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

Percentage Accurate: 93.9% → 99.8%

Time: 6.5s

Alternatives: 12

Speedup: 1.0×

• 26.1% 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: 93.9% 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 94.9%

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

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 98.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8:\\ \;\;\;\;\left(x \cdot \frac{x}{y}\right) \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 3:\\ \;\;\;\;\frac{1}{y} + \frac{x}{y} \cdot -1.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y \cdot 3}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.8)
   (* (* x (/ x y)) 0.3333333333333333)
   (if (<= x 3.0)
     (+ (/ 1.0 y) (* (/ x y) -1.3333333333333333))
     (/ x (/ (* y 3.0) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.8) {
		tmp = (x * (x / y)) * 0.3333333333333333;
	} else if (x <= 3.0) {
		tmp = (1.0 / y) + ((x / y) * -1.3333333333333333);
	} else {
		tmp = x / ((y * 3.0) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.8d0)) then
        tmp = (x * (x / y)) * 0.3333333333333333d0
    else if (x <= 3.0d0) then
        tmp = (1.0d0 / y) + ((x / y) * (-1.3333333333333333d0))
    else
        tmp = x / ((y * 3.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.8) {
		tmp = (x * (x / y)) * 0.3333333333333333;
	} else if (x <= 3.0) {
		tmp = (1.0 / y) + ((x / y) * -1.3333333333333333);
	} else {
		tmp = x / ((y * 3.0) / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.8:
		tmp = (x * (x / y)) * 0.3333333333333333
	elif x <= 3.0:
		tmp = (1.0 / y) + ((x / y) * -1.3333333333333333)
	else:
		tmp = x / ((y * 3.0) / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.8)
		tmp = Float64(Float64(x * Float64(x / y)) * 0.3333333333333333);
	elseif (x <= 3.0)
		tmp = Float64(Float64(1.0 / y) + Float64(Float64(x / y) * -1.3333333333333333));
	else
		tmp = Float64(x / Float64(Float64(y * 3.0) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.8)
		tmp = (x * (x / y)) * 0.3333333333333333;
	elseif (x <= 3.0)
		tmp = (1.0 / y) + ((x / y) * -1.3333333333333333);
	else
		tmp = x / ((y * 3.0) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.8], N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 3.0], N[(N[(1.0 / y), $MachinePrecision] + N[(N[(x / y), $MachinePrecision] * -1.3333333333333333), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(y * 3.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.79999999999999982

   1. Initial program 88.6%

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

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around inf 86.5%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{{x}^{2}}{y}} \]
   5. Step-by-step derivation

      1. *-commutative 86.5%

         \[\leadsto \color{blue}{\frac{{x}^{2}}{y} \cdot 0.3333333333333333} \]

      2. associate-*l/ 86.4%

         \[\leadsto \color{blue}{\frac{{x}^{2} \cdot 0.3333333333333333}{y}} \]

      3. unpow2 86.4%

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

   6. Simplified 86.4%

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

      1. associate-*l* 86.3%

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

      2. associate-*l/ 97.5%

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

      3. associate-*r* 97.6%

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

      4. *-commutative 97.6%

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

   8. Applied egg-rr 97.6%

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

   if -4.79999999999999982 < x < 3

   1. Initial program 99.5%

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

      1. times-frac 100.0%

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

      2. div-sub 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 98.8%

      \[\leadsto \color{blue}{\frac{1}{y} + -1.3333333333333333 \cdot \frac{x}{y}} \]

   if 3 < x

   1. Initial program 91.8%

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

   2. Taylor expanded in x around inf 90.0%

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

      1. unpow2 90.0%

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

   4. Simplified 90.0%

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

   5. Taylor expanded in x around 0 90.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{{x}^{2}}{y}} \]
   6. Step-by-step derivation

      1. *-commutative 90.0%

         \[\leadsto \color{blue}{\frac{{x}^{2}}{y} \cdot 0.3333333333333333} \]

      2. unpow2 90.0%

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

      3. associate-*l/ 97.8%

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

      4. associate-*r* 97.7%

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

   7. Simplified 97.7%

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

      1. associate-*l/ 89.9%

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

      2. associate-*l* 89.9%

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

      3. associate-/l* 89.9%

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

      4. div-inv 90.0%

         \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot \frac{1}{0.3333333333333333}}} \]

      5. metadata-eval 90.0%

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

      6. associate-/l* 97.9%

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

   9. Applied egg-rr 97.9%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8:\\ \;\;\;\;\left(x \cdot \frac{x}{y}\right) \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 3:\\ \;\;\;\;\frac{1}{y} + \frac{x}{y} \cdot -1.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y \cdot 3}{x}}\\ \end{array} \]

Alternative 3: 92.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.75 \lor \neg \left(x \leq 3\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{x \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.75) (not (<= x 3.0)))
   (* 0.3333333333333333 (/ (* x x) y))
   (/ (- 1.0 x) y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.75) || !(x <= 3.0)) {
		tmp = 0.3333333333333333 * ((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 <= (-3.75d0)) .or. (.not. (x <= 3.0d0))) then
        tmp = 0.3333333333333333d0 * ((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 <= -3.75) || !(x <= 3.0)) {
		tmp = 0.3333333333333333 * ((x * x) / y);
	} else {
		tmp = (1.0 - x) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3.75) or not (x <= 3.0):
		tmp = 0.3333333333333333 * ((x * x) / y)
	else:
		tmp = (1.0 - x) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.75) || !(x <= 3.0))
		tmp = Float64(0.3333333333333333 * Float64(Float64(x * 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 <= -3.75) || ~((x <= 3.0)))
		tmp = 0.3333333333333333 * ((x * x) / y);
	else
		tmp = (1.0 - x) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.75], N[Not[LessEqual[x, 3.0]], $MachinePrecision]], N[(0.3333333333333333 * N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.75 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. 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%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 88.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{{x}^{2}}{y}} \]
   5. Step-by-step derivation

      1. unpow2 88.4%

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

   6. Simplified 88.4%

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

   if -3.75 < x < 3

   1. Initial program 99.5%

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

      1. associate-/l* 99.8%

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

      2. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 97.7%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.75 \lor \neg \left(x \leq 3\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{x \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{y}\\ \end{array} \]

Alternative 4: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.75 \lor \neg \left(x \leq 3\right):\\ \;\;\;\;\left(x \cdot \frac{x}{y}\right) \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.75) (not (<= x 3.0)))
   (* (* x (/ x y)) 0.3333333333333333)
   (/ (- 1.0 x) y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.75) || !(x <= 3.0)) {
		tmp = (x * (x / y)) * 0.3333333333333333;
	} 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 <= (-3.75d0)) .or. (.not. (x <= 3.0d0))) then
        tmp = (x * (x / y)) * 0.3333333333333333d0
    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 <= -3.75) || !(x <= 3.0)) {
		tmp = (x * (x / y)) * 0.3333333333333333;
	} else {
		tmp = (1.0 - x) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3.75) or not (x <= 3.0):
		tmp = (x * (x / y)) * 0.3333333333333333
	else:
		tmp = (1.0 - x) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.75) || !(x <= 3.0))
		tmp = Float64(Float64(x * Float64(x / y)) * 0.3333333333333333);
	else
		tmp = Float64(Float64(1.0 - x) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -3.75) || ~((x <= 3.0)))
		tmp = (x * (x / y)) * 0.3333333333333333;
	else
		tmp = (1.0 - x) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.75], N[Not[LessEqual[x, 3.0]], $MachinePrecision]], N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.75 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. 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%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 88.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{{x}^{2}}{y}} \]
   5. Step-by-step derivation

      1. *-commutative 88.4%

         \[\leadsto \color{blue}{\frac{{x}^{2}}{y} \cdot 0.3333333333333333} \]

      2. associate-*l/ 88.3%

         \[\leadsto \color{blue}{\frac{{x}^{2} \cdot 0.3333333333333333}{y}} \]

      3. unpow2 88.3%

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

   6. Simplified 88.3%

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

      1. associate-*l* 88.3%

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

      2. associate-*l/ 97.6%

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

      3. associate-*r* 97.7%

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

      4. *-commutative 97.7%

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

   8. Applied egg-rr 97.7%

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

   if -3.75 < x < 3

   1. Initial program 99.5%

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

      1. associate-/l* 99.8%

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

      2. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 97.7%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.75 \lor \neg \left(x \leq 3\right):\\ \;\;\;\;\left(x \cdot \frac{x}{y}\right) \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{y}\\ \end{array} \]

Alternative 5: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.75:\\ \;\;\;\;\left(x \cdot \frac{x}{y}\right) \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 3:\\ \;\;\;\;\frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y \cdot 3}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.75)
   (* (* x (/ x y)) 0.3333333333333333)
   (if (<= x 3.0) (/ (- 1.0 x) y) (/ x (/ (* y 3.0) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.75) {
		tmp = (x * (x / y)) * 0.3333333333333333;
	} else if (x <= 3.0) {
		tmp = (1.0 - x) / y;
	} else {
		tmp = x / ((y * 3.0) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.75d0)) then
        tmp = (x * (x / y)) * 0.3333333333333333d0
    else if (x <= 3.0d0) then
        tmp = (1.0d0 - x) / y
    else
        tmp = x / ((y * 3.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.75) {
		tmp = (x * (x / y)) * 0.3333333333333333;
	} else if (x <= 3.0) {
		tmp = (1.0 - x) / y;
	} else {
		tmp = x / ((y * 3.0) / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.75:
		tmp = (x * (x / y)) * 0.3333333333333333
	elif x <= 3.0:
		tmp = (1.0 - x) / y
	else:
		tmp = x / ((y * 3.0) / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.75)
		tmp = Float64(Float64(x * Float64(x / y)) * 0.3333333333333333);
	elseif (x <= 3.0)
		tmp = Float64(Float64(1.0 - x) / y);
	else
		tmp = Float64(x / Float64(Float64(y * 3.0) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.75)
		tmp = (x * (x / y)) * 0.3333333333333333;
	elseif (x <= 3.0)
		tmp = (1.0 - x) / y;
	else
		tmp = x / ((y * 3.0) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.75], N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 3.0], N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision], N[(x / N[(N[(y * 3.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.75

   1. Initial program 88.6%

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

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around inf 86.5%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{{x}^{2}}{y}} \]
   5. Step-by-step derivation

      1. *-commutative 86.5%

         \[\leadsto \color{blue}{\frac{{x}^{2}}{y} \cdot 0.3333333333333333} \]

      2. associate-*l/ 86.4%

         \[\leadsto \color{blue}{\frac{{x}^{2} \cdot 0.3333333333333333}{y}} \]

      3. unpow2 86.4%

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

   6. Simplified 86.4%

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

      1. associate-*l* 86.3%

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

      2. associate-*l/ 97.5%

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

      3. associate-*r* 97.6%

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

      4. *-commutative 97.6%

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

   8. Applied egg-rr 97.6%

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

   if -3.75 < x < 3

   1. Initial program 99.5%

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

      1. associate-/l* 99.8%

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

      2. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 97.7%

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

   if 3 < x

   1. Initial program 91.8%

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

   2. Taylor expanded in x around inf 90.0%

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

      1. unpow2 90.0%

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

   4. Simplified 90.0%

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

   5. Taylor expanded in x around 0 90.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{{x}^{2}}{y}} \]
   6. Step-by-step derivation

      1. *-commutative 90.0%

         \[\leadsto \color{blue}{\frac{{x}^{2}}{y} \cdot 0.3333333333333333} \]

      2. unpow2 90.0%

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

      3. associate-*l/ 97.8%

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

      4. associate-*r* 97.7%

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

   7. Simplified 97.7%

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

      1. associate-*l/ 89.9%

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

      2. associate-*l* 89.9%

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

      3. associate-/l* 89.9%

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

      4. div-inv 90.0%

         \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot \frac{1}{0.3333333333333333}}} \]

      5. metadata-eval 90.0%

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

      6. associate-/l* 97.9%

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

   9. Applied egg-rr 97.9%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.75:\\ \;\;\;\;\left(x \cdot \frac{x}{y}\right) \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 3:\\ \;\;\;\;\frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y \cdot 3}{x}}\\ \end{array} \]

Alternative 6: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8:\\ \;\;\;\;\left(x \cdot \frac{x}{y}\right) \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 3:\\ \;\;\;\;\frac{1 + x \cdot -1.3333333333333333}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y \cdot 3}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.8)
   (* (* x (/ x y)) 0.3333333333333333)
   (if (<= x 3.0)
     (/ (+ 1.0 (* x -1.3333333333333333)) y)
     (/ x (/ (* y 3.0) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.8) {
		tmp = (x * (x / y)) * 0.3333333333333333;
	} else if (x <= 3.0) {
		tmp = (1.0 + (x * -1.3333333333333333)) / y;
	} else {
		tmp = x / ((y * 3.0) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.8d0)) then
        tmp = (x * (x / y)) * 0.3333333333333333d0
    else if (x <= 3.0d0) then
        tmp = (1.0d0 + (x * (-1.3333333333333333d0))) / y
    else
        tmp = x / ((y * 3.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.8) {
		tmp = (x * (x / y)) * 0.3333333333333333;
	} else if (x <= 3.0) {
		tmp = (1.0 + (x * -1.3333333333333333)) / y;
	} else {
		tmp = x / ((y * 3.0) / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.8:
		tmp = (x * (x / y)) * 0.3333333333333333
	elif x <= 3.0:
		tmp = (1.0 + (x * -1.3333333333333333)) / y
	else:
		tmp = x / ((y * 3.0) / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.8)
		tmp = Float64(Float64(x * Float64(x / y)) * 0.3333333333333333);
	elseif (x <= 3.0)
		tmp = Float64(Float64(1.0 + Float64(x * -1.3333333333333333)) / y);
	else
		tmp = Float64(x / Float64(Float64(y * 3.0) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.8)
		tmp = (x * (x / y)) * 0.3333333333333333;
	elseif (x <= 3.0)
		tmp = (1.0 + (x * -1.3333333333333333)) / y;
	else
		tmp = x / ((y * 3.0) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.8], N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 3.0], N[(N[(1.0 + N[(x * -1.3333333333333333), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(x / N[(N[(y * 3.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.79999999999999982

   1. Initial program 88.6%

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

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around inf 86.5%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{{x}^{2}}{y}} \]
   5. Step-by-step derivation

      1. *-commutative 86.5%

         \[\leadsto \color{blue}{\frac{{x}^{2}}{y} \cdot 0.3333333333333333} \]

      2. associate-*l/ 86.4%

         \[\leadsto \color{blue}{\frac{{x}^{2} \cdot 0.3333333333333333}{y}} \]

      3. unpow2 86.4%

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

   6. Simplified 86.4%

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

      1. associate-*l* 86.3%

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

      2. associate-*l/ 97.5%

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

      3. associate-*r* 97.6%

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

      4. *-commutative 97.6%

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

   8. Applied egg-rr 97.6%

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

   if -4.79999999999999982 < x < 3

   1. Initial program 99.5%

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

      1. times-frac 100.0%

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

      2. div-sub 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 98.8%

      \[\leadsto \color{blue}{\frac{1}{y} + -1.3333333333333333 \cdot \frac{x}{y}} \]

   5. Taylor expanded in y around 0 98.8%

      \[\leadsto \color{blue}{\frac{1 + -1.3333333333333333 \cdot x}{y}} \]

   if 3 < x

   1. Initial program 91.8%

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

   2. Taylor expanded in x around inf 90.0%

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

      1. unpow2 90.0%

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

   4. Simplified 90.0%

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

   5. Taylor expanded in x around 0 90.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{{x}^{2}}{y}} \]
   6. Step-by-step derivation

      1. *-commutative 90.0%

         \[\leadsto \color{blue}{\frac{{x}^{2}}{y} \cdot 0.3333333333333333} \]

      2. unpow2 90.0%

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

      3. associate-*l/ 97.8%

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

      4. associate-*r* 97.7%

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

   7. Simplified 97.7%

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

      1. associate-*l/ 89.9%

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

      2. associate-*l* 89.9%

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

      3. associate-/l* 89.9%

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

      4. div-inv 90.0%

         \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot \frac{1}{0.3333333333333333}}} \]

      5. metadata-eval 90.0%

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

      6. associate-/l* 97.9%

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

   9. Applied egg-rr 97.9%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8:\\ \;\;\;\;\left(x \cdot \frac{x}{y}\right) \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 3:\\ \;\;\;\;\frac{1 + x \cdot -1.3333333333333333}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y \cdot 3}{x}}\\ \end{array} \]

Alternative 7: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.9%

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

   1. associate-*l/ 99.5%

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

   2. *-commutative 99.5%

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

   3. *-commutative 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 8: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.9%

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

   1. associate-*l/ 99.5%

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

   2. *-commutative 99.5%

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

   3. associate-/r* 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 9: 57.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -0.75

   1. Initial program 88.6%

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

   2. Taylor expanded in x around 0 27.0%

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

      1. *-commutative 27.0%

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

   4. Simplified 27.0%

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

   5. Taylor expanded in x around inf 25.4%

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

   if -0.75 < x

   1. Initial program 96.7%

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

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 64.1%

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

4. Final simplification 55.1%

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

Alternative 10: 57.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 88.6%

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

      1. associate-*l/ 99.7%

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

      2. *-commutative 99.7%

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

      3. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around inf 97.6%

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

      1. *-commutative 97.6%

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

   6. Simplified 97.6%

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

   7. Taylor expanded in x around 0 25.3%

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

      1. associate-*r/ 25.3%

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

      2. neg-mul-1 25.3%

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

   9. Simplified 25.3%

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

   if -1 < x

   1. Initial program 96.7%

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

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 64.1%

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

4. Final simplification 55.1%

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

Alternative 11: 56.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.9%

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

   1. associate-/l* 99.7%

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

   2. *-commutative 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in x around 0 54.1%

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

5. Final simplification 54.1%

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

Alternative 12: 50.7% 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 94.9%

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

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 50.4%

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

5. Final simplification 50.4%

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

Developer target: 99.9% 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 2023199 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:bezierFromSweepQ1 from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (* (/ (- 1.0 x) y) (/ (- 3.0 x) 3.0))

  (/ (* (- 1.0 x) (- 3.0 x)) (* y 3.0)))