Diagrams.TwoD.Arc:bezierFromSweepQ1 from diagrams-lib-1.3.0.3

Percentage Accurate: 94.0% → 99.8%

Time: 9.6s

Alternatives: 8

Speedup: 1.1×

• 25.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (- 1.0 x) (- 3.0 x)) (* y 3.0)))

C

double code(double x, double y) {
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((1.0d0 - x) * (3.0d0 - x)) / (y * 3.0d0)
end function

Java

public static double code(double x, double y) {
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0);
}

Python

def code(x, y):
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0)

Julia

function code(x, y)
	return Float64(Float64(Float64(1.0 - x) * Float64(3.0 - x)) / Float64(y * 3.0))
end

MATLAB

function tmp = code(x, y)
	tmp = ((1.0 - x) * (3.0 - x)) / (y * 3.0);
end

Wolfram

code[x_, y_] := N[(N[(N[(1.0 - x), $MachinePrecision] * N[(3.0 - x), $MachinePrecision]), $MachinePrecision] / N[(y * 3.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (- 1.0 x) (- 3.0 x)) (* y 3.0)))

C

double code(double x, double y) {
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((1.0d0 - x) * (3.0d0 - x)) / (y * 3.0d0)
end function

Java

public static double code(double x, double y) {
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0);
}

Python

def code(x, y):
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0)

Julia

function code(x, y)
	return Float64(Float64(Float64(1.0 - x) * Float64(3.0 - x)) / Float64(y * 3.0))
end

MATLAB

function tmp = code(x, y)
	tmp = ((1.0 - x) * (3.0 - x)) / (y * 3.0);
end

Wolfram

code[x_, y_] := N[(N[(N[(1.0 - x), $MachinePrecision] * N[(3.0 - x), $MachinePrecision]), $MachinePrecision] / N[(y * 3.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(3 - x\right) \cdot \left(1 - x\right) \leq 5 \cdot 10^{+282}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.3333333333333333, -1.3333333333333333\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{3 \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 3.0 x) (- 1.0 x)) 5e+282)
   (/ (fma x (fma x 0.3333333333333333 -1.3333333333333333) 1.0) y)
   (* x (/ x (* 3.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (((3.0 - x) * (1.0 - x)) <= 5e+282) {
		tmp = fma(x, fma(x, 0.3333333333333333, -1.3333333333333333), 1.0) / y;
	} else {
		tmp = x * (x / (3.0 * y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(3.0 - x) * Float64(1.0 - x)) <= 5e+282)
		tmp = Float64(fma(x, fma(x, 0.3333333333333333, -1.3333333333333333), 1.0) / y);
	else
		tmp = Float64(x * Float64(x / Float64(3.0 * y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(3.0 - x), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], 5e+282], N[(N[(x * N[(x * 0.3333333333333333 + -1.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision], N[(x * N[(x / N[(3.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x)) < 4.99999999999999978e282

   1. Initial program 99.7%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(\left(1 - x\right) \cdot \left(3 - x\right)\right)}{y}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(\left(1 - x\right) \cdot \left(3 - x\right)\right)}{y}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \color{blue}{\left(\left(3 - x\right) \cdot \left(1 - x\right)\right)}}{y} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{3} \cdot \left(3 - x\right)\right) \cdot \left(1 - x\right)}}{y} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 - x\right) \cdot \left(\frac{1}{3} \cdot \left(3 - x\right)\right)}}{y} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 - x\right) \cdot \left(\frac{1}{3} \cdot \left(3 - x\right)\right)}}{y} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 - x\right)} \cdot \left(\frac{1}{3} \cdot \left(3 - x\right)\right)}{y} \]

      8. sub-neg N/A

         \[\leadsto \frac{\left(1 - x\right) \cdot \left(\frac{1}{3} \cdot \color{blue}{\left(3 + \left(\mathsf{neg}\left(x\right)\right)\right)}\right)}{y} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{\left(1 - x\right) \cdot \left(\frac{1}{3} \cdot \left(3 + \color{blue}{-1 \cdot x}\right)\right)}{y} \]

      10. +-commutative N/A

          \[\leadsto \frac{\left(1 - x\right) \cdot \left(\frac{1}{3} \cdot \color{blue}{\left(-1 \cdot x + 3\right)}\right)}{y} \]

      11. distribute-rgt-in N/A

          \[\leadsto \frac{\left(1 - x\right) \cdot \color{blue}{\left(\left(-1 \cdot x\right) \cdot \frac{1}{3} + 3 \cdot \frac{1}{3}\right)}}{y} \]

      12. mul-1-neg N/A

          \[\leadsto \frac{\left(1 - x\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \frac{1}{3} + 3 \cdot \frac{1}{3}\right)}{y} \]

      13. distribute-lft-neg-out N/A

          \[\leadsto \frac{\left(1 - x\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{3}\right)\right)} + 3 \cdot \frac{1}{3}\right)}{y} \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \frac{\left(1 - x\right) \cdot \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} + 3 \cdot \frac{1}{3}\right)}{y} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\left(1 - x\right) \cdot \left(x \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + \color{blue}{1}\right)}{y} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\left(1 - x\right) \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{neg}\left(\frac{1}{3}\right), 1\right)}}{y} \]

      17. metadata-eval 99.8

          \[\leadsto \frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, \color{blue}{-0.3333333333333333}, 1\right)}{y} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, -0.3333333333333333, 1\right)}{y}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{1 + x \cdot \left(\frac{1}{3} \cdot x - \frac{4}{3}\right)}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.8%

      \[\leadsto \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.3333333333333333, -1.3333333333333333\right), 1\right)}{y} \]

   if 4.99999999999999978e282 < (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x))

   1. Initial program 69.3%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{{x}^{2}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{x}^{2}}{y} \cdot \frac{1}{3}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y} \cdot \frac{1}{3} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{x}{y}\right)} \cdot \frac{1}{3} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{x}{y} \cdot \frac{1}{3}\right)} \]

      5. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{3} \cdot \frac{x}{y}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{3} \cdot \frac{x}{y}\right)} \]

      7. associate-*r/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{3} \cdot x}{y}} \]

      8. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{3} \cdot x}{y}} \]

      9. *-commutative N/A

         \[\leadsto x \cdot \frac{\color{blue}{x \cdot \frac{1}{3}}}{y} \]

      10. lower-*.f64 99.8

          \[\leadsto x \cdot \frac{\color{blue}{x \cdot 0.3333333333333333}}{y} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{x \cdot \frac{x \cdot 0.3333333333333333}{y}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 - x\right) \cdot \left(1 - x\right) \leq 5 \cdot 10^{+282}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.3333333333333333, -1.3333333333333333\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{3 \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(3 - x\right) \cdot \left(1 - x\right) \leq 5:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, -1.3333333333333333, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.3333333333333333, -1.3333333333333333\right) \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 3.0 x) (- 1.0 x)) 5.0)
   (/ (fma x -1.3333333333333333 1.0) y)
   (* (fma x 0.3333333333333333 -1.3333333333333333) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (((3.0 - x) * (1.0 - x)) <= 5.0) {
		tmp = fma(x, -1.3333333333333333, 1.0) / y;
	} else {
		tmp = fma(x, 0.3333333333333333, -1.3333333333333333) * (x / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(3.0 - x) * Float64(1.0 - x)) <= 5.0)
		tmp = Float64(fma(x, -1.3333333333333333, 1.0) / y);
	else
		tmp = Float64(fma(x, 0.3333333333333333, -1.3333333333333333) * Float64(x / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(3.0 - x), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], 5.0], N[(N[(x * -1.3333333333333333 + 1.0), $MachinePrecision] / y), $MachinePrecision], N[(N[(x * 0.3333333333333333 + -1.3333333333333333), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x)) < 5

   1. Initial program 99.7%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 96.4

         \[\leadsto \color{blue}{\frac{1}{y}} \]

   5. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\frac{1}{y}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-4}{3} \cdot \frac{x}{y} + \frac{1}{y}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-4}{3} \cdot x}{y}} + \frac{1}{y} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{-4}{3}}}{y} + \frac{1}{y} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{\frac{-4}{3}}{y}} + \frac{1}{y} \]

      4. metadata-eval N/A

         \[\leadsto x \cdot \frac{\color{blue}{\mathsf{neg}\left(\frac{4}{3}\right)}}{y} + \frac{1}{y} \]

      5. distribute-neg-frac N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{4}{3}}{y}\right)\right)} + \frac{1}{y} \]

      6. metadata-eval N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{4}{3} \cdot 1}}{y}\right)\right) + \frac{1}{y} \]

      7. associate-*r/ N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{4}{3} \cdot \frac{1}{y}}\right)\right) + \frac{1}{y} \]

      8. rgt-mult-inverse N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{y}\right)\right) + \frac{\color{blue}{x \cdot \frac{1}{x}}}{y} \]

      9. associate-*r/ N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{y}\right)\right) + \color{blue}{x \cdot \frac{\frac{1}{x}}{y}} \]

      10. associate-/r* N/A

          \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{y}\right)\right) + x \cdot \color{blue}{\frac{1}{x \cdot y}} \]

      11. distribute-lft-in N/A

          \[\leadsto \color{blue}{x \cdot \left(\left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{y}\right)\right) + \frac{1}{x \cdot y}\right)} \]

      12. neg-sub0 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(0 - \frac{4}{3} \cdot \frac{1}{y}\right)} + \frac{1}{x \cdot y}\right) \]

      13. associate-+l- N/A

          \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{4}{3} \cdot \frac{1}{y} - \frac{1}{x \cdot y}\right)\right)} \]

      14. neg-sub0 N/A

          \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{4}{3} \cdot \frac{1}{y} - \frac{1}{x \cdot y}\right)\right)\right)} \]

      15. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{4}{3} \cdot \frac{1}{y} - \frac{1}{x \cdot y}\right)\right)} \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y} - \frac{1}{x \cdot y}\right)} \]

      17. sub-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(\frac{4}{3} \cdot \frac{1}{y} + \left(\mathsf{neg}\left(\frac{1}{x \cdot y}\right)\right)\right)} \]

      18. sub-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(\frac{4}{3} \cdot \frac{1}{y} - \frac{1}{x \cdot y}\right)} \]

   8. Applied rewrites 97.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, -1.3333333333333333, 1\right)}{y}} \]

   if 5 < (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x))

   1. Initial program 84.1%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot \frac{1}{y} - \frac{4}{3} \cdot \frac{1}{x \cdot y}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{y} + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{x \cdot y}\right)\right)\right)} \]

      2. associate-*r/ N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{3} \cdot \frac{1}{y} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{4}{3} \cdot 1}{x \cdot y}}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{3} \cdot \frac{1}{y} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{4}{3}}}{x \cdot y}\right)\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot \frac{1}{y}\right) + {x}^{2} \cdot \left(\mathsf{neg}\left(\frac{\frac{4}{3}}{x \cdot y}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{y}\right) \cdot {x}^{2}} + {x}^{2} \cdot \left(\mathsf{neg}\left(\frac{\frac{4}{3}}{x \cdot y}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{1}{3} \cdot \left(\frac{1}{y} \cdot {x}^{2}\right)} + {x}^{2} \cdot \left(\mathsf{neg}\left(\frac{\frac{4}{3}}{x \cdot y}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{1 \cdot {x}^{2}}{y}} + {x}^{2} \cdot \left(\mathsf{neg}\left(\frac{\frac{4}{3}}{x \cdot y}\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\color{blue}{{x}^{2}}}{y} + {x}^{2} \cdot \left(\mathsf{neg}\left(\frac{\frac{4}{3}}{x \cdot y}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\color{blue}{x \cdot x}}{y} + {x}^{2} \cdot \left(\mathsf{neg}\left(\frac{\frac{4}{3}}{x \cdot y}\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \frac{1}{3} \cdot \color{blue}{\left(x \cdot \frac{x}{y}\right)} + {x}^{2} \cdot \left(\mathsf{neg}\left(\frac{\frac{4}{3}}{x \cdot y}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot x\right) \cdot \frac{x}{y}} + {x}^{2} \cdot \left(\mathsf{neg}\left(\frac{\frac{4}{3}}{x \cdot y}\right)\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \left(\frac{1}{3} \cdot x\right) \cdot \frac{x}{y} + {x}^{2} \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{4}{3}\right)}{x \cdot y}} \]

      13. metadata-eval N/A

          \[\leadsto \left(\frac{1}{3} \cdot x\right) \cdot \frac{x}{y} + {x}^{2} \cdot \frac{\color{blue}{\frac{-4}{3}}}{x \cdot y} \]

      14. associate-*r/ N/A

          \[\leadsto \left(\frac{1}{3} \cdot x\right) \cdot \frac{x}{y} + \color{blue}{\frac{{x}^{2} \cdot \frac{-4}{3}}{x \cdot y}} \]

      15. times-frac N/A

          \[\leadsto \left(\frac{1}{3} \cdot x\right) \cdot \frac{x}{y} + \color{blue}{\frac{{x}^{2}}{x} \cdot \frac{\frac{-4}{3}}{y}} \]

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \mathsf{fma}\left(x, 0.3333333333333333, -1.3333333333333333\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 - x\right) \cdot \left(1 - x\right) \leq 5:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, -1.3333333333333333, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.3333333333333333, -1.3333333333333333\right) \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(3 - x\right) \cdot \left(1 - x\right) \leq 5:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, -1.3333333333333333, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.3333333333333333}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 3.0 x) (- 1.0 x)) 5.0)
   (/ (fma x -1.3333333333333333 1.0) y)
   (* x (* x (/ 0.3333333333333333 y)))))

C

double code(double x, double y) {
	double tmp;
	if (((3.0 - x) * (1.0 - x)) <= 5.0) {
		tmp = fma(x, -1.3333333333333333, 1.0) / y;
	} else {
		tmp = x * (x * (0.3333333333333333 / y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(3.0 - x) * Float64(1.0 - x)) <= 5.0)
		tmp = Float64(fma(x, -1.3333333333333333, 1.0) / y);
	else
		tmp = Float64(x * Float64(x * Float64(0.3333333333333333 / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(3.0 - x), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], 5.0], N[(N[(x * -1.3333333333333333 + 1.0), $MachinePrecision] / y), $MachinePrecision], N[(x * N[(x * N[(0.3333333333333333 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x)) < 5

   1. Initial program 99.7%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 96.4

         \[\leadsto \color{blue}{\frac{1}{y}} \]

   5. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\frac{1}{y}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-4}{3} \cdot \frac{x}{y} + \frac{1}{y}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-4}{3} \cdot x}{y}} + \frac{1}{y} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{-4}{3}}}{y} + \frac{1}{y} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{\frac{-4}{3}}{y}} + \frac{1}{y} \]

      4. metadata-eval N/A

         \[\leadsto x \cdot \frac{\color{blue}{\mathsf{neg}\left(\frac{4}{3}\right)}}{y} + \frac{1}{y} \]

      5. distribute-neg-frac N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{4}{3}}{y}\right)\right)} + \frac{1}{y} \]

      6. metadata-eval N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{4}{3} \cdot 1}}{y}\right)\right) + \frac{1}{y} \]

      7. associate-*r/ N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{4}{3} \cdot \frac{1}{y}}\right)\right) + \frac{1}{y} \]

      8. rgt-mult-inverse N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{y}\right)\right) + \frac{\color{blue}{x \cdot \frac{1}{x}}}{y} \]

      9. associate-*r/ N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{y}\right)\right) + \color{blue}{x \cdot \frac{\frac{1}{x}}{y}} \]

      10. associate-/r* N/A

          \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{y}\right)\right) + x \cdot \color{blue}{\frac{1}{x \cdot y}} \]

      11. distribute-lft-in N/A

          \[\leadsto \color{blue}{x \cdot \left(\left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{y}\right)\right) + \frac{1}{x \cdot y}\right)} \]

      12. neg-sub0 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(0 - \frac{4}{3} \cdot \frac{1}{y}\right)} + \frac{1}{x \cdot y}\right) \]

      13. associate-+l- N/A

          \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{4}{3} \cdot \frac{1}{y} - \frac{1}{x \cdot y}\right)\right)} \]

      14. neg-sub0 N/A

          \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{4}{3} \cdot \frac{1}{y} - \frac{1}{x \cdot y}\right)\right)\right)} \]

      15. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{4}{3} \cdot \frac{1}{y} - \frac{1}{x \cdot y}\right)\right)} \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y} - \frac{1}{x \cdot y}\right)} \]

      17. sub-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(\frac{4}{3} \cdot \frac{1}{y} + \left(\mathsf{neg}\left(\frac{1}{x \cdot y}\right)\right)\right)} \]

      18. sub-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(\frac{4}{3} \cdot \frac{1}{y} - \frac{1}{x \cdot y}\right)} \]

   8. Applied rewrites 97.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, -1.3333333333333333, 1\right)}{y}} \]

   if 5 < (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x))

   1. Initial program 84.1%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{{x}^{2}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{x}^{2}}{y} \cdot \frac{1}{3}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y} \cdot \frac{1}{3} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{x}{y}\right)} \cdot \frac{1}{3} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{x}{y} \cdot \frac{1}{3}\right)} \]

      5. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{3} \cdot \frac{x}{y}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{3} \cdot \frac{x}{y}\right)} \]

      7. associate-*r/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{3} \cdot x}{y}} \]

      8. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{3} \cdot x}{y}} \]

      9. *-commutative N/A

         \[\leadsto x \cdot \frac{\color{blue}{x \cdot \frac{1}{3}}}{y} \]

      10. lower-*.f64 97.6

          \[\leadsto x \cdot \frac{\color{blue}{x \cdot 0.3333333333333333}}{y} \]

   5. Applied rewrites 97.6%

      \[\leadsto \color{blue}{x \cdot \frac{x \cdot 0.3333333333333333}{y}} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.6%

      \[\leadsto x \cdot \left(\frac{0.3333333333333333}{y} \cdot \color{blue}{x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 - x\right) \cdot \left(1 - x\right) \leq 5:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, -1.3333333333333333, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.3333333333333333}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, -0.3333333333333333, 0.3333333333333333\right) \cdot \frac{3 - x}{y} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (fma x -0.3333333333333333 0.3333333333333333) (/ (- 3.0 x) y)))

C

double code(double x, double y) {
	return fma(x, -0.3333333333333333, 0.3333333333333333) * ((3.0 - x) / y);
}

Julia

function code(x, y)
	return Float64(fma(x, -0.3333333333333333, 0.3333333333333333) * Float64(Float64(3.0 - x) / y))
end

Wolfram

code[x_, y_] := N[(N[(x * -0.3333333333333333 + 0.3333333333333333), $MachinePrecision] * N[(N[(3.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.7%

   \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(1 - x\right) \cdot \left(3 - x\right)}}{y \cdot 3} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(3 - x\right) \cdot \left(1 - x\right)}}{y \cdot 3} \]

   4. associate-/l* N/A

      \[\leadsto \color{blue}{\left(3 - x\right) \cdot \frac{1 - x}{y \cdot 3}} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\frac{1 - x}{y \cdot 3} \cdot \left(3 - x\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1 - x}{y \cdot 3} \cdot \left(3 - x\right)} \]

   7. lower-/.f64 99.5

      \[\leadsto \color{blue}{\frac{1 - x}{y \cdot 3}} \cdot \left(3 - x\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{1 - x}{\color{blue}{y \cdot 3}} \cdot \left(3 - x\right) \]

   9. *-commutative N/A

      \[\leadsto \frac{1 - x}{\color{blue}{3 \cdot y}} \cdot \left(3 - x\right) \]

   10. lower-*.f64 99.5

       \[\leadsto \frac{1 - x}{\color{blue}{3 \cdot y}} \cdot \left(3 - x\right) \]

4. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1 - x}{3 \cdot y} \cdot \left(3 - x\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 - x}{3 \cdot y}} \cdot \left(3 - x\right) \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{3 \cdot y}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(3 - x\right) \cdot \left(1 - x\right)}}{3 \cdot y} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\left(3 - x\right) \cdot \left(1 - x\right)}{\color{blue}{3 \cdot y}} \]

   6. *-commutative N/A

      \[\leadsto \frac{\left(3 - x\right) \cdot \left(1 - x\right)}{\color{blue}{y \cdot 3}} \]

   7. times-frac N/A

      \[\leadsto \color{blue}{\frac{3 - x}{y} \cdot \frac{1 - x}{3}} \]

   8. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{3 - x}{y} \cdot \frac{1 - x}{3}} \]

   9. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{3 - x}{y}} \cdot \frac{1 - x}{3} \]

   10. div-inv N/A

       \[\leadsto \frac{3 - x}{y} \cdot \color{blue}{\left(\left(1 - x\right) \cdot \frac{1}{3}\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{3 - x}{y} \cdot \left(\left(1 - x\right) \cdot \color{blue}{\frac{1}{3}}\right) \]

   12. lower-*.f64 99.6

       \[\leadsto \frac{3 - x}{y} \cdot \color{blue}{\left(\left(1 - x\right) \cdot 0.3333333333333333\right)} \]

6. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{3 - x}{y} \cdot \left(\left(1 - x\right) \cdot 0.3333333333333333\right)} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{3 - x}{y} \cdot \left(\left(1 - x\right) \cdot \frac{1}{3}\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(1 - x\right) \cdot \frac{1}{3}\right) \cdot \frac{3 - x}{y}} \]

   3. lower-*.f64 99.6

      \[\leadsto \color{blue}{\left(\left(1 - x\right) \cdot 0.3333333333333333\right) \cdot \frac{3 - x}{y}} \]

   4. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(1 - x\right) \cdot \frac{1}{3}\right)} \cdot \frac{3 - x}{y} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \left(1 - x\right)\right)} \cdot \frac{3 - x}{y} \]

   6. lift--.f64 N/A

      \[\leadsto \left(\frac{1}{3} \cdot \color{blue}{\left(1 - x\right)}\right) \cdot \frac{3 - x}{y} \]

   7. sub-neg N/A

      \[\leadsto \left(\frac{1}{3} \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right)}\right) \cdot \frac{3 - x}{y} \]

   8. +-commutative N/A

      \[\leadsto \left(\frac{1}{3} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + 1\right)}\right) \cdot \frac{3 - x}{y} \]

   9. distribute-lft-in N/A

      \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \left(\mathsf{neg}\left(x\right)\right) + \frac{1}{3} \cdot 1\right)} \cdot \frac{3 - x}{y} \]

   10. distribute-rgt-neg-in N/A

       \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot x\right)\right)} + \frac{1}{3} \cdot 1\right) \cdot \frac{3 - x}{y} \]

   11. *-commutative N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{1}{3}}\right)\right) + \frac{1}{3} \cdot 1\right) \cdot \frac{3 - x}{y} \]

   12. distribute-rgt-neg-in N/A

       \[\leadsto \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} + \frac{1}{3} \cdot 1\right) \cdot \frac{3 - x}{y} \]

   13. metadata-eval N/A

       \[\leadsto \left(x \cdot \color{blue}{\frac{-1}{3}} + \frac{1}{3} \cdot 1\right) \cdot \frac{3 - x}{y} \]

   14. metadata-eval N/A

       \[\leadsto \left(x \cdot \frac{-1}{3} + \color{blue}{\frac{1}{3}}\right) \cdot \frac{3 - x}{y} \]

   15. lower-fma.f64 99.6

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.3333333333333333, 0.3333333333333333\right)} \cdot \frac{3 - x}{y} \]

8. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.3333333333333333, 0.3333333333333333\right) \cdot \frac{3 - x}{y}} \]
9. Add Preprocessing

Alternative 5: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(3 - x\right) \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, x, 0.3333333333333333\right)}{y} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (- 3.0 x) (/ (fma -0.3333333333333333 x 0.3333333333333333) y)))

C

double code(double x, double y) {
	return (3.0 - x) * (fma(-0.3333333333333333, x, 0.3333333333333333) / y);
}

Julia

function code(x, y)
	return Float64(Float64(3.0 - x) * Float64(fma(-0.3333333333333333, x, 0.3333333333333333) / y))
end

Wolfram

code[x_, y_] := N[(N[(3.0 - x), $MachinePrecision] * N[(N[(-0.3333333333333333 * x + 0.3333333333333333), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.7%

   \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(1 - x\right) \cdot \left(3 - x\right)}}{y \cdot 3} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(3 - x\right) \cdot \left(1 - x\right)}}{y \cdot 3} \]

   4. associate-/l* N/A

      \[\leadsto \color{blue}{\left(3 - x\right) \cdot \frac{1 - x}{y \cdot 3}} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\frac{1 - x}{y \cdot 3} \cdot \left(3 - x\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1 - x}{y \cdot 3} \cdot \left(3 - x\right)} \]

   7. lower-/.f64 99.5

      \[\leadsto \color{blue}{\frac{1 - x}{y \cdot 3}} \cdot \left(3 - x\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{1 - x}{\color{blue}{y \cdot 3}} \cdot \left(3 - x\right) \]

   9. *-commutative N/A

      \[\leadsto \frac{1 - x}{\color{blue}{3 \cdot y}} \cdot \left(3 - x\right) \]

   10. lower-*.f64 99.5

       \[\leadsto \frac{1 - x}{\color{blue}{3 \cdot y}} \cdot \left(3 - x\right) \]

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{1 - x}{3 \cdot y} \cdot \left(3 - x\right)} \]

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

      \[\leadsto \left(\color{blue}{\frac{x}{y} \cdot \frac{-1}{3}} + \frac{1}{3} \cdot \frac{1}{y}\right) \cdot \left(3 - x\right) \]

   2. associate-*l/ N/A

      \[\leadsto \left(\color{blue}{\frac{x \cdot \frac{-1}{3}}{y}} + \frac{1}{3} \cdot \frac{1}{y}\right) \cdot \left(3 - x\right) \]

   3. associate-*r/ N/A

      \[\leadsto \left(\color{blue}{x \cdot \frac{\frac{-1}{3}}{y}} + \frac{1}{3} \cdot \frac{1}{y}\right) \cdot \left(3 - x\right) \]

   4. metadata-eval N/A

      \[\leadsto \left(x \cdot \frac{\color{blue}{\mathsf{neg}\left(\frac{1}{3}\right)}}{y} + \frac{1}{3} \cdot \frac{1}{y}\right) \cdot \left(3 - x\right) \]

   5. distribute-neg-frac N/A

      \[\leadsto \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{3}}{y}\right)\right)} + \frac{1}{3} \cdot \frac{1}{y}\right) \cdot \left(3 - x\right) \]

   6. metadata-eval N/A

      \[\leadsto \left(x \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{3} \cdot 1}}{y}\right)\right) + \frac{1}{3} \cdot \frac{1}{y}\right) \cdot \left(3 - x\right) \]

   7. associate-*r/ N/A

      \[\leadsto \left(x \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{3} \cdot \frac{1}{y}}\right)\right) + \frac{1}{3} \cdot \frac{1}{y}\right) \cdot \left(3 - x\right) \]

   8. distribute-rgt-neg-out N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(x \cdot \left(\frac{1}{3} \cdot \frac{1}{y}\right)\right)\right)} + \frac{1}{3} \cdot \frac{1}{y}\right) \cdot \left(3 - x\right) \]

   9. distribute-lft-neg-out N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{3} \cdot \frac{1}{y}\right)} + \frac{1}{3} \cdot \frac{1}{y}\right) \cdot \left(3 - x\right) \]

   10. distribute-lft1-in N/A

       \[\leadsto \color{blue}{\left(\left(\left(\mathsf{neg}\left(x\right)\right) + 1\right) \cdot \left(\frac{1}{3} \cdot \frac{1}{y}\right)\right)} \cdot \left(3 - x\right) \]

   11. +-commutative N/A

       \[\leadsto \left(\color{blue}{\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right)} \cdot \left(\frac{1}{3} \cdot \frac{1}{y}\right)\right) \cdot \left(3 - x\right) \]

   12. sub-neg N/A

       \[\leadsto \left(\color{blue}{\left(1 - x\right)} \cdot \left(\frac{1}{3} \cdot \frac{1}{y}\right)\right) \cdot \left(3 - x\right) \]

   13. associate-*r/ N/A

       \[\leadsto \left(\left(1 - x\right) \cdot \color{blue}{\frac{\frac{1}{3} \cdot 1}{y}}\right) \cdot \left(3 - x\right) \]

   14. metadata-eval N/A

       \[\leadsto \left(\left(1 - x\right) \cdot \frac{\color{blue}{\frac{1}{3}}}{y}\right) \cdot \left(3 - x\right) \]

   15. associate-/l* N/A

       \[\leadsto \color{blue}{\frac{\left(1 - x\right) \cdot \frac{1}{3}}{y}} \cdot \left(3 - x\right) \]

   16. *-commutative N/A

       \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot \left(1 - x\right)}}{y} \cdot \left(3 - x\right) \]

   17. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(1 - x\right)}{y}} \cdot \left(3 - x\right) \]

   18. sub-neg N/A

       \[\leadsto \frac{\frac{1}{3} \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right)}}{y} \cdot \left(3 - x\right) \]

   19. +-commutative N/A

       \[\leadsto \frac{\frac{1}{3} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + 1\right)}}{y} \cdot \left(3 - x\right) \]

   20. distribute-lft-in N/A

       \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot \left(\mathsf{neg}\left(x\right)\right) + \frac{1}{3} \cdot 1}}{y} \cdot \left(3 - x\right) \]

   21. mul-1-neg N/A

       \[\leadsto \frac{\frac{1}{3} \cdot \color{blue}{\left(-1 \cdot x\right)} + \frac{1}{3} \cdot 1}{y} \cdot \left(3 - x\right) \]

   22. associate-*r* N/A

       \[\leadsto \frac{\color{blue}{\left(\frac{1}{3} \cdot -1\right) \cdot x} + \frac{1}{3} \cdot 1}{y} \cdot \left(3 - x\right) \]

   23. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{\frac{-1}{3}} \cdot x + \frac{1}{3} \cdot 1}{y} \cdot \left(3 - x\right) \]

   24. metadata-eval N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot x + \color{blue}{\frac{1}{3}}}{y} \cdot \left(3 - x\right) \]

   25. lower-fma.f64 99.5

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.3333333333333333, x, 0.3333333333333333\right)}}{y} \cdot \left(3 - x\right) \]

7. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.3333333333333333, x, 0.3333333333333333\right)}{y}} \cdot \left(3 - x\right) \]

8. Final simplification 99.5%

   \[\leadsto \left(3 - x\right) \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, x, 0.3333333333333333\right)}{y} \]
9. Add Preprocessing

Alternative 6: 56.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.75:\\ \;\;\;\;x \cdot \frac{-1.3333333333333333}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.75) (* x (/ -1.3333333333333333 y)) (/ 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.75) {
		tmp = x * (-1.3333333333333333 / y);
	} else {
		tmp = 1.0 / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.75d0)) then
        tmp = x * ((-1.3333333333333333d0) / y)
    else
        tmp = 1.0d0 / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.75) {
		tmp = x * (-1.3333333333333333 / y);
	} else {
		tmp = 1.0 / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.75:
		tmp = x * (-1.3333333333333333 / y)
	else:
		tmp = 1.0 / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.75)
		tmp = Float64(x * Float64(-1.3333333333333333 / y));
	else
		tmp = Float64(1.0 / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.75)
		tmp = x * (-1.3333333333333333 / y);
	else
		tmp = 1.0 / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.75], N[(x * N[(-1.3333333333333333 / y), $MachinePrecision]), $MachinePrecision], N[(1.0 / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.75

   1. Initial program 83.7%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-4}{3} \cdot \frac{x}{y} + \frac{1}{y}} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{-4}{3} \cdot \frac{\color{blue}{1 \cdot x}}{y} + \frac{1}{y} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{-4}{3} \cdot \color{blue}{\left(\frac{1}{y} \cdot x\right)} + \frac{1}{y} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{-4}{3} \cdot \frac{1}{y}\right) \cdot x} + \frac{1}{y} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{4}{3}\right)\right)} \cdot \frac{1}{y}\right) \cdot x + \frac{1}{y} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{y}\right)\right)} \cdot x + \frac{1}{y} \]

      6. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{1}{y} \cdot \frac{4}{3}}\right)\right) \cdot x + \frac{1}{y} \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(\frac{1}{y} \cdot \left(\mathsf{neg}\left(\frac{4}{3}\right)\right)\right)} \cdot x + \frac{1}{y} \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{1}{y} \cdot \color{blue}{\frac{-4}{3}}\right) \cdot x + \frac{1}{y} \]

      9. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{1}{y} \cdot \left(\frac{-4}{3} \cdot x\right)} + \frac{1}{y} \]

      10. *-rgt-identity N/A

          \[\leadsto \frac{1}{y} \cdot \left(\frac{-4}{3} \cdot x\right) + \color{blue}{\frac{1}{y} \cdot 1} \]

      11. distribute-lft-out N/A

          \[\leadsto \color{blue}{\frac{1}{y} \cdot \left(\frac{-4}{3} \cdot x + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{y} \cdot \left(\frac{-4}{3} \cdot x + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{y}} \cdot \left(\frac{-4}{3} \cdot x + 1\right) \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{y} \cdot \left(\color{blue}{x \cdot \frac{-4}{3}} + 1\right) \]

      15. lower-fma.f64 29.2

          \[\leadsto \frac{1}{y} \cdot \color{blue}{\mathsf{fma}\left(x, -1.3333333333333333, 1\right)} \]

   5. Applied rewrites 29.2%

      \[\leadsto \color{blue}{\frac{1}{y} \cdot \mathsf{fma}\left(x, -1.3333333333333333, 1\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-4}{3} \cdot \color{blue}{\frac{x}{y}} \]
   7. Step-by-step derivation

   8. Applied rewrites 29.2%

      \[\leadsto \frac{x \cdot -1.3333333333333333}{\color{blue}{y}} \]
   9. Step-by-step derivation

   10. Applied rewrites 29.2%

       \[\leadsto \frac{-1.3333333333333333}{y} \cdot x \]

   if -0.75 < x

   1. Initial program 94.7%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 66.4

         \[\leadsto \color{blue}{\frac{1}{y}} \]

   5. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\frac{1}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.75:\\ \;\;\;\;x \cdot \frac{-1.3333333333333333}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(x, -1.3333333333333333, 1\right)}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (fma x -1.3333333333333333 1.0) y))

C

double code(double x, double y) {
	return fma(x, -1.3333333333333333, 1.0) / y;
}

Julia

function code(x, y)
	return Float64(fma(x, -1.3333333333333333, 1.0) / y)
end

Wolfram

code[x_, y_] := N[(N[(x * -1.3333333333333333 + 1.0), $MachinePrecision] / y), $MachinePrecision]

Derivation

1. Initial program 91.7%

   \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{y}} \]
4. Step-by-step derivation

   1. lower-/.f64 49.6

      \[\leadsto \color{blue}{\frac{1}{y}} \]

5. Applied rewrites 49.6%

   \[\leadsto \color{blue}{\frac{1}{y}} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-4}{3} \cdot \frac{x}{y} + \frac{1}{y}} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{-4}{3} \cdot x}{y}} + \frac{1}{y} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{x \cdot \frac{-4}{3}}}{y} + \frac{1}{y} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{x \cdot \frac{\frac{-4}{3}}{y}} + \frac{1}{y} \]

   4. metadata-eval N/A

      \[\leadsto x \cdot \frac{\color{blue}{\mathsf{neg}\left(\frac{4}{3}\right)}}{y} + \frac{1}{y} \]

   5. distribute-neg-frac N/A

      \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{4}{3}}{y}\right)\right)} + \frac{1}{y} \]

   6. metadata-eval N/A

      \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{4}{3} \cdot 1}}{y}\right)\right) + \frac{1}{y} \]

   7. associate-*r/ N/A

      \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{4}{3} \cdot \frac{1}{y}}\right)\right) + \frac{1}{y} \]

   8. rgt-mult-inverse N/A

      \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{y}\right)\right) + \frac{\color{blue}{x \cdot \frac{1}{x}}}{y} \]

   9. associate-*r/ N/A

      \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{y}\right)\right) + \color{blue}{x \cdot \frac{\frac{1}{x}}{y}} \]

   10. associate-/r* N/A

       \[\leadsto x \cdot \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{y}\right)\right) + x \cdot \color{blue}{\frac{1}{x \cdot y}} \]

   11. distribute-lft-in N/A

       \[\leadsto \color{blue}{x \cdot \left(\left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{y}\right)\right) + \frac{1}{x \cdot y}\right)} \]

   12. neg-sub0 N/A

       \[\leadsto x \cdot \left(\color{blue}{\left(0 - \frac{4}{3} \cdot \frac{1}{y}\right)} + \frac{1}{x \cdot y}\right) \]

   13. associate-+l- N/A

       \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{4}{3} \cdot \frac{1}{y} - \frac{1}{x \cdot y}\right)\right)} \]

   14. neg-sub0 N/A

       \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{4}{3} \cdot \frac{1}{y} - \frac{1}{x \cdot y}\right)\right)\right)} \]

   15. distribute-rgt-neg-in N/A

       \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{4}{3} \cdot \frac{1}{y} - \frac{1}{x \cdot y}\right)\right)} \]

   16. distribute-lft-neg-in N/A

       \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{4}{3} \cdot \frac{1}{y} - \frac{1}{x \cdot y}\right)} \]

   17. sub-neg N/A

       \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(\frac{4}{3} \cdot \frac{1}{y} + \left(\mathsf{neg}\left(\frac{1}{x \cdot y}\right)\right)\right)} \]

   18. sub-neg N/A

       \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(\frac{4}{3} \cdot \frac{1}{y} - \frac{1}{x \cdot y}\right)} \]

8. Applied rewrites 55.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, -1.3333333333333333, 1\right)}{y}} \]
9. Add Preprocessing

Alternative 8: 50.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 y))

C

double code(double x, double y) {
	return 1.0 / y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / y
end function

Java

public static double code(double x, double y) {
	return 1.0 / y;
}

Python

def code(x, y):
	return 1.0 / y

Julia

function code(x, y)
	return Float64(1.0 / y)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / y;
end

Wolfram

code[x_, y_] := N[(1.0 / y), $MachinePrecision]

Derivation

1. Initial program 91.7%

   \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{y}} \]
4. Step-by-step derivation

   1. lower-/.f64 49.6

      \[\leadsto \color{blue}{\frac{1}{y}} \]

5. Applied rewrites 49.6%

   \[\leadsto \color{blue}{\frac{1}{y}} \]
6. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{1 - x}{y} \cdot \frac{3 - x}{3} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (/ (- 1.0 x) y) (/ (- 3.0 x) 3.0)))

C

double code(double x, double y) {
	return ((1.0 - x) / y) * ((3.0 - x) / 3.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((1.0d0 - x) / y) * ((3.0d0 - x) / 3.0d0)
end function

Java

public static double code(double x, double y) {
	return ((1.0 - x) / y) * ((3.0 - x) / 3.0);
}

Python

def code(x, y):
	return ((1.0 - x) / y) * ((3.0 - x) / 3.0)

Julia

function code(x, y)
	return Float64(Float64(Float64(1.0 - x) / y) * Float64(Float64(3.0 - x) / 3.0))
end

MATLAB

function tmp = code(x, y)
	tmp = ((1.0 - x) / y) * ((3.0 - x) / 3.0);
end

Wolfram

code[x_, y_] := N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] * N[(N[(3.0 - x), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024220 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:bezierFromSweepQ1 from diagrams-lib-1.3.0.3"
  :precision binary64

  :alt
  (! :herbie-platform default (* (/ (- 1 x) y) (/ (- 3 x) 3)))

  (/ (* (- 1.0 x) (- 3.0 x)) (* y 3.0)))