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

Percentage Accurate: 93.7% → 99.8%

Time: 10.9s

Alternatives: 16

Speedup: 1.0×

• 24.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return ((1.0 - x) / y) * (1.0 - (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) * (1.0d0 - (x / 3.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.0%

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

   1. times-frac 99.9%

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

   2. div-sub 99.9%

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

   3. metadata-eval 99.9%

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

4. Applied egg-rr 99.9%

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

Alternative 2: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \lor \neg \left(x \leq 1.75\right):\\ \;\;\;\;x \cdot \left(\frac{0.3333333333333333}{y} \cdot \left(x - 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.75) (not (<= x 1.75)))
   (* x (* (/ 0.3333333333333333 y) (- x 3.0)))
   (/ (- 1.0 x) y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.75) || !(x <= 1.75)) {
		tmp = x * ((0.3333333333333333 / y) * (x - 3.0));
	} else {
		tmp = (1.0 - x) / 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 <= (-1.75d0)) .or. (.not. (x <= 1.75d0))) then
        tmp = x * ((0.3333333333333333d0 / y) * (x - 3.0d0))
    else
        tmp = (1.0d0 - x) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.75) || !(x <= 1.75)) {
		tmp = x * ((0.3333333333333333 / y) * (x - 3.0));
	} else {
		tmp = (1.0 - x) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.75) or not (x <= 1.75):
		tmp = x * ((0.3333333333333333 / y) * (x - 3.0))
	else:
		tmp = (1.0 - x) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.75) || !(x <= 1.75))
		tmp = Float64(x * Float64(Float64(0.3333333333333333 / y) * Float64(x - 3.0)));
	else
		tmp = Float64(Float64(1.0 - x) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.75) || ~((x <= 1.75)))
		tmp = x * ((0.3333333333333333 / y) * (x - 3.0));
	else
		tmp = (1.0 - x) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.75], N[Not[LessEqual[x, 1.75]], $MachinePrecision]], N[(x * N[(N[(0.3333333333333333 / y), $MachinePrecision] * N[(x - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75 or 1.75 < x

   1. Initial program 90.4%

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

      1. associate-/l* 99.7%

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

      2. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 96.5%

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

      1. neg-mul-1 96.5%

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

   7. Simplified 96.5%

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

   8. Taylor expanded in y around 0 87.1%

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

      1. *-commutative 87.1%

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

      2. associate-/l* 96.4%

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

   10. Simplified 96.4%

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

   11. Taylor expanded in x around 0 96.5%

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

       1. associate-*r/ 96.4%

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

       2. *-commutative 96.4%

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

       3. associate-*r/ 96.4%

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

       4. metadata-eval 96.4%

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

       5. associate-*r/ 96.4%

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

       6. distribute-rgt-out-- 96.4%

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

   13. Simplified 96.4%

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

   if -1.75 < x < 1.75

   1. Initial program 99.7%

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

      1. associate-/l* 99.7%

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

      2. *-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.9%

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

      2. un-div-inv 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-/l* 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 98.4%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \lor \neg \left(x \leq 1.75\right):\\ \;\;\;\;x \cdot \left(\frac{0.3333333333333333}{y} \cdot \left(x - 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \lor \neg \left(x \leq 1.75\right):\\ \;\;\;\;-0.3333333333333333 \cdot \left(\left(3 - x\right) \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.75) (not (<= x 1.75)))
   (* -0.3333333333333333 (* (- 3.0 x) (/ x y)))
   (/ (- 1.0 x) y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.75) || !(x <= 1.75)) {
		tmp = -0.3333333333333333 * ((3.0 - x) * (x / y));
	} else {
		tmp = (1.0 - x) / 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 <= (-1.75d0)) .or. (.not. (x <= 1.75d0))) then
        tmp = (-0.3333333333333333d0) * ((3.0d0 - x) * (x / y))
    else
        tmp = (1.0d0 - x) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.75) || !(x <= 1.75)) {
		tmp = -0.3333333333333333 * ((3.0 - x) * (x / y));
	} else {
		tmp = (1.0 - x) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.75) or not (x <= 1.75):
		tmp = -0.3333333333333333 * ((3.0 - x) * (x / y))
	else:
		tmp = (1.0 - x) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.75) || !(x <= 1.75))
		tmp = Float64(-0.3333333333333333 * Float64(Float64(3.0 - x) * Float64(x / y)));
	else
		tmp = Float64(Float64(1.0 - x) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.75) || ~((x <= 1.75)))
		tmp = -0.3333333333333333 * ((3.0 - x) * (x / y));
	else
		tmp = (1.0 - x) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.75], N[Not[LessEqual[x, 1.75]], $MachinePrecision]], N[(-0.3333333333333333 * N[(N[(3.0 - x), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75 or 1.75 < x

   1. Initial program 90.4%

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

      1. associate-/l* 99.7%

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

      2. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 96.5%

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

      1. neg-mul-1 96.5%

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

   7. Simplified 96.5%

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

   8. Taylor expanded in y around 0 87.1%

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

      1. *-commutative 87.1%

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

      2. associate-/l* 96.4%

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

   10. Simplified 96.4%

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

   if -1.75 < x < 1.75

   1. Initial program 99.7%

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

      1. associate-/l* 99.7%

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

      2. *-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.9%

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

      2. un-div-inv 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-/l* 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 98.4%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \lor \neg \left(x \leq 1.75\right):\\ \;\;\;\;-0.3333333333333333 \cdot \left(\left(3 - x\right) \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3:\\ \;\;\;\;x \cdot \left(\frac{0.3333333333333333}{y} \cdot \left(x - 3\right)\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\frac{3 + x \cdot -4}{y \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(\frac{x}{3} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.3)
   (* x (* (/ 0.3333333333333333 y) (- x 3.0)))
   (if (<= x 1.3)
     (/ (+ 3.0 (* x -4.0)) (* y 3.0))
     (* (/ x y) (+ (/ x 3.0) -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.3) {
		tmp = x * ((0.3333333333333333 / y) * (x - 3.0));
	} else if (x <= 1.3) {
		tmp = (3.0 + (x * -4.0)) / (y * 3.0);
	} else {
		tmp = (x / y) * ((x / 3.0) + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.3d0)) then
        tmp = x * ((0.3333333333333333d0 / y) * (x - 3.0d0))
    else if (x <= 1.3d0) then
        tmp = (3.0d0 + (x * (-4.0d0))) / (y * 3.0d0)
    else
        tmp = (x / y) * ((x / 3.0d0) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.3) {
		tmp = x * ((0.3333333333333333 / y) * (x - 3.0));
	} else if (x <= 1.3) {
		tmp = (3.0 + (x * -4.0)) / (y * 3.0);
	} else {
		tmp = (x / y) * ((x / 3.0) + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.3:
		tmp = x * ((0.3333333333333333 / y) * (x - 3.0))
	elif x <= 1.3:
		tmp = (3.0 + (x * -4.0)) / (y * 3.0)
	else:
		tmp = (x / y) * ((x / 3.0) + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.3)
		tmp = Float64(x * Float64(Float64(0.3333333333333333 / y) * Float64(x - 3.0)));
	elseif (x <= 1.3)
		tmp = Float64(Float64(3.0 + Float64(x * -4.0)) / Float64(y * 3.0));
	else
		tmp = Float64(Float64(x / y) * Float64(Float64(x / 3.0) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.3)
		tmp = x * ((0.3333333333333333 / y) * (x - 3.0));
	elseif (x <= 1.3)
		tmp = (3.0 + (x * -4.0)) / (y * 3.0);
	else
		tmp = (x / y) * ((x / 3.0) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.3], N[(x * N[(N[(0.3333333333333333 / y), $MachinePrecision] * N[(x - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.3], N[(N[(3.0 + N[(x * -4.0), $MachinePrecision]), $MachinePrecision] / N[(y * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(N[(x / 3.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2999999999999998

   1. Initial program 90.2%

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

      1. associate-/l* 99.8%

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

      2. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 96.3%

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

      1. neg-mul-1 96.3%

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

   7. Simplified 96.3%

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

   8. Taylor expanded in y around 0 86.5%

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

      1. *-commutative 86.5%

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

      2. associate-/l* 96.2%

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

   10. Simplified 96.2%

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

   11. Taylor expanded in x around 0 96.2%

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

       1. associate-*r/ 96.2%

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

       2. *-commutative 96.2%

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

       3. associate-*r/ 96.3%

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

       4. metadata-eval 96.3%

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

       5. associate-*r/ 96.3%

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

       6. distribute-rgt-out-- 96.3%

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

   13. Simplified 96.3%

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

   if -2.2999999999999998 < x < 1.30000000000000004

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 98.8%

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

      1. *-commutative 98.8%

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

   5. Simplified 98.8%

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

   if 1.30000000000000004 < x

   1. Initial program 90.7%

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

      1. times-frac 99.8%

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

      2. div-sub 99.8%

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

      3. metadata-eval 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf 96.7%

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

      1. neg-mul-1 96.6%

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

   7. Simplified 96.7%

      \[\leadsto \frac{\color{blue}{-x}}{y} \cdot \left(1 - \frac{x}{3}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3:\\ \;\;\;\;x \cdot \left(\frac{0.3333333333333333}{y} \cdot \left(x - 3\right)\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\frac{3 + x \cdot -4}{y \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(\frac{x}{3} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3:\\ \;\;\;\;x \cdot \left(\frac{0.3333333333333333}{y} \cdot \left(x - 3\right)\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\frac{3 + x \cdot -4}{y \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\left(3 - x\right) \cdot -0.3333333333333333}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.3)
   (* x (* (/ 0.3333333333333333 y) (- x 3.0)))
   (if (<= x 1.3)
     (/ (+ 3.0 (* x -4.0)) (* y 3.0))
     (* x (/ (* (- 3.0 x) -0.3333333333333333) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.3) {
		tmp = x * ((0.3333333333333333 / y) * (x - 3.0));
	} else if (x <= 1.3) {
		tmp = (3.0 + (x * -4.0)) / (y * 3.0);
	} else {
		tmp = x * (((3.0 - x) * -0.3333333333333333) / 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 <= (-2.3d0)) then
        tmp = x * ((0.3333333333333333d0 / y) * (x - 3.0d0))
    else if (x <= 1.3d0) then
        tmp = (3.0d0 + (x * (-4.0d0))) / (y * 3.0d0)
    else
        tmp = x * (((3.0d0 - x) * (-0.3333333333333333d0)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.3) {
		tmp = x * ((0.3333333333333333 / y) * (x - 3.0));
	} else if (x <= 1.3) {
		tmp = (3.0 + (x * -4.0)) / (y * 3.0);
	} else {
		tmp = x * (((3.0 - x) * -0.3333333333333333) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.3:
		tmp = x * ((0.3333333333333333 / y) * (x - 3.0))
	elif x <= 1.3:
		tmp = (3.0 + (x * -4.0)) / (y * 3.0)
	else:
		tmp = x * (((3.0 - x) * -0.3333333333333333) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.3)
		tmp = Float64(x * Float64(Float64(0.3333333333333333 / y) * Float64(x - 3.0)));
	elseif (x <= 1.3)
		tmp = Float64(Float64(3.0 + Float64(x * -4.0)) / Float64(y * 3.0));
	else
		tmp = Float64(x * Float64(Float64(Float64(3.0 - x) * -0.3333333333333333) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.3)
		tmp = x * ((0.3333333333333333 / y) * (x - 3.0));
	elseif (x <= 1.3)
		tmp = (3.0 + (x * -4.0)) / (y * 3.0);
	else
		tmp = x * (((3.0 - x) * -0.3333333333333333) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.3], N[(x * N[(N[(0.3333333333333333 / y), $MachinePrecision] * N[(x - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.3], N[(N[(3.0 + N[(x * -4.0), $MachinePrecision]), $MachinePrecision] / N[(y * 3.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(N[(3.0 - x), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2999999999999998

   1. Initial program 90.2%

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

      1. associate-/l* 99.8%

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

      2. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 96.3%

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

      1. neg-mul-1 96.3%

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

   7. Simplified 96.3%

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

   8. Taylor expanded in y around 0 86.5%

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

      1. *-commutative 86.5%

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

      2. associate-/l* 96.2%

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

   10. Simplified 96.2%

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

   11. Taylor expanded in x around 0 96.2%

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

       1. associate-*r/ 96.2%

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

       2. *-commutative 96.2%

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

       3. associate-*r/ 96.3%

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

       4. metadata-eval 96.3%

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

       5. associate-*r/ 96.3%

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

       6. distribute-rgt-out-- 96.3%

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

   13. Simplified 96.3%

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

   if -2.2999999999999998 < x < 1.30000000000000004

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 98.8%

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

      1. *-commutative 98.8%

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

   5. Simplified 98.8%

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

   if 1.30000000000000004 < x

   1. Initial program 90.7%

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

      1. associate-/l* 99.7%

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

      2. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 96.6%

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

      1. neg-mul-1 96.6%

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

   7. Simplified 96.6%

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

   8. Taylor expanded in y around 0 87.6%

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

      1. *-commutative 87.6%

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

      2. associate-/l* 96.6%

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

   10. Simplified 96.6%

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

       1. associate-*r* 96.6%

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

       2. associate-*r/ 87.7%

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

       3. *-commutative 87.7%

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

   12. Applied egg-rr 87.7%

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

       1. *-commutative 87.7%

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

       2. associate-/l* 96.7%

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

   14. Applied egg-rr 96.7%

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

Alternative 6: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75:\\ \;\;\;\;x \cdot \left(\frac{0.3333333333333333}{y} \cdot \left(x - 3\right)\right)\\ \mathbf{elif}\;x \leq 1.75:\\ \;\;\;\;\frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\left(3 - x\right) \cdot -0.3333333333333333}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.75)
   (* x (* (/ 0.3333333333333333 y) (- x 3.0)))
   (if (<= x 1.75)
     (/ (- 1.0 x) y)
     (* x (/ (* (- 3.0 x) -0.3333333333333333) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.75) {
		tmp = x * ((0.3333333333333333 / y) * (x - 3.0));
	} else if (x <= 1.75) {
		tmp = (1.0 - x) / y;
	} else {
		tmp = x * (((3.0 - x) * -0.3333333333333333) / 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 <= (-1.75d0)) then
        tmp = x * ((0.3333333333333333d0 / y) * (x - 3.0d0))
    else if (x <= 1.75d0) then
        tmp = (1.0d0 - x) / y
    else
        tmp = x * (((3.0d0 - x) * (-0.3333333333333333d0)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.75) {
		tmp = x * ((0.3333333333333333 / y) * (x - 3.0));
	} else if (x <= 1.75) {
		tmp = (1.0 - x) / y;
	} else {
		tmp = x * (((3.0 - x) * -0.3333333333333333) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.75:
		tmp = x * ((0.3333333333333333 / y) * (x - 3.0))
	elif x <= 1.75:
		tmp = (1.0 - x) / y
	else:
		tmp = x * (((3.0 - x) * -0.3333333333333333) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.75)
		tmp = Float64(x * Float64(Float64(0.3333333333333333 / y) * Float64(x - 3.0)));
	elseif (x <= 1.75)
		tmp = Float64(Float64(1.0 - x) / y);
	else
		tmp = Float64(x * Float64(Float64(Float64(3.0 - x) * -0.3333333333333333) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.75)
		tmp = x * ((0.3333333333333333 / y) * (x - 3.0));
	elseif (x <= 1.75)
		tmp = (1.0 - x) / y;
	else
		tmp = x * (((3.0 - x) * -0.3333333333333333) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.75], N[(x * N[(N[(0.3333333333333333 / y), $MachinePrecision] * N[(x - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.75], N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision], N[(x * N[(N[(N[(3.0 - x), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.75

   1. Initial program 90.2%

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

      1. associate-/l* 99.8%

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

      2. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 96.3%

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

      1. neg-mul-1 96.3%

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

   7. Simplified 96.3%

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

   8. Taylor expanded in y around 0 86.5%

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

      1. *-commutative 86.5%

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

      2. associate-/l* 96.2%

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

   10. Simplified 96.2%

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

   11. Taylor expanded in x around 0 96.2%

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

       1. associate-*r/ 96.2%

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

       2. *-commutative 96.2%

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

       3. associate-*r/ 96.3%

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

       4. metadata-eval 96.3%

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

       5. associate-*r/ 96.3%

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

       6. distribute-rgt-out-- 96.3%

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

   13. Simplified 96.3%

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

   if -1.75 < x < 1.75

   1. Initial program 99.7%

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

      1. associate-/l* 99.7%

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

      2. *-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.9%

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

      2. un-div-inv 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-/l* 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 98.4%

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

   if 1.75 < x

   1. Initial program 90.7%

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

      1. associate-/l* 99.7%

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

      2. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 96.6%

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

      1. neg-mul-1 96.6%

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

   7. Simplified 96.6%

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

   8. Taylor expanded in y around 0 87.6%

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

      1. *-commutative 87.6%

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

      2. associate-/l* 96.6%

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

   10. Simplified 96.6%

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

       1. associate-*r* 96.6%

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

       2. associate-*r/ 87.7%

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

       3. *-commutative 87.7%

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

   12. Applied egg-rr 87.7%

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

       1. *-commutative 87.7%

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

       2. associate-/l* 96.7%

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

   14. Applied egg-rr 96.7%

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

Alternative 7: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \lor \neg \left(x \leq 3\right):\\ \;\;\;\;x \cdot \left(x \cdot \frac{-0.3333333333333333}{-y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.8) (not (<= x 3.0)))
   (* x (* x (/ -0.3333333333333333 (- y))))
   (/ (- 1.0 x) y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.8) || !(x <= 3.0)) {
		tmp = x * (x * (-0.3333333333333333 / -y));
	} else {
		tmp = (1.0 - x) / 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 <= (-3.8d0)) .or. (.not. (x <= 3.0d0))) then
        tmp = x * (x * ((-0.3333333333333333d0) / -y))
    else
        tmp = (1.0d0 - x) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -3.8) || !(x <= 3.0)) {
		tmp = x * (x * (-0.3333333333333333 / -y));
	} else {
		tmp = (1.0 - x) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3.8) or not (x <= 3.0):
		tmp = x * (x * (-0.3333333333333333 / -y))
	else:
		tmp = (1.0 - x) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.8) || !(x <= 3.0))
		tmp = Float64(x * Float64(x * Float64(-0.3333333333333333 / Float64(-y))));
	else
		tmp = Float64(Float64(1.0 - x) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -3.8) || ~((x <= 3.0)))
		tmp = x * (x * (-0.3333333333333333 / -y));
	else
		tmp = (1.0 - x) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.8], N[Not[LessEqual[x, 3.0]], $MachinePrecision]], N[(x * N[(x * N[(-0.3333333333333333 / (-y)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7999999999999998 or 3 < x

   1. Initial program 90.4%

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

      1. associate-/l* 99.7%

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

      2. *-rgt-identity 99.7%

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

      3. remove-double-neg 99.7%

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

      4. distribute-lft-neg-out 99.7%

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

      5. neg-mul-1 99.7%

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

      6. times-frac 99.7%

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

      7. *-rgt-identity 99.7%

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

      8. associate-/l* 99.7%

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

      9. metadata-eval 99.7%

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

      10. *-commutative 99.7%

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

      11. sub-neg 99.7%

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

      12. +-commutative 99.7%

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

      13. distribute-lft-in 99.7%

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

      14. neg-mul-1 99.7%

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

      15. remove-double-neg 99.7%

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

      16. metadata-eval 99.7%

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

      17. distribute-lft-neg-out 99.7%

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

      18. *-commutative 99.7%

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

      19. distribute-lft-neg-in 99.7%

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

      20. associate-/r* 99.7%

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

      21. metadata-eval 99.7%

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

      22. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 96.3%

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

   6. Taylor expanded in x around inf 96.3%

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

      1. neg-mul-1 96.5%

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

   8. Simplified 96.3%

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

   if -3.7999999999999998 < x < 3

   1. Initial program 99.7%

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

      1. associate-/l* 99.7%

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

      2. *-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.9%

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

      2. un-div-inv 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-/l* 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 98.4%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \lor \neg \left(x \leq 3\right):\\ \;\;\;\;x \cdot \left(x \cdot \frac{-0.3333333333333333}{-y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \lor \neg \left(x \leq 3\right):\\ \;\;\;\;\frac{x \cdot \left(x \cdot 0.3333333333333333\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.8) (not (<= x 3.0)))
   (/ (* x (* x 0.3333333333333333)) y)
   (/ (- 1.0 x) y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.8) || !(x <= 3.0)) {
		tmp = (x * (x * 0.3333333333333333)) / y;
	} else {
		tmp = (1.0 - x) / 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 <= (-3.8d0)) .or. (.not. (x <= 3.0d0))) then
        tmp = (x * (x * 0.3333333333333333d0)) / y
    else
        tmp = (1.0d0 - x) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -3.8) || !(x <= 3.0)) {
		tmp = (x * (x * 0.3333333333333333)) / y;
	} else {
		tmp = (1.0 - x) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3.8) or not (x <= 3.0):
		tmp = (x * (x * 0.3333333333333333)) / y
	else:
		tmp = (1.0 - x) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.8) || !(x <= 3.0))
		tmp = Float64(Float64(x * Float64(x * 0.3333333333333333)) / y);
	else
		tmp = Float64(Float64(1.0 - x) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -3.8) || ~((x <= 3.0)))
		tmp = (x * (x * 0.3333333333333333)) / y;
	else
		tmp = (1.0 - x) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.8], N[Not[LessEqual[x, 3.0]], $MachinePrecision]], N[(N[(x * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7999999999999998 or 3 < x

   1. Initial program 90.4%

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

      1. associate-/l* 99.7%

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

      2. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 96.5%

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

      1. neg-mul-1 96.5%

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

   7. Simplified 96.5%

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

   8. Taylor expanded in y around 0 87.1%

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

      1. *-commutative 87.1%

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

      2. associate-/l* 96.4%

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

   10. Simplified 96.4%

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

       1. associate-*r* 96.5%

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

       2. associate-*r/ 87.1%

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

       3. *-commutative 87.1%

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

   12. Applied egg-rr 87.1%

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

   13. Taylor expanded in x around inf 86.9%

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

       1. *-commutative 86.9%

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

   15. Simplified 86.9%

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

   if -3.7999999999999998 < x < 3

   1. Initial program 99.7%

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

      1. associate-/l* 99.7%

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

      2. *-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.9%

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

      2. un-div-inv 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-/l* 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 98.4%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \lor \neg \left(x \leq 3\right):\\ \;\;\;\;\frac{x \cdot \left(x \cdot 0.3333333333333333\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.75:\\ \;\;\;\;\frac{x}{y} \cdot -1.3333333333333333\\ \mathbf{elif}\;x \leq 0.33:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.75)
   (* (/ x y) -1.3333333333333333)
   (if (<= x 0.33) (/ 1.0 y) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.75) {
		tmp = (x / y) * -1.3333333333333333;
	} else if (x <= 0.33) {
		tmp = 1.0 / y;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.75d0)) then
        tmp = (x / y) * (-1.3333333333333333d0)
    else if (x <= 0.33d0) then
        tmp = 1.0d0 / y
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.75) {
		tmp = (x / y) * -1.3333333333333333;
	} else if (x <= 0.33) {
		tmp = 1.0 / y;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.75:
		tmp = (x / y) * -1.3333333333333333
	elif x <= 0.33:
		tmp = 1.0 / y
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.75)
		tmp = Float64(Float64(x / y) * -1.3333333333333333);
	elseif (x <= 0.33)
		tmp = Float64(1.0 / y);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.75)
		tmp = (x / y) * -1.3333333333333333;
	elseif (x <= 0.33)
		tmp = 1.0 / y;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.75], N[(N[(x / y), $MachinePrecision] * -1.3333333333333333), $MachinePrecision], If[LessEqual[x, 0.33], N[(1.0 / y), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.75

   1. Initial program 90.2%

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

   3. Taylor expanded in x around 0 19.1%

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

      1. *-commutative 19.1%

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

   5. Simplified 19.1%

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

   6. Taylor expanded in x around inf 19.1%

      \[\leadsto \color{blue}{-1.3333333333333333 \cdot \frac{x}{y}} \]
   7. Step-by-step derivation

      1. *-commutative 19.1%

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

   8. Simplified 19.1%

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

   if -0.75 < x < 0.330000000000000016

   1. Initial program 99.7%

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

      1. associate-/l* 99.7%

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

      2. *-rgt-identity 99.7%

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

      3. remove-double-neg 99.7%

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

      4. distribute-lft-neg-out 99.7%

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

      5. neg-mul-1 99.7%

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

      6. times-frac 99.4%

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

      7. *-rgt-identity 99.4%

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

      8. associate-/l* 99.4%

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

      9. metadata-eval 99.4%

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

      10. *-commutative 99.4%

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

      11. sub-neg 99.4%

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

      12. +-commutative 99.4%

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

      13. distribute-lft-in 99.4%

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

      14. neg-mul-1 99.4%

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

      15. remove-double-neg 99.4%

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

      16. metadata-eval 99.4%

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

      17. distribute-lft-neg-out 99.4%

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

      18. *-commutative 99.4%

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

      19. distribute-lft-neg-in 99.4%

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

      20. associate-/r* 99.4%

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

      21. metadata-eval 99.4%

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

      22. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 98.3%

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

   if 0.330000000000000016 < x

   1. Initial program 90.7%

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

      1. times-frac 99.8%

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

      2. div-sub 99.8%

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

      3. metadata-eval 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 0.7%

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

      1. *-un-lft-identity 0.7%

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

      2. *-un-lft-identity 0.7%

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

      3. sub-neg 0.7%

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

      4. *-un-lft-identity 0.7%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 45.4%

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

      7. sqr-neg 45.4%

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

      8. sqrt-unprod 24.5%

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

      9. add-sqr-sqrt 24.5%

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

   7. Applied egg-rr 24.5%

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

      1. *-lft-identity 24.5%

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

   9. Simplified 24.5%

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

   10. Taylor expanded in x around inf 24.5%

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

Alternative 10: 63.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;x \leq 0.34:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) (/ x (- y)) (if (<= x 0.34) (/ 1.0 y) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / -y;
	} else if (x <= 0.34) {
		tmp = 1.0 / y;
	} else {
		tmp = x / 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 <= (-1.0d0)) then
        tmp = x / -y
    else if (x <= 0.34d0) then
        tmp = 1.0d0 / y
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / -y;
	} else if (x <= 0.34) {
		tmp = 1.0 / y;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = x / -y
	elif x <= 0.34:
		tmp = 1.0 / y
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x / Float64(-y));
	elseif (x <= 0.34)
		tmp = Float64(1.0 / y);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = x / -y;
	elseif (x <= 0.34)
		tmp = 1.0 / y;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(x / (-y)), $MachinePrecision], If[LessEqual[x, 0.34], N[(1.0 / y), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 90.2%

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

      1. times-frac 99.7%

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

      2. div-sub 99.7%

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

      3. metadata-eval 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around 0 19.1%

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

   6. Taylor expanded in x around inf 19.1%

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

      1. neg-mul-1 19.1%

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

      2. distribute-frac-neg2 19.1%

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

   8. Simplified 19.1%

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

   if -1 < x < 0.340000000000000024

   1. Initial program 99.7%

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

      1. associate-/l* 99.7%

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

      2. *-rgt-identity 99.7%

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

      3. remove-double-neg 99.7%

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

      4. distribute-lft-neg-out 99.7%

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

      5. neg-mul-1 99.7%

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

      6. times-frac 99.4%

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

      7. *-rgt-identity 99.4%

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

      8. associate-/l* 99.4%

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

      9. metadata-eval 99.4%

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

      10. *-commutative 99.4%

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

      11. sub-neg 99.4%

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

      12. +-commutative 99.4%

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

      13. distribute-lft-in 99.4%

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

      14. neg-mul-1 99.4%

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

      15. remove-double-neg 99.4%

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

      16. metadata-eval 99.4%

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

      17. distribute-lft-neg-out 99.4%

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

      18. *-commutative 99.4%

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

      19. distribute-lft-neg-in 99.4%

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

      20. associate-/r* 99.4%

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

      21. metadata-eval 99.4%

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

      22. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 98.3%

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

   if 0.340000000000000024 < x

   1. Initial program 90.7%

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

      1. times-frac 99.8%

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

      2. div-sub 99.8%

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

      3. metadata-eval 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 0.7%

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

      1. *-un-lft-identity 0.7%

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

      2. *-un-lft-identity 0.7%

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

      3. sub-neg 0.7%

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

      4. *-un-lft-identity 0.7%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 45.4%

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

      7. sqr-neg 45.4%

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

      8. sqrt-unprod 24.5%

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

      9. add-sqr-sqrt 24.5%

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

   7. Applied egg-rr 24.5%

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

      1. *-lft-identity 24.5%

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

   9. Simplified 24.5%

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

   10. Taylor expanded in x around inf 24.5%

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

Alternative 11: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.0%

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

   1. associate-/l* 99.7%

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

   2. *-commutative 99.7%

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

3. Simplified 99.7%

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

5. Final simplification 99.7%

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

Alternative 12: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - x) * ((x + (-3.0d0)) * ((-0.3333333333333333d0) / y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.0%

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

   1. associate-/l* 99.7%

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

   2. *-rgt-identity 99.7%

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

   3. remove-double-neg 99.7%

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

   4. distribute-lft-neg-out 99.7%

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

   5. neg-mul-1 99.7%

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

   6. times-frac 99.6%

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

   7. *-rgt-identity 99.6%

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

   8. associate-/l* 99.6%

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

   9. metadata-eval 99.6%

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

   10. *-commutative 99.6%

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

   11. sub-neg 99.6%

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

   12. +-commutative 99.6%

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

   13. distribute-lft-in 99.6%

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

   14. neg-mul-1 99.6%

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

   15. remove-double-neg 99.6%

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

   16. metadata-eval 99.6%

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

   17. distribute-lft-neg-out 99.6%

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

   18. *-commutative 99.6%

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

   19. distribute-lft-neg-in 99.6%

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

   20. associate-/r* 99.6%

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

   21. metadata-eval 99.6%

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

   22. metadata-eval 99.6%

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

3. Simplified 99.6%

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

Alternative 13: 63.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3:\\ \;\;\;\;\frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x}{y}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.0) {
		tmp = (1.0 - x) / y;
	} else {
		tmp = (1.0 + x) / 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 <= 3.0d0) then
        tmp = (1.0d0 - x) / y
    else
        tmp = (1.0d0 + x) / y
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 3

   1. Initial program 96.3%

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

      1. associate-/l* 99.7%

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

      2. *-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.8%

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

      2. un-div-inv 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-/l* 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 70.5%

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

   if 3 < x

   1. Initial program 90.7%

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

      1. times-frac 99.8%

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

      2. div-sub 99.8%

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

      3. metadata-eval 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 0.7%

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

      1. *-un-lft-identity 0.7%

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

      2. *-un-lft-identity 0.7%

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

      3. sub-neg 0.7%

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

      4. *-un-lft-identity 0.7%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 45.4%

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

      7. sqr-neg 45.4%

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

      8. sqrt-unprod 24.5%

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

      9. add-sqr-sqrt 24.5%

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

   7. Applied egg-rr 24.5%

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

      1. *-lft-identity 24.5%

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

   9. Simplified 24.5%

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

   10. Taylor expanded in x around 0 24.5%

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

       1. *-rgt-identity 24.5%

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

       2. *-lft-identity 24.5%

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

       3. associate-*l/ 24.5%

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

       4. distribute-lft-in 24.5%

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

       5. associate-*l/ 24.5%

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

       6. *-lft-identity 24.5%

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

   12. Simplified 24.5%

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

Alternative 14: 63.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.42:\\ \;\;\;\;\frac{x}{y} \cdot -1.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.42) (* (/ x y) -1.3333333333333333) (/ (+ 1.0 x) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.42) {
		tmp = (x / y) * -1.3333333333333333;
	} else {
		tmp = (1.0 + x) / 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 <= (-0.42d0)) then
        tmp = (x / y) * (-1.3333333333333333d0)
    else
        tmp = (1.0d0 + x) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.42) {
		tmp = (x / y) * -1.3333333333333333;
	} else {
		tmp = (1.0 + x) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.42:
		tmp = (x / y) * -1.3333333333333333
	else:
		tmp = (1.0 + x) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.42)
		tmp = Float64(Float64(x / y) * -1.3333333333333333);
	else
		tmp = Float64(Float64(1.0 + x) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.42)
		tmp = (x / y) * -1.3333333333333333;
	else
		tmp = (1.0 + x) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.42], N[(N[(x / y), $MachinePrecision] * -1.3333333333333333), $MachinePrecision], N[(N[(1.0 + x), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.419999999999999984

   1. Initial program 90.2%

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

   3. Taylor expanded in x around 0 19.1%

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

      1. *-commutative 19.1%

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

   5. Simplified 19.1%

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

   6. Taylor expanded in x around inf 19.1%

      \[\leadsto \color{blue}{-1.3333333333333333 \cdot \frac{x}{y}} \]
   7. Step-by-step derivation

      1. *-commutative 19.1%

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

   8. Simplified 19.1%

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

   if -0.419999999999999984 < x

   1. Initial program 96.7%

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

      1. times-frac 99.9%

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

      2. div-sub 99.9%

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

      3. metadata-eval 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 65.7%

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

      1. *-un-lft-identity 65.7%

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

      2. *-un-lft-identity 65.7%

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

      3. sub-neg 65.7%

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

      4. *-un-lft-identity 65.7%

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

      5. add-sqr-sqrt 29.6%

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

      6. sqrt-unprod 80.6%

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

      7. sqr-neg 80.6%

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

      8. sqrt-unprod 43.9%

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

      9. add-sqr-sqrt 73.5%

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

   7. Applied egg-rr 73.5%

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

      1. *-lft-identity 73.5%

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

   9. Simplified 73.5%

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

   10. Taylor expanded in x around 0 73.5%

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

       1. *-rgt-identity 73.5%

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

       2. *-lft-identity 73.5%

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

       3. associate-*l/ 73.5%

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

       4. distribute-lft-in 73.5%

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

       5. associate-*l/ 73.5%

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

       6. *-lft-identity 73.5%

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

   12. Simplified 73.5%

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

Alternative 15: 57.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.34:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 0.34) (/ 1.0 y) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.34) {
		tmp = 1.0 / y;
	} else {
		tmp = x / 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 <= 0.34d0) then
        tmp = 1.0d0 / y
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.34) {
		tmp = 1.0 / y;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.34:
		tmp = 1.0 / y
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.34)
		tmp = Float64(1.0 / y);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.34)
		tmp = 1.0 / y;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.34], N[(1.0 / y), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.340000000000000024

   1. Initial program 96.3%

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

      1. associate-/l* 99.7%

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

      2. *-rgt-identity 99.7%

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

      3. remove-double-neg 99.7%

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

      4. distribute-lft-neg-out 99.7%

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

      5. neg-mul-1 99.7%

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

      6. times-frac 99.6%

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

      7. *-rgt-identity 99.6%

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

      8. associate-/l* 99.6%

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

      9. metadata-eval 99.6%

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

      10. *-commutative 99.6%

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

      11. sub-neg 99.6%

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

      12. +-commutative 99.6%

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

      13. distribute-lft-in 99.6%

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

      14. neg-mul-1 99.6%

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

      15. remove-double-neg 99.6%

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

      16. metadata-eval 99.6%

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

      17. distribute-lft-neg-out 99.6%

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

      18. *-commutative 99.6%

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

      19. distribute-lft-neg-in 99.6%

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

      20. associate-/r* 99.5%

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

      21. metadata-eval 99.5%

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

      22. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 65.4%

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

   if 0.340000000000000024 < x

   1. Initial program 90.7%

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

      1. times-frac 99.8%

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

      2. div-sub 99.8%

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

      3. metadata-eval 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 0.7%

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

      1. *-un-lft-identity 0.7%

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

      2. *-un-lft-identity 0.7%

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

      3. sub-neg 0.7%

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

      4. *-un-lft-identity 0.7%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 45.4%

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

      7. sqr-neg 45.4%

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

      8. sqrt-unprod 24.5%

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

      9. add-sqr-sqrt 24.5%

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

   7. Applied egg-rr 24.5%

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

      1. *-lft-identity 24.5%

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

   9. Simplified 24.5%

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

   10. Taylor expanded in x around inf 24.5%

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

Alternative 16: 51.2% accurate, 3.7× 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 95.0%

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

   1. associate-/l* 99.7%

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

   2. *-rgt-identity 99.7%

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

   3. remove-double-neg 99.7%

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

   4. distribute-lft-neg-out 99.7%

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

   5. neg-mul-1 99.7%

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

   6. times-frac 99.6%

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

   7. *-rgt-identity 99.6%

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

   8. associate-/l* 99.6%

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

   9. metadata-eval 99.6%

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

   10. *-commutative 99.6%

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

   11. sub-neg 99.6%

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

   12. +-commutative 99.6%

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

   13. distribute-lft-in 99.6%

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

   14. neg-mul-1 99.6%

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

   15. remove-double-neg 99.6%

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

   16. metadata-eval 99.6%

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

   17. distribute-lft-neg-out 99.6%

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

   18. *-commutative 99.6%

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

   19. distribute-lft-neg-in 99.6%

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

   20. associate-/r* 99.6%

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

   21. metadata-eval 99.6%

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

   22. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around 0 50.5%

   \[\leadsto \color{blue}{\frac{1}{y}} \]
6. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× 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 2024160 
(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)))