot y}\]
  3. Recombined 3 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 1.7m) Debug log

Please include this information when filing a bug report:

herbie --seed '#(3483362936 519828621 3026289779 470622841 1282952580 3724100236)'
(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))