Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 100.0%

Time: 7.8s

Alternatives: 7

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 99.9%

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

Alternative 2: 74.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{+58} \lor \neg \left(y \leq 1.15 \cdot 10^{-71}\right):\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.05e+58) (not (<= y 1.15e-71)))
   (+ (* -2.0 (/ x y)) -1.0)
   (/ x (- x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.05e+58) || !(y <= 1.15e-71)) {
		tmp = (-2.0 * (x / y)) + -1.0;
	} else {
		tmp = x / (x - y);
	}
	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+58)) .or. (.not. (y <= 1.15d-71))) then
        tmp = ((-2.0d0) * (x / y)) + (-1.0d0)
    else
        tmp = x / (x - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.05e+58) || !(y <= 1.15e-71)) {
		tmp = (-2.0 * (x / y)) + -1.0;
	} else {
		tmp = x / (x - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.05e+58) or not (y <= 1.15e-71):
		tmp = (-2.0 * (x / y)) + -1.0
	else:
		tmp = x / (x - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.05e+58) || !(y <= 1.15e-71))
		tmp = Float64(Float64(-2.0 * Float64(x / y)) + -1.0);
	else
		tmp = Float64(x / Float64(x - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.05e+58) || ~((y <= 1.15e-71)))
		tmp = (-2.0 * (x / y)) + -1.0;
	else
		tmp = x / (x - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.05e+58], N[Not[LessEqual[y, 1.15e-71]], $MachinePrecision]], N[(N[(-2.0 * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(x / N[(x - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.05e58 or 1.1499999999999999e-71 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 80.9%

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

   if -2.05e58 < y < 1.1499999999999999e-71

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 81.4%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{+58} \lor \neg \left(y \leq 1.15 \cdot 10^{-71}\right):\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+67}:\\ \;\;\;\;\frac{y}{x - y}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-72}:\\ \;\;\;\;\frac{x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{-y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3e+67)
   (/ y (- x y))
   (if (<= y 3.4e-72) (/ x (- x y)) (/ (+ x y) (- y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3e+67) {
		tmp = y / (x - y);
	} else if (y <= 3.4e-72) {
		tmp = x / (x - y);
	} else {
		tmp = (x + y) / -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3d+67)) then
        tmp = y / (x - y)
    else if (y <= 3.4d-72) then
        tmp = x / (x - y)
    else
        tmp = (x + y) / -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3e+67) {
		tmp = y / (x - y);
	} else if (y <= 3.4e-72) {
		tmp = x / (x - y);
	} else {
		tmp = (x + y) / -y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3e+67:
		tmp = y / (x - y)
	elif y <= 3.4e-72:
		tmp = x / (x - y)
	else:
		tmp = (x + y) / -y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3e+67)
		tmp = Float64(y / Float64(x - y));
	elseif (y <= 3.4e-72)
		tmp = Float64(x / Float64(x - y));
	else
		tmp = Float64(Float64(x + y) / Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3e+67)
		tmp = y / (x - y);
	elseif (y <= 3.4e-72)
		tmp = x / (x - y);
	else
		tmp = (x + y) / -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3e+67], N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.4e-72], N[(x / N[(x - y), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] / (-y)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.0000000000000001e67

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 78.0%

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

   if -3.0000000000000001e67 < y < 3.3999999999999998e-72

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 81.4%

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

   if 3.3999999999999998e-72 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 82.7%

      \[\leadsto \frac{x + y}{\color{blue}{-1 \cdot y}} \]
   4. Step-by-step derivation

      1. neg-mul-1 82.7%

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

   5. Simplified 82.7%

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

Alternative 4: 74.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+56} \lor \neg \left(y \leq 1.15 \cdot 10^{-71}\right):\\ \;\;\;\;\frac{y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.4e+56) (not (<= y 1.15e-71))) (/ y (- x y)) (/ x (- x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.4e+56) || !(y <= 1.15e-71)) {
		tmp = y / (x - y);
	} else {
		tmp = x / (x - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.4d+56)) .or. (.not. (y <= 1.15d-71))) then
        tmp = y / (x - y)
    else
        tmp = x / (x - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.4e+56) || !(y <= 1.15e-71)) {
		tmp = y / (x - y);
	} else {
		tmp = x / (x - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.4e+56) or not (y <= 1.15e-71):
		tmp = y / (x - y)
	else:
		tmp = x / (x - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.4e+56) || !(y <= 1.15e-71))
		tmp = Float64(y / Float64(x - y));
	else
		tmp = Float64(x / Float64(x - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.4e+56) || ~((y <= 1.15e-71)))
		tmp = y / (x - y);
	else
		tmp = x / (x - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.4e+56], N[Not[LessEqual[y, 1.15e-71]], $MachinePrecision]], N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision], N[(x / N[(x - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.40000000000000004e56 or 1.1499999999999999e-71 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 80.2%

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

   if -1.40000000000000004e56 < y < 1.1499999999999999e-71

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 81.4%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+56} \lor \neg \left(y \leq 1.15 \cdot 10^{-71}\right):\\ \;\;\;\;\frac{y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{+92}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-71}:\\ \;\;\;\;\frac{x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.8e+92) -1.0 (if (<= y 1.1e-71) (/ x (- x y)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.8e+92) {
		tmp = -1.0;
	} else if (y <= 1.1e-71) {
		tmp = x / (x - y);
	} 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 <= (-8.8d+92)) then
        tmp = -1.0d0
    else if (y <= 1.1d-71) then
        tmp = x / (x - y)
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8.8e+92) {
		tmp = -1.0;
	} else if (y <= 1.1e-71) {
		tmp = x / (x - y);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.8e+92:
		tmp = -1.0
	elif y <= 1.1e-71:
		tmp = x / (x - y)
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.8e+92)
		tmp = -1.0;
	elseif (y <= 1.1e-71)
		tmp = Float64(x / Float64(x - y));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8.8e+92)
		tmp = -1.0;
	elseif (y <= 1.1e-71)
		tmp = x / (x - y);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.8e+92], -1.0, If[LessEqual[y, 1.1e-71], N[(x / N[(x - y), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -8.79999999999999969e92 or 1.09999999999999999e-71 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 81.9%

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

   if -8.79999999999999969e92 < y < 1.09999999999999999e-71

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 79.8%

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

Alternative 6: 73.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+92}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-71}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.5e+92) -1.0 (if (<= y 1.15e-71) 1.0 -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.5e+92) {
		tmp = -1.0;
	} else if (y <= 1.15e-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 (y <= (-3.5d+92)) then
        tmp = -1.0d0
    else if (y <= 1.15d-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 (y <= -3.5e+92) {
		tmp = -1.0;
	} else if (y <= 1.15e-71) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.5e+92:
		tmp = -1.0
	elif y <= 1.15e-71:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.5e+92)
		tmp = -1.0;
	elseif (y <= 1.15e-71)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.5e+92)
		tmp = -1.0;
	elseif (y <= 1.15e-71)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.5e+92], -1.0, If[LessEqual[y, 1.15e-71], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -3.49999999999999986e92 or 1.1499999999999999e-71 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 81.9%

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

   if -3.49999999999999986e92 < y < 1.1499999999999999e-71

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 79.2%

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

Alternative 7: 49.6% 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 99.9%

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

3. Taylor expanded in x around 0 46.8%

   \[\leadsto \color{blue}{-1} \]
4. Add Preprocessing

Developer Target 1: 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 2024180 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (/ 1 (- (/ x (+ x y)) (/ y (+ x y)))))

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