Linear.Projection:inversePerspective from linear-1.19.1.3, B

Percentage Accurate: 76.5% → 100.0%

Time: 4.0s

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: 76.5% 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 71.6%

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

   1. div-sub 71.1%

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

   2. sub-neg 71.1%

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

   3. associate-/r* 78.7%

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

   4. associate-/r* 78.7%

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

   5. *-inverses 78.7%

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

   6. metadata-eval 78.7%

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

   7. *-commutative 78.7%

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

   8. *-commutative 78.7%

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

   9. associate-*l* 78.7%

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

   10. associate-/r* 100.0%

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

   11. distribute-neg-frac 100.0%

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

   12. associate-/r* 100.0%

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

   13. *-inverses 100.0%

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

   14. metadata-eval 100.0%

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

   15. 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. Add Preprocessing

5. Final simplification 100.0%

   \[\leadsto \frac{0.5}{y} + \frac{-0.5}{x} \]
6. Add Preprocessing

Alternative 2: 74.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+15} \lor \neg \left(x \leq 1.85 \cdot 10^{+45}\right):\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.25e+15) (not (<= x 1.85e+45))) (/ 0.5 y) (/ -0.5 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.25e+15) || !(x <= 1.85e+45)) {
		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 ((x <= (-1.25d+15)) .or. (.not. (x <= 1.85d+45))) 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 ((x <= -1.25e+15) || !(x <= 1.85e+45)) {
		tmp = 0.5 / y;
	} else {
		tmp = -0.5 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.25e+15) or not (x <= 1.85e+45):
		tmp = 0.5 / y
	else:
		tmp = -0.5 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.25e+15) || !(x <= 1.85e+45))
		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 ((x <= -1.25e+15) || ~((x <= 1.85e+45)))
		tmp = 0.5 / y;
	else
		tmp = -0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.25e+15], N[Not[LessEqual[x, 1.85e+45]], $MachinePrecision]], N[(0.5 / y), $MachinePrecision], N[(-0.5 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.25e15 or 1.84999999999999989e45 < x

   1. Initial program 67.8%

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

      1. div-sub 67.8%

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

      2. sub-neg 67.8%

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

      3. associate-/r* 83.7%

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

      4. associate-/r* 83.7%

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

      5. *-inverses 83.7%

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

      6. metadata-eval 83.7%

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

      7. *-commutative 83.7%

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

      8. *-commutative 83.7%

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

      9. associate-*l* 83.7%

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

      10. associate-/r* 100.0%

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

      11. distribute-neg-frac 100.0%

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

      12. associate-/r* 100.0%

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

      13. *-inverses 100.0%

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

      14. metadata-eval 100.0%

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

      15. 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. Add Preprocessing

   5. Taylor expanded in y around 0 79.5%

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

   if -1.25e15 < x < 1.84999999999999989e45

   1. Initial program 74.5%

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

      1. div-sub 73.6%

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

      2. sub-neg 73.6%

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

      3. associate-/r* 75.0%

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

      4. associate-/r* 75.0%

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

      5. *-inverses 75.0%

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

      6. metadata-eval 75.0%

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

      7. *-commutative 75.0%

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

      8. *-commutative 75.0%

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

      9. associate-*l* 75.0%

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

      10. associate-/r* 100.0%

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

      11. distribute-neg-frac 100.0%

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

      12. associate-/r* 100.0%

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

      13. *-inverses 100.0%

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

      14. metadata-eval 100.0%

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

      15. 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. Add Preprocessing

   5. Taylor expanded in y around inf 75.1%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+15} \lor \neg \left(x \leq 1.85 \cdot 10^{+45}\right):\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 49.6% 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 71.6%

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

   1. div-sub 71.1%

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

   2. sub-neg 71.1%

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

   3. associate-/r* 78.7%

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

   4. associate-/r* 78.7%

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

   5. *-inverses 78.7%

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

   6. metadata-eval 78.7%

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

   7. *-commutative 78.7%

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

   8. *-commutative 78.7%

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

   9. associate-*l* 78.7%

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

   10. associate-/r* 100.0%

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

   11. distribute-neg-frac 100.0%

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

   12. associate-/r* 100.0%

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

   13. *-inverses 100.0%

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

   14. metadata-eval 100.0%

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

   15. 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. Add Preprocessing

5. Taylor expanded in y around inf 51.6%

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

6. Final simplification 51.6%

   \[\leadsto \frac{-0.5}{x} \]
7. Add Preprocessing

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 2024019 
(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)))