Linear.Projection:inversePerspective from linear-1.19.1.3, B

Percentage Accurate: 77.2% → 100.0%

Time: 6.6s

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: 77.2% 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 76.9%

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

   1. div-sub 76.6%

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

   2. sub-neg 76.6%

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

   3. associate-/r* 82.5%

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

   4. associate-/r* 82.5%

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

   5. *-inverses 82.5%

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

   6. metadata-eval 82.5%

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

   7. distribute-frac-neg 82.5%

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

   8. associate-/l/ 100.0%

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

   9. distribute-frac-neg 100.0%

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

   10. *-inverses 100.0%

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

   11. metadata-eval 100.0%

       \[\leadsto \frac{0.5}{y} + \frac{-\color{blue}{\left(--1\right)}}{x \cdot 2} \]

   12. metadata-eval 100.0%

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

   13. metadata-eval 100.0%

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

   14. *-commutative 100.0%

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

   15. associate-/r* 100.0%

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

   16. 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: 74.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+129} \lor \neg \left(x \leq -2.05 \cdot 10^{+106} \lor \neg \left(x \leq -2.8 \cdot 10^{-27}\right) \land x \leq 6.8 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.4e+129)
         (not
          (or (<= x -2.05e+106) (and (not (<= x -2.8e-27)) (<= x 6.8e-43)))))
   (/ 0.5 y)
   (/ -0.5 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.4e+129) || !((x <= -2.05e+106) || (!(x <= -2.8e-27) && (x <= 6.8e-43)))) {
		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 <= (-3.4d+129)) .or. (.not. (x <= (-2.05d+106)) .or. (.not. (x <= (-2.8d-27))) .and. (x <= 6.8d-43))) 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 <= -3.4e+129) || !((x <= -2.05e+106) || (!(x <= -2.8e-27) && (x <= 6.8e-43)))) {
		tmp = 0.5 / y;
	} else {
		tmp = -0.5 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3.4e+129) or not ((x <= -2.05e+106) or (not (x <= -2.8e-27) and (x <= 6.8e-43))):
		tmp = 0.5 / y
	else:
		tmp = -0.5 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.4e+129) || !((x <= -2.05e+106) || (!(x <= -2.8e-27) && (x <= 6.8e-43))))
		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 <= -3.4e+129) || ~(((x <= -2.05e+106) || (~((x <= -2.8e-27)) && (x <= 6.8e-43)))))
		tmp = 0.5 / y;
	else
		tmp = -0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.4e+129], N[Not[Or[LessEqual[x, -2.05e+106], And[N[Not[LessEqual[x, -2.8e-27]], $MachinePrecision], LessEqual[x, 6.8e-43]]]], $MachinePrecision]], N[(0.5 / y), $MachinePrecision], N[(-0.5 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.40000000000000018e129 or -2.0500000000000001e106 < x < -2.8e-27 or 6.8000000000000001e-43 < x

   1. Initial program 79.4%

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

      1. div-sub 79.4%

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

      2. sub-neg 79.4%

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

      3. associate-/r* 89.2%

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

      4. associate-/r* 89.2%

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

      5. *-inverses 89.2%

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

      6. metadata-eval 89.2%

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

      7. distribute-frac-neg 89.2%

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

      8. associate-/l/ 100.0%

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

      9. distribute-frac-neg 100.0%

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

      10. *-inverses 100.0%

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

      11. metadata-eval 100.0%

          \[\leadsto \frac{0.5}{y} + \frac{-\color{blue}{\left(--1\right)}}{x \cdot 2} \]

      12. metadata-eval 100.0%

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

      13. metadata-eval 100.0%

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

      14. *-commutative 100.0%

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

      15. associate-/r* 100.0%

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

      16. 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 75.4%

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

   if -3.40000000000000018e129 < x < -2.0500000000000001e106 or -2.8e-27 < x < 6.8000000000000001e-43

   1. Initial program 73.7%

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

      1. div-sub 72.9%

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

      2. sub-neg 72.9%

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

      3. associate-/r* 73.7%

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

      4. associate-/r* 73.7%

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

      5. *-inverses 73.7%

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

      6. metadata-eval 73.7%

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

      7. distribute-frac-neg 73.7%

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

      8. associate-/l/ 100.0%

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

      9. distribute-frac-neg 100.0%

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

      10. *-inverses 100.0%

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

      11. metadata-eval 100.0%

          \[\leadsto \frac{0.5}{y} + \frac{-\color{blue}{\left(--1\right)}}{x \cdot 2} \]

      12. metadata-eval 100.0%

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

      13. metadata-eval 100.0%

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

      14. *-commutative 100.0%

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

      15. associate-/r* 100.0%

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

      16. 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 88.5%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+129} \lor \neg \left(x \leq -2.05 \cdot 10^{+106} \lor \neg \left(x \leq -2.8 \cdot 10^{-27}\right) \land x \leq 6.8 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.6% 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 76.9%

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

   1. div-sub 76.6%

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

   2. sub-neg 76.6%

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

   3. associate-/r* 82.5%

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

   4. associate-/r* 82.5%

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

   5. *-inverses 82.5%

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

   6. metadata-eval 82.5%

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

   7. distribute-frac-neg 82.5%

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

   8. associate-/l/ 100.0%

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

   9. distribute-frac-neg 100.0%

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

   10. *-inverses 100.0%

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

   11. metadata-eval 100.0%

       \[\leadsto \frac{0.5}{y} + \frac{-\color{blue}{\left(--1\right)}}{x \cdot 2} \]

   12. metadata-eval 100.0%

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

   13. metadata-eval 100.0%

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

   14. *-commutative 100.0%

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

   15. associate-/r* 100.0%

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

   16. 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 53.8%

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

6. Final simplification 53.8%

   \[\leadsto \frac{-0.5}{x} \]
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 2024046 
(FPCore (x y)
  :name "Linear.Projection:inversePerspective from linear-1.19.1.3, B"
  :precision binary64

  :alt
  (- (/ 0.5 y) (/ 0.5 x))

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