Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 100.0%

Time: 4.1s

Alternatives: 4

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{x + y}{x - y} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 74.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+45}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-12}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{elif}\;y \leq 62000:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 200000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.5e+45)
   -1.0
   (if (<= y 2.7e-12)
     (+ 1.0 (* 2.0 (/ y x)))
     (if (<= y 62000.0) -1.0 (if (<= y 200000000.0) 1.0 -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.5e+45) {
		tmp = -1.0;
	} else if (y <= 2.7e-12) {
		tmp = 1.0 + (2.0 * (y / x));
	} else if (y <= 62000.0) {
		tmp = -1.0;
	} else if (y <= 200000000.0) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.5d+45)) then
        tmp = -1.0d0
    else if (y <= 2.7d-12) then
        tmp = 1.0d0 + (2.0d0 * (y / x))
    else if (y <= 62000.0d0) then
        tmp = -1.0d0
    else if (y <= 200000000.0d0) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.5e+45) {
		tmp = -1.0;
	} else if (y <= 2.7e-12) {
		tmp = 1.0 + (2.0 * (y / x));
	} else if (y <= 62000.0) {
		tmp = -1.0;
	} else if (y <= 200000000.0) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.5e+45:
		tmp = -1.0
	elif y <= 2.7e-12:
		tmp = 1.0 + (2.0 * (y / x))
	elif y <= 62000.0:
		tmp = -1.0
	elif y <= 200000000.0:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.5e+45)
		tmp = -1.0;
	elseif (y <= 2.7e-12)
		tmp = Float64(1.0 + Float64(2.0 * Float64(y / x)));
	elseif (y <= 62000.0)
		tmp = -1.0;
	elseif (y <= 200000000.0)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.5e+45)
		tmp = -1.0;
	elseif (y <= 2.7e-12)
		tmp = 1.0 + (2.0 * (y / x));
	elseif (y <= 62000.0)
		tmp = -1.0;
	elseif (y <= 200000000.0)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.5e+45], -1.0, If[LessEqual[y, 2.7e-12], N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 62000.0], -1.0, If[LessEqual[y, 200000000.0], 1.0, -1.0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.4999999999999998e45 or 2.6999999999999998e-12 < y < 62000 or 2e8 < y

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 83.9%

      \[\leadsto \color{blue}{-1} \]

   if -4.4999999999999998e45 < y < 2.6999999999999998e-12

   1. Initial program 99.9%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 80.7%

      \[\leadsto \color{blue}{1 + 2 \cdot \frac{y}{x}} \]

   if 62000 < y < 2e8

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

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

Alternative 3: 74.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+43}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 59000:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 64000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2e+43)
   -1.0
   (if (<= y 8e-13)
     1.0
     (if (<= y 59000.0) -1.0 (if (<= y 64000000.0) 1.0 -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2e+43) {
		tmp = -1.0;
	} else if (y <= 8e-13) {
		tmp = 1.0;
	} else if (y <= 59000.0) {
		tmp = -1.0;
	} else if (y <= 64000000.0) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2d+43)) then
        tmp = -1.0d0
    else if (y <= 8d-13) then
        tmp = 1.0d0
    else if (y <= 59000.0d0) then
        tmp = -1.0d0
    else if (y <= 64000000.0d0) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2e+43) {
		tmp = -1.0;
	} else if (y <= 8e-13) {
		tmp = 1.0;
	} else if (y <= 59000.0) {
		tmp = -1.0;
	} else if (y <= 64000000.0) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2e+43:
		tmp = -1.0
	elif y <= 8e-13:
		tmp = 1.0
	elif y <= 59000.0:
		tmp = -1.0
	elif y <= 64000000.0:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2e+43)
		tmp = -1.0;
	elseif (y <= 8e-13)
		tmp = 1.0;
	elseif (y <= 59000.0)
		tmp = -1.0;
	elseif (y <= 64000000.0)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2e+43)
		tmp = -1.0;
	elseif (y <= 8e-13)
		tmp = 1.0;
	elseif (y <= 59000.0)
		tmp = -1.0;
	elseif (y <= 64000000.0)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2e+43], -1.0, If[LessEqual[y, 8e-13], 1.0, If[LessEqual[y, 59000.0], -1.0, If[LessEqual[y, 64000000.0], 1.0, -1.0]]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.00000000000000003e43 or 8.0000000000000002e-13 < y < 59000 or 6.4e7 < y

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 83.3%

      \[\leadsto \color{blue}{-1} \]

   if -2.00000000000000003e43 < y < 8.0000000000000002e-13 or 59000 < y < 6.4e7

   1. Initial program 99.9%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 80.0%

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

Alternative 4: 49.4% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

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

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 100.0%

   \[\frac{x + y}{x - y} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 53.2%

   \[\leadsto \color{blue}{-1} \]
4. Add Preprocessing

Developer target: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024107 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (/ 1.0 (- (/ x (+ x y)) (/ y (+ x y))))

  (/ (+ x y) (- x y)))