Average Error: 1.9 → 1.7
Time: 1.5m
Precision: 64
Internal Precision: 384
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\log a \cdot \left(t - 1.0\right) - b \le -9.14322058741918 \cdot 10^{+32}:\\ \;\;\;\;\left(\sqrt[3]{\frac{x \cdot {e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}{y}} \cdot \left(\sqrt[3]{x \cdot {e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}} \cdot \sqrt[3]{\frac{1}{y}}\right)\right) \cdot {\left(\frac{x \cdot {e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}{y}\right)}^{\frac{1}{3}}\\ \mathbf{if}\;\log a \cdot \left(t - 1.0\right) - b \le 1.3237648067054203 \cdot 10^{+24}:\\ \;\;\;\;\frac{x \cdot \left(\left({a}^{\left(t - 1.0\right)} \cdot {z}^{y}\right) \cdot e^{-b}\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{x \cdot {e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}{y}} \cdot \left(\sqrt[3]{x \cdot {e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}} \cdot \sqrt[3]{\frac{1}{y}}\right)\right) \cdot {\left(\frac{x \cdot {e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}{y}\right)}^{\frac{1}{3}}\\ \end{array}\]

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 (- (* (log a) (- t 1.0)) b) < -9.14322058741918e+32 or 1.3237648067054203e+24 < (- (* (log a) (- t 1.0)) b)

    1. Initial program 0

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

    3. Applied *-un-lft-identity0

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

    4. Applied exp-prod0

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

    5. Applied simplify0

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

    7. Applied add-cube-cbrt0

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

    9. Applied pow1/30

      \[\leadsto \left(\sqrt[3]{\frac{x \cdot {e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}{y}} \cdot \sqrt[3]{\frac{x \cdot {e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}{y}}\right) \cdot \color{blue}{{\left(\frac{x \cdot {e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}{y}\right)}^{\frac{1}{3}}}\]
    10. Using strategy rm

    11. Applied div-inv0.0

      \[\leadsto \left(\sqrt[3]{\frac{x \cdot {e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}{y}} \cdot \sqrt[3]{\color{blue}{\left(x \cdot {e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}\right) \cdot \frac{1}{y}}}\right) \cdot {\left(\frac{x \cdot {e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}{y}\right)}^{\frac{1}{3}}\]

    12. Applied cbrt-prod0.0

      \[\leadsto \left(\sqrt[3]{\frac{x \cdot {e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}{y}} \cdot \color{blue}{\left(\sqrt[3]{x \cdot {e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}} \cdot \sqrt[3]{\frac{1}{y}}\right)}\right) \cdot {\left(\frac{x \cdot {e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}{y}\right)}^{\frac{1}{3}}\]

    if -9.14322058741918e+32 < (- (* (log a) (- t 1.0)) b) < 1.3237648067054203e+24

    1. Initial program 5.8

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

    3. Applied sub-neg5.8

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

    4. Applied exp-sum6.4

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

    5. Applied simplify5.1

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

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' 
(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))