Linear.Projection:inversePerspective from linear-1.19.1.3, B

Percentage Accurate: 77.3% → 100.0%

Time: 3.3s

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: 77.3% 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 75.6%

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

   1. associate-/l/ 87.9%

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

   2. div-sub 87.9%

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

   3. *-inverses 87.9%

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

   4. metadata-eval 87.9%

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

   5. div-sub 87.9%

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

   6. associate-/l/ 81.6%

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

   7. associate-/r* 100.0%

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

   8. associate-/r* 100.0%

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

   9. *-inverses 100.0%

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

   10. metadata-eval 100.0%

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

   11. metadata-eval 100.0%

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

   12. metadata-eval 100.0%

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

   13. associate-*l/ 100.0%

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

   14. *-commutative 100.0%

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

   15. associate-/r* 100.0%

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

   16. associate-*l/ 100.0%

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

   17. metadata-eval 100.0%

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

   18. 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: 73.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{+42} \lor \neg \left(y \leq 53000 \lor \neg \left(y \leq 2.8 \cdot 10^{+93}\right) \land y \leq 1.02 \cdot 10^{+115}\right):\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -9.8e+42)
         (not (or (<= y 53000.0) (and (not (<= y 2.8e+93)) (<= y 1.02e+115)))))
   (/ -0.5 x)
   (/ 0.5 y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -9.8e+42) || !((y <= 53000.0) || (!(y <= 2.8e+93) && (y <= 1.02e+115)))) {
		tmp = -0.5 / x;
	} else {
		tmp = 0.5 / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-9.8d+42)) .or. (.not. (y <= 53000.0d0) .or. (.not. (y <= 2.8d+93)) .and. (y <= 1.02d+115))) then
        tmp = (-0.5d0) / x
    else
        tmp = 0.5d0 / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -9.8e+42) || !((y <= 53000.0) || (!(y <= 2.8e+93) && (y <= 1.02e+115)))) {
		tmp = -0.5 / x;
	} else {
		tmp = 0.5 / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -9.8e+42) or not ((y <= 53000.0) or (not (y <= 2.8e+93) and (y <= 1.02e+115))):
		tmp = -0.5 / x
	else:
		tmp = 0.5 / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -9.8e+42) || !((y <= 53000.0) || (!(y <= 2.8e+93) && (y <= 1.02e+115))))
		tmp = Float64(-0.5 / x);
	else
		tmp = Float64(0.5 / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -9.8e+42) || ~(((y <= 53000.0) || (~((y <= 2.8e+93)) && (y <= 1.02e+115)))))
		tmp = -0.5 / x;
	else
		tmp = 0.5 / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -9.8e+42], N[Not[Or[LessEqual[y, 53000.0], And[N[Not[LessEqual[y, 2.8e+93]], $MachinePrecision], LessEqual[y, 1.02e+115]]]], $MachinePrecision]], N[(-0.5 / x), $MachinePrecision], N[(0.5 / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.8000000000000004e42 or 53000 < y < 2.79999999999999989e93 or 1.02e115 < y

   1. Initial program 74.2%

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

      1. associate-/l/ 99.9%

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

      2. div-sub 99.9%

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

      3. *-inverses 99.9%

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

      4. metadata-eval 99.9%

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

      5. div-sub 99.9%

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

      6. associate-/l/ 89.4%

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

      7. associate-/r* 100.0%

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

      8. associate-/r* 100.0%

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

      9. *-inverses 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-*l/ 100.0%

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

      14. *-commutative 100.0%

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

      15. associate-/r* 100.0%

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

      16. associate-*l/ 100.0%

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

      17. metadata-eval 100.0%

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

      18. 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 85.9%

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

   if -9.8000000000000004e42 < y < 53000 or 2.79999999999999989e93 < y < 1.02e115

   1. Initial program 76.6%

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

      1. associate-/l/ 79.6%

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

      2. div-sub 79.6%

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

      3. *-inverses 79.6%

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

      4. metadata-eval 79.6%

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

      5. div-sub 79.6%

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

      6. associate-/l/ 76.2%

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

      7. associate-/r* 100.0%

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

      8. associate-/r* 100.0%

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

      9. *-inverses 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-*l/ 100.0%

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

      14. *-commutative 100.0%

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

      15. associate-/r* 100.0%

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

      16. associate-*l/ 100.0%

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

      17. metadata-eval 100.0%

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

      18. 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 78.9%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{+42} \lor \neg \left(y \leq 53000 \lor \neg \left(y \leq 2.8 \cdot 10^{+93}\right) \land y \leq 1.02 \cdot 10^{+115}\right):\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{y}\\ \end{array} \]

Alternative 3: 51.3% 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 75.6%

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

   1. associate-/l/ 87.9%

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

   2. div-sub 87.9%

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

   3. *-inverses 87.9%

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

   4. metadata-eval 87.9%

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

   5. div-sub 87.9%

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

   6. associate-/l/ 81.6%

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

   7. associate-/r* 100.0%

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

   8. associate-/r* 100.0%

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

   9. *-inverses 100.0%

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

   10. metadata-eval 100.0%

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

   11. metadata-eval 100.0%

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

   12. metadata-eval 100.0%

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

   13. associate-*l/ 100.0%

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

   14. *-commutative 100.0%

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

   15. associate-/r* 100.0%

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

   16. associate-*l/ 100.0%

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

   17. metadata-eval 100.0%

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

   18. 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 48.3%

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

5. Final simplification 48.3%

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

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 2023301 
(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)))