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. Final simplification 100.0%

   \[\leadsto \frac{x + y}{x - y} \]

Alternative 2: 75.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -2 \cdot \frac{x}{y} + -1\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-20}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+18} \lor \neg \left(y \leq 1.5 \cdot 10^{+49}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (* -2.0 (/ x y)) -1.0)))
   (if (<= y -6.5e-24)
     t_0
     (if (<= y 4.2e-20)
       (+ 1.0 (* 2.0 (/ y x)))
       (if (or (<= y 2.1e+18) (not (<= y 1.5e+49))) t_0 1.0)))))

C

double code(double x, double y) {
	double t_0 = (-2.0 * (x / y)) + -1.0;
	double tmp;
	if (y <= -6.5e-24) {
		tmp = t_0;
	} else if (y <= 4.2e-20) {
		tmp = 1.0 + (2.0 * (y / x));
	} else if ((y <= 2.1e+18) || !(y <= 1.5e+49)) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = ((-2.0d0) * (x / y)) + (-1.0d0)
    if (y <= (-6.5d-24)) then
        tmp = t_0
    else if (y <= 4.2d-20) then
        tmp = 1.0d0 + (2.0d0 * (y / x))
    else if ((y <= 2.1d+18) .or. (.not. (y <= 1.5d+49))) then
        tmp = t_0
    else
        tmp = 1.0d0
    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 tmp;
	if (y <= -6.5e-24) {
		tmp = t_0;
	} else if (y <= 4.2e-20) {
		tmp = 1.0 + (2.0 * (y / x));
	} else if ((y <= 2.1e+18) || !(y <= 1.5e+49)) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (-2.0 * (x / y)) + -1.0
	tmp = 0
	if y <= -6.5e-24:
		tmp = t_0
	elif y <= 4.2e-20:
		tmp = 1.0 + (2.0 * (y / x))
	elif (y <= 2.1e+18) or not (y <= 1.5e+49):
		tmp = t_0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(-2.0 * Float64(x / y)) + -1.0)
	tmp = 0.0
	if (y <= -6.5e-24)
		tmp = t_0;
	elseif (y <= 4.2e-20)
		tmp = Float64(1.0 + Float64(2.0 * Float64(y / x)));
	elseif ((y <= 2.1e+18) || !(y <= 1.5e+49))
		tmp = t_0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (-2.0 * (x / y)) + -1.0;
	tmp = 0.0;
	if (y <= -6.5e-24)
		tmp = t_0;
	elseif (y <= 4.2e-20)
		tmp = 1.0 + (2.0 * (y / x));
	elseif ((y <= 2.1e+18) || ~((y <= 1.5e+49)))
		tmp = t_0;
	else
		tmp = 1.0;
	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]}, If[LessEqual[y, -6.5e-24], t$95$0, If[LessEqual[y, 4.2e-20], N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 2.1e+18], N[Not[LessEqual[y, 1.5e+49]], $MachinePrecision]], t$95$0, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.5e-24 or 4.1999999999999998e-20 < y < 2.1e18 or 1.5000000000000001e49 < y

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]

   2. Taylor expanded in x around 0 79.7%

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

   if -6.5e-24 < y < 4.1999999999999998e-20

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]

   2. Taylor expanded in y around 0 80.4%

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

   if 2.1e18 < y < 1.5000000000000001e49

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]

   2. Taylor expanded in x around inf 100.0%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-24}:\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-20}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+18} \lor \neg \left(y \leq 1.5 \cdot 10^{+49}\right):\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 74.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-22}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-14}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+17}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+50}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.5e-22)
   -1.0
   (if (<= y 1.9e-14)
     (+ 1.0 (* 2.0 (/ y x)))
     (if (<= y 5.2e+17) -1.0 (if (<= y 2e+50) 1.0 -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.5e-22) {
		tmp = -1.0;
	} else if (y <= 1.9e-14) {
		tmp = 1.0 + (2.0 * (y / x));
	} else if (y <= 5.2e+17) {
		tmp = -1.0;
	} else if (y <= 2e+50) {
		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 <= (-2.5d-22)) then
        tmp = -1.0d0
    else if (y <= 1.9d-14) then
        tmp = 1.0d0 + (2.0d0 * (y / x))
    else if (y <= 5.2d+17) then
        tmp = -1.0d0
    else if (y <= 2d+50) 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 <= -2.5e-22) {
		tmp = -1.0;
	} else if (y <= 1.9e-14) {
		tmp = 1.0 + (2.0 * (y / x));
	} else if (y <= 5.2e+17) {
		tmp = -1.0;
	} else if (y <= 2e+50) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.5e-22:
		tmp = -1.0
	elif y <= 1.9e-14:
		tmp = 1.0 + (2.0 * (y / x))
	elif y <= 5.2e+17:
		tmp = -1.0
	elif y <= 2e+50:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.5e-22)
		tmp = -1.0;
	elseif (y <= 1.9e-14)
		tmp = Float64(1.0 + Float64(2.0 * Float64(y / x)));
	elseif (y <= 5.2e+17)
		tmp = -1.0;
	elseif (y <= 2e+50)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.5e-22)
		tmp = -1.0;
	elseif (y <= 1.9e-14)
		tmp = 1.0 + (2.0 * (y / x));
	elseif (y <= 5.2e+17)
		tmp = -1.0;
	elseif (y <= 2e+50)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.5e-22], -1.0, If[LessEqual[y, 1.9e-14], N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.2e+17], -1.0, If[LessEqual[y, 2e+50], 1.0, -1.0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.49999999999999977e-22 or 1.9000000000000001e-14 < y < 5.2e17 or 2.0000000000000002e50 < y

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]

   2. Taylor expanded in x around 0 78.6%

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

   if -2.49999999999999977e-22 < y < 1.9000000000000001e-14

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]

   2. Taylor expanded in y around 0 80.4%

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

   if 5.2e17 < y < 2.0000000000000002e50

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]

   2. Taylor expanded in x around inf 100.0%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-22}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-14}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+17}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+50}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4: 73.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-23}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-12}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+18}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+51}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.2e-23)
   -1.0
   (if (<= y 3.4e-12)
     1.0
     (if (<= y 4e+18) -1.0 (if (<= y 3.4e+51) 1.0 -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.2e-23) {
		tmp = -1.0;
	} else if (y <= 3.4e-12) {
		tmp = 1.0;
	} else if (y <= 4e+18) {
		tmp = -1.0;
	} else if (y <= 3.4e+51) {
		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 <= (-3.2d-23)) then
        tmp = -1.0d0
    else if (y <= 3.4d-12) then
        tmp = 1.0d0
    else if (y <= 4d+18) then
        tmp = -1.0d0
    else if (y <= 3.4d+51) 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 <= -3.2e-23) {
		tmp = -1.0;
	} else if (y <= 3.4e-12) {
		tmp = 1.0;
	} else if (y <= 4e+18) {
		tmp = -1.0;
	} else if (y <= 3.4e+51) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.2e-23:
		tmp = -1.0
	elif y <= 3.4e-12:
		tmp = 1.0
	elif y <= 4e+18:
		tmp = -1.0
	elif y <= 3.4e+51:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.2e-23)
		tmp = -1.0;
	elseif (y <= 3.4e-12)
		tmp = 1.0;
	elseif (y <= 4e+18)
		tmp = -1.0;
	elseif (y <= 3.4e+51)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.2e-23)
		tmp = -1.0;
	elseif (y <= 3.4e-12)
		tmp = 1.0;
	elseif (y <= 4e+18)
		tmp = -1.0;
	elseif (y <= 3.4e+51)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.2e-23], -1.0, If[LessEqual[y, 3.4e-12], 1.0, If[LessEqual[y, 4e+18], -1.0, If[LessEqual[y, 3.4e+51], 1.0, -1.0]]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.19999999999999976e-23 or 3.4000000000000001e-12 < y < 4e18 or 3.39999999999999984e51 < y

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]

   2. Taylor expanded in x around 0 78.6%

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

   if -3.19999999999999976e-23 < y < 3.4000000000000001e-12 or 4e18 < y < 3.39999999999999984e51

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]

   2. Taylor expanded in x around inf 80.6%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-23}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-12}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+18}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+51}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 5: 49.0% 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. Taylor expanded in x around 0 49.0%

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

3. Final simplification 49.0%

   \[\leadsto -1 \]

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 2023271 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (/ 1.0 (- (/ x (+ x y)) (/ y (+ x y))))

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