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

Percentage Accurate: 93.3% → 99.8%

Time: 6.2s

Alternatives: 11

Speedup: 1.1×

• 24.3% 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: 93.3% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - x\right) \cdot \left(3 - x\right) \leq 10^{+220}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -1.3333333333333333\right), x, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.3333333333333333}{y} \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 1.0 x) (- 3.0 x)) 1e+220)
   (/ (fma (fma 0.3333333333333333 x -1.3333333333333333) x 1.0) y)
   (* (* (/ 0.3333333333333333 y) x) x)))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 - x) * (3.0 - x)) <= 1e+220) {
		tmp = fma(fma(0.3333333333333333, x, -1.3333333333333333), x, 1.0) / y;
	} else {
		tmp = ((0.3333333333333333 / y) * x) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(1.0 - x) * Float64(3.0 - x)) <= 1e+220)
		tmp = Float64(fma(fma(0.3333333333333333, x, -1.3333333333333333), x, 1.0) / y);
	else
		tmp = Float64(Float64(Float64(0.3333333333333333 / y) * x) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(1.0 - x), $MachinePrecision] * N[(3.0 - x), $MachinePrecision]), $MachinePrecision], 1e+220], N[(N[(N[(0.3333333333333333 * x + -1.3333333333333333), $MachinePrecision] * x + 1.0), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(0.3333333333333333 / y), $MachinePrecision] * x), $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)) < 1e220

   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 1e220 < (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x))

   1. Initial program 80.7%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   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. *-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 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.8%

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

   9. Applied rewrites 99.8%

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

Alternative 2: 57.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.75)
		tmp = Float64(-1.3333333333333333 * Float64(x / y));
	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 * (x / y);
	else
		tmp = y ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -0.75

   1. Initial program 85.3%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 30.1%

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

   10. Applied rewrites 30.1%

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

   if -0.75 < x

   1. Initial program 96.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 63.9

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

   5. Applied rewrites 63.9%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.75:\\ \;\;\;\;-1.3333333333333333 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;{y}^{-1}\\ \end{array} \]
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 93.1%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 47.5

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

5. Applied rewrites 47.5%

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

6. Final simplification 47.5%

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

Alternative 4: 98.7% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 1.0 x) (- 3.0 x)) 5.0)
   (/ (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)) <= 5.0) {
		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)) <= 5.0)
		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], 5.0], 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)) < 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 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 99.9

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

   5. Applied rewrites 99.9%

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

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

   1. Initial program 87.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 98.6%

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

Alternative 5: 98.1% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 1.0 x) (- 3.0 x)) 5.0)
   (/ (fma -1.3333333333333333 x 1.0) y)
   (* (/ x (* 3.0 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 = (x / (3.0 * 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(x / Float64(3.0 * 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[(x / N[(3.0 * 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 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 99.9

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

   5. Applied rewrites 99.9%

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

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

   1. Initial program 87.7%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   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. *-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 97.3

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

   5. Applied rewrites 97.3%

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

   7. Applied rewrites 97.3%

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

   9. Applied rewrites 97.3%

      \[\leadsto \frac{x}{3 \cdot y} \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.1% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 1.0 x) (- 3.0 x)) 5.0)
   (/ (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)) <= 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(1.0 - x) * Float64(3.0 - x)) <= 5.0)
		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], 5.0], 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)) < 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 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 99.9

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

   5. Applied rewrites 99.9%

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

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

   1. Initial program 87.7%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   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. *-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 97.3

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

   5. Applied rewrites 97.3%

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

   7. Applied rewrites 97.3%

      \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(0.3333333333333333 \cdot x\right)} \]
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 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 99.9

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

   5. Applied rewrites 99.9%

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

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

   1. Initial program 87.7%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   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. *-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 97.3

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

   5. Applied rewrites 97.3%

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

   7. Applied rewrites 97.3%

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

Alternative 8: 99.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

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) / 3.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.1%

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

4. Applied rewrites 99.6%

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

Alternative 9: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.1%

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in x around 0

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

   1. lower--.f64 99.6

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

7. Applied rewrites 99.6%

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

Alternative 10: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.1%

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in x around 0

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

   1. lower--.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in x around 0

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

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

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. div-add-rev N/A

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

   7. lower-/.f64 N/A

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

   8. lower-fma.f64 99.5

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

10. Applied rewrites 99.5%

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

Alternative 11: 57.0% 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 93.1%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-4}{3} \cdot \frac{x}{y} + \frac{1}{y}} \]
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 53.8

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

5. Applied rewrites 53.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.3333333333333333, x, 1\right)}{y}} \]
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 2024342 
(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)))