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

Percentage Accurate: 93.3% → 99.8%

Time: 7.1s

Alternatives: 8

Speedup: 1.1×

• 23.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.3% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(3 - x\right) \cdot \left(1 - x\right) \leq 2 \cdot 10^{+141}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.3333333333333333, -1.3333333333333333\right), x, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot 3} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 3.0 x) (- 1.0 x)) 2e+141)
   (/ (fma (fma x 0.3333333333333333 -1.3333333333333333) x 1.0) y)
   (* (/ x (* y 3.0)) x)))

C

double code(double x, double y) {
	double tmp;
	if (((3.0 - x) * (1.0 - x)) <= 2e+141) {
		tmp = fma(fma(x, 0.3333333333333333, -1.3333333333333333), x, 1.0) / y;
	} else {
		tmp = (x / (y * 3.0)) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(3.0 - x) * Float64(1.0 - x)) <= 2e+141)
		tmp = Float64(fma(fma(x, 0.3333333333333333, -1.3333333333333333), x, 1.0) / y);
	else
		tmp = Float64(Float64(x / Float64(y * 3.0)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(3.0 - x), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], 2e+141], N[(N[(N[(x * 0.3333333333333333 + -1.3333333333333333), $MachinePrecision] * x + 1.0), $MachinePrecision] / y), $MachinePrecision], N[(N[(x / N[(y * 3.0), $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)) < 2.00000000000000003e141

   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. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. mul-1-neg N/A

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

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

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

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

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

      17. metadata-eval N/A

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

      18. lower-fma.f64 N/A

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

      19. metadata-eval N/A

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

      20. lower--.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.9%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 99.9%

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

   if 2.00000000000000003e141 < (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x))

   1. Initial program 87.5%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 - x\right) \cdot \left(1 - x\right) \leq 2 \cdot 10^{+141}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.3333333333333333, -1.3333333333333333\right), x, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot 3} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.7× speedup

Math

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

FPCore

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

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 / y) * fma(x, 0.3333333333333333, -1.3333333333333333);
	}
	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(Float64(x / y) * fma(x, 0.3333333333333333, -1.3333333333333333));
	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 / y), $MachinePrecision] * N[(x * 0.3333333333333333 + -1.3333333333333333), $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 y around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. mul-1-neg N/A

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

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

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

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

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

      17. metadata-eval N/A

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

      18. lower-fma.f64 N/A

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

      19. metadata-eval N/A

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

      20. lower--.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 98.4%

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

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

   1. Initial program 90.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot \frac{1}{y} - \frac{4}{3} \cdot \frac{1}{x \cdot y}\right)} \]
   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 98.2%

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

4. Final simplification 98.3%

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

Alternative 3: 98.3% accurate, 0.7× speedup

Math

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

FPCore

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

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 / (y * 3.0)) * x;
	}
	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(Float64(x / Float64(y * 3.0)) * x);
	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 / N[(y * 3.0), $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 y around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. mul-1-neg N/A

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

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

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

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

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

      17. metadata-eval N/A

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

      18. lower-fma.f64 N/A

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

      19. metadata-eval N/A

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

      20. lower--.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 98.4%

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

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

   1. Initial program 90.2%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 96.7

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

   5. Applied rewrites 96.7%

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

   7. Applied rewrites 96.8%

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

4. Final simplification 97.7%

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

Alternative 4: 98.3% accurate, 0.7× speedup

Math

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

FPCore

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

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 = ((0.3333333333333333 / y) * x) * x;
	}
	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(Float64(Float64(0.3333333333333333 / y) * x) * x);
	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[(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)) < 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 y around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. mul-1-neg N/A

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

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

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

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

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

      17. metadata-eval N/A

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

      18. lower-fma.f64 N/A

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

      19. metadata-eval N/A

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

      20. lower--.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 98.4%

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

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

   1. Initial program 90.2%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 96.7

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

   5. Applied rewrites 96.7%

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

   7. Applied rewrites 96.7%

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

4. Final simplification 97.7%

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

Alternative 5: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. associate-*l/ N/A

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

   8. *-commutative N/A

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

   9. lower-/.f64 N/A

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

   10. sub-neg N/A

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

   11. mul-1-neg N/A

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

   12. +-commutative N/A

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

   13. distribute-lft-in N/A

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

   14. mul-1-neg N/A

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

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

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

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

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

   17. metadata-eval N/A

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

   18. lower-fma.f64 N/A

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

   19. metadata-eval N/A

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

   20. lower--.f64 99.6

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 99.8%

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

9. Final simplification 99.8%

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

Alternative 6: 57.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -0.75

   1. Initial program 90.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot \frac{1}{y} - \frac{4}{3} \cdot \frac{1}{x \cdot y}\right)} \]
   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 98.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 27.5%

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

   if -0.75 < x

   1. Initial program 97.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 74.0

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

   5. Applied rewrites 74.0%

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

Alternative 7: 57.2% 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 95.7%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. associate-*l/ N/A

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

   8. *-commutative N/A

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

   9. lower-/.f64 N/A

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

   10. sub-neg N/A

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

   11. mul-1-neg N/A

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

   12. +-commutative N/A

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

   13. distribute-lft-in N/A

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

   14. mul-1-neg N/A

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

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

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

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

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

   17. metadata-eval N/A

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

   18. lower-fma.f64 N/A

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

   19. metadata-eval N/A

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

   20. lower--.f64 99.6

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 99.8%

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

10. Applied rewrites 95.9%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 63.6%

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

Alternative 8: 51.6% 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 95.7%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 58.4

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

5. Applied rewrites 58.4%

   \[\leadsto \color{blue}{\frac{1}{y}} \]
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 2024254 
(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)))