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

Percentage Accurate: 93.6% → 99.8%

Time: 6.4s

Alternatives: 12

Speedup: 1.1×

• 24.8% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - x\right) \cdot \left(3 - x\right) \leq 2 \cdot 10^{+61}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -1.3333333333333333\right), x, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} \cdot x\right) \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 1.0 x) (- 3.0 x)) 2e+61)
   (/ (fma (fma 0.3333333333333333 x -1.3333333333333333) x 1.0) y)
   (* (* (/ x y) x) 0.3333333333333333)))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 - x) * (3.0 - x)) <= 2e+61) {
		tmp = fma(fma(0.3333333333333333, x, -1.3333333333333333), x, 1.0) / y;
	} else {
		tmp = ((x / y) * x) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(1.0 - x) * Float64(3.0 - x)) <= 2e+61)
		tmp = Float64(fma(fma(0.3333333333333333, x, -1.3333333333333333), x, 1.0) / y);
	else
		tmp = Float64(Float64(Float64(x / y) * x) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(1.0 - x), $MachinePrecision] * N[(3.0 - x), $MachinePrecision]), $MachinePrecision], 2e+61], N[(N[(N[(0.3333333333333333 * x + -1.3333333333333333), $MachinePrecision] * x + 1.0), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x)) < 1.9999999999999999e61

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 12.5

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

   5. Applied rewrites 12.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 12.5

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

   7. Applied rewrites 12.5%

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

   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 1.9999999999999999e61 < (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x))

   1. Initial program 85.0%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 99.8

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

   7. Applied rewrites 99.8%

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

Alternative 2: 57.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.75) {
		tmp = (-1.3333333333333333 / y) * x;
	} else {
		tmp = pow(y, -1.0);
	}
	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 = ((-1.3333333333333333d0) / y) * x
    else
        tmp = y ** (-1.0d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -0.75

   1. Initial program 79.4%

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot \frac{1}{y} - \frac{4}{3} \cdot \frac{1}{x \cdot y}\right)} \]
   6. 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-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*l/ N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      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

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 29.6%

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

   if -0.75 < x

   1. Initial program 98.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 68.1

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

   5. Applied rewrites 68.1%

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

4. Final simplification 57.8%

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

Alternative 3: 51.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ {y}^{-1} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (pow y -1.0))

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return Math.pow(y, -1.0);
}

Python

def code(x, y):
	return math.pow(y, -1.0)

Julia

function code(x, y)
	return y ^ -1.0
end

MATLAB

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

Wolfram

code[x_, y_] := N[Power[y, -1.0], $MachinePrecision]

Derivation

1. Initial program 93.2%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 51.3

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

5. Applied rewrites 51.3%

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

6. Final simplification 51.3%

   \[\leadsto {y}^{-1} \]
7. Add Preprocessing

Alternative 4: 98.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(1.0 - x), $MachinePrecision] * N[(3.0 - x), $MachinePrecision]), $MachinePrecision], 5.0], N[(N[(-1.3333333333333333 * x + 1.0), $MachinePrecision] / y), $MachinePrecision], N[(N[(0.3333333333333333 * x + -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 99.6%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 5.1

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

   7. Applied rewrites 5.1%

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

   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 98.3

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

   10. Applied rewrites 98.3%

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

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

   1. Initial program 86.7%

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

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      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 98.8%

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

4. Final simplification 98.5%

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

Alternative 5: 99.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.2%

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

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{\frac{3 - x}{y}}{\frac{3}{1 - x}}} \]
5. Add Preprocessing

Alternative 6: 98.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      1. unpow2 N/A

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

      2. lower-*.f64 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 5.1

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

   7. Applied rewrites 5.1%

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

   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 98.3

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

   10. Applied rewrites 98.3%

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

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

   1. Initial program 86.7%

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 97.0

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

   7. Applied rewrites 97.0%

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

Alternative 7: 98.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      1. unpow2 N/A

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

      2. lower-*.f64 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 5.1

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

   7. Applied rewrites 5.1%

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

   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 98.3

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

   10. Applied rewrites 98.3%

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

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

   1. Initial program 86.7%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   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 96.9

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

   5. Applied rewrites 96.9%

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

4. Final simplification 97.6%

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

Alternative 8: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{3 - x}{y} \cdot \left(1 - x\right)\right) \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (* (/ (- 3.0 x) y) (- 1.0 x)) 0.3333333333333333))

C

double code(double x, double y) {
	return (((3.0 - x) / y) * (1.0 - x)) * 0.3333333333333333;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (((3.0d0 - x) / y) * (1.0d0 - x)) * 0.3333333333333333d0
end function

Java

public static double code(double x, double y) {
	return (((3.0 - x) / y) * (1.0 - x)) * 0.3333333333333333;
}

Python

def code(x, y):
	return (((3.0 - x) / y) * (1.0 - x)) * 0.3333333333333333

Julia

function code(x, y)
	return Float64(Float64(Float64(Float64(3.0 - x) / y) * Float64(1.0 - x)) * 0.3333333333333333)
end

MATLAB

function tmp = code(x, y)
	tmp = (((3.0 - x) / y) * (1.0 - x)) * 0.3333333333333333;
end

Wolfram

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

Derivation

1. Initial program 93.2%

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

4. Applied rewrites 99.6%

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

Alternative 9: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.2%

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

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

   12. distribute-lft-in N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. lower-fma.f64 N/A

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

   17. metadata-eval N/A

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

   18. 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. Step-by-step derivation

7. Applied rewrites 99.6%

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

8. Final simplification 99.6%

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

Alternative 10: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.2%

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

4. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. metadata-eval N/A

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

   8. associate-/r/ N/A

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

   9. frac-2neg N/A

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

   10. associate-/r/ N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. sub-neg N/A

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

   14. distribute-neg-in N/A

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

   15. metadata-eval N/A

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

   16. remove-double-neg N/A

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

   17. distribute-lft-in N/A

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

   18. metadata-eval N/A

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

   19. +-commutative N/A

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

   20. lift-fma.f64 N/A

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

   21. lower-*.f64 99.5

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

6. Applied rewrites 99.5%

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

Alternative 11: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.2%

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

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

   12. distribute-lft-in N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. lower-fma.f64 N/A

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

   17. metadata-eval N/A

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

   18. 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. Final simplification 99.5%

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

Alternative 12: 57.1% 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 93.2%

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

3. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 44.2

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

5. Applied rewrites 44.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. div-inv N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 44.2

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

7. Applied rewrites 44.2%

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

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 57.6

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

10. Applied rewrites 57.6%

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.3333333333333333, x, 1\right)}}{y} \]
11. 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 2024298 
(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)))