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

Percentage Accurate: 93.6% → 99.8%

Time: 7.1s

Alternatives: 6

Speedup: 1.1×

• 25.3% 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: 93.6% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(3 - x\right) \cdot \left(1 - x\right) \leq 5 \cdot 10^{+62}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -1.3333333333333333\right), x, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} \cdot 0.3333333333333333\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 3.0 x) (- 1.0 x)) 5e+62)
   (/ (fma (fma 0.3333333333333333 x -1.3333333333333333) x 1.0) y)
   (* (* (/ x y) 0.3333333333333333) x)))

C

double code(double x, double y) {
	double tmp;
	if (((3.0 - x) * (1.0 - x)) <= 5e+62) {
		tmp = fma(fma(0.3333333333333333, x, -1.3333333333333333), x, 1.0) / y;
	} else {
		tmp = ((x / y) * 0.3333333333333333) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(3.0 - x) * Float64(1.0 - x)) <= 5e+62)
		tmp = Float64(fma(fma(0.3333333333333333, x, -1.3333333333333333), x, 1.0) / y);
	else
		tmp = Float64(Float64(Float64(x / y) * 0.3333333333333333) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(3.0 - x), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], 5e+62], N[(N[(N[(0.3333333333333333 * x + -1.3333333333333333), $MachinePrecision] * x + 1.0), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x)) < 5.00000000000000029e62

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-out-- N/A

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

      2. *-lft-identity N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. unpow2 N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*l* N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

         \[\leadsto \frac{x \cdot x + -4 \cdot \frac{\color{blue}{{x}^{2}}}{x}}{y \cdot 3} \]

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. *-rgt-identity N/A

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

      13. associate-*r/ N/A

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

      14. rgt-mult-inverse N/A

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

      15. *-rgt-identity N/A

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

      16. distribute-rgt-in N/A

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

      17. metadata-eval N/A

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

      18. sub-neg N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lower--.f64 12.9

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

   5. Applied rewrites 12.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 12.9

         \[\leadsto \frac{\color{blue}{\left(\left(x - 4\right) \cdot x\right) \cdot 0.3333333333333333}}{y} \]

   7. Applied rewrites 12.9%

      \[\leadsto \color{blue}{\frac{\left(\left(x - 4\right) \cdot x\right) \cdot 0.3333333333333333}{y}} \]

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 99.9

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

   10. Applied rewrites 99.9%

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

   if 5.00000000000000029e62 < (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x))

   1. Initial program 85.2%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   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. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 - x\right) \cdot \left(1 - x\right) \leq 5 \cdot 10^{+62}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -1.3333333333333333\right), x, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} \cdot 0.3333333333333333\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (((3.0 - x) * (1.0 - x)) <= 5.0) {
		tmp = fma(-1.3333333333333333, x, 1.0) / y;
	} else {
		tmp = (x / y) * fma(0.3333333333333333, x, -1.3333333333333333);
	}
	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(-1.3333333333333333, x, 1.0) / y);
	else
		tmp = Float64(Float64(x / y) * fma(0.3333333333333333, x, -1.3333333333333333));
	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[(-1.3333333333333333 * x + 1.0), $MachinePrecision] / y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(0.3333333333333333 * x + -1.3333333333333333), $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 99.6%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-out-- N/A

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

      2. *-lft-identity N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. unpow2 N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*l* N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

         \[\leadsto \frac{x \cdot x + -4 \cdot \frac{\color{blue}{{x}^{2}}}{x}}{y \cdot 3} \]

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. *-rgt-identity N/A

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

      13. associate-*r/ N/A

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

      14. rgt-mult-inverse N/A

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

      15. *-rgt-identity N/A

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

      16. distribute-rgt-in N/A

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

      17. metadata-eval N/A

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

      18. sub-neg N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lower--.f64 4.8

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

   5. Applied rewrites 4.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 4.8

         \[\leadsto \frac{\color{blue}{\left(\left(x - 4\right) \cdot x\right) \cdot 0.3333333333333333}}{y} \]

   7. Applied rewrites 4.8%

      \[\leadsto \color{blue}{\frac{\left(\left(x - 4\right) \cdot x\right) \cdot 0.3333333333333333}{y}} \]

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 97.7

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

   10. Applied rewrites 97.7%

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

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

   1. Initial program 86.6%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   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. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-*l/ N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

      15. times-frac N/A

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

   5. Applied rewrites 98.6%

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

4. Final simplification 98.2%

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

Alternative 3: 98.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (((3.0 - x) * (1.0 - x)) <= 5.0) {
		tmp = fma(-1.3333333333333333, x, 1.0) / y;
	} else {
		tmp = ((x / y) * 0.3333333333333333) * x;
	}
	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(-1.3333333333333333, x, 1.0) / y);
	else
		tmp = Float64(Float64(Float64(x / y) * 0.3333333333333333) * x);
	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[(-1.3333333333333333 * x + 1.0), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * x), $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 99.6%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-out-- N/A

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

      2. *-lft-identity N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. unpow2 N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*l* N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

         \[\leadsto \frac{x \cdot x + -4 \cdot \frac{\color{blue}{{x}^{2}}}{x}}{y \cdot 3} \]

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. *-rgt-identity N/A

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

      13. associate-*r/ N/A

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

      14. rgt-mult-inverse N/A

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

      15. *-rgt-identity N/A

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

      16. distribute-rgt-in N/A

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

      17. metadata-eval N/A

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

      18. sub-neg N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lower--.f64 4.8

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

   5. Applied rewrites 4.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 4.8

         \[\leadsto \frac{\color{blue}{\left(\left(x - 4\right) \cdot x\right) \cdot 0.3333333333333333}}{y} \]

   7. Applied rewrites 4.8%

      \[\leadsto \color{blue}{\frac{\left(\left(x - 4\right) \cdot x\right) \cdot 0.3333333333333333}{y}} \]

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 97.7

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

   10. Applied rewrites 97.7%

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

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

   1. Initial program 86.6%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   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. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 97.3

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

   5. Applied rewrites 97.3%

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

4. Final simplification 97.5%

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

Alternative 4: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(1 - x\right) \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, x, 1\right)}{y} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (- 1.0 x) (/ (fma -0.3333333333333333 x 1.0) y)))

C

double code(double x, double y) {
	return (1.0 - x) * (fma(-0.3333333333333333, x, 1.0) / y);
}

Julia

function code(x, y)
	return Float64(Float64(1.0 - x) * Float64(fma(-0.3333333333333333, x, 1.0) / y))
end

Wolfram

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

Derivation

1. Initial program 92.8%

   \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
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. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. associate-*l/ N/A

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

   8. *-commutative N/A

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

   9. lower-/.f64 N/A

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

   10. sub-neg N/A

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

   11. mul-1-neg N/A

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

   12. +-commutative N/A

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

   13. distribute-lft-in N/A

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

   14. mul-1-neg N/A

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

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

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

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

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

   17. metadata-eval N/A

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

   18. lower-fma.f64 N/A

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

   19. metadata-eval N/A

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

   20. lower--.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 99.8%

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

9. Final simplification 99.8%

   \[\leadsto \left(1 - x\right) \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, x, 1\right)}{y} \]
10. Add Preprocessing

Alternative 5: 57.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.8%

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

3. Taylor expanded in x around inf

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

   1. distribute-rgt-out-- N/A

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

   2. *-lft-identity N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. unpow2 N/A

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

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

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

   6. metadata-eval N/A

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

   7. associate-*l* N/A

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

   8. associate-*l/ N/A

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

   9. *-lft-identity N/A

      \[\leadsto \frac{x \cdot x + -4 \cdot \frac{\color{blue}{{x}^{2}}}{x}}{y \cdot 3} \]

   10. unpow2 N/A

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

   11. associate-/l* N/A

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

   12. *-rgt-identity N/A

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

   13. associate-*r/ N/A

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

   14. rgt-mult-inverse N/A

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

   15. *-rgt-identity N/A

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

   16. distribute-rgt-in N/A

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

   17. metadata-eval N/A

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

   18. sub-neg N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 N/A

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

   21. lower--.f64 47.0

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

5. Applied rewrites 47.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 47.4

      \[\leadsto \frac{\color{blue}{\left(\left(x - 4\right) \cdot x\right) \cdot 0.3333333333333333}}{y} \]

7. Applied rewrites 47.4%

   \[\leadsto \color{blue}{\frac{\left(\left(x - 4\right) \cdot x\right) \cdot 0.3333333333333333}{y}} \]

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 53.5

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

10. Applied rewrites 53.5%

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

Alternative 6: 51.6% 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 92.8%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 48.7

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

5. Applied rewrites 48.7%

   \[\leadsto \color{blue}{\frac{1}{y}} \]
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 2024331 
(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)))