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

Percentage Accurate: 93.8% → 99.8%

Time: 9.2s

Alternatives: 9

Speedup: 1.0×

• 25.0% 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: 93.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(3 - x\right) \cdot \left(1 - x\right) \leq 2 \cdot 10^{+102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.3333333333333333, -1.3333333333333333\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{3 \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 3.0 x) (- 1.0 x)) 2e+102)
   (/ (fma x (fma x 0.3333333333333333 -1.3333333333333333) 1.0) y)
   (* x (/ x (* 3.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (((3.0 - x) * (1.0 - x)) <= 2e+102) {
		tmp = fma(x, fma(x, 0.3333333333333333, -1.3333333333333333), 1.0) / y;
	} else {
		tmp = x * (x / (3.0 * y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(3.0 - x) * Float64(1.0 - x)) <= 2e+102)
		tmp = Float64(fma(x, fma(x, 0.3333333333333333, -1.3333333333333333), 1.0) / y);
	else
		tmp = Float64(x * Float64(x / Float64(3.0 * y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(3.0 - x), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], 2e+102], N[(N[(x * N[(x * 0.3333333333333333 + -1.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision], N[(x * N[(x / N[(3.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x)) < 1.99999999999999995e102

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. *-lft-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 100.0%

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

   if 1.99999999999999995e102 < (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x))

   1. Initial program 89.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 - x\right) \cdot \left(1 - x\right) \leq 2 \cdot 10^{+102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.3333333333333333, -1.3333333333333333\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{3 \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. *-lft-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 98.3%

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

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

   1. Initial program 90.8%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. associate-*r* N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

      15. times-frac N/A

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

   5. Applied rewrites 98.7%

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

4. Final simplification 98.5%

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

Alternative 3: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(3 - x\right) \cdot \left(1 - x\right) \leq 20:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.3333333333333333, x, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{3 \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 3.0 x) (- 1.0 x)) 20.0)
   (/ (fma -1.3333333333333333 x 1.0) y)
   (* x (/ x (* 3.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (((3.0 - x) * (1.0 - x)) <= 20.0) {
		tmp = fma(-1.3333333333333333, x, 1.0) / y;
	} else {
		tmp = x * (x / (3.0 * y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(3.0 - x) * Float64(1.0 - x)) <= 20.0)
		tmp = Float64(fma(-1.3333333333333333, x, 1.0) / y);
	else
		tmp = Float64(x * Float64(x / Float64(3.0 * y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(3.0 - x), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], 20.0], N[(N[(-1.3333333333333333 * x + 1.0), $MachinePrecision] / y), $MachinePrecision], N[(x * N[(x / N[(3.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x)) < 20

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. *-lft-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 97.7%

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

   if 20 < (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x))

   1. Initial program 90.7%

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

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 98.6

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

   5. Applied rewrites 98.6%

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

   7. Applied rewrites 98.7%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 - x\right) \cdot \left(1 - x\right) \leq 20:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.3333333333333333, x, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{3 \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(3 - x\right) \cdot \left(1 - x\right) \leq 20:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.3333333333333333, x, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.3333333333333333}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 3.0 x) (- 1.0 x)) 20.0)
   (/ (fma -1.3333333333333333 x 1.0) y)
   (* x (* x (/ 0.3333333333333333 y)))))

C

double code(double x, double y) {
	double tmp;
	if (((3.0 - x) * (1.0 - x)) <= 20.0) {
		tmp = fma(-1.3333333333333333, x, 1.0) / y;
	} else {
		tmp = x * (x * (0.3333333333333333 / y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(3.0 - x) * Float64(1.0 - x)) <= 20.0)
		tmp = Float64(fma(-1.3333333333333333, x, 1.0) / y);
	else
		tmp = Float64(x * Float64(x * Float64(0.3333333333333333 / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(3.0 - x), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], 20.0], N[(N[(-1.3333333333333333 * x + 1.0), $MachinePrecision] / y), $MachinePrecision], N[(x * N[(x * N[(0.3333333333333333 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x)) < 20

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. *-lft-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 97.7%

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

   if 20 < (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x))

   1. Initial program 90.7%

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

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 98.6

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

   5. Applied rewrites 98.6%

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

   7. Applied rewrites 98.6%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 - x\right) \cdot \left(1 - x\right) \leq 20:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.3333333333333333, x, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.3333333333333333}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. clear-num N/A

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

   6. frac-times N/A

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

   7. *-lft-identity N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 6: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 99.6

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 7: 56.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -0.75

   1. Initial program 88.4%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. associate-*r* N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

      15. times-frac N/A

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 30.6%

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

   10. Applied rewrites 30.6%

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

   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 71.6

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

   5. Applied rewrites 71.6%

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

Alternative 8: 56.3% 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 95.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. clear-num N/A

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

   6. frac-times N/A

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

   7. *-lft-identity N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in y around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

7. Applied rewrites 95.5%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 59.3%

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

Alternative 9: 50.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.3%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 52.4

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

5. Applied rewrites 52.4%

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

Developer Target 1: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024238 
(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)))