Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 100.0%

Time: 3.5s

Alternatives: 6

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. log1p-expm1-u 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x + y}{x - y}\right)\right)} \]

3. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x + y}{x - y}\right)\right)} \]
4. Step-by-step derivation

   1. *-un-lft-identity 100.0%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{1 \cdot \left(x + y\right)}}{x - y}\right)\right) \]

   2. associate-*l/ 99.9%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{1}{x - y} \cdot \left(x + y\right)}\right)\right) \]

   3. log1p-expm1-u 99.8%

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

   4. +-commutative 99.8%

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

   5. distribute-rgt-in 99.8%

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

   6. un-div-inv 99.9%

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

   7. un-div-inv 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 74.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -65000000 \lor \neg \left(x \leq 130\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -65000000.0) (not (<= x 130.0))) (+ 1.0 (* 2.0 (/ y x))) -1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -65000000.0) || !(x <= 130.0)) {
		tmp = 1.0 + (2.0 * (y / x));
	} 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 ((x <= (-65000000.0d0)) .or. (.not. (x <= 130.0d0))) then
        tmp = 1.0d0 + (2.0d0 * (y / x))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -65000000.0) || !(x <= 130.0)) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -65000000.0) or not (x <= 130.0):
		tmp = 1.0 + (2.0 * (y / x))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -65000000.0) || !(x <= 130.0))
		tmp = Float64(1.0 + Float64(2.0 * Float64(y / x)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -65000000.0) || ~((x <= 130.0)))
		tmp = 1.0 + (2.0 * (y / x));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -65000000.0], N[Not[LessEqual[x, 130.0]], $MachinePrecision]], N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if x < -6.5e7 or 130 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 79.8%

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

   if -6.5e7 < x < 130

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 78.8%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -65000000 \lor \neg \left(x \leq 130\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3: 75.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+17} \lor \neg \left(x \leq 17\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.9e+17) (not (<= x 17.0)))
   (+ 1.0 (* 2.0 (/ y x)))
   (+ (* -2.0 (/ x y)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.9e+17) || !(x <= 17.0)) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = (-2.0 * (x / y)) + -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 ((x <= (-2.9d+17)) .or. (.not. (x <= 17.0d0))) then
        tmp = 1.0d0 + (2.0d0 * (y / x))
    else
        tmp = ((-2.0d0) * (x / y)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2.9e+17) || !(x <= 17.0)) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = (-2.0 * (x / y)) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.9e+17) or not (x <= 17.0):
		tmp = 1.0 + (2.0 * (y / x))
	else:
		tmp = (-2.0 * (x / y)) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.9e+17) || !(x <= 17.0))
		tmp = Float64(1.0 + Float64(2.0 * Float64(y / x)));
	else
		tmp = Float64(Float64(-2.0 * Float64(x / y)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.9e+17) || ~((x <= 17.0)))
		tmp = 1.0 + (2.0 * (y / x));
	else
		tmp = (-2.0 * (x / y)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.9e+17], N[Not[LessEqual[x, 17.0]], $MachinePrecision]], N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.9e17 or 17 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 79.8%

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

   if -2.9e17 < x < 17

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 79.7%

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{y} - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+17} \lor \neg \left(x \leq 17\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \end{array} \]

Alternative 4: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 5: 74.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+55}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 96:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6e+55) 1.0 (if (<= x 96.0) -1.0 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6e+55) {
		tmp = 1.0;
	} else if (x <= 96.0) {
		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 (x <= (-6d+55)) then
        tmp = 1.0d0
    else if (x <= 96.0d0) 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 (x <= -6e+55) {
		tmp = 1.0;
	} else if (x <= 96.0) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6e+55:
		tmp = 1.0
	elif x <= 96.0:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6e+55)
		tmp = 1.0;
	elseif (x <= 96.0)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6e+55)
		tmp = 1.0;
	elseif (x <= 96.0)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6e+55], 1.0, If[LessEqual[x, 96.0], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -6.00000000000000033e55 or 96 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 79.0%

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

   if -6.00000000000000033e55 < x < 96

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 78.0%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+55}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 96:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 50.1% 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 49.8%

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

3. Final simplification 49.8%

   \[\leadsto -1 \]

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 2023318 
(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)))