Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 98.3%

Time: 5.0s

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: 98.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ e^{\mathsf{log1p}\left(\frac{x + y}{x - y}\right)} + -1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (exp (log1p (/ (+ x y) (- x y)))) -1.0))

C

double code(double x, double y) {
	return exp(log1p(((x + y) / (x - y)))) + -1.0;
}

Java

public static double code(double x, double y) {
	return Math.exp(Math.log1p(((x + y) / (x - y)))) + -1.0;
}

Python

def code(x, y):
	return math.exp(math.log1p(((x + y) / (x - y)))) + -1.0

Julia

function code(x, y)
	return Float64(exp(log1p(Float64(Float64(x + y) / Float64(x - y)))) + -1.0)
end

Wolfram

code[x_, y_] := N[(N[Exp[N[Log[1 + N[(N[(x + y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto e^{\mathsf{log1p}\left(\frac{x + y}{x - y}\right)} + -1 \]

Alternative 2: 72.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -34000000 \lor \neg \left(y \leq 4 \cdot 10^{+15} \lor \neg \left(y \leq 1.35 \cdot 10^{+141}\right) \land y \leq 1.2 \cdot 10^{+199}\right):\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -34000000.0)
         (not (or (<= y 4e+15) (and (not (<= y 1.35e+141)) (<= y 1.2e+199)))))
   (+ (* -2.0 (/ x y)) -1.0)
   (+ 1.0 (* 2.0 (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -34000000.0) || !((y <= 4e+15) || (!(y <= 1.35e+141) && (y <= 1.2e+199)))) {
		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 <= (-34000000.0d0)) .or. (.not. (y <= 4d+15) .or. (.not. (y <= 1.35d+141)) .and. (y <= 1.2d+199))) 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 <= -34000000.0) || !((y <= 4e+15) || (!(y <= 1.35e+141) && (y <= 1.2e+199)))) {
		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 <= -34000000.0) or not ((y <= 4e+15) or (not (y <= 1.35e+141) and (y <= 1.2e+199))):
		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 <= -34000000.0) || !((y <= 4e+15) || (!(y <= 1.35e+141) && (y <= 1.2e+199))))
		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 <= -34000000.0) || ~(((y <= 4e+15) || (~((y <= 1.35e+141)) && (y <= 1.2e+199)))))
		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, -34000000.0], N[Not[Or[LessEqual[y, 4e+15], And[N[Not[LessEqual[y, 1.35e+141]], $MachinePrecision], LessEqual[y, 1.2e+199]]]], $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 < -3.4e7 or 4e15 < y < 1.35e141 or 1.20000000000000007e199 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 85.6%

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

   if -3.4e7 < y < 4e15 or 1.35e141 < y < 1.20000000000000007e199

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 82.3%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -34000000 \lor \neg \left(y \leq 4 \cdot 10^{+15} \lor \neg \left(y \leq 1.35 \cdot 10^{+141}\right) \land y \leq 1.2 \cdot 10^{+199}\right):\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \end{array} \]

Alternative 3: 72.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -42000:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2500000000000 \lor \neg \left(y \leq 1.4 \cdot 10^{+142}\right) \land y \leq 1.25 \cdot 10^{+199}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -42000.0)
   -1.0
   (if (or (<= y 2500000000000.0) (and (not (<= y 1.4e+142)) (<= y 1.25e+199)))
     (+ 1.0 (* 2.0 (/ y x)))
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -42000.0) {
		tmp = -1.0;
	} else if ((y <= 2500000000000.0) || (!(y <= 1.4e+142) && (y <= 1.25e+199))) {
		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 <= (-42000.0d0)) then
        tmp = -1.0d0
    else if ((y <= 2500000000000.0d0) .or. (.not. (y <= 1.4d+142)) .and. (y <= 1.25d+199)) 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 <= -42000.0) {
		tmp = -1.0;
	} else if ((y <= 2500000000000.0) || (!(y <= 1.4e+142) && (y <= 1.25e+199))) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -42000.0:
		tmp = -1.0
	elif (y <= 2500000000000.0) or (not (y <= 1.4e+142) and (y <= 1.25e+199)):
		tmp = 1.0 + (2.0 * (y / x))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -42000.0)
		tmp = -1.0;
	elseif ((y <= 2500000000000.0) || (!(y <= 1.4e+142) && (y <= 1.25e+199)))
		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 <= -42000.0)
		tmp = -1.0;
	elseif ((y <= 2500000000000.0) || (~((y <= 1.4e+142)) && (y <= 1.25e+199)))
		tmp = 1.0 + (2.0 * (y / x));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -42000.0], -1.0, If[Or[LessEqual[y, 2500000000000.0], And[N[Not[LessEqual[y, 1.4e+142]], $MachinePrecision], LessEqual[y, 1.25e+199]]], N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -42000 or 2.5e12 < y < 1.4e142 or 1.25e199 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 84.7%

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

   if -42000 < y < 2.5e12 or 1.4e142 < y < 1.25e199

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 82.3%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -42000:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2500000000000 \lor \neg \left(y \leq 1.4 \cdot 10^{+142}\right) \land y \leq 1.25 \cdot 10^{+199}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4: 71.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+14}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+142}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+199}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5e+14)
   -1.0
   (if (<= y 4.4e+16)
     1.0
     (if (<= y 1.4e+142) -1.0 (if (<= y 1.2e+199) 1.0 -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5e+14) {
		tmp = -1.0;
	} else if (y <= 4.4e+16) {
		tmp = 1.0;
	} else if (y <= 1.4e+142) {
		tmp = -1.0;
	} else if (y <= 1.2e+199) {
		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+14)) then
        tmp = -1.0d0
    else if (y <= 4.4d+16) then
        tmp = 1.0d0
    else if (y <= 1.4d+142) then
        tmp = -1.0d0
    else if (y <= 1.2d+199) 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+14) {
		tmp = -1.0;
	} else if (y <= 4.4e+16) {
		tmp = 1.0;
	} else if (y <= 1.4e+142) {
		tmp = -1.0;
	} else if (y <= 1.2e+199) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5e+14:
		tmp = -1.0
	elif y <= 4.4e+16:
		tmp = 1.0
	elif y <= 1.4e+142:
		tmp = -1.0
	elif y <= 1.2e+199:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5e+14)
		tmp = -1.0;
	elseif (y <= 4.4e+16)
		tmp = 1.0;
	elseif (y <= 1.4e+142)
		tmp = -1.0;
	elseif (y <= 1.2e+199)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5e+14)
		tmp = -1.0;
	elseif (y <= 4.4e+16)
		tmp = 1.0;
	elseif (y <= 1.4e+142)
		tmp = -1.0;
	elseif (y <= 1.2e+199)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5e+14], -1.0, If[LessEqual[y, 4.4e+16], 1.0, If[LessEqual[y, 1.4e+142], -1.0, If[LessEqual[y, 1.2e+199], 1.0, -1.0]]]]

Derivation

1. Split input into 2 regimes

2. if y < -5e14 or 4.4e16 < y < 1.4e142 or 1.20000000000000007e199 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 84.7%

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

   if -5e14 < y < 4.4e16 or 1.4e142 < y < 1.20000000000000007e199

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 81.4%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+14}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+142}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+199}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 5: 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. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

3. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x + y}{x - y}\right)} - 1} \]
4. Step-by-step derivation

   1. expm1-def 100.0%

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

   2. expm1-log1p-u 99.9%

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

   3. 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 6: 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 7: 49.6% 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 49.2%

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

3. Final simplification 49.2%

   \[\leadsto -1 \]

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 2023320 
(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)))