Linear.Projection:inversePerspective from linear-1.19.1.3, B

Percentage Accurate: 76.8% → 100.0%

Time: 2.6s

Alternatives: 3

Speedup: 1.3×

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: 76.8% 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, 1.3× 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 78.0%

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

   1. div-sub 77.3%

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

   2. associate-/r* 82.4%

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

   3. associate-/r* 82.4%

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

   4. *-inverses 82.4%

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

   5. metadata-eval 82.4%

      \[\leadsto \frac{\color{blue}{0.5}}{y} - \frac{y}{\left(x \cdot 2\right) \cdot y} \]

   6. associate-/l/ 100.0%

      \[\leadsto \frac{0.5}{y} - \color{blue}{\frac{\frac{y}{y}}{x \cdot 2}} \]

   7. *-inverses 100.0%

      \[\leadsto \frac{0.5}{y} - \frac{\color{blue}{1}}{x \cdot 2} \]

   8. *-inverses 100.0%

      \[\leadsto \frac{0.5}{y} - \frac{\color{blue}{\frac{x}{x}}}{x \cdot 2} \]

   9. *-commutative 100.0%

      \[\leadsto \frac{0.5}{y} - \frac{\frac{x}{x}}{\color{blue}{2 \cdot x}} \]

   10. associate-/r* 100.0%

       \[\leadsto \frac{0.5}{y} - \color{blue}{\frac{\frac{\frac{x}{x}}{2}}{x}} \]

   11. *-inverses 100.0%

       \[\leadsto \frac{0.5}{y} - \frac{\frac{\color{blue}{1}}{2}}{x} \]

   12. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \frac{0.5}{y} - \frac{0.5}{x} \]

Alternative 2: 74.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-39}:\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{elif}\;y \leq 10^{-42}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4e-39) (/ -0.5 x) (if (<= y 1e-42) (/ 0.5 y) (/ -0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4e-39) {
		tmp = -0.5 / x;
	} else if (y <= 1e-42) {
		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 <= (-4d-39)) then
        tmp = (-0.5d0) / x
    else if (y <= 1d-42) 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 <= -4e-39) {
		tmp = -0.5 / x;
	} else if (y <= 1e-42) {
		tmp = 0.5 / y;
	} else {
		tmp = -0.5 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4e-39:
		tmp = -0.5 / x
	elif y <= 1e-42:
		tmp = 0.5 / y
	else:
		tmp = -0.5 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4e-39)
		tmp = Float64(-0.5 / x);
	elseif (y <= 1e-42)
		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 <= -4e-39)
		tmp = -0.5 / x;
	elseif (y <= 1e-42)
		tmp = 0.5 / y;
	else
		tmp = -0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4e-39], N[(-0.5 / x), $MachinePrecision], If[LessEqual[y, 1e-42], N[(0.5 / y), $MachinePrecision], N[(-0.5 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.99999999999999972e-39 or 1.00000000000000004e-42 < y

   1. Initial program 80.4%

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

      1. div-sub 80.4%

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

      2. associate-/r* 87.8%

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

      3. associate-/r* 87.8%

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

      4. *-inverses 87.8%

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

      5. metadata-eval 87.8%

         \[\leadsto \frac{\color{blue}{0.5}}{y} - \frac{y}{\left(x \cdot 2\right) \cdot y} \]

      6. associate-/l/ 100.0%

         \[\leadsto \frac{0.5}{y} - \color{blue}{\frac{\frac{y}{y}}{x \cdot 2}} \]

      7. *-inverses 100.0%

         \[\leadsto \frac{0.5}{y} - \frac{\color{blue}{1}}{x \cdot 2} \]

      8. *-inverses 100.0%

         \[\leadsto \frac{0.5}{y} - \frac{\color{blue}{\frac{x}{x}}}{x \cdot 2} \]

      9. *-commutative 100.0%

         \[\leadsto \frac{0.5}{y} - \frac{\frac{x}{x}}{\color{blue}{2 \cdot x}} \]

      10. associate-/r* 100.0%

          \[\leadsto \frac{0.5}{y} - \color{blue}{\frac{\frac{\frac{x}{x}}{2}}{x}} \]

      11. *-inverses 100.0%

          \[\leadsto \frac{0.5}{y} - \frac{\frac{\color{blue}{1}}{2}}{x} \]

      12. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 76.3%

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

   if -3.99999999999999972e-39 < y < 1.00000000000000004e-42

   1. Initial program 75.0%

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

      1. div-sub 73.5%

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

      2. associate-/r* 75.5%

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

      3. associate-/r* 75.5%

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

      4. *-inverses 75.5%

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

      5. metadata-eval 75.5%

         \[\leadsto \frac{\color{blue}{0.5}}{y} - \frac{y}{\left(x \cdot 2\right) \cdot y} \]

      6. associate-/l/ 100.0%

         \[\leadsto \frac{0.5}{y} - \color{blue}{\frac{\frac{y}{y}}{x \cdot 2}} \]

      7. *-inverses 100.0%

         \[\leadsto \frac{0.5}{y} - \frac{\color{blue}{1}}{x \cdot 2} \]

      8. *-inverses 100.0%

         \[\leadsto \frac{0.5}{y} - \frac{\color{blue}{\frac{x}{x}}}{x \cdot 2} \]

      9. *-commutative 100.0%

         \[\leadsto \frac{0.5}{y} - \frac{\frac{x}{x}}{\color{blue}{2 \cdot x}} \]

      10. associate-/r* 100.0%

          \[\leadsto \frac{0.5}{y} - \color{blue}{\frac{\frac{\frac{x}{x}}{2}}{x}} \]

      11. *-inverses 100.0%

          \[\leadsto \frac{0.5}{y} - \frac{\frac{\color{blue}{1}}{2}}{x} \]

      12. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 81.6%

      \[\leadsto \color{blue}{\frac{0.5}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-39}:\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{elif}\;y \leq 10^{-42}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x}\\ \end{array} \]

Alternative 3: 50.4% accurate, 3.0× 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 78.0%

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

   1. div-sub 77.3%

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

   2. associate-/r* 82.4%

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

   3. associate-/r* 82.4%

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

   4. *-inverses 82.4%

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

   5. metadata-eval 82.4%

      \[\leadsto \frac{\color{blue}{0.5}}{y} - \frac{y}{\left(x \cdot 2\right) \cdot y} \]

   6. associate-/l/ 100.0%

      \[\leadsto \frac{0.5}{y} - \color{blue}{\frac{\frac{y}{y}}{x \cdot 2}} \]

   7. *-inverses 100.0%

      \[\leadsto \frac{0.5}{y} - \frac{\color{blue}{1}}{x \cdot 2} \]

   8. *-inverses 100.0%

      \[\leadsto \frac{0.5}{y} - \frac{\color{blue}{\frac{x}{x}}}{x \cdot 2} \]

   9. *-commutative 100.0%

      \[\leadsto \frac{0.5}{y} - \frac{\frac{x}{x}}{\color{blue}{2 \cdot x}} \]

   10. associate-/r* 100.0%

       \[\leadsto \frac{0.5}{y} - \color{blue}{\frac{\frac{\frac{x}{x}}{2}}{x}} \]

   11. *-inverses 100.0%

       \[\leadsto \frac{0.5}{y} - \frac{\frac{\color{blue}{1}}{2}}{x} \]

   12. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in y around inf 50.7%

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

5. Final simplification 50.7%

   \[\leadsto \frac{-0.5}{x} \]

Developer target: 100.0% accurate, 1.3× 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]

Reproduce

herbie shell --seed 2023224 
(FPCore (x y)
  :name "Linear.Projection:inversePerspective from linear-1.19.1.3, B"
  :precision binary64

  :herbie-target
  (- (/ 0.5 y) (/ 0.5 x))

  (/ (- x y) (* (* x 2.0) y)))