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

Percentage Accurate: 93.4% → 99.7%

Time: 4.3s

Alternatives: 10

Speedup: 1.1×

• 25.1% 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: 93.4% 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: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - x\right) \cdot \left(3 - x\right) \leq 10^{+23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -1.3333333333333333\right), x, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-0.3333333333333333}{y} \cdot x\right) \cdot \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 1.0 x) (- 3.0 x)) 1e+23)
   (/ (fma (fma 0.3333333333333333 x -1.3333333333333333) x 1.0) y)
   (* (* (/ -0.3333333333333333 y) x) (- 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 - x) * (3.0 - x)) <= 1e+23) {
		tmp = fma(fma(0.3333333333333333, x, -1.3333333333333333), x, 1.0) / y;
	} else {
		tmp = ((-0.3333333333333333 / y) * x) * (1.0 - x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(1.0 - x) * Float64(3.0 - x)) <= 1e+23)
		tmp = Float64(fma(fma(0.3333333333333333, x, -1.3333333333333333), x, 1.0) / y);
	else
		tmp = Float64(Float64(Float64(-0.3333333333333333 / y) * x) * Float64(1.0 - x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(1.0 - x), $MachinePrecision] * N[(3.0 - x), $MachinePrecision]), $MachinePrecision], 1e+23], N[(N[(N[(0.3333333333333333 * x + -1.3333333333333333), $MachinePrecision] * x + 1.0), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(-0.3333333333333333 / y), $MachinePrecision] * x), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x)) < 9.9999999999999992e22

   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 0

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

      1. *-commutative N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. div-add-rev N/A

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

      8. associate-*l/ N/A

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

      9. div-add-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-fma.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if 9.9999999999999992e22 < (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x))

   1. Initial program 83.9%

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

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.7%

      \[\leadsto \left(\frac{-0.3333333333333333}{y} \cdot x\right) \cdot \left(\color{blue}{1} - x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 58.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -0.75

   1. Initial program 87.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 31.3%

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

   if -0.75 < x

   1. Initial program 93.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 66.5

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

   5. Applied rewrites 66.5%

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

4. Final simplification 57.5%

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

Alternative 3: 51.4% 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 92.0%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 50.8

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

5. Applied rewrites 50.8%

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

6. Final simplification 50.8%

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

Alternative 4: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - x\right) \cdot \left(3 - x\right) \leq 10^{+286}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -1.3333333333333333\right), x, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, x, -1.3333333333333333\right)}{y} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 1.0 x) (- 3.0 x)) 1e+286)
   (/ (fma (fma 0.3333333333333333 x -1.3333333333333333) x 1.0) y)
   (* (/ (fma 0.3333333333333333 x -1.3333333333333333) y) x)))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 - x) * (3.0 - x)) <= 1e+286) {
		tmp = fma(fma(0.3333333333333333, x, -1.3333333333333333), x, 1.0) / y;
	} else {
		tmp = (fma(0.3333333333333333, x, -1.3333333333333333) / y) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(1.0 - x) * Float64(3.0 - x)) <= 1e+286)
		tmp = Float64(fma(fma(0.3333333333333333, x, -1.3333333333333333), x, 1.0) / y);
	else
		tmp = Float64(Float64(fma(0.3333333333333333, x, -1.3333333333333333) / y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(1.0 - x), $MachinePrecision] * N[(3.0 - x), $MachinePrecision]), $MachinePrecision], 1e+286], N[(N[(N[(0.3333333333333333 * x + -1.3333333333333333), $MachinePrecision] * x + 1.0), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(0.3333333333333333 * x + -1.3333333333333333), $MachinePrecision] / y), $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)) < 1.00000000000000003e286

   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 0

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

      1. *-commutative N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. div-add-rev N/A

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

      8. associate-*l/ N/A

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

      9. div-add-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-fma.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 1.00000000000000003e286 < (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x))

   1. Initial program 74.4%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.9%

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

Alternative 5: 98.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 1.0 x) (- 3.0 x)) 10.0)
   (/ (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)) <= 10.0) {
		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)) <= 10.0)
		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], 10.0], 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)) < 10

   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 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. lower-fma.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if 10 < (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x))

   1. Initial program 84.5%

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

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 98.2%

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

Alternative 6: 98.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 1.0 x) (- 3.0 x)) 10.0)
   (/ (fma -1.3333333333333333 x 1.0) y)
   (* (/ (fma 0.3333333333333333 x -1.3333333333333333) y) x)))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 - x) * (3.0 - x)) <= 10.0) {
		tmp = fma(-1.3333333333333333, x, 1.0) / y;
	} else {
		tmp = (fma(0.3333333333333333, x, -1.3333333333333333) / y) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(1.0 - x) * Float64(3.0 - x)) <= 10.0)
		tmp = Float64(fma(-1.3333333333333333, x, 1.0) / y);
	else
		tmp = Float64(Float64(fma(0.3333333333333333, x, -1.3333333333333333) / y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(1.0 - x), $MachinePrecision] * N[(3.0 - x), $MachinePrecision]), $MachinePrecision], 10.0], N[(N[(-1.3333333333333333 * x + 1.0), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(0.3333333333333333 * x + -1.3333333333333333), $MachinePrecision] / y), $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)) < 10

   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 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. lower-fma.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if 10 < (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x))

   1. Initial program 84.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 98.2%

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

Alternative 7: 98.5% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 1.0 x) (- 3.0 x)) 10.0)
   (/ (fma -1.3333333333333333 x 1.0) y)
   (* (/ x y) (* 0.3333333333333333 x))))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 - x) * (3.0 - x)) <= 10.0) {
		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(1.0 - x) * Float64(3.0 - x)) <= 10.0)
		tmp = Float64(fma(-1.3333333333333333, x, 1.0) / y);
	else
		tmp = Float64(Float64(x / y) * Float64(0.3333333333333333 * x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(1.0 - x), $MachinePrecision] * N[(3.0 - x), $MachinePrecision]), $MachinePrecision], 10.0], N[(N[(-1.3333333333333333 * x + 1.0), $MachinePrecision] / y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x)) < 10

   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 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. lower-fma.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if 10 < (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x))

   1. Initial program 84.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 96.2

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

   5. Applied rewrites 96.2%

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

   7. Applied rewrites 97.4%

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

Alternative 8: 98.4% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 1.0 x) (- 3.0 x)) 10.0)
   (/ (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)) <= 10.0) {
		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)) <= 10.0)
		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], 10.0], 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)) < 10

   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 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. lower-fma.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if 10 < (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x))

   1. Initial program 84.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 96.2

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

   5. Applied rewrites 96.2%

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

Alternative 9: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.0%

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

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

5. Applied rewrites 99.8%

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

Alternative 10: 57.6% 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 92.0%

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

3. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. div-add-rev N/A

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

   3. lower-/.f64 N/A

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

   4. lower-fma.f64 57.0

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

5. Applied rewrites 57.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.3333333333333333, x, 1\right)}{y}} \]
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 2024332 
(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)))