Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 100.0%

Time: 4.1s

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: 74.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.35 \cdot 10^{+56} \lor \neg \left(x \leq -1.22 \cdot 10^{-7} \lor \neg \left(x \leq -2.7 \cdot 10^{-60}\right) \land x \leq 2.7 \cdot 10^{-17}\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.35e+56)
         (not (or (<= x -1.22e-7) (and (not (<= x -2.7e-60)) (<= x 2.7e-17)))))
   (+ 1.0 (* 2.0 (/ y x)))
   (+ (* -2.0 (/ x y)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.35e+56) || !((x <= -1.22e-7) || (!(x <= -2.7e-60) && (x <= 2.7e-17)))) {
		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 <= (-3.35d+56)) .or. (.not. (x <= (-1.22d-7)) .or. (.not. (x <= (-2.7d-60))) .and. (x <= 2.7d-17))) 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 <= -3.35e+56) || !((x <= -1.22e-7) || (!(x <= -2.7e-60) && (x <= 2.7e-17)))) {
		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 <= -3.35e+56) or not ((x <= -1.22e-7) or (not (x <= -2.7e-60) and (x <= 2.7e-17))):
		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 <= -3.35e+56) || !((x <= -1.22e-7) || (!(x <= -2.7e-60) && (x <= 2.7e-17))))
		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 <= -3.35e+56) || ~(((x <= -1.22e-7) || (~((x <= -2.7e-60)) && (x <= 2.7e-17)))))
		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, -3.35e+56], N[Not[Or[LessEqual[x, -1.22e-7], And[N[Not[LessEqual[x, -2.7e-60]], $MachinePrecision], LessEqual[x, 2.7e-17]]]], $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 < -3.35e56 or -1.2200000000000001e-7 < x < -2.7e-60 or 2.7000000000000001e-17 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 79.0%

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

   if -3.35e56 < x < -1.2200000000000001e-7 or -2.7e-60 < x < 2.7000000000000001e-17

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 80.1%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.35 \cdot 10^{+56} \lor \neg \left(x \leq -1.22 \cdot 10^{-7} \lor \neg \left(x \leq -2.7 \cdot 10^{-60}\right) \land x \leq 2.7 \cdot 10^{-17}\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+57}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{+27}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-59}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-74}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.6e+57)
   1.0
   (if (<= x -2.15e+27)
     -1.0
     (if (<= x -5.8e-59) 1.0 (if (<= x 2.5e-74) -1.0 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.6e+57) {
		tmp = 1.0;
	} else if (x <= -2.15e+27) {
		tmp = -1.0;
	} else if (x <= -5.8e-59) {
		tmp = 1.0;
	} else if (x <= 2.5e-74) {
		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 <= (-5.6d+57)) then
        tmp = 1.0d0
    else if (x <= (-2.15d+27)) then
        tmp = -1.0d0
    else if (x <= (-5.8d-59)) then
        tmp = 1.0d0
    else if (x <= 2.5d-74) 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 <= -5.6e+57) {
		tmp = 1.0;
	} else if (x <= -2.15e+27) {
		tmp = -1.0;
	} else if (x <= -5.8e-59) {
		tmp = 1.0;
	} else if (x <= 2.5e-74) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.6e+57:
		tmp = 1.0
	elif x <= -2.15e+27:
		tmp = -1.0
	elif x <= -5.8e-59:
		tmp = 1.0
	elif x <= 2.5e-74:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.6e+57)
		tmp = 1.0;
	elseif (x <= -2.15e+27)
		tmp = -1.0;
	elseif (x <= -5.8e-59)
		tmp = 1.0;
	elseif (x <= 2.5e-74)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.6e+57)
		tmp = 1.0;
	elseif (x <= -2.15e+27)
		tmp = -1.0;
	elseif (x <= -5.8e-59)
		tmp = 1.0;
	elseif (x <= 2.5e-74)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.6e+57], 1.0, If[LessEqual[x, -2.15e+27], -1.0, If[LessEqual[x, -5.8e-59], 1.0, If[LessEqual[x, 2.5e-74], -1.0, 1.0]]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.59999999999999999e57 or -2.15000000000000004e27 < x < -5.80000000000000033e-59 or 2.49999999999999999e-74 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 74.6%

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

   if -5.59999999999999999e57 < x < -2.15000000000000004e27 or -5.80000000000000033e-59 < x < 2.49999999999999999e-74

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 83.1%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+57}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{+27}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-59}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-74}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{-22}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+22}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.05e-22) -1.0 (if (<= y 1.9e+22) (+ 1.0 (* 2.0 (/ y x))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.05e-22) {
		tmp = -1.0;
	} else if (y <= 1.9e+22) {
		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 (y <= (-2.05d-22)) then
        tmp = -1.0d0
    else if (y <= 1.9d+22) 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 (y <= -2.05e-22) {
		tmp = -1.0;
	} else if (y <= 1.9e+22) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.05e-22:
		tmp = -1.0
	elif y <= 1.9e+22:
		tmp = 1.0 + (2.0 * (y / x))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.05e-22)
		tmp = -1.0;
	elseif (y <= 1.9e+22)
		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 (y <= -2.05e-22)
		tmp = -1.0;
	elseif (y <= 1.9e+22)
		tmp = 1.0 + (2.0 * (y / x));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.05e-22], -1.0, If[LessEqual[y, 1.9e+22], N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -2.05e-22 or 1.9000000000000002e22 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 77.1%

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

   if -2.05e-22 < y < 1.9000000000000002e22

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 74.6%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{-22}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+22}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
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 49.9%

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

4. Final simplification 49.9%

   \[\leadsto -1 \]
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 2024029 
(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)))