Linear.Projection:inversePerspective from linear-1.19.1.3, C

Percentage Accurate: 76.9% → 99.5%

Time: 5.8s

Alternatives: 5

Speedup: 1.4×

Specification

Math

\[\begin{array}{l} \\ \frac{x + y}{\left(x \cdot 2\right) \cdot y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (+ x y) (* (* x 2.0) y)))

C

double code(double x, double y) {
	return (x + y) / ((x * 2.0) * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x + y) / ((x * 2.0d0) * y)
end function

Java

public static double code(double x, double y) {
	return (x + y) / ((x * 2.0) * y);
}

Python

def code(x, y):
	return (x + y) / ((x * 2.0) * y)

Julia

function code(x, y)
	return Float64(Float64(x + y) / Float64(Float64(x * 2.0) * y))
end

MATLAB

function tmp = code(x, y)
	tmp = (x + y) / ((x * 2.0) * y);
end

Wolfram

code[x_, y_] := N[(N[(x + y), $MachinePrecision] / N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]

Initial Program: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + y}{\left(x \cdot 2\right) \cdot y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (+ x y) (* (* x 2.0) y)))

C

double code(double x, double y) {
	return (x + y) / ((x * 2.0) * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x + y) / ((x * 2.0d0) * y)
end function

Java

public static double code(double x, double y) {
	return (x + y) / ((x * 2.0) * y);
}

Python

def code(x, y):
	return (x + y) / ((x * 2.0) * y)

Julia

function code(x, y)
	return Float64(Float64(x + y) / Float64(Float64(x * 2.0) * y))
end

MATLAB

function tmp = code(x, y)
	tmp = (x + y) / ((x * 2.0) * y);
end

Wolfram

code[x_, y_] := N[(N[(x + y), $MachinePrecision] / N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-74}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{y}{x}, 0.5\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{x}{y}, 0.5\right)}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.05e-74) (/ (fma 0.5 (/ y x) 0.5) y) (/ (fma 0.5 (/ x y) 0.5) x)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.05e-74) {
		tmp = fma(0.5, (y / x), 0.5) / y;
	} else {
		tmp = fma(0.5, (x / y), 0.5) / x;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.05e-74)
		tmp = Float64(fma(0.5, Float64(y / x), 0.5) / y);
	else
		tmp = Float64(fma(0.5, Float64(x / y), 0.5) / x);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.05e-74], N[(N[(0.5 * N[(y / x), $MachinePrecision] + 0.5), $MachinePrecision] / y), $MachinePrecision], N[(N[(0.5 * N[(x / y), $MachinePrecision] + 0.5), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.05000000000000016e-74

   1. Initial program 77.3%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{2} + \frac{1}{2} \cdot \frac{y}{x}}{y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, \frac{1}{2}\right)}}{y} \]

      4. lower-/.f64 99.8

         \[\leadsto \frac{\mathsf{fma}\left(0.5, \color{blue}{\frac{y}{x}}, 0.5\right)}{y} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, \frac{y}{x}, 0.5\right)}{y}} \]

   if -2.05000000000000016e-74 < x

   1. Initial program 75.4%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{2} + \frac{1}{2} \cdot \frac{x}{y}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{x}{y}, \frac{1}{2}\right)}}{x} \]

      4. lower-/.f64 88.9

         \[\leadsto \frac{\mathsf{fma}\left(0.5, \color{blue}{\frac{x}{y}}, 0.5\right)}{x} \]

   5. Applied rewrites 88.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, \frac{x}{y}, 0.5\right)}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 90.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+142}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-194}:\\ \;\;\;\;\frac{x + y}{y \cdot \left(x \cdot 2\right)}\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-205}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.9e+142)
   (/ 0.5 y)
   (if (<= x -1.1e-194)
     (/ (+ x y) (* y (* x 2.0)))
     (if (<= x -3.6e-205) (/ 0.5 y) (/ 0.5 x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.9e+142) {
		tmp = 0.5 / y;
	} else if (x <= -1.1e-194) {
		tmp = (x + y) / (y * (x * 2.0));
	} else if (x <= -3.6e-205) {
		tmp = 0.5 / y;
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.9d+142)) then
        tmp = 0.5d0 / y
    else if (x <= (-1.1d-194)) then
        tmp = (x + y) / (y * (x * 2.0d0))
    else if (x <= (-3.6d-205)) then
        tmp = 0.5d0 / y
    else
        tmp = 0.5d0 / x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.9e+142) {
		tmp = 0.5 / y;
	} else if (x <= -1.1e-194) {
		tmp = (x + y) / (y * (x * 2.0));
	} else if (x <= -3.6e-205) {
		tmp = 0.5 / y;
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.9e+142:
		tmp = 0.5 / y
	elif x <= -1.1e-194:
		tmp = (x + y) / (y * (x * 2.0))
	elif x <= -3.6e-205:
		tmp = 0.5 / y
	else:
		tmp = 0.5 / x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.9e+142)
		tmp = Float64(0.5 / y);
	elseif (x <= -1.1e-194)
		tmp = Float64(Float64(x + y) / Float64(y * Float64(x * 2.0)));
	elseif (x <= -3.6e-205)
		tmp = Float64(0.5 / y);
	else
		tmp = Float64(0.5 / x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.9e+142)
		tmp = 0.5 / y;
	elseif (x <= -1.1e-194)
		tmp = (x + y) / (y * (x * 2.0));
	elseif (x <= -3.6e-205)
		tmp = 0.5 / y;
	else
		tmp = 0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.9e+142], N[(0.5 / y), $MachinePrecision], If[LessEqual[x, -1.1e-194], N[(N[(x + y), $MachinePrecision] / N[(y * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.6e-205], N[(0.5 / y), $MachinePrecision], N[(0.5 / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.90000000000000013e142 or -1.1000000000000001e-194 < x < -3.5999999999999998e-205

   1. Initial program 63.4%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 75.1

         \[\leadsto \color{blue}{\frac{0.5}{y}} \]

   5. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\frac{0.5}{y}} \]

   if -2.90000000000000013e142 < x < -1.1000000000000001e-194

   1. Initial program 89.7%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   2. Add Preprocessing

   if -3.5999999999999998e-205 < x

   1. Initial program 73.3%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 58.1

         \[\leadsto \color{blue}{\frac{0.5}{x}} \]

   5. Applied rewrites 58.1%

      \[\leadsto \color{blue}{\frac{0.5}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+142}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-194}:\\ \;\;\;\;\frac{x + y}{y \cdot \left(x \cdot 2\right)}\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-205}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+158}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{x}{y}, 0.5\right)}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -7.6e+158) (/ 0.5 y) (/ (fma 0.5 (/ x y) 0.5) x)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -7.6e+158) {
		tmp = 0.5 / y;
	} else {
		tmp = fma(0.5, (x / y), 0.5) / x;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -7.6e+158)
		tmp = Float64(0.5 / y);
	else
		tmp = Float64(fma(0.5, Float64(x / y), 0.5) / x);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -7.6e+158], N[(0.5 / y), $MachinePrecision], N[(N[(0.5 * N[(x / y), $MachinePrecision] + 0.5), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.5999999999999997e158

   1. Initial program 68.1%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 78.2

         \[\leadsto \color{blue}{\frac{0.5}{y}} \]

   5. Applied rewrites 78.2%

      \[\leadsto \color{blue}{\frac{0.5}{y}} \]

   if -7.5999999999999997e158 < x

   1. Initial program 77.0%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{2} + \frac{1}{2} \cdot \frac{x}{y}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{x}{y}, \frac{1}{2}\right)}}{x} \]

      4. lower-/.f64 89.7

         \[\leadsto \frac{\mathsf{fma}\left(0.5, \color{blue}{\frac{x}{y}}, 0.5\right)}{x} \]

   5. Applied rewrites 89.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, \frac{x}{y}, 0.5\right)}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 82.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-103}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -4.5e-103) (/ 0.5 y) (/ 0.5 x)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4.5e-103) {
		tmp = 0.5 / y;
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.5d-103)) then
        tmp = 0.5d0 / y
    else
        tmp = 0.5d0 / x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -4.5e-103) {
		tmp = 0.5 / y;
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -4.5e-103:
		tmp = 0.5 / y
	else:
		tmp = 0.5 / x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -4.5e-103)
		tmp = Float64(0.5 / y);
	else
		tmp = Float64(0.5 / x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.5e-103)
		tmp = 0.5 / y;
	else
		tmp = 0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -4.5e-103], N[(0.5 / y), $MachinePrecision], N[(0.5 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.5e-103

   1. Initial program 79.6%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 56.0

         \[\leadsto \color{blue}{\frac{0.5}{y}} \]

   5. Applied rewrites 56.0%

      \[\leadsto \color{blue}{\frac{0.5}{y}} \]

   if -4.5e-103 < x

   1. Initial program 74.3%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 60.7

         \[\leadsto \color{blue}{\frac{0.5}{x}} \]

   5. Applied rewrites 60.7%

      \[\leadsto \color{blue}{\frac{0.5}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 50.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{0.5}{x} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ 0.5 x))

C

assert(x < y);
double code(double x, double y) {
	return 0.5 / x;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.5d0 / x
end function

Java

assert x < y;
public static double code(double x, double y) {
	return 0.5 / x;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return 0.5 / x

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(0.5 / x)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 0.5 / x;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(0.5 / x), $MachinePrecision]

Derivation

1. Initial program 75.9%

   \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{2}}{x}} \]
4. Step-by-step derivation

   1. lower-/.f64 56.2

      \[\leadsto \color{blue}{\frac{0.5}{x}} \]

5. Applied rewrites 56.2%

   \[\leadsto \color{blue}{\frac{0.5}{x}} \]
6. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{x} + \frac{0.5}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (/ 0.5 x) (/ 0.5 y)))

C

double code(double x, double y) {
	return (0.5 / x) + (0.5 / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (0.5d0 / x) + (0.5d0 / y)
end function

Java

public static double code(double x, double y) {
	return (0.5 / x) + (0.5 / y);
}

Python

def code(x, y):
	return (0.5 / x) + (0.5 / y)

Julia

function code(x, y)
	return Float64(Float64(0.5 / x) + Float64(0.5 / y))
end

MATLAB

function tmp = code(x, y)
	tmp = (0.5 / x) + (0.5 / y);
end

Wolfram

code[x_, y_] := N[(N[(0.5 / x), $MachinePrecision] + N[(0.5 / y), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024221 
(FPCore (x y)
  :name "Linear.Projection:inversePerspective from linear-1.19.1.3, C"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (/ 1/2 x) (/ 1/2 y)))

  (/ (+ x y) (* (* x 2.0) y)))