Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I

1. Split input into 2 regimes

2. if (*.f64 y z) < 9.9999999999999999e144

   1. Initial program 98.2%

      \[x \cdot \left(1 - y \cdot z\right) \]

   if 9.9999999999999999e144 < (*.f64 y z)

   1. Initial program 84.4%

      \[x \cdot \left(1 - y \cdot z\right) \]
   2. Step-by-step derivation

      1. flip--20.5%

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

      2. associate-*r/20.5%

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

      3. metadata-eval20.5%

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

      4. pow220.5%

         \[\leadsto \frac{x \cdot \left(1 - \color{blue}{{\left(y \cdot z\right)}^{2}}\right)}{1 + y \cdot z} \]

   3. Applied egg-rr20.5%

      \[\leadsto \color{blue}{\frac{x \cdot \left(1 - {\left(y \cdot z\right)}^{2}\right)}{1 + y \cdot z}} \]
   4. Step-by-step derivation

      1. *-commutative20.5%

         \[\leadsto \frac{\color{blue}{\left(1 - {\left(y \cdot z\right)}^{2}\right) \cdot x}}{1 + y \cdot z} \]

      2. associate-/l*19.6%

         \[\leadsto \color{blue}{\frac{1 - {\left(y \cdot z\right)}^{2}}{\frac{1 + y \cdot z}{x}}} \]

   5. Simplified19.6%

      \[\leadsto \color{blue}{\frac{1 - {\left(y \cdot z\right)}^{2}}{\frac{1 + y \cdot z}{x}}} \]
   6. Step-by-step derivation

      1. associate-/r/20.5%

         \[\leadsto \color{blue}{\frac{1 - {\left(y \cdot z\right)}^{2}}{1 + y \cdot z} \cdot x} \]

      2. metadata-eval20.5%

         \[\leadsto \frac{\color{blue}{1 \cdot 1} - {\left(y \cdot z\right)}^{2}}{1 + y \cdot z} \cdot x \]

      3. unpow220.5%

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

      4. flip--84.4%

         \[\leadsto \color{blue}{\left(1 - y \cdot z\right)} \cdot x \]

      5. sub-neg84.4%

         \[\leadsto \color{blue}{\left(1 + \left(-y \cdot z\right)\right)} \cdot x \]

      6. distribute-rgt-neg-out84.4%

         \[\leadsto \left(1 + \color{blue}{y \cdot \left(-z\right)}\right) \cdot x \]

      7. +-commutative84.4%

         \[\leadsto \color{blue}{\left(y \cdot \left(-z\right) + 1\right)} \cdot x \]

      8. distribute-rgt1-in84.4%

         \[\leadsto \color{blue}{x + \left(y \cdot \left(-z\right)\right) \cdot x} \]

      9. distribute-rgt-neg-out84.4%

         \[\leadsto x + \color{blue}{\left(-y \cdot z\right)} \cdot x \]

      10. distribute-lft-neg-out84.4%

          \[\leadsto x + \color{blue}{\left(-\left(y \cdot z\right) \cdot x\right)} \]

      11. distribute-rgt-neg-out84.4%

          \[\leadsto x + \color{blue}{\left(y \cdot z\right) \cdot \left(-x\right)} \]

      12. add-sqr-sqrt40.9%

          \[\leadsto x + \left(y \cdot z\right) \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)} \]

      13. sqrt-unprod32.9%

          \[\leadsto x + \left(y \cdot z\right) \cdot \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} \]

      14. sqr-neg32.9%

          \[\leadsto x + \left(y \cdot z\right) \cdot \sqrt{\color{blue}{x \cdot x}} \]

      15. sqrt-unprod0.2%

          \[\leadsto x + \left(y \cdot z\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \]

      16. add-sqr-sqrt0.5%

          \[\leadsto x + \left(y \cdot z\right) \cdot \color{blue}{x} \]

      17. cancel-sign-sub0.5%

          \[\leadsto \color{blue}{x - \left(-y \cdot z\right) \cdot x} \]

      18. distribute-rgt-neg-out0.5%

          \[\leadsto x - \color{blue}{\left(y \cdot \left(-z\right)\right)} \cdot x \]

      19. *-commutative0.5%

          \[\leadsto x - \color{blue}{x \cdot \left(y \cdot \left(-z\right)\right)} \]

      20. associate-*r*0.5%

          \[\leadsto x - \color{blue}{\left(x \cdot y\right) \cdot \left(-z\right)} \]

   7. Applied egg-rr99.7%

      \[\leadsto \color{blue}{x + \left(-y \cdot x\right) \cdot z} \]
   8. Step-by-step derivation

      1. distribute-lft-neg-out99.7%

         \[\leadsto x + \color{blue}{\left(-\left(y \cdot x\right) \cdot z\right)} \]

      2. unsub-neg99.7%

         \[\leadsto \color{blue}{x - \left(y \cdot x\right) \cdot z} \]

      3. *-commutative99.7%

         \[\leadsto x - \color{blue}{z \cdot \left(y \cdot x\right)} \]

   9. Simplified99.7%

      \[\leadsto \color{blue}{x - z \cdot \left(y \cdot x\right)} \]

   10. Taylor expanded in z around inf 97.3%

       \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(z \cdot x\right)\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg97.3%

          \[\leadsto \color{blue}{-y \cdot \left(z \cdot x\right)} \]

       2. distribute-rgt-neg-in97.3%

          \[\leadsto \color{blue}{y \cdot \left(-z \cdot x\right)} \]

       3. *-commutative97.3%

          \[\leadsto y \cdot \left(-\color{blue}{x \cdot z}\right) \]

       4. distribute-rgt-neg-out97.3%

          \[\leadsto y \cdot \color{blue}{\left(x \cdot \left(-z\right)\right)} \]

       5. associate-*l*99.7%

          \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot \left(-z\right)} \]

   12. Simplified99.7%

       \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot \left(-z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq 10^{+145}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if (*.f64 y z) < -1 or 0.5 < (*.f64 y z)

   1. Initial program 91.7%

      \[x \cdot \left(1 - y \cdot z\right) \]

   2. Taylor expanded in y around inf 87.8%

      \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \left(y \cdot z\right)\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg87.8%

         \[\leadsto x \cdot \color{blue}{\left(-y \cdot z\right)} \]

      2. distribute-rgt-neg-out87.8%

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(-z\right)\right)} \]

   4. Simplified87.8%

      \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(-z\right)\right)} \]

   if -1 < (*.f64 y z) < 0.5

   1. Initial program 100.0%

      \[x \cdot \left(1 - y \cdot z\right) \]

   2. Taylor expanded in y around 0 97.7%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -1 \lor \neg \left(y \cdot z \leq 0.5\right):\\ \;\;\;\;\left(y \cdot z\right) \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

1. Split input into 2 regimes

2. if (*.f64 y z) < -1 or 0.5 < (*.f64 y z)

   1. Initial program 91.7%

      \[x \cdot \left(1 - y \cdot z\right) \]

   2. Taylor expanded in y around inf 89.9%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(z \cdot x\right)\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg89.9%

         \[\leadsto \color{blue}{-y \cdot \left(z \cdot x\right)} \]

      2. distribute-rgt-neg-in89.9%

         \[\leadsto \color{blue}{y \cdot \left(-z \cdot x\right)} \]

      3. distribute-rgt-neg-in89.9%

         \[\leadsto y \cdot \color{blue}{\left(z \cdot \left(-x\right)\right)} \]

   4. Simplified89.9%

      \[\leadsto \color{blue}{y \cdot \left(z \cdot \left(-x\right)\right)} \]

   if -1 < (*.f64 y z) < 0.5

   1. Initial program 100.0%

      \[x \cdot \left(1 - y \cdot z\right) \]

   2. Taylor expanded in y around 0 97.7%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -1 \lor \neg \left(y \cdot z \leq 0.5\right):\\ \;\;\;\;y \cdot \left(z \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

1. Split input into 3 regimes

2. if (*.f64 y z) < -1

   1. Initial program 92.8%

      \[x \cdot \left(1 - y \cdot z\right) \]
   2. Step-by-step derivation

      1. flip--53.7%

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

      2. associate-*r/48.6%

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

      3. metadata-eval48.6%

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

      4. pow248.6%

         \[\leadsto \frac{x \cdot \left(1 - \color{blue}{{\left(y \cdot z\right)}^{2}}\right)}{1 + y \cdot z} \]

   3. Applied egg-rr48.6%

      \[\leadsto \color{blue}{\frac{x \cdot \left(1 - {\left(y \cdot z\right)}^{2}\right)}{1 + y \cdot z}} \]
   4. Step-by-step derivation

      1. *-commutative48.6%

         \[\leadsto \frac{\color{blue}{\left(1 - {\left(y \cdot z\right)}^{2}\right) \cdot x}}{1 + y \cdot z} \]

      2. associate-/l*53.4%

         \[\leadsto \color{blue}{\frac{1 - {\left(y \cdot z\right)}^{2}}{\frac{1 + y \cdot z}{x}}} \]

   5. Simplified53.4%

      \[\leadsto \color{blue}{\frac{1 - {\left(y \cdot z\right)}^{2}}{\frac{1 + y \cdot z}{x}}} \]
   6. Step-by-step derivation

      1. associate-/r/53.7%

         \[\leadsto \color{blue}{\frac{1 - {\left(y \cdot z\right)}^{2}}{1 + y \cdot z} \cdot x} \]

      2. metadata-eval53.7%

         \[\leadsto \frac{\color{blue}{1 \cdot 1} - {\left(y \cdot z\right)}^{2}}{1 + y \cdot z} \cdot x \]

      3. unpow253.7%

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

      4. flip--92.8%

         \[\leadsto \color{blue}{\left(1 - y \cdot z\right)} \cdot x \]

      5. sub-neg92.8%

         \[\leadsto \color{blue}{\left(1 + \left(-y \cdot z\right)\right)} \cdot x \]

      6. distribute-rgt-neg-out92.8%

         \[\leadsto \left(1 + \color{blue}{y \cdot \left(-z\right)}\right) \cdot x \]

      7. +-commutative92.8%

         \[\leadsto \color{blue}{\left(y \cdot \left(-z\right) + 1\right)} \cdot x \]

      8. distribute-rgt1-in92.8%

         \[\leadsto \color{blue}{x + \left(y \cdot \left(-z\right)\right) \cdot x} \]

      9. distribute-rgt-neg-out92.8%

         \[\leadsto x + \color{blue}{\left(-y \cdot z\right)} \cdot x \]

      10. distribute-lft-neg-out92.8%

          \[\leadsto x + \color{blue}{\left(-\left(y \cdot z\right) \cdot x\right)} \]

      11. distribute-rgt-neg-out92.8%

          \[\leadsto x + \color{blue}{\left(y \cdot z\right) \cdot \left(-x\right)} \]

      12. add-sqr-sqrt56.3%

          \[\leadsto x + \left(y \cdot z\right) \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)} \]

      13. sqrt-unprod43.4%

          \[\leadsto x + \left(y \cdot z\right) \cdot \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} \]

      14. sqr-neg43.4%

          \[\leadsto x + \left(y \cdot z\right) \cdot \sqrt{\color{blue}{x \cdot x}} \]

      15. sqrt-unprod0.3%

          \[\leadsto x + \left(y \cdot z\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \]

      16. add-sqr-sqrt0.9%

          \[\leadsto x + \left(y \cdot z\right) \cdot \color{blue}{x} \]

      17. cancel-sign-sub0.9%

          \[\leadsto \color{blue}{x - \left(-y \cdot z\right) \cdot x} \]

      18. distribute-rgt-neg-out0.9%

          \[\leadsto x - \color{blue}{\left(y \cdot \left(-z\right)\right)} \cdot x \]

      19. *-commutative0.9%

          \[\leadsto x - \color{blue}{x \cdot \left(y \cdot \left(-z\right)\right)} \]

      20. associate-*r*1.0%

          \[\leadsto x - \color{blue}{\left(x \cdot y\right) \cdot \left(-z\right)} \]

   7. Applied egg-rr93.1%

      \[\leadsto \color{blue}{x + \left(-y \cdot x\right) \cdot z} \]
   8. Step-by-step derivation

      1. distribute-lft-neg-out93.1%

         \[\leadsto x + \color{blue}{\left(-\left(y \cdot x\right) \cdot z\right)} \]

      2. unsub-neg93.1%

         \[\leadsto \color{blue}{x - \left(y \cdot x\right) \cdot z} \]

      3. *-commutative93.1%

         \[\leadsto x - \color{blue}{z \cdot \left(y \cdot x\right)} \]

   9. Simplified93.1%

      \[\leadsto \color{blue}{x - z \cdot \left(y \cdot x\right)} \]

   10. Taylor expanded in z around inf 87.7%

       \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(z \cdot x\right)\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg87.7%

          \[\leadsto \color{blue}{-y \cdot \left(z \cdot x\right)} \]

       2. distribute-rgt-neg-in87.7%

          \[\leadsto \color{blue}{y \cdot \left(-z \cdot x\right)} \]

       3. *-commutative87.7%

          \[\leadsto y \cdot \left(-\color{blue}{x \cdot z}\right) \]

       4. distribute-rgt-neg-out87.7%

          \[\leadsto y \cdot \color{blue}{\left(x \cdot \left(-z\right)\right)} \]

       5. associate-*l*87.9%

          \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot \left(-z\right)} \]

   12. Simplified87.9%

       \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot \left(-z\right)} \]

   if -1 < (*.f64 y z) < 0.5

   1. Initial program 100.0%

      \[x \cdot \left(1 - y \cdot z\right) \]

   2. Taylor expanded in y around 0 97.7%

      \[\leadsto \color{blue}{x} \]

   if 0.5 < (*.f64 y z)

   1. Initial program 90.8%

      \[x \cdot \left(1 - y \cdot z\right) \]

   2. Taylor expanded in y around inf 91.7%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(z \cdot x\right)\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg91.7%

         \[\leadsto \color{blue}{-y \cdot \left(z \cdot x\right)} \]

      2. distribute-rgt-neg-in91.7%

         \[\leadsto \color{blue}{y \cdot \left(-z \cdot x\right)} \]

      3. distribute-rgt-neg-in91.7%

         \[\leadsto y \cdot \color{blue}{\left(z \cdot \left(-x\right)\right)} \]

   4. Simplified91.7%

      \[\leadsto \color{blue}{y \cdot \left(z \cdot \left(-x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -1:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{elif}\;y \cdot z \leq 0.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-x\right)\right)\\ \end{array} \]

1. Initial program 96.2%

   \[x \cdot \left(1 - y \cdot z\right) \]

2. Taylor expanded in y around 0 54.3%

   \[\leadsto \color{blue}{x} \]

3. Final simplification54.3%

   \[\leadsto x \]

herbie shell --seed 2023166 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))