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

Percentage Accurate: 93.7% → 99.8%

Time: 6.4s

Alternatives: 12

Speedup: 1.0×

• 24.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.8%

   \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
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. Final simplification 99.8%

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

Alternative 2: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \lor \neg \left(x \leq 1.3\right):\\ \;\;\;\;\left(3 - x\right) \cdot \left(-0.3333333333333333 \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot -1.3333333333333333}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.2) (not (<= x 1.3)))
   (* (- 3.0 x) (* -0.3333333333333333 (/ x y)))
   (/ (+ 1.0 (* x -1.3333333333333333)) y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.2) || !(x <= 1.3)) {
		tmp = (3.0 - x) * (-0.3333333333333333 * (x / 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 <= (-2.2d0)) .or. (.not. (x <= 1.3d0))) then
        tmp = (3.0d0 - x) * ((-0.3333333333333333d0) * (x / 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 <= -2.2) || !(x <= 1.3)) {
		tmp = (3.0 - x) * (-0.3333333333333333 * (x / y));
	} else {
		tmp = (1.0 + (x * -1.3333333333333333)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.2) or not (x <= 1.3):
		tmp = (3.0 - x) * (-0.3333333333333333 * (x / y))
	else:
		tmp = (1.0 + (x * -1.3333333333333333)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.2) || !(x <= 1.3))
		tmp = Float64(Float64(3.0 - x) * Float64(-0.3333333333333333 * Float64(x / 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 <= -2.2) || ~((x <= 1.3)))
		tmp = (3.0 - x) * (-0.3333333333333333 * (x / y));
	else
		tmp = (1.0 + (x * -1.3333333333333333)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.2], N[Not[LessEqual[x, 1.3]], $MachinePrecision]], N[(N[(3.0 - x), $MachinePrecision] * N[(-0.3333333333333333 * N[(x / y), $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 < -2.2000000000000002 or 1.30000000000000004 < x

   1. Initial program 87.9%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   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. Taylor expanded in x around inf 96.5%

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

   if -2.2000000000000002 < x < 1.30000000000000004

   1. Initial program 99.6%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   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 99.2%

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

   5. Taylor expanded in y around 0 99.2%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \lor \neg \left(x \leq 1.3\right):\\ \;\;\;\;\left(3 - x\right) \cdot \left(-0.3333333333333333 \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot -1.3333333333333333}{y}\\ \end{array} \]

Alternative 3: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \lor \neg \left(x \leq 1.3\right):\\ \;\;\;\;\left(3 - x\right) \cdot \left(-0.3333333333333333 \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} + \frac{x}{y} \cdot -1.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.2) (not (<= x 1.3)))
   (* (- 3.0 x) (* -0.3333333333333333 (/ x y)))
   (+ (/ 1.0 y) (* (/ x y) -1.3333333333333333))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.2) || !(x <= 1.3)) {
		tmp = (3.0 - x) * (-0.3333333333333333 * (x / y));
	} else {
		tmp = (1.0 / y) + ((x / y) * -1.3333333333333333);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-2.2d0)) .or. (.not. (x <= 1.3d0))) then
        tmp = (3.0d0 - x) * ((-0.3333333333333333d0) * (x / y))
    else
        tmp = (1.0d0 / y) + ((x / y) * (-1.3333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2.2) || !(x <= 1.3)) {
		tmp = (3.0 - x) * (-0.3333333333333333 * (x / y));
	} else {
		tmp = (1.0 / y) + ((x / y) * -1.3333333333333333);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.2) or not (x <= 1.3):
		tmp = (3.0 - x) * (-0.3333333333333333 * (x / y))
	else:
		tmp = (1.0 / y) + ((x / y) * -1.3333333333333333)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.2) || !(x <= 1.3))
		tmp = Float64(Float64(3.0 - x) * Float64(-0.3333333333333333 * Float64(x / y)));
	else
		tmp = Float64(Float64(1.0 / y) + Float64(Float64(x / y) * -1.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.2) || ~((x <= 1.3)))
		tmp = (3.0 - x) * (-0.3333333333333333 * (x / y));
	else
		tmp = (1.0 / y) + ((x / y) * -1.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.2], N[Not[LessEqual[x, 1.3]], $MachinePrecision]], N[(N[(3.0 - x), $MachinePrecision] * N[(-0.3333333333333333 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] + N[(N[(x / y), $MachinePrecision] * -1.3333333333333333), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2000000000000002 or 1.30000000000000004 < x

   1. Initial program 87.9%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   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. Taylor expanded in x around inf 96.5%

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

   if -2.2000000000000002 < x < 1.30000000000000004

   1. Initial program 99.6%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   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 99.2%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \lor \neg \left(x \leq 1.3\right):\\ \;\;\;\;\left(3 - x\right) \cdot \left(-0.3333333333333333 \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} + \frac{x}{y} \cdot -1.3333333333333333\\ \end{array} \]

Alternative 4: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \lor \neg \left(x \leq 5\right):\\ \;\;\;\;x \cdot \left(\frac{x}{y} \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.7) (not (<= x 5.0)))
   (* x (* (/ x y) 0.3333333333333333))
   (/ 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.7) || !(x <= 5.0)) {
		tmp = x * ((x / y) * 0.3333333333333333);
	} 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 <= (-1.7d0)) .or. (.not. (x <= 5.0d0))) then
        tmp = x * ((x / y) * 0.3333333333333333d0)
    else
        tmp = 1.0d0 / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.7) || !(x <= 5.0)) {
		tmp = x * ((x / y) * 0.3333333333333333);
	} 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 * ((x / y) * 0.3333333333333333)
	else:
		tmp = 1.0 / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.7) || !(x <= 5.0))
		tmp = Float64(x * Float64(Float64(x / y) * 0.3333333333333333));
	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 * ((x / y) * 0.3333333333333333);
	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[(x * N[(N[(x / y), $MachinePrecision] * 0.3333333333333333), $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 87.9%

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

   2. Taylor expanded in x around inf 84.4%

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

      1. unpow2 84.4%

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

   4. Simplified 84.4%

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

      1. associate-/l* 96.3%

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

      2. associate-/r/ 96.2%

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

      3. metadata-eval 96.2%

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

      4. div-inv 96.2%

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

      5. clear-num 96.2%

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

      6. add-sqr-sqrt 54.6%

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

      7. sqrt-unprod 38.9%

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

      8. frac-times 38.3%

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

      9. metadata-eval 38.3%

         \[\leadsto \frac{x}{\frac{1}{\sqrt{\frac{\color{blue}{0.1111111111111111}}{y \cdot y}}}} \cdot x \]

      10. metadata-eval 38.3%

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

      11. frac-times 38.9%

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

      12. sqrt-unprod 0.3%

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

      13. add-sqr-sqrt 0.6%

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

      14. clear-num 0.6%

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

      15. associate-/l* 0.6%

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

      16. associate-*r/ 0.6%

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

      17. add-sqr-sqrt 0.3%

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

      18. sqrt-unprod 38.9%

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

      19. frac-times 38.3%

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

      20. metadata-eval 38.3%

          \[\leadsto \left(x \cdot \sqrt{\frac{\color{blue}{0.1111111111111111}}{y \cdot y}}\right) \cdot x \]

      21. metadata-eval 38.3%

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

      22. frac-times 38.9%

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

      23. sqrt-unprod 54.6%

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

      24. add-sqr-sqrt 96.2%

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

   6. Applied egg-rr 96.2%

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

   7. Taylor expanded in x around 0 96.3%

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

   if -1.69999999999999996 < x < 5

   1. Initial program 99.6%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   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 98.5%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \lor \neg \left(x \leq 5\right):\\ \;\;\;\;x \cdot \left(\frac{x}{y} \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y}\\ \end{array} \]

Alternative 5: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \lor \neg \left(x \leq 5\right):\\ \;\;\;\;\frac{x}{\frac{y}{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.7) (not (<= x 5.0)))
   (* (/ x (/ y x)) 0.3333333333333333)
   (/ 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.7) || !(x <= 5.0)) {
		tmp = (x / (y / x)) * 0.3333333333333333;
	} 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 <= (-1.7d0)) .or. (.not. (x <= 5.0d0))) then
        tmp = (x / (y / x)) * 0.3333333333333333d0
    else
        tmp = 1.0d0 / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.7) || !(x <= 5.0)) {
		tmp = (x / (y / x)) * 0.3333333333333333;
	} 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 / (y / x)) * 0.3333333333333333
	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 / Float64(y / x)) * 0.3333333333333333);
	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 / (y / x)) * 0.3333333333333333;
	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 / N[(y / x), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(1.0 / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.69999999999999996 or 5 < x

   1. Initial program 87.9%

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

   2. Taylor expanded in x around inf 84.4%

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

      1. unpow2 84.4%

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

   4. Simplified 84.4%

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

      1. associate-/r* 84.4%

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

      2. div-inv 84.4%

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

      3. associate-/l* 96.3%

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

      4. metadata-eval 96.3%

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

   6. Applied egg-rr 96.3%

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

   if -1.69999999999999996 < x < 5

   1. Initial program 99.6%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   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 98.5%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \lor \neg \left(x \leq 5\right):\\ \;\;\;\;\frac{x}{\frac{y}{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y}\\ \end{array} \]

Alternative 6: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \lor \neg \left(x \leq 3\right):\\ \;\;\;\;\frac{x}{\frac{y}{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot -1.3333333333333333}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -4.5) (not (<= x 3.0)))
   (* (/ x (/ y x)) 0.3333333333333333)
   (/ (+ 1.0 (* x -1.3333333333333333)) y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -4.5) || !(x <= 3.0)) {
		tmp = (x / (y / x)) * 0.3333333333333333;
	} 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 <= (-4.5d0)) .or. (.not. (x <= 3.0d0))) then
        tmp = (x / (y / x)) * 0.3333333333333333d0
    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 <= -4.5) || !(x <= 3.0)) {
		tmp = (x / (y / x)) * 0.3333333333333333;
	} else {
		tmp = (1.0 + (x * -1.3333333333333333)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -4.5) or not (x <= 3.0):
		tmp = (x / (y / x)) * 0.3333333333333333
	else:
		tmp = (1.0 + (x * -1.3333333333333333)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -4.5) || !(x <= 3.0))
		tmp = Float64(Float64(x / Float64(y / x)) * 0.3333333333333333);
	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 <= -4.5) || ~((x <= 3.0)))
		tmp = (x / (y / x)) * 0.3333333333333333;
	else
		tmp = (1.0 + (x * -1.3333333333333333)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -4.5], N[Not[LessEqual[x, 3.0]], $MachinePrecision]], N[(N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(1.0 + N[(x * -1.3333333333333333), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.5 or 3 < x

   1. Initial program 87.9%

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

   2. Taylor expanded in x around inf 84.4%

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

      1. unpow2 84.4%

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

   4. Simplified 84.4%

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

      1. associate-/r* 84.4%

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

      2. div-inv 84.4%

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

      3. associate-/l* 96.3%

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

      4. metadata-eval 96.3%

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

   6. Applied egg-rr 96.3%

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

   if -4.5 < x < 3

   1. Initial program 99.6%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   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 99.2%

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

   5. Taylor expanded in y around 0 99.2%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \lor \neg \left(x \leq 3\right):\\ \;\;\;\;\frac{x}{\frac{y}{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot -1.3333333333333333}{y}\\ \end{array} \]

Alternative 7: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7:\\ \;\;\;\;x \cdot \left(\frac{x}{y} \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;x \leq 5.2:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.7)
   (* x (* (/ x y) 0.3333333333333333))
   (if (<= x 5.2) (/ 1.0 y) (* (/ x y) (* x 0.3333333333333333)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.7) {
		tmp = x * ((x / y) * 0.3333333333333333);
	} else if (x <= 5.2) {
		tmp = 1.0 / y;
	} else {
		tmp = (x / y) * (x * 0.3333333333333333);
	}
	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 * ((x / y) * 0.3333333333333333d0)
    else if (x <= 5.2d0) then
        tmp = 1.0d0 / y
    else
        tmp = (x / y) * (x * 0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.7) {
		tmp = x * ((x / y) * 0.3333333333333333);
	} else if (x <= 5.2) {
		tmp = 1.0 / y;
	} else {
		tmp = (x / y) * (x * 0.3333333333333333);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.7:
		tmp = x * ((x / y) * 0.3333333333333333)
	elif x <= 5.2:
		tmp = 1.0 / y
	else:
		tmp = (x / y) * (x * 0.3333333333333333)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.7)
		tmp = Float64(x * Float64(Float64(x / y) * 0.3333333333333333));
	elseif (x <= 5.2)
		tmp = Float64(1.0 / y);
	else
		tmp = Float64(Float64(x / y) * Float64(x * 0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.7)
		tmp = x * ((x / y) * 0.3333333333333333);
	elseif (x <= 5.2)
		tmp = 1.0 / y;
	else
		tmp = (x / y) * (x * 0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.7], N[(x * N[(N[(x / y), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.2], N[(1.0 / y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.69999999999999996

   1. Initial program 88.8%

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

   2. Taylor expanded in x around inf 85.7%

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

      1. unpow2 85.7%

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

   4. Simplified 85.7%

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

      1. associate-/l* 96.5%

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

      2. associate-/r/ 96.5%

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

      3. metadata-eval 96.5%

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

      4. div-inv 96.6%

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

      5. clear-num 96.5%

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

      6. add-sqr-sqrt 50.8%

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

      7. sqrt-unprod 37.1%

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

      8. frac-times 37.1%

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

      9. metadata-eval 37.1%

         \[\leadsto \frac{x}{\frac{1}{\sqrt{\frac{\color{blue}{0.1111111111111111}}{y \cdot y}}}} \cdot x \]

      10. metadata-eval 37.1%

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

      11. frac-times 37.1%

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

      12. sqrt-unprod 0.3%

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

      13. add-sqr-sqrt 0.5%

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

      14. clear-num 0.5%

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

      15. associate-/l* 0.5%

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

      16. associate-*r/ 0.5%

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

      17. add-sqr-sqrt 0.3%

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

      18. sqrt-unprod 37.1%

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

      19. frac-times 37.1%

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

      20. metadata-eval 37.1%

          \[\leadsto \left(x \cdot \sqrt{\frac{\color{blue}{0.1111111111111111}}{y \cdot y}}\right) \cdot x \]

      21. metadata-eval 37.1%

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

      22. frac-times 37.1%

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

      23. sqrt-unprod 50.8%

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

      24. add-sqr-sqrt 96.6%

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

   6. Applied egg-rr 96.6%

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

   7. Taylor expanded in x around 0 96.6%

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

   if -1.69999999999999996 < x < 5.20000000000000018

   1. Initial program 99.6%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   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 98.5%

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

   if 5.20000000000000018 < x

   1. Initial program 86.7%

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

   2. Taylor expanded in x around inf 82.9%

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

      1. unpow2 82.9%

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

   4. Simplified 82.9%

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

      1. times-frac 95.9%

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

      2. div-inv 95.9%

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

      3. metadata-eval 95.9%

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

   6. Applied egg-rr 95.9%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7:\\ \;\;\;\;x \cdot \left(\frac{x}{y} \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;x \leq 5.2:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.3333333333333333\right)\\ \end{array} \]

Alternative 8: 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 93.8%

   \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
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 9: 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 93.8%

   \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
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* 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 10: 57.7% 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 88.8%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   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 0 33.5%

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

   5. Taylor expanded in x around inf 33.5%

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

   if -0.75 < x

   1. Initial program 95.7%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   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 70.2%

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

4. Final simplification 60.3%

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

Alternative 11: 57.6% 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 88.8%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   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. Taylor expanded in x around inf 96.8%

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

      1. metadata-eval 96.8%

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

      2. distribute-lft-neg-in 96.8%

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

      3. *-commutative 96.8%

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

      4. associate-*l/ 96.8%

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

      5. associate-*r/ 96.8%

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

      6. distribute-rgt-neg-in 96.8%

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

      7. distribute-neg-frac 96.8%

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

      8. metadata-eval 96.8%

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

   6. Simplified 96.8%

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

   7. Taylor expanded in x around 0 32.4%

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

      1. associate-*r/ 32.4%

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

      2. neg-mul-1 32.4%

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

   9. Simplified 32.4%

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

   if -1 < x

   1. Initial program 95.7%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   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 70.2%

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

4. Final simplification 60.0%

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

Alternative 12: 51.3% 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 93.8%

   \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
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 0 52.6%

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

5. Final simplification 52.6%

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

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 2023240 
(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)))