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

Percentage Accurate: 93.6% → 99.8%

Time: 10.0s

Alternatives: 11

Speedup: 1.1×

• 25.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.6% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - x\right) \cdot \left(3 - x\right) \leq 5 \cdot 10^{+34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.3333333333333333, x, -1.3333333333333333\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 1.0 x) (- 3.0 x)) 5e+34)
   (/ (fma x (fma 0.3333333333333333 x -1.3333333333333333) 1.0) y)
   (* x (/ x (* y 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 - x) * (3.0 - x)) <= 5e+34) {
		tmp = fma(x, fma(0.3333333333333333, x, -1.3333333333333333), 1.0) / y;
	} else {
		tmp = x * (x / (y * 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(1.0 - x) * Float64(3.0 - x)) <= 5e+34)
		tmp = Float64(fma(x, fma(0.3333333333333333, x, -1.3333333333333333), 1.0) / y);
	else
		tmp = Float64(x * Float64(x / Float64(y * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(1.0 - x), $MachinePrecision] * N[(3.0 - x), $MachinePrecision]), $MachinePrecision], 5e+34], N[(N[(x * N[(0.3333333333333333 * x + -1.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision], N[(x * N[(x / N[(y * 3.0), $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)) < 4.9999999999999998e34

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 99.6

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. distribute-lft-in N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 99.9

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

   8. Simplified 99.9%

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

   if 4.9999999999999998e34 < (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x))

   1. Initial program 81.9%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 81.9

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

   5. Simplified 81.9%

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

      1. lift-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 99.8

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 99.8

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

   7. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - x\right) \cdot \left(3 - x\right) \leq 5 \cdot 10^{+34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.3333333333333333, x, -1.3333333333333333\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.7% accurate, 0.7× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 98.3

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

   5. Simplified 98.3%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 98.6

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

   8. Simplified 98.6%

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

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

   1. Initial program 83.4%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 99.8

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.8

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

   4. Applied egg-rr 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. div-inv N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. div-inv N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 99.6

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around inf

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

   9. Simplified 82.0%

      \[\leadsto \color{blue}{\frac{x \cdot \mathsf{fma}\left(x, 0.3333333333333333, -1.3333333333333333\right)}{y}} \]
   10. Step-by-step derivation

       1. lift-fma.f64 N/A

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

       2. associate-/l* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 98.1

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

   11. Applied egg-rr 98.1%

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

4. Final simplification 98.4%

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

Alternative 3: 98.7% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 1.0 x) (- 3.0 x)) 5.0)
   (/ (fma x -1.3333333333333333 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(x, -1.3333333333333333, 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(x, -1.3333333333333333, 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[(x * -1.3333333333333333 + 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. lower-fma.f64 98.3

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

   5. Simplified 98.3%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 98.6

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

   8. Simplified 98.6%

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

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

   1. Initial program 83.4%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      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 98.1%

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

4. Final simplification 98.4%

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

Alternative 4: 98.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (((1.0 - x) * (3.0 - x)) <= 5.0) {
		tmp = fma(x, -1.3333333333333333, 1.0) / y;
	} else {
		tmp = x * (x / (y * 3.0));
	}
	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(x, -1.3333333333333333, 1.0) / y);
	else
		tmp = Float64(x * Float64(x / Float64(y * 3.0)));
	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[(x * -1.3333333333333333 + 1.0), $MachinePrecision] / y), $MachinePrecision], N[(x * N[(x / N[(y * 3.0), $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. lower-fma.f64 98.3

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

   5. Simplified 98.3%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 98.6

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

   8. Simplified 98.6%

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

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

   1. Initial program 83.4%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 80.2

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

   5. Simplified 80.2%

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

      1. lift-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 96.5

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 96.5

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

   7. Applied egg-rr 96.5%

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

4. Final simplification 97.5%

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

Alternative 5: 98.1% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 1.0 x) (- 3.0 x)) 5.0)
   (/ (fma x -1.3333333333333333 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(x, -1.3333333333333333, 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(x, -1.3333333333333333, 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[(x * -1.3333333333333333 + 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. lower-fma.f64 98.3

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

   5. Simplified 98.3%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 98.6

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

   8. Simplified 98.6%

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

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

   1. Initial program 83.4%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 80.2

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

   5. Simplified 80.2%

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

      1. times-frac N/A

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

      2. associate-*r/ N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval 96.4

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

   7. Applied egg-rr 96.4%

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

4. Final simplification 97.5%

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

Alternative 6: 98.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 98.3

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

   5. Simplified 98.3%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 98.6

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

   8. Simplified 98.6%

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

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

   1. Initial program 83.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 96.4

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

   5. Simplified 96.4%

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

Alternative 7: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.3%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 99.6

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.6

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

4. Applied egg-rr 99.6%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. div-inv N/A

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

   7. lift-/.f64 N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. div-inv N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 99.6

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

6. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\left(\left(1 - x\right) \cdot 0.3333333333333333\right) \cdot \left(\frac{1}{y} \cdot \left(3 - x\right)\right)} \]
7. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-/.f64 N/A

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

   11. un-div-inv N/A

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

   12. associate-*r/ N/A

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

   13. lower-/.f64 N/A

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

   14. lift--.f64 N/A

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

   15. sub-neg N/A

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

   16. +-commutative N/A

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

   17. distribute-rgt-in N/A

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

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

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

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

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

   20. metadata-eval N/A

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

   21. lower-fma.f64 N/A

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

   22. metadata-eval 99.8

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

8. Applied egg-rr 99.8%

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

Alternative 8: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.3%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 99.6

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.6

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

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

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

   4. associate-*l/ N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

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

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

   9. distribute-lft1-in N/A

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

   10. +-commutative N/A

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

   11. sub-neg N/A

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

   12. associate-*r/ N/A

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

   13. metadata-eval N/A

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

   14. associate-/l* N/A

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

   15. *-commutative N/A

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

   16. lower-/.f64 N/A

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

   17. sub-neg N/A

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

   18. +-commutative N/A

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

   19. distribute-lft-in N/A

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

   20. mul-1-neg N/A

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

   21. associate-*r* N/A

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

   22. metadata-eval N/A

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

   23. metadata-eval N/A

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

   24. lower-fma.f64 99.5

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

7. Simplified 99.5%

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

Alternative 9: 57.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -0.75

   1. Initial program 83.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 83.2

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

   5. Simplified 83.2%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left(1 - 4 \cdot \frac{1}{x}\right)}}{y \cdot 3} \]
   7. 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. unpow2 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

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

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

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

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

      9. associate-*l* N/A

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

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

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

      11. lft-mult-inverse N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. distribute-rgt-in N/A

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

      15. metadata-eval N/A

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

      16. sub-neg N/A

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

      17. lower-*.f64 N/A

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

      18. sub-neg N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 81.6

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

   8. Simplified 81.6%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 21.9

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

   11. Simplified 21.9%

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

   if -0.75 < x

   1. Initial program 94.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 67.7

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

   5. Simplified 67.7%

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

4. Final simplification 54.6%

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

Alternative 10: 57.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.3%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 54.0

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

5. Simplified 54.0%

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

6. Taylor expanded in y around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 54.2

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

8. Simplified 54.2%

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

Alternative 11: 51.5% 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 91.3%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 49.8

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

5. Simplified 49.8%

   \[\leadsto \color{blue}{\frac{1}{y}} \]
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 2024207 
(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)))