Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 99.9%

Time: 4.3s

Alternatives: 6

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: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{x - y}{x + y}\right)}^{-1} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (pow (/ (- x y) (+ x y)) -1.0))

C

double code(double x, double y) {
	return pow(((x - y) / (x + y)), -1.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x - y) / (x + y)) ** (-1.0d0)
end function

Java

public static double code(double x, double y) {
	return Math.pow(((x - y) / (x + y)), -1.0);
}

Python

def code(x, y):
	return math.pow(((x - y) / (x + y)), -1.0)

Julia

function code(x, y)
	return Float64(Float64(x - y) / Float64(x + y)) ^ -1.0
end

MATLAB

function tmp = code(x, y)
	tmp = ((x - y) / (x + y)) ^ -1.0;
end

Wolfram

code[x_, y_] := N[Power[N[(N[(x - y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{x + y}{x - y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. clear-num 100.0%

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

   2. inv-pow 100.0%

      \[\leadsto \color{blue}{{\left(\frac{x - y}{x + y}\right)}^{-1}} \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{\left(\frac{x - y}{x + y}\right)}^{-1}} \]
5. Add Preprocessing

Alternative 2: 74.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.75 \cdot 10^{+24} \lor \neg \left(y \leq 1.1 \cdot 10^{-56}\right):\\ \;\;\;\;-1 + -2 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.75e+24) (not (<= y 1.1e-56)))
   (+ -1.0 (* -2.0 (/ x y)))
   (+ 1.0 (* 2.0 (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.75e+24) || !(y <= 1.1e-56)) {
		tmp = -1.0 + (-2.0 * (x / y));
	} else {
		tmp = 1.0 + (2.0 * (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-4.75d+24)) .or. (.not. (y <= 1.1d-56))) then
        tmp = (-1.0d0) + ((-2.0d0) * (x / y))
    else
        tmp = 1.0d0 + (2.0d0 * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4.75e+24) || !(y <= 1.1e-56)) {
		tmp = -1.0 + (-2.0 * (x / y));
	} else {
		tmp = 1.0 + (2.0 * (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.75e+24) or not (y <= 1.1e-56):
		tmp = -1.0 + (-2.0 * (x / y))
	else:
		tmp = 1.0 + (2.0 * (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.75e+24) || !(y <= 1.1e-56))
		tmp = Float64(-1.0 + Float64(-2.0 * Float64(x / y)));
	else
		tmp = Float64(1.0 + Float64(2.0 * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4.75e+24) || ~((y <= 1.1e-56)))
		tmp = -1.0 + (-2.0 * (x / y));
	else
		tmp = 1.0 + (2.0 * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.75e+24], N[Not[LessEqual[y, 1.1e-56]], $MachinePrecision]], N[(-1.0 + N[(-2.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.7500000000000001e24 or 1.10000000000000002e-56 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 78.7%

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

   if -4.7500000000000001e24 < y < 1.10000000000000002e-56

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 81.9%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.75 \cdot 10^{+24} \lor \neg \left(y \leq 1.1 \cdot 10^{-56}\right):\\ \;\;\;\;-1 + -2 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+54}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-56}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9e+54) -1.0 (if (<= y 2.8e-56) (+ 1.0 (* 2.0 (/ y x))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9e+54) {
		tmp = -1.0;
	} else if (y <= 2.8e-56) {
		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 <= (-9d+54)) then
        tmp = -1.0d0
    else if (y <= 2.8d-56) 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 <= -9e+54) {
		tmp = -1.0;
	} else if (y <= 2.8e-56) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9e+54:
		tmp = -1.0
	elif y <= 2.8e-56:
		tmp = 1.0 + (2.0 * (y / x))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9e+54)
		tmp = -1.0;
	elseif (y <= 2.8e-56)
		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 <= -9e+54)
		tmp = -1.0;
	elseif (y <= 2.8e-56)
		tmp = 1.0 + (2.0 * (y / x));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9e+54], -1.0, If[LessEqual[y, 2.8e-56], N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -8.99999999999999968e54 or 2.79999999999999993e-56 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 79.1%

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

   if -8.99999999999999968e54 < y < 2.79999999999999993e-56

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 80.5%

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

Alternative 4: 73.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-22}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-58}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5e-22) -1.0 (if (<= y 1.4e-58) 1.0 -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5e-22) {
		tmp = -1.0;
	} else if (y <= 1.4e-58) {
		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 <= (-5d-22)) then
        tmp = -1.0d0
    else if (y <= 1.4d-58) 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 <= -5e-22) {
		tmp = -1.0;
	} else if (y <= 1.4e-58) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5e-22:
		tmp = -1.0
	elif y <= 1.4e-58:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5e-22)
		tmp = -1.0;
	elseif (y <= 1.4e-58)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5e-22)
		tmp = -1.0;
	elseif (y <= 1.4e-58)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5e-22], -1.0, If[LessEqual[y, 1.4e-58], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -4.99999999999999954e-22 or 1.4e-58 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 76.6%

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

   if -4.99999999999999954e-22 < y < 1.4e-58

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 82.8%

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

Alternative 5: 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 6: 49.4% 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 48.8%

   \[\leadsto \color{blue}{-1} \]
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 2024085 
(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)))