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. Final simplification 100.0%

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

Alternative 2: 73.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+18} \lor \neg \left(x \leq -1.6 \cdot 10^{-51} \lor \neg \left(x \leq -3.55 \cdot 10^{-90}\right) \land x \leq 3 \cdot 10^{-67}\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.75e+18)
         (not (or (<= x -1.6e-51) (and (not (<= x -3.55e-90)) (<= x 3e-67)))))
   (+ 1.0 (* 2.0 (/ y x)))
   -1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.75e+18) || !((x <= -1.6e-51) || (!(x <= -3.55e-90) && (x <= 3e-67)))) {
		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 <= (-1.75d+18)) .or. (.not. (x <= (-1.6d-51)) .or. (.not. (x <= (-3.55d-90))) .and. (x <= 3d-67))) 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 <= -1.75e+18) || !((x <= -1.6e-51) || (!(x <= -3.55e-90) && (x <= 3e-67)))) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.75e+18) or not ((x <= -1.6e-51) or (not (x <= -3.55e-90) and (x <= 3e-67))):
		tmp = 1.0 + (2.0 * (y / x))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.75e+18) || !((x <= -1.6e-51) || (!(x <= -3.55e-90) && (x <= 3e-67))))
		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 <= -1.75e+18) || ~(((x <= -1.6e-51) || (~((x <= -3.55e-90)) && (x <= 3e-67)))))
		tmp = 1.0 + (2.0 * (y / x));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.75e+18], N[Not[Or[LessEqual[x, -1.6e-51], And[N[Not[LessEqual[x, -3.55e-90]], $MachinePrecision], LessEqual[x, 3e-67]]]], $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 < -1.75e18 or -1.6e-51 < x < -3.5500000000000001e-90 or 3.00000000000000032e-67 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 78.1%

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

   if -1.75e18 < x < -1.6e-51 or -3.5500000000000001e-90 < x < 3.00000000000000032e-67

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 85.9%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+18} \lor \neg \left(x \leq -1.6 \cdot 10^{-51} \lor \neg \left(x \leq -3.55 \cdot 10^{-90}\right) \land x \leq 3 \cdot 10^{-67}\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+17} \lor \neg \left(x \leq -3.1 \cdot 10^{-53} \lor \neg \left(x \leq -5.7 \cdot 10^{-92}\right) \land x \leq 3.2 \cdot 10^{-65}\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.7e+17)
         (not (or (<= x -3.1e-53) (and (not (<= x -5.7e-92)) (<= x 3.2e-65)))))
   (+ 1.0 (* 2.0 (/ y x)))
   (+ (* -2.0 (/ x y)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.7e+17) || !((x <= -3.1e-53) || (!(x <= -5.7e-92) && (x <= 3.2e-65)))) {
		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 <= (-3.7d+17)) .or. (.not. (x <= (-3.1d-53)) .or. (.not. (x <= (-5.7d-92))) .and. (x <= 3.2d-65))) 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 <= -3.7e+17) || !((x <= -3.1e-53) || (!(x <= -5.7e-92) && (x <= 3.2e-65)))) {
		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 <= -3.7e+17) or not ((x <= -3.1e-53) or (not (x <= -5.7e-92) and (x <= 3.2e-65))):
		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 <= -3.7e+17) || !((x <= -3.1e-53) || (!(x <= -5.7e-92) && (x <= 3.2e-65))))
		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 <= -3.7e+17) || ~(((x <= -3.1e-53) || (~((x <= -5.7e-92)) && (x <= 3.2e-65)))))
		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, -3.7e+17], N[Not[Or[LessEqual[x, -3.1e-53], And[N[Not[LessEqual[x, -5.7e-92]], $MachinePrecision], LessEqual[x, 3.2e-65]]]], $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 < -3.7e17 or -3.10000000000000015e-53 < x < -5.70000000000000009e-92 or 3.1999999999999999e-65 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 78.1%

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

   if -3.7e17 < x < -3.10000000000000015e-53 or -5.70000000000000009e-92 < x < 3.1999999999999999e-65

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 86.2%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+17} \lor \neg \left(x \leq -3.1 \cdot 10^{-53} \lor \neg \left(x \leq -5.7 \cdot 10^{-92}\right) \land x \leq 3.2 \cdot 10^{-65}\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+18}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-51}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-90}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-71}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1e+18)
   1.0
   (if (<= x -4e-51) -1.0 (if (<= x -4e-90) 1.0 (if (<= x 5e-71) -1.0 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1e+18) {
		tmp = 1.0;
	} else if (x <= -4e-51) {
		tmp = -1.0;
	} else if (x <= -4e-90) {
		tmp = 1.0;
	} else if (x <= 5e-71) {
		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 <= (-1d+18)) then
        tmp = 1.0d0
    else if (x <= (-4d-51)) then
        tmp = -1.0d0
    else if (x <= (-4d-90)) then
        tmp = 1.0d0
    else if (x <= 5d-71) 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 <= -1e+18) {
		tmp = 1.0;
	} else if (x <= -4e-51) {
		tmp = -1.0;
	} else if (x <= -4e-90) {
		tmp = 1.0;
	} else if (x <= 5e-71) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1e+18:
		tmp = 1.0
	elif x <= -4e-51:
		tmp = -1.0
	elif x <= -4e-90:
		tmp = 1.0
	elif x <= 5e-71:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1e+18)
		tmp = 1.0;
	elseif (x <= -4e-51)
		tmp = -1.0;
	elseif (x <= -4e-90)
		tmp = 1.0;
	elseif (x <= 5e-71)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1e+18)
		tmp = 1.0;
	elseif (x <= -4e-51)
		tmp = -1.0;
	elseif (x <= -4e-90)
		tmp = 1.0;
	elseif (x <= 5e-71)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1e+18], 1.0, If[LessEqual[x, -4e-51], -1.0, If[LessEqual[x, -4e-90], 1.0, If[LessEqual[x, 5e-71], -1.0, 1.0]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1e18 or -4e-51 < x < -3.99999999999999998e-90 or 4.99999999999999998e-71 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 77.3%

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

   if -1e18 < x < -4e-51 or -3.99999999999999998e-90 < x < 4.99999999999999998e-71

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 85.9%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+18}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-51}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-90}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-71}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 48.8% 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 48.7%

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

4. Final simplification 48.7%

   \[\leadsto -1 \]
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 2024059 
(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)))