Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 100.0%

Time: 4.8s

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, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. log1p-expm1-u 99.9%

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

3. Applied egg-rr 99.9%

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

   1. log1p-expm1-u 99.9%

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

   2. clear-num 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 75.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{-37} \lor \neg \left(y \leq 6.4 \cdot 10^{+67}\right):\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6.2e-37) (not (<= y 6.4e+67)))
   (+ (* -2.0 (/ x y)) -1.0)
   (+ 1.0 (* 2.0 (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6.2e-37) || !(y <= 6.4e+67)) {
		tmp = (-2.0 * (x / y)) + -1.0;
	} else {
		tmp = 1.0 + (2.0 * (y / 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 <= (-6.2d-37)) .or. (.not. (y <= 6.4d+67))) then
        tmp = ((-2.0d0) * (x / y)) + (-1.0d0)
    else
        tmp = 1.0d0 + (2.0d0 * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -6.2e-37) || !(y <= 6.4e+67)) {
		tmp = (-2.0 * (x / y)) + -1.0;
	} else {
		tmp = 1.0 + (2.0 * (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -6.2e-37) or not (y <= 6.4e+67):
		tmp = (-2.0 * (x / y)) + -1.0
	else:
		tmp = 1.0 + (2.0 * (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6.2e-37) || !(y <= 6.4e+67))
		tmp = Float64(Float64(-2.0 * Float64(x / y)) + -1.0);
	else
		tmp = Float64(1.0 + Float64(2.0 * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -6.2e-37) || ~((y <= 6.4e+67)))
		tmp = (-2.0 * (x / y)) + -1.0;
	else
		tmp = 1.0 + (2.0 * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6.2e-37], N[Not[LessEqual[y, 6.4e+67]], $MachinePrecision]], N[(N[(-2.0 * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.19999999999999987e-37 or 6.39999999999999965e67 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 78.5%

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

   if -6.19999999999999987e-37 < y < 6.39999999999999965e67

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 80.4%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{-37} \lor \neg \left(y \leq 6.4 \cdot 10^{+67}\right):\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \end{array} \]

Alternative 3: 74.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-30}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+67}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.4e-30) -1.0 (if (<= y 1.45e+67) (+ 1.0 (* 2.0 (/ y x))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.4e-30) {
		tmp = -1.0;
	} else if (y <= 1.45e+67) {
		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 (y <= (-1.4d-30)) then
        tmp = -1.0d0
    else if (y <= 1.45d+67) 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 (y <= -1.4e-30) {
		tmp = -1.0;
	} else if (y <= 1.45e+67) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.4e-30:
		tmp = -1.0
	elif y <= 1.45e+67:
		tmp = 1.0 + (2.0 * (y / x))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.4e-30)
		tmp = -1.0;
	elseif (y <= 1.45e+67)
		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 (y <= -1.4e-30)
		tmp = -1.0;
	elseif (y <= 1.45e+67)
		tmp = 1.0 + (2.0 * (y / x));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.4e-30], -1.0, If[LessEqual[y, 1.45e+67], N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.39999999999999994e-30 or 1.45000000000000012e67 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 77.6%

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

   if -1.39999999999999994e-30 < y < 1.45000000000000012e67

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 80.4%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-30}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+67}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4: 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 99.9%

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

2. Final simplification 99.9%

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

Alternative 5: 73.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-30}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+67}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5e-30) -1.0 (if (<= y 2e+67) 1.0 -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5e-30) {
		tmp = -1.0;
	} else if (y <= 2e+67) {
		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 <= (-5d-30)) then
        tmp = -1.0d0
    else if (y <= 2d+67) 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 <= -5e-30) {
		tmp = -1.0;
	} else if (y <= 2e+67) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5e-30:
		tmp = -1.0
	elif y <= 2e+67:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5e-30)
		tmp = -1.0;
	elseif (y <= 2e+67)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5e-30)
		tmp = -1.0;
	elseif (y <= 2e+67)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5e-30], -1.0, If[LessEqual[y, 2e+67], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -4.99999999999999972e-30 or 1.99999999999999997e67 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 77.6%

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

   if -4.99999999999999972e-30 < y < 1.99999999999999997e67

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 79.2%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-30}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+67}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 6: 50.2% 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 99.9%

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

2. Taylor expanded in x around 0 46.8%

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

3. Final simplification 46.8%

   \[\leadsto -1 \]

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 2023185 
(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)))