Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 99.9%

Time: 2.9s

Alternatives: 5

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: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \left(3 \cdot \frac{x + y}{x - y}\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* 0.3333333333333333 (* 3.0 (/ (+ x y) (- x y)))))

C

double code(double x, double y) {
	return 0.3333333333333333 * (3.0 * ((x + y) / (x - y)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0 * (3.0d0 * ((x + y) / (x - y)))
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333 * (3.0 * ((x + y) / (x - y)));
}

Python

def code(x, y):
	return 0.3333333333333333 * (3.0 * ((x + y) / (x - y)))

Julia

function code(x, y)
	return Float64(0.3333333333333333 * Float64(3.0 * Float64(Float64(x + y) / Float64(x - y))))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333 * (3.0 * ((x + y) / (x - y)));
end

Wolfram

code[x_, y_] := N[(0.3333333333333333 * N[(3.0 * N[(N[(x + y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. add-log-exp 100.0%

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

3. Applied egg-rr 100.0%

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

   1. add-cbrt-cube 100.0%

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

   2. pow1/3 98.9%

      \[\leadsto \log \color{blue}{\left({\left(\left(e^{\frac{x + y}{x - y}} \cdot e^{\frac{x + y}{x - y}}\right) \cdot e^{\frac{x + y}{x - y}}\right)}^{0.3333333333333333}\right)} \]

   3. log-pow 100.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \log \left(\left(e^{\frac{x + y}{x - y}} \cdot e^{\frac{x + y}{x - y}}\right) \cdot e^{\frac{x + y}{x - y}}\right)} \]

   4. pow3 100.0%

      \[\leadsto 0.3333333333333333 \cdot \log \color{blue}{\left({\left(e^{\frac{x + y}{x - y}}\right)}^{3}\right)} \]

   5. log-pow 100.0%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(e^{\frac{x + y}{x - y}}\right)\right)} \]

   6. add-log-exp 100.0%

      \[\leadsto 0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\frac{x + y}{x - y}}\right) \]

5. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(3 \cdot \frac{x + y}{x - y}\right)} \]

6. Final simplification 100.0%

   \[\leadsto 0.3333333333333333 \cdot \left(3 \cdot \frac{x + y}{x - y}\right) \]

Alternative 2: 74.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-80} \lor \neg \left(y \leq 1.12 \cdot 10^{+72}\right):\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1e-80) (not (<= y 1.12e+72))) (+ (* -2.0 (/ x y)) -1.0) 1.0))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1e-80) || !(y <= 1.12e+72)) {
		tmp = (-2.0 * (x / y)) + -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1d-80)) .or. (.not. (y <= 1.12d+72))) then
        tmp = ((-2.0d0) * (x / y)) + (-1.0d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1e-80) || !(y <= 1.12e+72)) {
		tmp = (-2.0 * (x / y)) + -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1e-80) or not (y <= 1.12e+72):
		tmp = (-2.0 * (x / y)) + -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1e-80) || !(y <= 1.12e+72))
		tmp = Float64(Float64(-2.0 * Float64(x / y)) + -1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1e-80) || ~((y <= 1.12e+72)))
		tmp = (-2.0 * (x / y)) + -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1e-80], N[Not[LessEqual[y, 1.12e+72]], $MachinePrecision]], N[(N[(-2.0 * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if y < -9.99999999999999961e-81 or 1.12000000000000001e72 < y

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]

   2. Taylor expanded in x around 0 77.0%

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

   if -9.99999999999999961e-81 < y < 1.12000000000000001e72

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]

   2. Taylor expanded in x around inf 80.1%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-80} \lor \neg \left(y \leq 1.12 \cdot 10^{+72}\right):\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 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]

Derivation

1. Initial program 100.0%

   \[\frac{x + y}{x - y} \]

2. Final simplification 100.0%

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

Alternative 4: 74.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-80}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+72}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1e-80) -1.0 (if (<= y 1.2e+72) 1.0 -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1e-80) {
		tmp = -1.0;
	} else if (y <= 1.2e+72) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1d-80)) then
        tmp = -1.0d0
    else if (y <= 1.2d+72) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1e-80) {
		tmp = -1.0;
	} else if (y <= 1.2e+72) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1e-80:
		tmp = -1.0
	elif y <= 1.2e+72:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1e-80)
		tmp = -1.0;
	elseif (y <= 1.2e+72)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1e-80)
		tmp = -1.0;
	elseif (y <= 1.2e+72)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1e-80], -1.0, If[LessEqual[y, 1.2e+72], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -9.99999999999999961e-81 or 1.20000000000000005e72 < y

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]

   2. Taylor expanded in x around 0 76.3%

      \[\leadsto \color{blue}{-1} \]

   if -9.99999999999999961e-81 < y < 1.20000000000000005e72

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]

   2. Taylor expanded in x around inf 80.1%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-80}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+72}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 5: 49.3% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

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

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 100.0%

   \[\frac{x + y}{x - y} \]

2. Taylor expanded in x around 0 46.9%

   \[\leadsto \color{blue}{-1} \]

3. Final simplification 46.9%

   \[\leadsto -1 \]

Developer target: 100.0% accurate, 0.5× 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 2023290 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (/ 1.0 (- (/ x (+ x y)) (/ y (+ x y))))

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