Linear.Projection:inversePerspective from linear-1.19.1.3, C

Percentage Accurate: 77.5% → 100.0%

Time: 4.5s

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: 77.5% 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 79.0%

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

   1. +-commutative 79.0%

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

   2. remove-double-neg 79.0%

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

   3. unsub-neg 79.0%

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

   4. div-sub 78.5%

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

   5. associate-/l/ 84.4%

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

   6. *-inverses 84.4%

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

   7. metadata-eval 84.4%

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

   8. distribute-neg-frac 84.4%

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

   9. distribute-frac-neg2 84.4%

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

   10. distribute-rgt-neg-in 84.4%

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

   11. metadata-eval 84.4%

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

   12. distribute-neg-frac 84.4%

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

   13. associate-/r* 100.0%

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

   14. distribute-neg-frac 100.0%

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

   15. associate-/r* 100.0%

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

   16. *-inverses 100.0%

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

   17. metadata-eval 100.0%

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

   18. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around inf 100.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{1}{x} + 0.5 \cdot \frac{1}{y}} \]
6. Step-by-step derivation

   1. associate-*r/ 100.0%

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

   2. metadata-eval 100.0%

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

   3. +-commutative 100.0%

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

   4. associate-*r/ 100.0%

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

   5. metadata-eval 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 62.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{-66}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.2e-66) (/ 0.5 y) (/ 0.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.2e-66) {
		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 <= 1.2d-66) 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 <= 1.2e-66) {
		tmp = 0.5 / y;
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.2e-66:
		tmp = 0.5 / y
	else:
		tmp = 0.5 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.2e-66)
		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 <= 1.2e-66)
		tmp = 0.5 / y;
	else
		tmp = 0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.2e-66], N[(0.5 / y), $MachinePrecision], N[(0.5 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.20000000000000013e-66

   1. Initial program 77.6%

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

      1. +-commutative 77.6%

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

      2. remove-double-neg 77.6%

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

      3. unsub-neg 77.6%

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

      4. div-sub 76.9%

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

      5. associate-/l/ 82.6%

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

      6. *-inverses 82.6%

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

      7. metadata-eval 82.6%

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

      8. distribute-neg-frac 82.6%

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

      9. distribute-frac-neg2 82.6%

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

      10. distribute-rgt-neg-in 82.6%

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

      11. metadata-eval 82.6%

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

      12. distribute-neg-frac 82.6%

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

      13. associate-/r* 100.0%

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

      14. distribute-neg-frac 100.0%

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

      15. associate-/r* 100.0%

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

      16. *-inverses 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 55.4%

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

   if 1.20000000000000013e-66 < y

   1. Initial program 83.1%

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

      1. +-commutative 83.1%

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

      2. remove-double-neg 83.1%

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

      3. unsub-neg 83.1%

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

      4. div-sub 83.0%

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

      5. associate-/l/ 89.5%

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

      6. *-inverses 89.5%

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

      7. metadata-eval 89.5%

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

      8. distribute-neg-frac 89.5%

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

      9. distribute-frac-neg2 89.5%

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

      10. distribute-rgt-neg-in 89.5%

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

      11. metadata-eval 89.5%

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

      12. distribute-neg-frac 89.5%

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

      13. associate-/r* 100.0%

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

      14. distribute-neg-frac 100.0%

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

      15. associate-/r* 100.0%

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

      16. *-inverses 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 76.9%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{-66}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.1% 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 79.0%

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

   1. +-commutative 79.0%

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

   2. remove-double-neg 79.0%

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

   3. unsub-neg 79.0%

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

   4. div-sub 78.5%

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

   5. associate-/l/ 84.4%

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

   6. *-inverses 84.4%

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

   7. metadata-eval 84.4%

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

   8. distribute-neg-frac 84.4%

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

   9. distribute-frac-neg2 84.4%

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

   10. distribute-rgt-neg-in 84.4%

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

   11. metadata-eval 84.4%

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

   12. distribute-neg-frac 84.4%

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

   13. associate-/r* 100.0%

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

   14. distribute-neg-frac 100.0%

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

   15. associate-/r* 100.0%

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

   16. *-inverses 100.0%

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

   17. metadata-eval 100.0%

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

   18. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 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

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

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 2024095 
(FPCore (x y)
  :name "Linear.Projection:inversePerspective from linear-1.19.1.3, C"
  :precision binary64

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

  (/ (+ x y) (* (* x 2.0) y)))