Average Error: 2.0 → 1.0
Time: 2.9m
Precision: 64
Internal Precision: 128
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{\sqrt{e^{\left(\log a \cdot t - b\right) + \left(\log z \cdot y - 1.0 \cdot \log a\right)}}}{\sqrt[3]{y}} \cdot \left(x \cdot \frac{\sqrt{{e}^{\left(\left(\log z \cdot y + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right)\]

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. Initial program 2.0

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

  3. Applied *-un-lft-identity2.0

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

  4. Applied *-un-lft-identity2.0

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

  5. Applied log-prod2.0

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

  6. Applied distribute-rgt-in2.0

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

  7. Applied associate-+l+2.0

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

  8. Applied associate--l+2.0

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

  9. Applied exp-sum2.0

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

  10. Applied associate-*r*2.0

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

  11. Applied times-frac1.9

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

  12. Simplified1.9

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

  14. Applied add-cube-cbrt1.9

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

  15. Applied add-sqr-sqrt1.9

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

  16. Applied times-frac1.9

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

  17. Applied associate-*r*1.0

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

  18. Taylor expanded around inf 1.0

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

  19. Simplified1.0

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

  21. Applied *-un-lft-identity1.0

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

  22. Applied *-un-lft-identity1.0

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

  23. Applied *-un-lft-identity1.0

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

  24. Applied associate-*l*1.0

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

  25. Applied distribute-lft-out1.0

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

  26. Applied distribute-lft-out--1.0

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

  27. Applied exp-prod1.0

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

  28. Simplified1.0

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

  29. Final simplification1.0

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

Reproduce

herbie shell --seed 2019072 
(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))