Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 100.0%

Time: 2.9s

Alternatives: 4

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, 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 2: 74.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 2 \cdot \frac{y}{x}\\ \mathbf{if}\;x \leq -8.6 \cdot 10^{-16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 10^{-105}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-89}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-57}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 2.0 (/ y x)))))
   (if (<= x -8.6e-16)
     t_0
     (if (<= x 1e-105)
       -1.0
       (if (<= x 2.25e-89) 1.0 (if (<= x 3.7e-57) -1.0 t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (2.0 * (y / x));
	double tmp;
	if (x <= -8.6e-16) {
		tmp = t_0;
	} else if (x <= 1e-105) {
		tmp = -1.0;
	} else if (x <= 2.25e-89) {
		tmp = 1.0;
	} else if (x <= 3.7e-57) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (2.0d0 * (y / x))
    if (x <= (-8.6d-16)) then
        tmp = t_0
    else if (x <= 1d-105) then
        tmp = -1.0d0
    else if (x <= 2.25d-89) then
        tmp = 1.0d0
    else if (x <= 3.7d-57) then
        tmp = -1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (2.0 * (y / x));
	double tmp;
	if (x <= -8.6e-16) {
		tmp = t_0;
	} else if (x <= 1e-105) {
		tmp = -1.0;
	} else if (x <= 2.25e-89) {
		tmp = 1.0;
	} else if (x <= 3.7e-57) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (2.0 * (y / x))
	tmp = 0
	if x <= -8.6e-16:
		tmp = t_0
	elif x <= 1e-105:
		tmp = -1.0
	elif x <= 2.25e-89:
		tmp = 1.0
	elif x <= 3.7e-57:
		tmp = -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(2.0 * Float64(y / x)))
	tmp = 0.0
	if (x <= -8.6e-16)
		tmp = t_0;
	elseif (x <= 1e-105)
		tmp = -1.0;
	elseif (x <= 2.25e-89)
		tmp = 1.0;
	elseif (x <= 3.7e-57)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (2.0 * (y / x));
	tmp = 0.0;
	if (x <= -8.6e-16)
		tmp = t_0;
	elseif (x <= 1e-105)
		tmp = -1.0;
	elseif (x <= 2.25e-89)
		tmp = 1.0;
	elseif (x <= 3.7e-57)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.6e-16], t$95$0, If[LessEqual[x, 1e-105], -1.0, If[LessEqual[x, 2.25e-89], 1.0, If[LessEqual[x, 3.7e-57], -1.0, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.5999999999999997e-16 or 3.7e-57 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 73.3%

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

   if -8.5999999999999997e-16 < x < 9.99999999999999965e-106 or 2.25e-89 < x < 3.7e-57

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 83.0%

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

   if 9.99999999999999965e-106 < x < 2.25e-89

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 100.0%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{-16}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq 10^{-105}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-89}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-57}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \end{array} \]

Alternative 3: 73.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-105}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.18 \cdot 10^{-88}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-57}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1e-10)
   1.0
   (if (<= x 2.15e-105)
     -1.0
     (if (<= x 1.18e-88) 1.0 (if (<= x 6.2e-57) -1.0 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1e-10) {
		tmp = 1.0;
	} else if (x <= 2.15e-105) {
		tmp = -1.0;
	} else if (x <= 1.18e-88) {
		tmp = 1.0;
	} else if (x <= 6.2e-57) {
		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 <= (-1d-10)) then
        tmp = 1.0d0
    else if (x <= 2.15d-105) then
        tmp = -1.0d0
    else if (x <= 1.18d-88) then
        tmp = 1.0d0
    else if (x <= 6.2d-57) 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 <= -1e-10) {
		tmp = 1.0;
	} else if (x <= 2.15e-105) {
		tmp = -1.0;
	} else if (x <= 1.18e-88) {
		tmp = 1.0;
	} else if (x <= 6.2e-57) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1e-10:
		tmp = 1.0
	elif x <= 2.15e-105:
		tmp = -1.0
	elif x <= 1.18e-88:
		tmp = 1.0
	elif x <= 6.2e-57:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1e-10)
		tmp = 1.0;
	elseif (x <= 2.15e-105)
		tmp = -1.0;
	elseif (x <= 1.18e-88)
		tmp = 1.0;
	elseif (x <= 6.2e-57)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1e-10)
		tmp = 1.0;
	elseif (x <= 2.15e-105)
		tmp = -1.0;
	elseif (x <= 1.18e-88)
		tmp = 1.0;
	elseif (x <= 6.2e-57)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1e-10], 1.0, If[LessEqual[x, 2.15e-105], -1.0, If[LessEqual[x, 1.18e-88], 1.0, If[LessEqual[x, 6.2e-57], -1.0, 1.0]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.00000000000000004e-10 or 2.14999999999999982e-105 < x < 1.18000000000000004e-88 or 6.19999999999999952e-57 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 73.3%

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

   if -1.00000000000000004e-10 < x < 2.14999999999999982e-105 or 1.18000000000000004e-88 < x < 6.19999999999999952e-57

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 83.0%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-105}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.18 \cdot 10^{-88}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-57}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 49.4% 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 50.3%

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

3. Final simplification 50.3%

   \[\leadsto -1 \]

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 2023275 
(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)))