Linear.Projection:inversePerspective from linear-1.19.1.3, C

Percentage Accurate: 76.7% → 99.9%

Time: 7.2s

Alternatives: 6

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: 76.7% 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: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{+20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{0.5}{x}, y, 0.5\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{y} \cdot \left(y + x\right)}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 6.8e+20) (/ (fma (/ 0.5 x) y 0.5) y) (/ (* (/ 0.5 y) (+ y x)) x)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 6.8e+20) {
		tmp = fma((0.5 / x), y, 0.5) / y;
	} else {
		tmp = ((0.5 / y) * (y + x)) / x;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 6.8e+20)
		tmp = Float64(fma(Float64(0.5 / x), y, 0.5) / y);
	else
		tmp = Float64(Float64(Float64(0.5 / y) * Float64(y + x)) / 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, 6.8e+20], N[(N[(N[(0.5 / x), $MachinePrecision] * y + 0.5), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(0.5 / y), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.8e20

   1. Initial program 72.6%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   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. 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 91.8

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

   5. Applied rewrites 91.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, \frac{y}{x}, 0.5\right)}{y}} \]
   6. Step-by-step derivation

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. lower-fma.f64 91.8

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

   7. Applied rewrites 91.8%

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

   if 6.8e20 < y

   1. Initial program 73.9%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x + y}}{\left(x \cdot 2\right) \cdot y} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x + y}{\color{blue}{\left(x \cdot 2\right)} \cdot y} \]

      3. associate-/r* N/A

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

      4. div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. div-inv N/A

          \[\leadsto \frac{\color{blue}{\left(\left(x + y\right) \cdot \frac{1}{2}\right)} \cdot \frac{1}{y}}{x} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\left(\left(x + y\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \frac{1}{y}}{x} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\left(\left(x + y\right) \cdot \color{blue}{{2}^{-1}}\right) \cdot \frac{1}{y}}{x} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\left(x + y\right) \cdot {2}^{-1}\right)} \cdot \frac{1}{y}}{x} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\left(\left(x + y\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \frac{1}{y}}{x} \]

      15. lower-/.f64 99.8

          \[\leadsto \frac{\left(\left(x + y\right) \cdot 0.5\right) \cdot \color{blue}{\frac{1}{y}}}{x} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\left(\left(x + y\right) \cdot 0.5\right) \cdot \frac{1}{y}}{x}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x + y\right)} \cdot \frac{1}{2}\right) \cdot \frac{1}{y}}{x} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x + y\right) \cdot \frac{1}{2}\right)} \cdot \frac{1}{y}}{x} \]

      3. un-div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(x + y\right) \cdot \frac{1}{2}}{y}}}{x} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(x + y\right) \cdot \frac{1}{2}}}{y}}{x} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(x + y\right)}}{y}}{x} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{y} \cdot \left(x + y\right)}}{x} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{y}} \cdot \left(x + y\right)}{x} \]

      8. lower-*.f64 99.8

         \[\leadsto \frac{\color{blue}{\frac{0.5}{y} \cdot \left(x + y\right)}}{x} \]

   6. Applied rewrites 99.8%

      \[\leadsto \frac{\color{blue}{\frac{0.5}{y} \cdot \left(x + y\right)}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{+20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{0.5}{x}, y, 0.5\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{y} \cdot \left(y + x\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{-174}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-137}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+122}:\\ \;\;\;\;\frac{y + x}{y \cdot \left(x \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.75e-174)
   (/ 0.5 y)
   (if (<= y 1.55e-137)
     (/ 0.5 x)
     (if (<= y 3.5e+122) (/ (+ y x) (* y (* x 2.0))) (/ 0.5 x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.75e-174) {
		tmp = 0.5 / y;
	} else if (y <= 1.55e-137) {
		tmp = 0.5 / x;
	} else if (y <= 3.5e+122) {
		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 (y <= 1.75d-174) then
        tmp = 0.5d0 / y
    else if (y <= 1.55d-137) then
        tmp = 0.5d0 / x
    else if (y <= 3.5d+122) 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 (y <= 1.75e-174) {
		tmp = 0.5 / y;
	} else if (y <= 1.55e-137) {
		tmp = 0.5 / x;
	} else if (y <= 3.5e+122) {
		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 y <= 1.75e-174:
		tmp = 0.5 / y
	elif y <= 1.55e-137:
		tmp = 0.5 / x
	elif y <= 3.5e+122:
		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 (y <= 1.75e-174)
		tmp = Float64(0.5 / y);
	elseif (y <= 1.55e-137)
		tmp = Float64(0.5 / x);
	elseif (y <= 3.5e+122)
		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 (y <= 1.75e-174)
		tmp = 0.5 / y;
	elseif (y <= 1.55e-137)
		tmp = 0.5 / x;
	elseif (y <= 3.5e+122)
		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[y, 1.75e-174], N[(0.5 / y), $MachinePrecision], If[LessEqual[y, 1.55e-137], N[(0.5 / x), $MachinePrecision], If[LessEqual[y, 3.5e+122], 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 y < 1.74999999999999994e-174

   1. Initial program 70.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 61.2

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

   5. Applied rewrites 61.2%

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

   if 1.74999999999999994e-174 < y < 1.54999999999999989e-137 or 3.50000000000000014e122 < y

   1. Initial program 62.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 79.5

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

   5. Applied rewrites 79.5%

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

   if 1.54999999999999989e-137 < y < 3.50000000000000014e122

   1. Initial program 94.8%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{-174}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-137}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+122}:\\ \;\;\;\;\frac{y + x}{y \cdot \left(x \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{+122}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{0.5}{x}, y, 0.5\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.9e+122) (/ (fma (/ 0.5 x) y 0.5) y) (/ 0.5 x)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.9e+122) {
		tmp = fma((0.5 / x), y, 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 <= 3.9e+122)
		tmp = Float64(fma(Float64(0.5 / x), y, 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, 3.9e+122], N[(N[(N[(0.5 / x), $MachinePrecision] * y + 0.5), $MachinePrecision] / y), $MachinePrecision], N[(0.5 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.8999999999999999e122

   1. Initial program 74.1%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   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 92.0

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

   5. Applied rewrites 92.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, \frac{y}{x}, 0.5\right)}{y}} \]
   6. Step-by-step derivation

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. lower-fma.f64 92.0

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

   7. Applied rewrites 92.0%

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

   if 3.8999999999999999e122 < y

   1. Initial program 64.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 94.0

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

   5. Applied rewrites 94.0%

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

Alternative 4: 95.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{+122}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{y}{x}, 0.5\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.9e+122) (/ (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 <= 3.9e+122) {
		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 <= 3.9e+122)
		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, 3.9e+122], 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 < 3.8999999999999999e122

   1. Initial program 74.1%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   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 92.0

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

   5. Applied rewrites 92.0%

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

   if 3.8999999999999999e122 < y

   1. Initial program 64.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 94.0

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

   5. Applied rewrites 94.0%

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

Alternative 5: 81.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{-174}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 1.75e-174) (/ 0.5 y) (/ 0.5 x)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.75e-174) {
		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 (y <= 1.75d-174) 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 (y <= 1.75e-174) {
		tmp = 0.5 / y;
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.75e-174:
		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 (y <= 1.75e-174)
		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 (y <= 1.75e-174)
		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[y, 1.75e-174], N[(0.5 / y), $MachinePrecision], N[(0.5 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.74999999999999994e-174

   1. Initial program 70.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 61.2

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

   5. Applied rewrites 61.2%

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

   if 1.74999999999999994e-174 < y

   1. Initial program 77.5%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   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 63.1

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

   5. Applied rewrites 63.1%

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

Alternative 6: 50.0% 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 72.8%

   \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
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 47.5

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

5. Applied rewrites 47.5%

   \[\leadsto \color{blue}{\frac{0.5}{x}} \]
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 2024216 
(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)))