Error

Target

Derivation

1. Split input into 2 regimes

2. if x < 6.162182387522347e-283

   1. Initial program 1.6

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

   2. Applied simplify18.5

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

   4. Applied add-exp-log18.5

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

   5. Applied div-exp18.5

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

   6. Applied pow-exp10.9

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

   8. Applied add-exp-log10.9

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

   9. Applied pow-exp10.9

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

   10. Applied simplify2.3

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

   if 6.162182387522347e-283 < x

   1. Initial program 2.1

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

   2. Applied simplify22.5

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

   4. Applied add-exp-log22.5

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

   5. Applied div-exp22.5

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

   6. Applied pow-exp17.0

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

   8. Applied add-exp-log17.0

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

   9. Applied pow-exp17.0

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

   10. Applied add-exp-log18.1

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

   11. Applied prod-exp16.5

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

   12. Applied simplify3.8

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

Runtime

Time bar (total: 5.5m)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit +o reduce:binary-search
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B"

  :herbie-target
  (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))

  (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))