Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 100.0%

Time: 7.3s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x y) :precision binary64 (/ (+ x y) (- x y)))

C

double code(double x, double y) {
	return (x + y) / (x - y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x + y) / (x - y)
end function

Java

public static double code(double x, double y) {
	return (x + y) / (x - y);
}

Python

def code(x, y):
	return (x + y) / (x - y)

Julia

function code(x, y)
	return Float64(Float64(x + y) / Float64(x - y))
end

MATLAB

function tmp = code(x, y)
	tmp = (x + y) / (x - y);
end

Wolfram

code[x_, y_] := N[(N[(x + y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (/ (+ x y) (- x y)))

C

double code(double x, double y) {
	return (x + y) / (x - y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x + y) / (x - y)
end function

Java

public static double code(double x, double y) {
	return (x + y) / (x - y);
}

Python

def code(x, y):
	return (x + y) / (x - y)

Julia

function code(x, y)
	return Float64(Float64(x + y) / Float64(x - y))
end

MATLAB

function tmp = code(x, y)
	tmp = (x + y) / (x - y);
end

Wolfram

code[x_, y_] := N[(N[(x + y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (- (/ y (- x y)) (/ x (- y x))))

C

double code(double x, double y) {
	return (y / (x - y)) - (x / (y - x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (y / (x - y)) - (x / (y - x))
end function

Java

public static double code(double x, double y) {
	return (y / (x - y)) - (x / (y - x));
}

Python

def code(x, y):
	return (y / (x - y)) - (x / (y - x))

Julia

function code(x, y)
	return Float64(Float64(y / Float64(x - y)) - Float64(x / Float64(y - x)))
end

MATLAB

function tmp = code(x, y)
	tmp = (y / (x - y)) - (x / (y - x));
end

Wolfram

code[x_, y_] := N[(N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision] - N[(x / N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\frac{x + y}{x - y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift--.f64 N/A

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

   5. flip3-- N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{3} - {y}^{3}}{x \cdot x + \left(y \cdot y + x \cdot y\right)}}} \cdot \left(x + y\right) \]

   6. clear-num N/A

      \[\leadsto \color{blue}{\frac{x \cdot x + \left(y \cdot y + x \cdot y\right)}{{x}^{3} - {y}^{3}}} \cdot \left(x + y\right) \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot x + \left(y \cdot y + x \cdot y\right)}{{x}^{3} - {y}^{3}} \cdot \color{blue}{\left(x + y\right)} \]

   8. distribute-lft-in N/A

      \[\leadsto \color{blue}{\frac{x \cdot x + \left(y \cdot y + x \cdot y\right)}{{x}^{3} - {y}^{3}} \cdot x + \frac{x \cdot x + \left(y \cdot y + x \cdot y\right)}{{x}^{3} - {y}^{3}} \cdot y} \]

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x \cdot x + \left(y \cdot y + x \cdot y\right)}{{x}^{3} - {y}^{3}}, x, \frac{x \cdot x + \left(y \cdot y + x \cdot y\right)}{{x}^{3} - {y}^{3}} \cdot y\right)} \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{y - x}, x, \frac{-1}{y - x} \cdot y\right)} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*l/ N/A

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

   6. neg-mul-1 N/A

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

   7. distribute-frac-neg N/A

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

   8. unsub-neg N/A

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

   9. lower--.f64 N/A

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

6. Applied rewrites 100.0%

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

7. Final simplification 100.0%

   \[\leadsto \frac{y}{x - y} - \frac{x}{y - x} \]
8. Add Preprocessing

Alternative 2: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y + x}{x - y} \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(\frac{-2}{y}, x, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{x}, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (+ y x) (- x y)) -0.5)
   (fma (/ -2.0 y) x -1.0)
   (fma (/ 2.0 x) y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((y + x) / (x - y)) <= -0.5) {
		tmp = fma((-2.0 / y), x, -1.0);
	} else {
		tmp = fma((2.0 / x), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y + x) / Float64(x - y)) <= -0.5)
		tmp = fma(Float64(-2.0 / y), x, -1.0);
	else
		tmp = fma(Float64(2.0 / x), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y + x), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(-2.0 / y), $MachinePrecision] * x + -1.0), $MachinePrecision], N[(N[(2.0 / x), $MachinePrecision] * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 x y)) < -0.5

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{y} - 1} \]
   4. Step-by-step derivation

      1. sub-neg N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\frac{\color{blue}{2}}{y}\right), x, -1\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(2\right)}{y}}, x, -1\right) \]

      13. metadata-eval N/A

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

      14. lower-/.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if -0.5 < (/.f64 (+.f64 x y) (-.f64 x y))

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\left(1 + \frac{y}{x}\right) - -1 \cdot \frac{y}{x}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. distribute-rgt1-in N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. *-lft-identity N/A

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

      11. associate-*l/ N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{2}}{x}, y, 1\right) \]

      16. lower-/.f64 98.7

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{2}{x}}, y, 1\right) \]

   5. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{x}, y, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{x - y} \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(\frac{-2}{y}, x, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{x}, y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{x}{x + y} - \frac{y}{x + y}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 (- (/ x (+ x y)) (/ y (+ x y)))))

C

double code(double x, double y) {
	return 1.0 / ((x / (x + y)) - (y / (x + y)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / ((x / (x + y)) - (y / (x + y)))
end function

Java

public static double code(double x, double y) {
	return 1.0 / ((x / (x + y)) - (y / (x + y)));
}

Python

def code(x, y):
	return 1.0 / ((x / (x + y)) - (y / (x + y)))

Julia

function code(x, y)
	return Float64(1.0 / Float64(Float64(x / Float64(x + y)) - Float64(y / Float64(x + y))))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / ((x / (x + y)) - (y / (x + y)));
end

Wolfram

code[x_, y_] := N[(1.0 / N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] - N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024230 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (/ 1 (- (/ x (+ x y)) (/ y (+ x y)))))

  (/ (+ x y) (- x y)))