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

Percentage Accurate: 93.8% → 99.6%

Time: 8.2s

Alternatives: 10

Speedup: 1.1×

• 25.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.3%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

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

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

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

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

   13. lower-*.f64 N/A

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

   14. metadata-eval 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lift-neg.f64 N/A

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

   7. neg-mul-1 N/A

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

   8. *-commutative N/A

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

   9. div-inv N/A

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

   10. times-frac N/A

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

   11. un-div-inv N/A

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

   12. clear-num N/A

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

   13. /-rgt-identity N/A

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

   14. neg-mul-1 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. sub-neg N/A

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

   21. distribute-neg-in N/A

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

   22. metadata-eval N/A

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

   23. remove-double-neg N/A

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

   24. lower-+.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - x\right) \cdot \left(3 - x\right) \leq 10:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, x, 3\right)}{y \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.3333333333333333, -1.3333333333333333\right) \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 1.0 x) (- 3.0 x)) 10.0)
   (/ (fma -4.0 x 3.0) (* y 3.0))
   (* (fma x 0.3333333333333333 -1.3333333333333333) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 - x) * (3.0 - x)) <= 10.0) {
		tmp = fma(-4.0, x, 3.0) / (y * 3.0);
	} 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)) <= 10.0)
		tmp = Float64(fma(-4.0, x, 3.0) / Float64(y * 3.0));
	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], 10.0], N[(N[(-4.0 * x + 3.0), $MachinePrecision] / N[(y * 3.0), $MachinePrecision]), $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)) < 10

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if 10 < (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x))

   1. Initial program 90.7%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. associate-*r* N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

      15. times-frac N/A

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

   5. Applied rewrites 99.2%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - x\right) \cdot \left(3 - x\right) \leq 10:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, x, 3\right)}{y \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.3333333333333333, -1.3333333333333333\right) \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 1.0 x) (- 3.0 x)) 10.0)
   (* (/ -1.0 y) (+ -1.0 x))
   (* (fma x 0.3333333333333333 -1.3333333333333333) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 - x) * (3.0 - x)) <= 10.0) {
		tmp = (-1.0 / y) * (-1.0 + x);
	} 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)) <= 10.0)
		tmp = Float64(Float64(-1.0 / y) * Float64(-1.0 + x));
	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], 10.0], N[(N[(-1.0 / y), $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $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)) < 10

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lift-neg.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. un-div-inv N/A

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

      12. clear-num N/A

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

      13. /-rgt-identity N/A

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

      14. neg-mul-1 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. sub-neg N/A

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

      21. distribute-neg-in N/A

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

      22. metadata-eval N/A

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

      23. remove-double-neg N/A

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

      24. lower-+.f64 99.6

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 98.5

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

   9. Applied rewrites 98.5%

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

   if 10 < (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x))

   1. Initial program 90.7%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. associate-*r* N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

      15. times-frac N/A

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

   5. Applied rewrites 99.2%

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

4. Final simplification 98.8%

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

Alternative 4: 98.4% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 1.0 x) (- 3.0 x)) 10.0)
   (* (/ -1.0 y) (+ -1.0 x))
   (* (/ (fma 0.3333333333333333 x -1.3333333333333333) y) x)))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 - x) * (3.0 - x)) <= 10.0) {
		tmp = (-1.0 / y) * (-1.0 + x);
	} else {
		tmp = (fma(0.3333333333333333, x, -1.3333333333333333) / y) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(1.0 - x) * Float64(3.0 - x)) <= 10.0)
		tmp = Float64(Float64(-1.0 / y) * Float64(-1.0 + x));
	else
		tmp = Float64(Float64(fma(0.3333333333333333, x, -1.3333333333333333) / y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(1.0 - x), $MachinePrecision] * N[(3.0 - x), $MachinePrecision]), $MachinePrecision], 10.0], N[(N[(-1.0 / y), $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.3333333333333333 * x + -1.3333333333333333), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x)) < 10

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lift-neg.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. un-div-inv N/A

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

      12. clear-num N/A

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

      13. /-rgt-identity N/A

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

      14. neg-mul-1 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. sub-neg N/A

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

      21. distribute-neg-in N/A

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

      22. metadata-eval N/A

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

      23. remove-double-neg N/A

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

      24. lower-+.f64 99.6

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 98.5

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

   9. Applied rewrites 98.5%

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

   if 10 < (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x))

   1. Initial program 90.7%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. associate-*r* N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

      15. times-frac N/A

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 99.2%

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

4. Final simplification 98.8%

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

Alternative 5: 97.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 1.0 x) (- 3.0 x)) 10.0)
   (* (/ -1.0 y) (+ -1.0 x))
   (* (* (/ x y) 0.3333333333333333) x)))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 - x) * (3.0 - x)) <= 10.0) {
		tmp = (-1.0 / y) * (-1.0 + x);
	} else {
		tmp = ((x / y) * 0.3333333333333333) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((1.0d0 - x) * (3.0d0 - x)) <= 10.0d0) then
        tmp = ((-1.0d0) / y) * ((-1.0d0) + x)
    else
        tmp = ((x / y) * 0.3333333333333333d0) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((1.0 - x) * (3.0 - x)) <= 10.0) {
		tmp = (-1.0 / y) * (-1.0 + x);
	} else {
		tmp = ((x / y) * 0.3333333333333333) * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((1.0 - x) * (3.0 - x)) <= 10.0:
		tmp = (-1.0 / y) * (-1.0 + x)
	else:
		tmp = ((x / y) * 0.3333333333333333) * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(1.0 - x) * Float64(3.0 - x)) <= 10.0)
		tmp = Float64(Float64(-1.0 / y) * Float64(-1.0 + x));
	else
		tmp = Float64(Float64(Float64(x / y) * 0.3333333333333333) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((1.0 - x) * (3.0 - x)) <= 10.0)
		tmp = (-1.0 / y) * (-1.0 + x);
	else
		tmp = ((x / y) * 0.3333333333333333) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(1.0 - x), $MachinePrecision] * N[(3.0 - x), $MachinePrecision]), $MachinePrecision], 10.0], N[(N[(-1.0 / y), $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x)) < 10

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lift-neg.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. un-div-inv N/A

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

      12. clear-num N/A

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

      13. /-rgt-identity N/A

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

      14. neg-mul-1 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. sub-neg N/A

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

      21. distribute-neg-in N/A

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

      22. metadata-eval N/A

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

      23. remove-double-neg N/A

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

      24. lower-+.f64 99.6

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 98.5

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

   9. Applied rewrites 98.5%

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

   if 10 < (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x))

   1. Initial program 90.7%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 98.4

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

   5. Applied rewrites 98.4%

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

Alternative 6: 97.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (- 1.0 x) (- 3.0 x)) 10.0)
   (* (/ -1.0 y) (+ -1.0 x))
   (* (/ (* 0.3333333333333333 x) y) x)))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 - x) * (3.0 - x)) <= 10.0) {
		tmp = (-1.0 / y) * (-1.0 + x);
	} else {
		tmp = ((0.3333333333333333 * x) / y) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((1.0d0 - x) * (3.0d0 - x)) <= 10.0d0) then
        tmp = ((-1.0d0) / y) * ((-1.0d0) + x)
    else
        tmp = ((0.3333333333333333d0 * x) / y) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((1.0 - x) * (3.0 - x)) <= 10.0) {
		tmp = (-1.0 / y) * (-1.0 + x);
	} else {
		tmp = ((0.3333333333333333 * x) / y) * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((1.0 - x) * (3.0 - x)) <= 10.0:
		tmp = (-1.0 / y) * (-1.0 + x)
	else:
		tmp = ((0.3333333333333333 * x) / y) * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(1.0 - x) * Float64(3.0 - x)) <= 10.0)
		tmp = Float64(Float64(-1.0 / y) * Float64(-1.0 + x));
	else
		tmp = Float64(Float64(Float64(0.3333333333333333 * x) / y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((1.0 - x) * (3.0 - x)) <= 10.0)
		tmp = (-1.0 / y) * (-1.0 + x);
	else
		tmp = ((0.3333333333333333 * x) / y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(1.0 - x), $MachinePrecision] * N[(3.0 - x), $MachinePrecision]), $MachinePrecision], 10.0], N[(N[(-1.0 / y), $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.3333333333333333 * x), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x)) < 10

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lift-neg.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. un-div-inv N/A

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

      12. clear-num N/A

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

      13. /-rgt-identity N/A

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

      14. neg-mul-1 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. sub-neg N/A

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

      21. distribute-neg-in N/A

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

      22. metadata-eval N/A

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

      23. remove-double-neg N/A

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

      24. lower-+.f64 99.6

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 98.5

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

   9. Applied rewrites 98.5%

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

   if 10 < (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 3 binary64) x))

   1. Initial program 90.7%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. associate-*r* N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

      15. times-frac N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.4%

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

   10. Applied rewrites 98.4%

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

4. Final simplification 98.5%

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

Alternative 7: 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 95.3%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. associate-*l/ N/A

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

   8. *-commutative N/A

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

   9. lower-/.f64 N/A

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

   10. sub-neg N/A

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

   11. mul-1-neg N/A

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

   12. +-commutative N/A

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

   13. distribute-lft-in N/A

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

   14. mul-1-neg N/A

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

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

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

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

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

   17. metadata-eval N/A

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

   18. lower-fma.f64 N/A

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

   19. metadata-eval N/A

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

   20. lower--.f64 99.6

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

5. Applied rewrites 99.6%

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

Alternative 8: 57.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -0.75

   1. Initial program 93.3%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. associate-*r* N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

      15. times-frac N/A

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 36.1%

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

   if -0.75 < x

   1. Initial program 96.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 72.4

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

   5. Applied rewrites 72.4%

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

Alternative 9: 56.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.3%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

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

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

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

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

   13. lower-*.f64 N/A

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

   14. metadata-eval 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lift-neg.f64 N/A

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

   7. neg-mul-1 N/A

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

   8. *-commutative N/A

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

   9. div-inv N/A

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

   10. times-frac N/A

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

   11. un-div-inv N/A

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

   12. clear-num N/A

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

   13. /-rgt-identity N/A

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

   14. neg-mul-1 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. sub-neg N/A

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

   21. distribute-neg-in N/A

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

   22. metadata-eval N/A

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

   23. remove-double-neg N/A

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

   24. lower-+.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in x around 0

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

   1. lower-/.f64 61.2

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

9. Applied rewrites 61.2%

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

Alternative 10: 51.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.3%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 53.3

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

5. Applied rewrites 53.3%

   \[\leadsto \color{blue}{\frac{1}{y}} \]
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 2024268 
(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)))