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

Percentage Accurate: 93.7% → 99.8%

Time: 8.9s

Alternatives: 9

Speedup: 1.0×

• 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: 93.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(3 - x\right) \cdot \left(1 - x\right) \leq 5 \cdot 10^{+181}:\\ \;\;\;\;\frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, -0.3333333333333333, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{3 \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 3.0 x) (- 1.0 x)) 5e+181)
   (/ (* (- 1.0 x) (fma x -0.3333333333333333 1.0)) y)
   (* x (/ x (* 3.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (((3.0 - x) * (1.0 - x)) <= 5e+181) {
		tmp = ((1.0 - x) * fma(x, -0.3333333333333333, 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)) <= 5e+181)
		tmp = Float64(Float64(Float64(1.0 - x) * fma(x, -0.3333333333333333, 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], 5e+181], N[(N[(N[(1.0 - x), $MachinePrecision] * N[(x * -0.3333333333333333 + 1.0), $MachinePrecision]), $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)) < 5.0000000000000003e181

   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 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. 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

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-rgt-in N/A

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

      12. mul-1-neg N/A

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

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

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

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

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. metadata-eval 99.9

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

   5. Applied rewrites 99.9%

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

   if 5.0000000000000003e181 < (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x))

   1. Initial program 84.6%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 84.6

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

   5. Applied rewrites 84.6%

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

      1. lift-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 99.8

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 - x\right) \cdot \left(1 - x\right) \leq 5 \cdot 10^{+181}:\\ \;\;\;\;\frac{\left(1 - x\right) \cdot \mathsf{fma}\left(x, -0.3333333333333333, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{3 \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(3 - x\right) \cdot \left(1 - x\right) \leq 5 \cdot 10^{+181}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.3333333333333333, -1.3333333333333333\right), x, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{3 \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 3.0 x) (- 1.0 x)) 5e+181)
   (/ (fma (fma x 0.3333333333333333 -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)) <= 5e+181) {
		tmp = fma(fma(x, 0.3333333333333333, -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)) <= 5e+181)
		tmp = Float64(fma(fma(x, 0.3333333333333333, -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], 5e+181], N[(N[(N[(x * 0.3333333333333333 + -1.3333333333333333), $MachinePrecision] * 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)) < 5.0000000000000003e181

   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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. distribute-rgt-in N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 99.9

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

   8. Applied rewrites 99.9%

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

   if 5.0000000000000003e181 < (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x))

   1. Initial program 84.6%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 84.6

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

   5. Applied rewrites 84.6%

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

      1. lift-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 99.8

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 - x\right) \cdot \left(1 - x\right) \leq 5 \cdot 10^{+181}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.3333333333333333, -1.3333333333333333\right), x, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{3 \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 3.0 x) (- 1.0 x)) 5.0)
   (/ (fma -1.3333333333333333 x 1.0) y)
   (* x (/ (fma x 0.3333333333333333 -1.3333333333333333) 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 = x * (fma(x, 0.3333333333333333, -1.3333333333333333) / 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(x * Float64(fma(x, 0.3333333333333333, -1.3333333333333333) / 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[(x * N[(N[(x * 0.3333333333333333 + -1.3333333333333333), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 98.8

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

   8. Applied rewrites 98.8%

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

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

   1. Initial program 88.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 88.4

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

   5. Applied rewrites 88.4%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval 88.4

          \[\leadsto \mathsf{fma}\left(x, x + -4, 3\right) \cdot \frac{\color{blue}{0.3333333333333333}}{y} \]

   7. Applied rewrites 88.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x + -4, 3\right) \cdot \frac{0.3333333333333333}{y}} \]

   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 98.9%

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

4. Final simplification 98.9%

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

Alternative 4: 98.2% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 3.0 x) (- 1.0 x)) 5.0)
   (/ (fma -1.3333333333333333 x 1.0) y)
   (* x (/ (* x 0.3333333333333333) 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 = 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)) <= 5.0)
		tmp = Float64(fma(-1.3333333333333333, x, 1.0) / y);
	else
		tmp = Float64(x * Float64(Float64(x * 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], 5.0], N[(N[(-1.3333333333333333 * x + 1.0), $MachinePrecision] / y), $MachinePrecision], N[(x * N[(N[(x * 0.3333333333333333), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 98.8

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

   8. Applied rewrites 98.8%

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

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

   1. Initial program 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

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{{x}^{2}}{y}} \]
   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 98.1

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

   5. Applied rewrites 98.1%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 - x\right) \cdot \left(1 - x\right) \leq 5:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.3333333333333333, x, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x \cdot 0.3333333333333333}{y}\\ \end{array} \]
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 94.2%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. clear-num N/A

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

   7. frac-times N/A

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

   8. *-lft-identity N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. 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

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

FPCore

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

C

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

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)) * 0.3333333333333333d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-rgt-identity N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. metadata-eval 99.6

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

4. Applied rewrites 99.6%

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

Alternative 7: 57.6% accurate, 1.2× speedup

Math

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

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(Float64(x * -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[(N[(x * -1.3333333333333333), $MachinePrecision] / y), $MachinePrecision], N[(1.0 / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.75

   1. Initial program 88.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 29.0

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

   5. Applied rewrites 29.0%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 29.0

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

   8. Applied rewrites 29.0%

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

   if -0.75 < x

   1. Initial program 96.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 69.0

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

   5. Applied rewrites 69.0%

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

4. Final simplification 58.7%

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

Alternative 8: 57.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 58.4

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

5. Applied rewrites 58.4%

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

6. Taylor expanded in y around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. associate-*r* N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 58.6

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

8. Applied rewrites 58.6%

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

Alternative 9: 50.8% 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 94.2%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 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 2024219 
(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)))