Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 100.0%

Time: 5.1s

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. Final simplification 100.0%

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

Alternative 2: 72.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+89} \lor \neg \left(x \leq 2 \cdot 10^{-52} \lor \neg \left(x \leq 1.06 \cdot 10^{+43}\right) \land x \leq 1.7 \cdot 10^{+82}\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.5e+89)
         (not (or (<= x 2e-52) (and (not (<= x 1.06e+43)) (<= x 1.7e+82)))))
   (+ 1.0 (* 2.0 (/ y x)))
   -1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.5e+89) || !((x <= 2e-52) || (!(x <= 1.06e+43) && (x <= 1.7e+82)))) {
		tmp = 1.0 + (2.0 * (y / x));
	} 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 ((x <= (-1.5d+89)) .or. (.not. (x <= 2d-52) .or. (.not. (x <= 1.06d+43)) .and. (x <= 1.7d+82))) then
        tmp = 1.0d0 + (2.0d0 * (y / x))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.5e+89) || !((x <= 2e-52) || (!(x <= 1.06e+43) && (x <= 1.7e+82)))) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.5e+89) or not ((x <= 2e-52) or (not (x <= 1.06e+43) and (x <= 1.7e+82))):
		tmp = 1.0 + (2.0 * (y / x))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.5e+89) || !((x <= 2e-52) || (!(x <= 1.06e+43) && (x <= 1.7e+82))))
		tmp = Float64(1.0 + Float64(2.0 * Float64(y / x)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.5e+89) || ~(((x <= 2e-52) || (~((x <= 1.06e+43)) && (x <= 1.7e+82)))))
		tmp = 1.0 + (2.0 * (y / x));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.5e+89], N[Not[Or[LessEqual[x, 2e-52], And[N[Not[LessEqual[x, 1.06e+43]], $MachinePrecision], LessEqual[x, 1.7e+82]]]], $MachinePrecision]], N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if x < -1.50000000000000006e89 or 2e-52 < x < 1.06000000000000006e43 or 1.69999999999999997e82 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 82.2%

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

   if -1.50000000000000006e89 < x < 2e-52 or 1.06000000000000006e43 < x < 1.69999999999999997e82

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 80.8%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+89} \lor \neg \left(x \leq 2 \cdot 10^{-52} \lor \neg \left(x \leq 1.06 \cdot 10^{+43}\right) \land x \leq 1.7 \cdot 10^{+82}\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+89} \lor \neg \left(x \leq 1.05 \cdot 10^{-50} \lor \neg \left(x \leq 2.8 \cdot 10^{+41}\right) \land x \leq 2.15 \cdot 10^{+83}\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.1e+89)
         (not (or (<= x 1.05e-50) (and (not (<= x 2.8e+41)) (<= x 2.15e+83)))))
   (+ 1.0 (* 2.0 (/ y x)))
   (+ (* -2.0 (/ x y)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.1e+89) || !((x <= 1.05e-50) || (!(x <= 2.8e+41) && (x <= 2.15e+83)))) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = (-2.0 * (x / y)) + -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 ((x <= (-1.1d+89)) .or. (.not. (x <= 1.05d-50) .or. (.not. (x <= 2.8d+41)) .and. (x <= 2.15d+83))) then
        tmp = 1.0d0 + (2.0d0 * (y / x))
    else
        tmp = ((-2.0d0) * (x / y)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.1e+89) || !((x <= 1.05e-50) || (!(x <= 2.8e+41) && (x <= 2.15e+83)))) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = (-2.0 * (x / y)) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.1e+89) or not ((x <= 1.05e-50) or (not (x <= 2.8e+41) and (x <= 2.15e+83))):
		tmp = 1.0 + (2.0 * (y / x))
	else:
		tmp = (-2.0 * (x / y)) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.1e+89) || !((x <= 1.05e-50) || (!(x <= 2.8e+41) && (x <= 2.15e+83))))
		tmp = Float64(1.0 + Float64(2.0 * Float64(y / x)));
	else
		tmp = Float64(Float64(-2.0 * Float64(x / y)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.1e+89) || ~(((x <= 1.05e-50) || (~((x <= 2.8e+41)) && (x <= 2.15e+83)))))
		tmp = 1.0 + (2.0 * (y / x));
	else
		tmp = (-2.0 * (x / y)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.1e+89], N[Not[Or[LessEqual[x, 1.05e-50], And[N[Not[LessEqual[x, 2.8e+41]], $MachinePrecision], LessEqual[x, 2.15e+83]]]], $MachinePrecision]], N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1e89 or 1.05e-50 < x < 2.7999999999999999e41 or 2.15e83 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 82.2%

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

   if -1.1e89 < x < 1.05e-50 or 2.7999999999999999e41 < x < 2.15e83

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 82.5%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+89} \lor \neg \left(x \leq 1.05 \cdot 10^{-50} \lor \neg \left(x \leq 2.8 \cdot 10^{+41}\right) \land x \leq 2.15 \cdot 10^{+83}\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.7 \cdot 10^{+88}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-50}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.96 \cdot 10^{+43}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 10^{+77}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.7e+88)
   1.0
   (if (<= x 6e-50) -1.0 (if (<= x 1.96e+43) 1.0 (if (<= x 1e+77) -1.0 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.7e+88) {
		tmp = 1.0;
	} else if (x <= 6e-50) {
		tmp = -1.0;
	} else if (x <= 1.96e+43) {
		tmp = 1.0;
	} else if (x <= 1e+77) {
		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 (x <= (-5.7d+88)) then
        tmp = 1.0d0
    else if (x <= 6d-50) then
        tmp = -1.0d0
    else if (x <= 1.96d+43) then
        tmp = 1.0d0
    else if (x <= 1d+77) 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 (x <= -5.7e+88) {
		tmp = 1.0;
	} else if (x <= 6e-50) {
		tmp = -1.0;
	} else if (x <= 1.96e+43) {
		tmp = 1.0;
	} else if (x <= 1e+77) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.7e+88:
		tmp = 1.0
	elif x <= 6e-50:
		tmp = -1.0
	elif x <= 1.96e+43:
		tmp = 1.0
	elif x <= 1e+77:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.7e+88)
		tmp = 1.0;
	elseif (x <= 6e-50)
		tmp = -1.0;
	elseif (x <= 1.96e+43)
		tmp = 1.0;
	elseif (x <= 1e+77)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.7e+88)
		tmp = 1.0;
	elseif (x <= 6e-50)
		tmp = -1.0;
	elseif (x <= 1.96e+43)
		tmp = 1.0;
	elseif (x <= 1e+77)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.7e+88], 1.0, If[LessEqual[x, 6e-50], -1.0, If[LessEqual[x, 1.96e+43], 1.0, If[LessEqual[x, 1e+77], -1.0, 1.0]]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.70000000000000021e88 or 5.99999999999999981e-50 < x < 1.9600000000000001e43 or 9.99999999999999983e76 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 80.8%

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

   if -5.70000000000000021e88 < x < 5.99999999999999981e-50 or 1.9600000000000001e43 < x < 9.99999999999999983e76

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 81.2%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.7 \cdot 10^{+88}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-50}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.96 \cdot 10^{+43}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 10^{+77}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.2% 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 55.9%

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

4. Final simplification 55.9%

   \[\leadsto -1 \]
5. 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 2024066 
(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)))