Linear.Projection:inversePerspective from linear-1.19.1.3, C

Percentage Accurate: 77.8% → 100.0%

Time: 5.0s

Alternatives: 5

Speedup: 1.0×

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: 77.8% 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: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 53.4

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

5. Applied rewrites 53.4%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{0.5}{y} - \frac{-0.5}{x}} \]
9. Add Preprocessing

Alternative 2: 68.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-172}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+71}:\\ \;\;\;\;\frac{y + x}{\left(2 \cdot x\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.2e-172)
   (/ 0.5 y)
   (if (<= y 1.2e+71) (/ (+ y x) (* (* 2.0 x) y)) (/ 0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.2e-172) {
		tmp = 0.5 / y;
	} else if (y <= 1.2e+71) {
		tmp = (y + x) / ((2.0 * x) * y);
	} else {
		tmp = 0.5 / 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 <= 5.2d-172) then
        tmp = 0.5d0 / y
    else if (y <= 1.2d+71) then
        tmp = (y + x) / ((2.0d0 * x) * y)
    else
        tmp = 0.5d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.2e-172) {
		tmp = 0.5 / y;
	} else if (y <= 1.2e+71) {
		tmp = (y + x) / ((2.0 * x) * y);
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.2e-172:
		tmp = 0.5 / y
	elif y <= 1.2e+71:
		tmp = (y + x) / ((2.0 * x) * y)
	else:
		tmp = 0.5 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.2e-172)
		tmp = Float64(0.5 / y);
	elseif (y <= 1.2e+71)
		tmp = Float64(Float64(y + x) / Float64(Float64(2.0 * x) * y));
	else
		tmp = Float64(0.5 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.2e-172)
		tmp = 0.5 / y;
	elseif (y <= 1.2e+71)
		tmp = (y + x) / ((2.0 * x) * y);
	else
		tmp = 0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.2e-172], N[(0.5 / y), $MachinePrecision], If[LessEqual[y, 1.2e+71], N[(N[(y + x), $MachinePrecision] / N[(N[(2.0 * x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(0.5 / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.1999999999999996e-172

   1. Initial program 71.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 55.5

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

   5. Applied rewrites 55.5%

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

   if 5.1999999999999996e-172 < y < 1.1999999999999999e71

   1. Initial program 88.5%

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

   if 1.1999999999999999e71 < y

   1. Initial program 65.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 87.7

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

   5. Applied rewrites 87.7%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-172}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+71}:\\ \;\;\;\;\frac{y + x}{\left(2 \cdot x\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{-169}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+71}:\\ \;\;\;\;\left(y + x\right) \cdot \frac{0.5}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.02e-169)
   (/ 0.5 y)
   (if (<= y 1.2e+71) (* (+ y x) (/ 0.5 (* y x))) (/ 0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.02e-169) {
		tmp = 0.5 / y;
	} else if (y <= 1.2e+71) {
		tmp = (y + x) * (0.5 / (y * x));
	} else {
		tmp = 0.5 / 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 <= 1.02d-169) then
        tmp = 0.5d0 / y
    else if (y <= 1.2d+71) then
        tmp = (y + x) * (0.5d0 / (y * x))
    else
        tmp = 0.5d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.02e-169) {
		tmp = 0.5 / y;
	} else if (y <= 1.2e+71) {
		tmp = (y + x) * (0.5 / (y * x));
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.02e-169:
		tmp = 0.5 / y
	elif y <= 1.2e+71:
		tmp = (y + x) * (0.5 / (y * x))
	else:
		tmp = 0.5 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.02e-169)
		tmp = Float64(0.5 / y);
	elseif (y <= 1.2e+71)
		tmp = Float64(Float64(y + x) * Float64(0.5 / Float64(y * x)));
	else
		tmp = Float64(0.5 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.02e-169)
		tmp = 0.5 / y;
	elseif (y <= 1.2e+71)
		tmp = (y + x) * (0.5 / (y * x));
	else
		tmp = 0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.02e-169], N[(0.5 / y), $MachinePrecision], If[LessEqual[y, 1.2e+71], N[(N[(y + x), $MachinePrecision] * N[(0.5 / N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.01999999999999996e-169

   1. Initial program 71.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 55.5

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

   5. Applied rewrites 55.5%

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

   if 1.01999999999999996e-169 < y < 1.1999999999999999e71

   1. Initial program 88.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. associate-/r* N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 88.5

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 88.5

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

   4. Applied rewrites 88.5%

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

   if 1.1999999999999999e71 < y

   1. Initial program 65.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 87.7

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

   5. Applied rewrites 87.7%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{-169}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+71}:\\ \;\;\;\;\left(y + x\right) \cdot \frac{0.5}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-123}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 8.5e-123) (/ 0.5 y) (/ 0.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8.5e-123) {
		tmp = 0.5 / y;
	} else {
		tmp = 0.5 / 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 <= 8.5d-123) then
        tmp = 0.5d0 / y
    else
        tmp = 0.5d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 8.5e-123) {
		tmp = 0.5 / y;
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8.5e-123:
		tmp = 0.5 / y
	else:
		tmp = 0.5 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8.5e-123)
		tmp = Float64(0.5 / y);
	else
		tmp = Float64(0.5 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8.5e-123)
		tmp = 0.5 / y;
	else
		tmp = 0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8.5e-123], N[(0.5 / y), $MachinePrecision], N[(0.5 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.4999999999999995e-123

   1. Initial program 71.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 56.4

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

   5. Applied rewrites 56.4%

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

   if 8.4999999999999995e-123 < y

   1. Initial program 77.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 73.0

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

   5. Applied rewrites 73.0%

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

Alternative 5: 50.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 53.4

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

5. Applied rewrites 53.4%

   \[\leadsto \color{blue}{\frac{0.5}{x}} \]
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 2024235 
(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)))