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

Percentage Accurate: 94.0% → 99.8%

Time: 9.8s

Alternatives: 9

Speedup: 1.0×

• 24.7% 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: 94.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - x\right) \cdot \left(3 - x\right) \leq 2 \cdot 10^{+58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.3333333333333333, -1.3333333333333333\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 1.0 x) (- 3.0 x)) 2e+58)
   (/ (fma x (fma x 0.3333333333333333 -1.3333333333333333) 1.0) y)
   (* (/ x y) (* x 0.3333333333333333))))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 - x) * (3.0 - x)) <= 2e+58) {
		tmp = fma(x, fma(x, 0.3333333333333333, -1.3333333333333333), 1.0) / y;
	} else {
		tmp = (x / y) * (x * 0.3333333333333333);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(1.0 - x) * Float64(3.0 - x)) <= 2e+58)
		tmp = Float64(fma(x, fma(x, 0.3333333333333333, -1.3333333333333333), 1.0) / y);
	else
		tmp = Float64(Float64(x / y) * Float64(x * 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(1.0 - x), $MachinePrecision] * N[(3.0 - x), $MachinePrecision]), $MachinePrecision], 2e+58], N[(N[(x * N[(x * 0.3333333333333333 + -1.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x)) < 1.99999999999999989e58

   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-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. +-lft-identity N/A

         \[\leadsto \left(\frac{3 - x}{y} \cdot \frac{1}{3}\right) \cdot \color{blue}{\left(0 + \left(1 - x\right)\right)} \]

      6. distribute-rgt-in N/A

         \[\leadsto \color{blue}{0 \cdot \left(\frac{3 - x}{y} \cdot \frac{1}{3}\right) + \left(1 - x\right) \cdot \left(\frac{3 - x}{y} \cdot \frac{1}{3}\right)} \]

      7. mul0-lft N/A

         \[\leadsto \color{blue}{0} + \left(1 - x\right) \cdot \left(\frac{3 - x}{y} \cdot \frac{1}{3}\right) \]

      8. associate-*l* N/A

         \[\leadsto 0 + \color{blue}{\left(\left(1 - x\right) \cdot \frac{3 - x}{y}\right) \cdot \frac{1}{3}} \]

      9. associate-/l* N/A

         \[\leadsto 0 + \color{blue}{\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y}} \cdot \frac{1}{3} \]

      10. *-commutative N/A

          \[\leadsto 0 + \color{blue}{\frac{1}{3} \cdot \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y}} \]

      11. +-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y} + 0} \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y} \cdot \frac{1}{3}} + 0 \]

      13. associate-/l* N/A

          \[\leadsto \color{blue}{\left(\left(1 - x\right) \cdot \frac{3 - x}{y}\right)} \cdot \frac{1}{3} + 0 \]

      14. associate-*l* N/A

          \[\leadsto \color{blue}{\left(1 - x\right) \cdot \left(\frac{3 - x}{y} \cdot \frac{1}{3}\right)} + 0 \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(1 - x, \frac{3 - x}{y} \cdot \frac{1}{3}, 0\right)} \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - x, \frac{\mathsf{fma}\left(x, -0.3333333333333333, 1\right)}{y}, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. accelerator-lowering-fma.f64 99.9

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 100.0

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

   10. Simplified 100.0%

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

   if 1.99999999999999989e58 < (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x))

   1. Initial program 87.4%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

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

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

      5. metadata-eval N/A

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

      6. associate-*l* N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

         \[\leadsto \frac{{x}^{2} + -4 \cdot \frac{\color{blue}{{x}^{2}}}{x}}{y \cdot 3} \]

      9. unpow2 N/A

         \[\leadsto \frac{{x}^{2} + -4 \cdot \frac{\color{blue}{x \cdot x}}{x}}{y \cdot 3} \]

      10. associate-/l* N/A

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

      11. *-rgt-identity N/A

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

      12. associate-*r/ N/A

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

      13. rgt-mult-inverse N/A

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

      14. *-rgt-identity N/A

          \[\leadsto \frac{{x}^{2} + -4 \cdot \color{blue}{x}}{y \cdot 3} \]

      15. unpow2 N/A

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

      16. distribute-rgt-in N/A

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

      17. metadata-eval N/A

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

      18. sub-neg N/A

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

      19. *-lowering-*.f64 N/A

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

      20. sub-neg N/A

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

      21. metadata-eval N/A

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

      22. +-commutative N/A

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

      23. +-lowering-+.f64 87.4

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

   5. Simplified 87.4%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 87.4%

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

      1. times-frac N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 99.8

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

   10. Applied egg-rr 99.8%

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

Alternative 2: 98.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.6%

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

   3. 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. accelerator-lowering-fma.f64 99.6

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

   5. Simplified 99.6%

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

   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. /-lowering-/.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. accelerator-lowering-fma.f64 100.0

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

   8. Simplified 100.0%

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

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

   1. Initial program 88.2%

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

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

      2. metadata-eval N/A

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

      3. distribute-rgt-out-- N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. *-commutative N/A

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

      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. Simplified 99.4%

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

4. Final simplification 99.7%

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

Alternative 3: 98.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 1.0 x) (- 3.0 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 (((1.0 - x) * (3.0 - 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(1.0 - x) * Float64(3.0 - 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[(1.0 - x), $MachinePrecision] * N[(3.0 - 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. accelerator-lowering-fma.f64 99.6

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

   5. Simplified 99.6%

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

   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. /-lowering-/.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. accelerator-lowering-fma.f64 100.0

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

   8. Simplified 100.0%

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

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

   1. Initial program 88.2%

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. +-lft-identity N/A

         \[\leadsto \left(\frac{3 - x}{y} \cdot \frac{1}{3}\right) \cdot \color{blue}{\left(0 + \left(1 - x\right)\right)} \]

      6. distribute-rgt-in N/A

         \[\leadsto \color{blue}{0 \cdot \left(\frac{3 - x}{y} \cdot \frac{1}{3}\right) + \left(1 - x\right) \cdot \left(\frac{3 - x}{y} \cdot \frac{1}{3}\right)} \]

      7. mul0-lft N/A

         \[\leadsto \color{blue}{0} + \left(1 - x\right) \cdot \left(\frac{3 - x}{y} \cdot \frac{1}{3}\right) \]

      8. associate-*l* N/A

         \[\leadsto 0 + \color{blue}{\left(\left(1 - x\right) \cdot \frac{3 - x}{y}\right) \cdot \frac{1}{3}} \]

      9. associate-/l* N/A

         \[\leadsto 0 + \color{blue}{\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y}} \cdot \frac{1}{3} \]

      10. *-commutative N/A

          \[\leadsto 0 + \color{blue}{\frac{1}{3} \cdot \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y}} \]

      11. +-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y} + 0} \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y} \cdot \frac{1}{3}} + 0 \]

      13. associate-/l* N/A

          \[\leadsto \color{blue}{\left(\left(1 - x\right) \cdot \frac{3 - x}{y}\right)} \cdot \frac{1}{3} + 0 \]

      14. associate-*l* N/A

          \[\leadsto \color{blue}{\left(1 - x\right) \cdot \left(\frac{3 - x}{y} \cdot \frac{1}{3}\right)} + 0 \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(1 - x, \frac{3 - x}{y} \cdot \frac{1}{3}, 0\right)} \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - x, \frac{\mathsf{fma}\left(x, -0.3333333333333333, 1\right)}{y}, 0\right)} \]

   6. Taylor expanded in x around inf

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

   8. Simplified 99.4%

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

Alternative 4: 98.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(1.0 - x), $MachinePrecision] * N[(3.0 - x), $MachinePrecision]), $MachinePrecision], 5.0], N[(N[(-1.3333333333333333 * x + 1.0), $MachinePrecision] / y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x * 0.3333333333333333), $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 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. accelerator-lowering-fma.f64 99.6

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

   5. Simplified 99.6%

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

   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. /-lowering-/.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. accelerator-lowering-fma.f64 100.0

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

   8. Simplified 100.0%

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

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

   1. Initial program 88.2%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

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

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

      5. metadata-eval N/A

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

      6. associate-*l* N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

         \[\leadsto \frac{{x}^{2} + -4 \cdot \frac{\color{blue}{{x}^{2}}}{x}}{y \cdot 3} \]

      9. unpow2 N/A

         \[\leadsto \frac{{x}^{2} + -4 \cdot \frac{\color{blue}{x \cdot x}}{x}}{y \cdot 3} \]

      10. associate-/l* N/A

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

      11. *-rgt-identity N/A

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

      12. associate-*r/ N/A

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

      13. rgt-mult-inverse N/A

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

      14. *-rgt-identity N/A

          \[\leadsto \frac{{x}^{2} + -4 \cdot \color{blue}{x}}{y \cdot 3} \]

      15. unpow2 N/A

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

      16. distribute-rgt-in N/A

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

      17. metadata-eval N/A

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

      18. sub-neg N/A

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

      19. *-lowering-*.f64 N/A

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

      20. sub-neg N/A

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

      21. metadata-eval N/A

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

      22. +-commutative N/A

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

      23. +-lowering-+.f64 87.9

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

   5. Simplified 87.9%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 87.3%

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

      1. times-frac N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 98.9

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

   10. Applied egg-rr 98.9%

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

Alternative 5: 98.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   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. accelerator-lowering-fma.f64 99.6

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

   5. Simplified 99.6%

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

   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. /-lowering-/.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. accelerator-lowering-fma.f64 100.0

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

   8. Simplified 100.0%

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

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

   1. Initial program 88.2%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

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

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

      5. metadata-eval N/A

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

      6. associate-*l* N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

         \[\leadsto \frac{{x}^{2} + -4 \cdot \frac{\color{blue}{{x}^{2}}}{x}}{y \cdot 3} \]

      9. unpow2 N/A

         \[\leadsto \frac{{x}^{2} + -4 \cdot \frac{\color{blue}{x \cdot x}}{x}}{y \cdot 3} \]

      10. associate-/l* N/A

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

      11. *-rgt-identity N/A

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

      12. associate-*r/ N/A

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

      13. rgt-mult-inverse N/A

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

      14. *-rgt-identity N/A

          \[\leadsto \frac{{x}^{2} + -4 \cdot \color{blue}{x}}{y \cdot 3} \]

      15. unpow2 N/A

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

      16. distribute-rgt-in N/A

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

      17. metadata-eval N/A

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

      18. sub-neg N/A

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

      19. *-lowering-*.f64 N/A

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

      20. sub-neg N/A

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

      21. metadata-eval N/A

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

      22. +-commutative N/A

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

      23. +-lowering-+.f64 87.9

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

   5. Simplified 87.9%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 87.3%

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

      1. times-frac N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 98.8

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

   10. Applied egg-rr 98.8%

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

4. Final simplification 99.4%

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

Alternative 6: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1 - x, \frac{\mathsf{fma}\left(x, -0.3333333333333333, 1\right)}{y}, 0\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (fma (- 1.0 x) (/ (fma x -0.3333333333333333 1.0) y) 0.0))

C

double code(double x, double y) {
	return fma((1.0 - x), (fma(x, -0.3333333333333333, 1.0) / y), 0.0);
}

Julia

function code(x, y)
	return fma(Float64(1.0 - x), Float64(fma(x, -0.3333333333333333, 1.0) / y), 0.0)
end

Wolfram

code[x_, y_] := N[(N[(1.0 - x), $MachinePrecision] * N[(N[(x * -0.3333333333333333 + 1.0), $MachinePrecision] / y), $MachinePrecision] + 0.0), $MachinePrecision]

Derivation

1. Initial program 93.9%

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

3. Taylor expanded in y around 0

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. +-lft-identity N/A

      \[\leadsto \left(\frac{3 - x}{y} \cdot \frac{1}{3}\right) \cdot \color{blue}{\left(0 + \left(1 - x\right)\right)} \]

   6. distribute-rgt-in N/A

      \[\leadsto \color{blue}{0 \cdot \left(\frac{3 - x}{y} \cdot \frac{1}{3}\right) + \left(1 - x\right) \cdot \left(\frac{3 - x}{y} \cdot \frac{1}{3}\right)} \]

   7. mul0-lft N/A

      \[\leadsto \color{blue}{0} + \left(1 - x\right) \cdot \left(\frac{3 - x}{y} \cdot \frac{1}{3}\right) \]

   8. associate-*l* N/A

      \[\leadsto 0 + \color{blue}{\left(\left(1 - x\right) \cdot \frac{3 - x}{y}\right) \cdot \frac{1}{3}} \]

   9. associate-/l* N/A

      \[\leadsto 0 + \color{blue}{\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y}} \cdot \frac{1}{3} \]

   10. *-commutative N/A

       \[\leadsto 0 + \color{blue}{\frac{1}{3} \cdot \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y}} \]

   11. +-commutative N/A

       \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y} + 0} \]

   12. *-commutative N/A

       \[\leadsto \color{blue}{\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y} \cdot \frac{1}{3}} + 0 \]

   13. associate-/l* N/A

       \[\leadsto \color{blue}{\left(\left(1 - x\right) \cdot \frac{3 - x}{y}\right)} \cdot \frac{1}{3} + 0 \]

   14. associate-*l* N/A

       \[\leadsto \color{blue}{\left(1 - x\right) \cdot \left(\frac{3 - x}{y} \cdot \frac{1}{3}\right)} + 0 \]

   15. accelerator-lowering-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(1 - x, \frac{3 - x}{y} \cdot \frac{1}{3}, 0\right)} \]

5. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1 - x, \frac{\mathsf{fma}\left(x, -0.3333333333333333, 1\right)}{y}, 0\right)} \]
6. Add Preprocessing

Alternative 7: 57.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.9%

   \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
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. accelerator-lowering-fma.f64 57.5

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

5. Simplified 57.5%

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

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. /-lowering-/.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. accelerator-lowering-fma.f64 57.7

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

8. Simplified 57.7%

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

Alternative 8: 57.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.9%

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

3. Taylor expanded in y around 0

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. +-lft-identity N/A

      \[\leadsto \left(\frac{3 - x}{y} \cdot \frac{1}{3}\right) \cdot \color{blue}{\left(0 + \left(1 - x\right)\right)} \]

   6. distribute-rgt-in N/A

      \[\leadsto \color{blue}{0 \cdot \left(\frac{3 - x}{y} \cdot \frac{1}{3}\right) + \left(1 - x\right) \cdot \left(\frac{3 - x}{y} \cdot \frac{1}{3}\right)} \]

   7. mul0-lft N/A

      \[\leadsto \color{blue}{0} + \left(1 - x\right) \cdot \left(\frac{3 - x}{y} \cdot \frac{1}{3}\right) \]

   8. associate-*l* N/A

      \[\leadsto 0 + \color{blue}{\left(\left(1 - x\right) \cdot \frac{3 - x}{y}\right) \cdot \frac{1}{3}} \]

   9. associate-/l* N/A

      \[\leadsto 0 + \color{blue}{\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y}} \cdot \frac{1}{3} \]

   10. *-commutative N/A

       \[\leadsto 0 + \color{blue}{\frac{1}{3} \cdot \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y}} \]

   11. +-commutative N/A

       \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y} + 0} \]

   12. *-commutative N/A

       \[\leadsto \color{blue}{\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y} \cdot \frac{1}{3}} + 0 \]

   13. associate-/l* N/A

       \[\leadsto \color{blue}{\left(\left(1 - x\right) \cdot \frac{3 - x}{y}\right)} \cdot \frac{1}{3} + 0 \]

   14. associate-*l* N/A

       \[\leadsto \color{blue}{\left(1 - x\right) \cdot \left(\frac{3 - x}{y} \cdot \frac{1}{3}\right)} + 0 \]

   15. accelerator-lowering-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(1 - x, \frac{3 - x}{y} \cdot \frac{1}{3}, 0\right)} \]

5. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1 - x, \frac{\mathsf{fma}\left(x, -0.3333333333333333, 1\right)}{y}, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. associate-*r/ N/A

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

   3. /-lowering-/.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. --lowering--.f64 N/A

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

   6. accelerator-lowering-fma.f64 94.0

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

7. Applied egg-rr 94.0%

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

8. Taylor expanded in x around 0

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

10. Simplified 57.3%

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

    1. *-rgt-identity N/A

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

    2. --lowering--.f64 57.3

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

12. Applied egg-rr 57.3%

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

Alternative 9: 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 93.9%

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

3. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 51.6

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

5. Simplified 51.6%

   \[\leadsto \color{blue}{\frac{1}{y}} \]
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 2024195 
(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)))