Linear.Projection:inversePerspective from linear-1.19.1.3, C

Percentage Accurate: 76.6% → 95.3%

Time: 5.5s

Alternatives: 4

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: 76.6% 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: 95.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{+134}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{y}{x}, 0.5\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.2e+134) (/ (fma 0.5 (/ y x) 0.5) y) (/ 0.5 x)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.2e+134) {
		tmp = fma(0.5, (y / x), 0.5) / y;
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.2e+134)
		tmp = Float64(fma(0.5, Float64(y / x), 0.5) / y);
	else
		tmp = Float64(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[y, 3.2e+134], N[(N[(0.5 * N[(y / x), $MachinePrecision] + 0.5), $MachinePrecision] / y), $MachinePrecision], N[(0.5 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.2000000000000001e134

   1. Initial program 78.6%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 90.9

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

   5. Simplified 90.9%

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

   if 3.2000000000000001e134 < y

   1. Initial program 65.9%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 84.9

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

   5. Simplified 84.9%

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

Alternative 2: 90.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+148}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-183}:\\ \;\;\;\;\frac{y + x}{y \cdot \left(x \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -6.2e+148)
   (/ 0.5 y)
   (if (<= x -2.8e-183) (/ (+ y x) (* y (* x 2.0))) (/ 0.5 x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -6.2e+148) {
		tmp = 0.5 / y;
	} else if (x <= -2.8e-183) {
		tmp = (y + x) / (y * (x * 2.0));
	} 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 <= (-6.2d+148)) then
        tmp = 0.5d0 / y
    else if (x <= (-2.8d-183)) then
        tmp = (y + x) / (y * (x * 2.0d0))
    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 <= -6.2e+148) {
		tmp = 0.5 / y;
	} else if (x <= -2.8e-183) {
		tmp = (y + x) / (y * (x * 2.0));
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -6.2e+148:
		tmp = 0.5 / y
	elif x <= -2.8e-183:
		tmp = (y + x) / (y * (x * 2.0))
	else:
		tmp = 0.5 / x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -6.2e+148)
		tmp = Float64(0.5 / y);
	elseif (x <= -2.8e-183)
		tmp = Float64(Float64(y + x) / Float64(y * Float64(x * 2.0)));
	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 <= -6.2e+148)
		tmp = 0.5 / y;
	elseif (x <= -2.8e-183)
		tmp = (y + x) / (y * (x * 2.0));
	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, -6.2e+148], N[(0.5 / y), $MachinePrecision], If[LessEqual[x, -2.8e-183], N[(N[(y + x), $MachinePrecision] / N[(y * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.19999999999999951e148

   1. Initial program 83.6%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 97.4

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

   5. Simplified 97.4%

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

   if -6.19999999999999951e148 < x < -2.79999999999999985e-183

   1. Initial program 85.5%

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

   if -2.79999999999999985e-183 < x

   1. Initial program 72.0%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 56.3

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

   5. Simplified 56.3%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+148}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-183}:\\ \;\;\;\;\frac{y + x}{y \cdot \left(x \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-21}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -3.6e-21) (/ 0.5 y) (/ 0.5 x)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.6e-21) {
		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 <= (-3.6d-21)) 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 <= -3.6e-21) {
		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 <= -3.6e-21:
		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 <= -3.6e-21)
		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 <= -3.6e-21)
		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, -3.6e-21], N[(0.5 / y), $MachinePrecision], N[(0.5 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.59999999999999989e-21

   1. Initial program 84.5%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 77.7

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

   5. Simplified 77.7%

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

   if -3.59999999999999989e-21 < x

   1. Initial program 74.1%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 58.5

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

   5. Simplified 58.5%

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

Alternative 4: 49.8% 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 77.2%

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

3. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 48.4

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

5. Simplified 48.4%

   \[\leadsto \color{blue}{\frac{0.5}{x}} \]
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 2024204 
(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)))