Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 100.0%

Time: 3.0s

Alternatives: 5

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: 75.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -2 \cdot \frac{x}{y} + -1\\ t_1 := 1 + 2 \cdot \frac{y}{x}\\ \mathbf{if}\;y \leq -63:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-46}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (* -2.0 (/ x y)) -1.0)) (t_1 (+ 1.0 (* 2.0 (/ y x)))))
   (if (<= y -63.0)
     t_0
     (if (<= y 8.2e-70)
       t_1
       (if (<= y 5.6e-46) -1.0 (if (<= y 1.35e+25) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = (-2.0 * (x / y)) + -1.0;
	double t_1 = 1.0 + (2.0 * (y / x));
	double tmp;
	if (y <= -63.0) {
		tmp = t_0;
	} else if (y <= 8.2e-70) {
		tmp = t_1;
	} else if (y <= 5.6e-46) {
		tmp = -1.0;
	} else if (y <= 1.35e+25) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((-2.0d0) * (x / y)) + (-1.0d0)
    t_1 = 1.0d0 + (2.0d0 * (y / x))
    if (y <= (-63.0d0)) then
        tmp = t_0
    else if (y <= 8.2d-70) then
        tmp = t_1
    else if (y <= 5.6d-46) then
        tmp = -1.0d0
    else if (y <= 1.35d+25) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (-2.0 * (x / y)) + -1.0;
	double t_1 = 1.0 + (2.0 * (y / x));
	double tmp;
	if (y <= -63.0) {
		tmp = t_0;
	} else if (y <= 8.2e-70) {
		tmp = t_1;
	} else if (y <= 5.6e-46) {
		tmp = -1.0;
	} else if (y <= 1.35e+25) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (-2.0 * (x / y)) + -1.0
	t_1 = 1.0 + (2.0 * (y / x))
	tmp = 0
	if y <= -63.0:
		tmp = t_0
	elif y <= 8.2e-70:
		tmp = t_1
	elif y <= 5.6e-46:
		tmp = -1.0
	elif y <= 1.35e+25:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(-2.0 * Float64(x / y)) + -1.0)
	t_1 = Float64(1.0 + Float64(2.0 * Float64(y / x)))
	tmp = 0.0
	if (y <= -63.0)
		tmp = t_0;
	elseif (y <= 8.2e-70)
		tmp = t_1;
	elseif (y <= 5.6e-46)
		tmp = -1.0;
	elseif (y <= 1.35e+25)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (-2.0 * (x / y)) + -1.0;
	t_1 = 1.0 + (2.0 * (y / x));
	tmp = 0.0;
	if (y <= -63.0)
		tmp = t_0;
	elseif (y <= 8.2e-70)
		tmp = t_1;
	elseif (y <= 5.6e-46)
		tmp = -1.0;
	elseif (y <= 1.35e+25)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(-2.0 * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -63.0], t$95$0, If[LessEqual[y, 8.2e-70], t$95$1, If[LessEqual[y, 5.6e-46], -1.0, If[LessEqual[y, 1.35e+25], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -63 or 1.35e25 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 78.0%

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

   if -63 < y < 8.19999999999999955e-70 or 5.5999999999999997e-46 < y < 1.35e25

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 79.8%

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

   if 8.19999999999999955e-70 < y < 5.5999999999999997e-46

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -63:\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-70}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-46}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+25}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 2 \cdot \frac{y}{x}\\ \mathbf{if}\;x \leq -1.15 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.32 \cdot 10^{-154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-166}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 38000:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 2.0 (/ y x)))))
   (if (<= x -1.15e-9)
     t_0
     (if (<= x -1.32e-154)
       -1.0
       (if (<= x -3.4e-166) 1.0 (if (<= x 38000.0) -1.0 t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (2.0 * (y / x));
	double tmp;
	if (x <= -1.15e-9) {
		tmp = t_0;
	} else if (x <= -1.32e-154) {
		tmp = -1.0;
	} else if (x <= -3.4e-166) {
		tmp = 1.0;
	} else if (x <= 38000.0) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (2.0d0 * (y / x))
    if (x <= (-1.15d-9)) then
        tmp = t_0
    else if (x <= (-1.32d-154)) then
        tmp = -1.0d0
    else if (x <= (-3.4d-166)) then
        tmp = 1.0d0
    else if (x <= 38000.0d0) then
        tmp = -1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (2.0 * (y / x));
	double tmp;
	if (x <= -1.15e-9) {
		tmp = t_0;
	} else if (x <= -1.32e-154) {
		tmp = -1.0;
	} else if (x <= -3.4e-166) {
		tmp = 1.0;
	} else if (x <= 38000.0) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (2.0 * (y / x))
	tmp = 0
	if x <= -1.15e-9:
		tmp = t_0
	elif x <= -1.32e-154:
		tmp = -1.0
	elif x <= -3.4e-166:
		tmp = 1.0
	elif x <= 38000.0:
		tmp = -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(2.0 * Float64(y / x)))
	tmp = 0.0
	if (x <= -1.15e-9)
		tmp = t_0;
	elseif (x <= -1.32e-154)
		tmp = -1.0;
	elseif (x <= -3.4e-166)
		tmp = 1.0;
	elseif (x <= 38000.0)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (2.0 * (y / x));
	tmp = 0.0;
	if (x <= -1.15e-9)
		tmp = t_0;
	elseif (x <= -1.32e-154)
		tmp = -1.0;
	elseif (x <= -3.4e-166)
		tmp = 1.0;
	elseif (x <= 38000.0)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.15e-9], t$95$0, If[LessEqual[x, -1.32e-154], -1.0, If[LessEqual[x, -3.4e-166], 1.0, If[LessEqual[x, 38000.0], -1.0, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.15e-9 or 38000 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 80.9%

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

   if -1.15e-9 < x < -1.31999999999999994e-154 or -3.3999999999999997e-166 < x < 38000

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 75.9%

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

   if -1.31999999999999994e-154 < x < -3.3999999999999997e-166

   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 4: 74.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -10000:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{-70}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-46}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -10000.0)
   -1.0
   (if (<= y 8.6e-70)
     1.0
     (if (<= y 5.2e-46) -1.0 (if (<= y 6.6e+24) 1.0 -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -10000.0) {
		tmp = -1.0;
	} else if (y <= 8.6e-70) {
		tmp = 1.0;
	} else if (y <= 5.2e-46) {
		tmp = -1.0;
	} else if (y <= 6.6e+24) {
		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 <= (-10000.0d0)) then
        tmp = -1.0d0
    else if (y <= 8.6d-70) then
        tmp = 1.0d0
    else if (y <= 5.2d-46) then
        tmp = -1.0d0
    else if (y <= 6.6d+24) 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 <= -10000.0) {
		tmp = -1.0;
	} else if (y <= 8.6e-70) {
		tmp = 1.0;
	} else if (y <= 5.2e-46) {
		tmp = -1.0;
	} else if (y <= 6.6e+24) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -10000.0:
		tmp = -1.0
	elif y <= 8.6e-70:
		tmp = 1.0
	elif y <= 5.2e-46:
		tmp = -1.0
	elif y <= 6.6e+24:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -10000.0)
		tmp = -1.0;
	elseif (y <= 8.6e-70)
		tmp = 1.0;
	elseif (y <= 5.2e-46)
		tmp = -1.0;
	elseif (y <= 6.6e+24)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -10000.0)
		tmp = -1.0;
	elseif (y <= 8.6e-70)
		tmp = 1.0;
	elseif (y <= 5.2e-46)
		tmp = -1.0;
	elseif (y <= 6.6e+24)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -10000.0], -1.0, If[LessEqual[y, 8.6e-70], 1.0, If[LessEqual[y, 5.2e-46], -1.0, If[LessEqual[y, 6.6e+24], 1.0, -1.0]]]]

Derivation

1. Split input into 2 regimes

2. if y < -1e4 or 8.6e-70 < y < 5.2000000000000004e-46 or 6.5999999999999998e24 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 77.6%

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

   if -1e4 < y < 8.6e-70 or 5.2000000000000004e-46 < y < 6.5999999999999998e24

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 79.1%

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

Alternative 5: 49.3% 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 46.5%

   \[\leadsto \color{blue}{-1} \]
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 2024091 
(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)))