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

Percentage Accurate: 93.6% → 99.8%

Time: 8.5s

Alternatives: 12

Speedup: 1.0×

• 25.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.8%

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

5. Final simplification 99.9%

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

Alternative 2: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \lor \neg \left(x \leq 1.7\right):\\ \;\;\;\;\frac{x}{3} \cdot \frac{x + -4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot -1.3333333333333333}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.7) (not (<= x 1.7)))
   (* (/ x 3.0) (/ (+ x -4.0) y))
   (/ (+ 1.0 (* x -1.3333333333333333)) y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.7) || !(x <= 1.7)) {
		tmp = (x / 3.0) * ((x + -4.0) / y);
	} else {
		tmp = (1.0 + (x * -1.3333333333333333)) / 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 <= (-1.7d0)) .or. (.not. (x <= 1.7d0))) then
        tmp = (x / 3.0d0) * ((x + (-4.0d0)) / y)
    else
        tmp = (1.0d0 + (x * (-1.3333333333333333d0))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.7) || !(x <= 1.7)) {
		tmp = (x / 3.0) * ((x + -4.0) / y);
	} else {
		tmp = (1.0 + (x * -1.3333333333333333)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.7) or not (x <= 1.7):
		tmp = (x / 3.0) * ((x + -4.0) / y)
	else:
		tmp = (1.0 + (x * -1.3333333333333333)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.7) || !(x <= 1.7))
		tmp = Float64(Float64(x / 3.0) * Float64(Float64(x + -4.0) / y));
	else
		tmp = Float64(Float64(1.0 + Float64(x * -1.3333333333333333)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.7) || ~((x <= 1.7)))
		tmp = (x / 3.0) * ((x + -4.0) / y);
	else
		tmp = (1.0 + (x * -1.3333333333333333)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.7], N[Not[LessEqual[x, 1.7]], $MachinePrecision]], N[(N[(x / 3.0), $MachinePrecision] * N[(N[(x + -4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x * -1.3333333333333333), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.69999999999999996 or 1.69999999999999996 < x

   1. Initial program 91.7%

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

   3. Taylor expanded in x around inf 90.7%

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

      1. +-commutative 90.7%

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

      2. unpow2 90.7%

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

      3. distribute-rgt-out 90.7%

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

   5. Simplified 90.7%

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

      1. *-commutative 90.7%

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

      2. times-frac 98.8%

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

   7. Applied egg-rr 98.8%

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

   if -1.69999999999999996 < x < 1.69999999999999996

   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/ 99.5%

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

      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. Add Preprocessing

   5. Taylor expanded in x around 0 99.4%

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

   6. Taylor expanded in y around 0 99.4%

      \[\leadsto \color{blue}{\frac{1 + -1.3333333333333333 \cdot x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \lor \neg \left(x \leq 1.7\right):\\ \;\;\;\;\frac{x}{3} \cdot \frac{x + -4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot -1.3333333333333333}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \lor \neg \left(x \leq 1.7\right):\\ \;\;\;\;\frac{x}{3} \cdot \frac{x + -4}{y}\\ \mathbf{else}:\\ \;\;\;\;-1.3333333333333333 \cdot \frac{x}{y} + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.7) (not (<= x 1.7)))
   (* (/ x 3.0) (/ (+ x -4.0) y))
   (+ (* -1.3333333333333333 (/ x y)) (/ 1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.7) || !(x <= 1.7)) {
		tmp = (x / 3.0) * ((x + -4.0) / y);
	} else {
		tmp = (-1.3333333333333333 * (x / y)) + (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 <= (-1.7d0)) .or. (.not. (x <= 1.7d0))) then
        tmp = (x / 3.0d0) * ((x + (-4.0d0)) / y)
    else
        tmp = ((-1.3333333333333333d0) * (x / y)) + (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.7) || !(x <= 1.7)) {
		tmp = (x / 3.0) * ((x + -4.0) / y);
	} else {
		tmp = (-1.3333333333333333 * (x / y)) + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.7) or not (x <= 1.7):
		tmp = (x / 3.0) * ((x + -4.0) / y)
	else:
		tmp = (-1.3333333333333333 * (x / y)) + (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.7) || !(x <= 1.7))
		tmp = Float64(Float64(x / 3.0) * Float64(Float64(x + -4.0) / y));
	else
		tmp = Float64(Float64(-1.3333333333333333 * Float64(x / y)) + Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.7) || ~((x <= 1.7)))
		tmp = (x / 3.0) * ((x + -4.0) / y);
	else
		tmp = (-1.3333333333333333 * (x / y)) + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.7], N[Not[LessEqual[x, 1.7]], $MachinePrecision]], N[(N[(x / 3.0), $MachinePrecision] * N[(N[(x + -4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(-1.3333333333333333 * N[(x / y), $MachinePrecision]), $MachinePrecision] + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.69999999999999996 or 1.69999999999999996 < x

   1. Initial program 91.7%

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

   3. Taylor expanded in x around inf 90.7%

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

      1. +-commutative 90.7%

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

      2. unpow2 90.7%

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

      3. distribute-rgt-out 90.7%

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

   5. Simplified 90.7%

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

      1. *-commutative 90.7%

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

      2. times-frac 98.8%

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

   7. Applied egg-rr 98.8%

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

   if -1.69999999999999996 < x < 1.69999999999999996

   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/ 99.5%

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

      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. Add Preprocessing

   5. Taylor expanded in x around 0 99.4%

      \[\leadsto \color{blue}{-1.3333333333333333 \cdot \frac{x}{y} + \frac{1}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \lor \neg \left(x \leq 1.7\right):\\ \;\;\;\;\frac{x}{3} \cdot \frac{x + -4}{y}\\ \mathbf{else}:\\ \;\;\;\;-1.3333333333333333 \cdot \frac{x}{y} + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7:\\ \;\;\;\;\frac{x}{3} \cdot \frac{x + -4}{y}\\ \mathbf{elif}\;x \leq 1.7:\\ \;\;\;\;-1.3333333333333333 \cdot \frac{x}{y} + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{3 \cdot \frac{y}{x + -4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.7)
   (* (/ x 3.0) (/ (+ x -4.0) y))
   (if (<= x 1.7)
     (+ (* -1.3333333333333333 (/ x y)) (/ 1.0 y))
     (/ x (* 3.0 (/ y (+ x -4.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.7) {
		tmp = (x / 3.0) * ((x + -4.0) / y);
	} else if (x <= 1.7) {
		tmp = (-1.3333333333333333 * (x / y)) + (1.0 / y);
	} else {
		tmp = x / (3.0 * (y / (x + -4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.7d0)) then
        tmp = (x / 3.0d0) * ((x + (-4.0d0)) / y)
    else if (x <= 1.7d0) then
        tmp = ((-1.3333333333333333d0) * (x / y)) + (1.0d0 / y)
    else
        tmp = x / (3.0d0 * (y / (x + (-4.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.7) {
		tmp = (x / 3.0) * ((x + -4.0) / y);
	} else if (x <= 1.7) {
		tmp = (-1.3333333333333333 * (x / y)) + (1.0 / y);
	} else {
		tmp = x / (3.0 * (y / (x + -4.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.7:
		tmp = (x / 3.0) * ((x + -4.0) / y)
	elif x <= 1.7:
		tmp = (-1.3333333333333333 * (x / y)) + (1.0 / y)
	else:
		tmp = x / (3.0 * (y / (x + -4.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.7)
		tmp = Float64(Float64(x / 3.0) * Float64(Float64(x + -4.0) / y));
	elseif (x <= 1.7)
		tmp = Float64(Float64(-1.3333333333333333 * Float64(x / y)) + Float64(1.0 / y));
	else
		tmp = Float64(x / Float64(3.0 * Float64(y / Float64(x + -4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.7)
		tmp = (x / 3.0) * ((x + -4.0) / y);
	elseif (x <= 1.7)
		tmp = (-1.3333333333333333 * (x / y)) + (1.0 / y);
	else
		tmp = x / (3.0 * (y / (x + -4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.7], N[(N[(x / 3.0), $MachinePrecision] * N[(N[(x + -4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.7], N[(N[(-1.3333333333333333 * N[(x / y), $MachinePrecision]), $MachinePrecision] + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(3.0 * N[(y / N[(x + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.69999999999999996

   1. Initial program 92.0%

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

   3. Taylor expanded in x around inf 92.1%

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

      1. +-commutative 92.1%

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

      2. unpow2 92.1%

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

      3. distribute-rgt-out 92.0%

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

   5. Simplified 92.0%

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

      1. *-commutative 92.0%

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

      2. times-frac 99.7%

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

   7. Applied egg-rr 99.7%

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

   if -1.69999999999999996 < x < 1.69999999999999996

   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/ 99.5%

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

      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. Add Preprocessing

   5. Taylor expanded in x around 0 99.4%

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

   if 1.69999999999999996 < x

   1. Initial program 91.3%

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

   3. Taylor expanded in x around inf 89.4%

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

      1. +-commutative 89.4%

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

      2. unpow2 89.4%

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

      3. distribute-rgt-out 89.4%

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

   5. Simplified 89.4%

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

      1. *-commutative 89.4%

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

      2. times-frac 97.9%

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

   7. Applied egg-rr 97.9%

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

      1. clear-num 97.8%

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

      2. frac-times 97.9%

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

      3. *-un-lft-identity 97.9%

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

   9. Applied egg-rr 97.9%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{x + -4} \cdot 3}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7:\\ \;\;\;\;\frac{x}{3} \cdot \frac{x + -4}{y}\\ \mathbf{elif}\;x \leq 1.7:\\ \;\;\;\;-1.3333333333333333 \cdot \frac{x}{y} + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{3 \cdot \frac{y}{x + -4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \lor \neg \left(x \leq 5\right):\\ \;\;\;\;\frac{x}{3} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.7) (not (<= x 5.0))) (* (/ x 3.0) (/ x y)) (/ 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.7) || !(x <= 5.0)) {
		tmp = (x / 3.0) * (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 <= (-1.7d0)) .or. (.not. (x <= 5.0d0))) then
        tmp = (x / 3.0d0) * (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 <= -1.7) || !(x <= 5.0)) {
		tmp = (x / 3.0) * (x / y);
	} else {
		tmp = 1.0 / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.7) or not (x <= 5.0):
		tmp = (x / 3.0) * (x / y)
	else:
		tmp = 1.0 / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.7) || !(x <= 5.0))
		tmp = Float64(Float64(x / 3.0) * 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 <= -1.7) || ~((x <= 5.0)))
		tmp = (x / 3.0) * (x / y);
	else
		tmp = 1.0 / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.7], N[Not[LessEqual[x, 5.0]], $MachinePrecision]], N[(N[(x / 3.0), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(1.0 / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.69999999999999996 or 5 < x

   1. Initial program 91.7%

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

   3. Taylor expanded in x around inf 90.7%

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

      1. +-commutative 90.7%

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

      2. unpow2 90.7%

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

      3. distribute-rgt-out 90.7%

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

   5. Simplified 90.7%

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

      1. *-commutative 90.7%

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

      2. times-frac 98.8%

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

   7. Applied egg-rr 98.8%

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

   8. Taylor expanded in x around inf 97.3%

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

   if -1.69999999999999996 < x < 5

   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/ 99.5%

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

      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. Add Preprocessing

   5. Taylor expanded in x around 0 97.9%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \lor \neg \left(x \leq 5\right):\\ \;\;\;\;\frac{x}{3} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.8) || !(x <= 3.0)) {
		tmp = (x / 3.0) * (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 <= (-3.8d0)) .or. (.not. (x <= 3.0d0))) then
        tmp = (x / 3.0d0) * (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 <= -3.8) || !(x <= 3.0)) {
		tmp = (x / 3.0) * (x / 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 / 3.0) * (x / 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 / 3.0) * 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 <= -3.8) || ~((x <= 3.0)))
		tmp = (x / 3.0) * (x / 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 / 3.0), $MachinePrecision] * N[(x / y), $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 91.7%

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

   3. Taylor expanded in x around inf 90.7%

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

      1. +-commutative 90.7%

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

      2. unpow2 90.7%

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

      3. distribute-rgt-out 90.7%

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

   5. Simplified 90.7%

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

      1. *-commutative 90.7%

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

      2. times-frac 98.8%

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

   7. Applied egg-rr 98.8%

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

   8. Taylor expanded in x around inf 97.3%

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

   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. 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. Add Preprocessing

   5. Taylor expanded in x around 0 98.1%

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

4. Final simplification 97.7%

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

Alternative 7: 97.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.8) || !(x <= 3.0)) {
		tmp = x / (3.0 * (y / x));
	} 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 / (3.0d0 * (y / x))
    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 / (3.0 * (y / x));
	} 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 / (3.0 * (y / x))
	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(3.0 * Float64(y / x)));
	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 / (3.0 * (y / x));
	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[(3.0 * N[(y / x), $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 91.7%

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

   3. Taylor expanded in x around inf 90.7%

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

      1. +-commutative 90.7%

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

      2. unpow2 90.7%

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

      3. distribute-rgt-out 90.7%

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

   5. Simplified 90.7%

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

      1. *-commutative 90.7%

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

      2. times-frac 98.8%

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

   7. Applied egg-rr 98.8%

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

   8. Taylor expanded in x around inf 97.3%

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

      1. clear-num 97.3%

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

      2. frac-times 97.3%

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

      3. *-un-lft-identity 97.3%

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

   10. Applied egg-rr 97.3%

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

   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. 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. Add Preprocessing

   5. Taylor expanded in x around 0 98.1%

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

4. Final simplification 97.7%

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

Alternative 8: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.6 \lor \neg \left(x \leq 3\right):\\ \;\;\;\;\frac{x}{3 \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot -1.3333333333333333}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -4.6) (not (<= x 3.0)))
   (/ x (* 3.0 (/ y x)))
   (/ (+ 1.0 (* x -1.3333333333333333)) y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -4.6) || !(x <= 3.0)) {
		tmp = x / (3.0 * (y / x));
	} else {
		tmp = (1.0 + (x * -1.3333333333333333)) / 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 <= (-4.6d0)) .or. (.not. (x <= 3.0d0))) then
        tmp = x / (3.0d0 * (y / x))
    else
        tmp = (1.0d0 + (x * (-1.3333333333333333d0))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -4.6) || !(x <= 3.0)) {
		tmp = x / (3.0 * (y / x));
	} else {
		tmp = (1.0 + (x * -1.3333333333333333)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -4.6) or not (x <= 3.0):
		tmp = x / (3.0 * (y / x))
	else:
		tmp = (1.0 + (x * -1.3333333333333333)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -4.6) || !(x <= 3.0))
		tmp = Float64(x / Float64(3.0 * Float64(y / x)));
	else
		tmp = Float64(Float64(1.0 + Float64(x * -1.3333333333333333)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -4.6) || ~((x <= 3.0)))
		tmp = x / (3.0 * (y / x));
	else
		tmp = (1.0 + (x * -1.3333333333333333)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -4.6], N[Not[LessEqual[x, 3.0]], $MachinePrecision]], N[(x / N[(3.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x * -1.3333333333333333), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.5999999999999996 or 3 < x

   1. Initial program 91.7%

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

   3. Taylor expanded in x around inf 90.7%

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

      1. +-commutative 90.7%

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

      2. unpow2 90.7%

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

      3. distribute-rgt-out 90.7%

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

   5. Simplified 90.7%

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

      1. *-commutative 90.7%

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

      2. times-frac 98.8%

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

   7. Applied egg-rr 98.8%

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

   8. Taylor expanded in x around inf 97.3%

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

      1. clear-num 97.3%

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

      2. frac-times 97.3%

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

      3. *-un-lft-identity 97.3%

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

   10. Applied egg-rr 97.3%

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

   if -4.5999999999999996 < 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/ 99.5%

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

      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. Add Preprocessing

   5. Taylor expanded in x around 0 99.4%

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

   6. Taylor expanded in y around 0 99.4%

      \[\leadsto \color{blue}{\frac{1 + -1.3333333333333333 \cdot x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6 \lor \neg \left(x \leq 3\right):\\ \;\;\;\;\frac{x}{3 \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot -1.3333333333333333}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3:\\ \;\;\;\;\left(3 - x\right) \cdot \left(x \cdot \frac{-0.3333333333333333}{y}\right)\\ \mathbf{elif}\;x \leq 3:\\ \;\;\;\;\frac{1 + x \cdot -1.3333333333333333}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{3 \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.3)
   (* (- 3.0 x) (* x (/ -0.3333333333333333 y)))
   (if (<= x 3.0)
     (/ (+ 1.0 (* x -1.3333333333333333)) y)
     (/ x (* 3.0 (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.3) {
		tmp = (3.0 - x) * (x * (-0.3333333333333333 / y));
	} else if (x <= 3.0) {
		tmp = (1.0 + (x * -1.3333333333333333)) / y;
	} else {
		tmp = x / (3.0 * (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.3d0)) then
        tmp = (3.0d0 - x) * (x * ((-0.3333333333333333d0) / y))
    else if (x <= 3.0d0) then
        tmp = (1.0d0 + (x * (-1.3333333333333333d0))) / y
    else
        tmp = x / (3.0d0 * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.3) {
		tmp = (3.0 - x) * (x * (-0.3333333333333333 / y));
	} else if (x <= 3.0) {
		tmp = (1.0 + (x * -1.3333333333333333)) / y;
	} else {
		tmp = x / (3.0 * (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.3:
		tmp = (3.0 - x) * (x * (-0.3333333333333333 / y))
	elif x <= 3.0:
		tmp = (1.0 + (x * -1.3333333333333333)) / y
	else:
		tmp = x / (3.0 * (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.3)
		tmp = Float64(Float64(3.0 - x) * Float64(x * Float64(-0.3333333333333333 / y)));
	elseif (x <= 3.0)
		tmp = Float64(Float64(1.0 + Float64(x * -1.3333333333333333)) / y);
	else
		tmp = Float64(x / Float64(3.0 * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.3)
		tmp = (3.0 - x) * (x * (-0.3333333333333333 / y));
	elseif (x <= 3.0)
		tmp = (1.0 + (x * -1.3333333333333333)) / y;
	else
		tmp = x / (3.0 * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.3], N[(N[(3.0 - x), $MachinePrecision] * N[(x * N[(-0.3333333333333333 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.0], N[(N[(1.0 + N[(x * -1.3333333333333333), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(x / N[(3.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2999999999999998

   1. Initial program 92.0%

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

      1. associate-*l/ 99.7%

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

      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. Add Preprocessing

   5. Taylor expanded in x around inf 97.7%

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

      1. associate-*r/ 97.7%

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

      2. *-commutative 97.7%

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

      3. associate-*r/ 97.6%

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

   7. Simplified 97.6%

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

   if -2.2999999999999998 < 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/ 99.5%

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

      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. Add Preprocessing

   5. Taylor expanded in x around 0 99.4%

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

   6. Taylor expanded in y around 0 99.4%

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

   if 3 < x

   1. Initial program 91.3%

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

   3. Taylor expanded in x around inf 89.4%

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

      1. +-commutative 89.4%

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

      2. unpow2 89.4%

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

      3. distribute-rgt-out 89.4%

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

   5. Simplified 89.4%

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

      1. *-commutative 89.4%

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

      2. times-frac 97.9%

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

   7. Applied egg-rr 97.9%

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

   8. Taylor expanded in x around inf 97.1%

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

      1. clear-num 97.1%

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

      2. frac-times 97.2%

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

      3. *-un-lft-identity 97.2%

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

   10. Applied egg-rr 97.2%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3:\\ \;\;\;\;\left(3 - x\right) \cdot \left(x \cdot \frac{-0.3333333333333333}{y}\right)\\ \mathbf{elif}\;x \leq 3:\\ \;\;\;\;\frac{1 + x \cdot -1.3333333333333333}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{3 \cdot \frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.5% 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 95.8%

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

   1. associate-*l/ 99.6%

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

   2. *-commutative 99.6%

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

   3. *-commutative 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 11: 58.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -0.75

   1. Initial program 92.0%

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

   3. Taylor expanded in x around inf 92.1%

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

      1. +-commutative 92.1%

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

      2. unpow2 92.1%

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

      3. distribute-rgt-out 92.0%

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

   5. Simplified 92.0%

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

   6. Taylor expanded in x around 0 28.6%

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

   if -0.75 < x

   1. Initial program 96.9%

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

   5. Taylor expanded in x around 0 67.1%

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

4. Final simplification 58.1%

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

Alternative 12: 51.5% 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 95.8%

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

   1. associate-*l/ 99.6%

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

   2. *-commutative 99.6%

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

   3. *-commutative 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around 0 52.6%

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

6. Final simplification 52.6%

   \[\leadsto \frac{1}{y} \]
7. Add Preprocessing

Developer target: 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 2024021 
(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)))