Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 100.0%

Time: 6.8s

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, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. add-cube-cbrt 99.9%

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

   2. pow3 99.9%

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

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{x + y}{x - y}}\right)}^{3}} \]
5. Step-by-step derivation

   1. rem-cube-cbrt 100.0%

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

   2. expm1-log1p-u 99.2%

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

   3. expm1-udef 99.2%

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

   4. log1p-udef 99.2%

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

   5. rem-exp-log 100.0%

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

   6. +-commutative 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

   \[\leadsto \left(\frac{x + y}{x - y} + 1\right) + -1 \]
8. Add Preprocessing

Alternative 2: 73.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+76} \lor \neg \left(x \leq -1.2 \cdot 10^{-34} \lor \neg \left(x \leq -5.4 \cdot 10^{-111}\right) \land x \leq 9 \cdot 10^{-28}\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.15e+76)
         (not (or (<= x -1.2e-34) (and (not (<= x -5.4e-111)) (<= x 9e-28)))))
   (+ 1.0 (* 2.0 (/ y x)))
   -1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.15e+76) || !((x <= -1.2e-34) || (!(x <= -5.4e-111) && (x <= 9e-28)))) {
		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 <= (-2.15d+76)) .or. (.not. (x <= (-1.2d-34)) .or. (.not. (x <= (-5.4d-111))) .and. (x <= 9d-28))) 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 <= -2.15e+76) || !((x <= -1.2e-34) || (!(x <= -5.4e-111) && (x <= 9e-28)))) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.15e+76) or not ((x <= -1.2e-34) or (not (x <= -5.4e-111) and (x <= 9e-28))):
		tmp = 1.0 + (2.0 * (y / x))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.15e+76) || !((x <= -1.2e-34) || (!(x <= -5.4e-111) && (x <= 9e-28))))
		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 <= -2.15e+76) || ~(((x <= -1.2e-34) || (~((x <= -5.4e-111)) && (x <= 9e-28)))))
		tmp = 1.0 + (2.0 * (y / x));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.15e+76], N[Not[Or[LessEqual[x, -1.2e-34], And[N[Not[LessEqual[x, -5.4e-111]], $MachinePrecision], LessEqual[x, 9e-28]]]], $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 < -2.14999999999999989e76 or -1.19999999999999996e-34 < x < -5.39999999999999977e-111 or 8.9999999999999996e-28 < x

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 76.8%

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

   if -2.14999999999999989e76 < x < -1.19999999999999996e-34 or -5.39999999999999977e-111 < x < 8.9999999999999996e-28

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 88.2%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+76} \lor \neg \left(x \leq -1.2 \cdot 10^{-34} \lor \neg \left(x \leq -5.4 \cdot 10^{-111}\right) \land x \leq 9 \cdot 10^{-28}\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+75} \lor \neg \left(x \leq -4.6 \cdot 10^{-95}\right) \land \left(x \leq -5.4 \cdot 10^{-111} \lor \neg \left(x \leq 1.2 \cdot 10^{-28}\right)\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -5.6e+75)
         (and (not (<= x -4.6e-95))
              (or (<= x -5.4e-111) (not (<= x 1.2e-28)))))
   (+ 1.0 (* 2.0 (/ y x)))
   (+ (* -2.0 (/ x y)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -5.6e+75) || (!(x <= -4.6e-95) && ((x <= -5.4e-111) || !(x <= 1.2e-28)))) {
		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 <= (-5.6d+75)) .or. (.not. (x <= (-4.6d-95))) .and. (x <= (-5.4d-111)) .or. (.not. (x <= 1.2d-28))) 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 <= -5.6e+75) || (!(x <= -4.6e-95) && ((x <= -5.4e-111) || !(x <= 1.2e-28)))) {
		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 <= -5.6e+75) or (not (x <= -4.6e-95) and ((x <= -5.4e-111) or not (x <= 1.2e-28))):
		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 <= -5.6e+75) || (!(x <= -4.6e-95) && ((x <= -5.4e-111) || !(x <= 1.2e-28))))
		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 <= -5.6e+75) || (~((x <= -4.6e-95)) && ((x <= -5.4e-111) || ~((x <= 1.2e-28)))))
		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, -5.6e+75], And[N[Not[LessEqual[x, -4.6e-95]], $MachinePrecision], Or[LessEqual[x, -5.4e-111], N[Not[LessEqual[x, 1.2e-28]], $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 < -5.60000000000000023e75 or -4.59999999999999998e-95 < x < -5.39999999999999977e-111 or 1.2000000000000001e-28 < x

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 79.1%

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

   if -5.60000000000000023e75 < x < -4.59999999999999998e-95 or -5.39999999999999977e-111 < x < 1.2000000000000001e-28

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 85.1%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+75} \lor \neg \left(x \leq -4.6 \cdot 10^{-95}\right) \land \left(x \leq -5.4 \cdot 10^{-111} \lor \neg \left(x \leq 1.2 \cdot 10^{-28}\right)\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+81}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-34}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-111}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 10^{-36}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.8e+81)
   1.0
   (if (<= x -1.4e-34)
     -1.0
     (if (<= x -5.4e-111) 1.0 (if (<= x 1e-36) -1.0 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.8e+81) {
		tmp = 1.0;
	} else if (x <= -1.4e-34) {
		tmp = -1.0;
	} else if (x <= -5.4e-111) {
		tmp = 1.0;
	} else if (x <= 1e-36) {
		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 <= (-1.8d+81)) then
        tmp = 1.0d0
    else if (x <= (-1.4d-34)) then
        tmp = -1.0d0
    else if (x <= (-5.4d-111)) then
        tmp = 1.0d0
    else if (x <= 1d-36) 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 <= -1.8e+81) {
		tmp = 1.0;
	} else if (x <= -1.4e-34) {
		tmp = -1.0;
	} else if (x <= -5.4e-111) {
		tmp = 1.0;
	} else if (x <= 1e-36) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.8e+81:
		tmp = 1.0
	elif x <= -1.4e-34:
		tmp = -1.0
	elif x <= -5.4e-111:
		tmp = 1.0
	elif x <= 1e-36:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.8e+81)
		tmp = 1.0;
	elseif (x <= -1.4e-34)
		tmp = -1.0;
	elseif (x <= -5.4e-111)
		tmp = 1.0;
	elseif (x <= 1e-36)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.8e+81)
		tmp = 1.0;
	elseif (x <= -1.4e-34)
		tmp = -1.0;
	elseif (x <= -5.4e-111)
		tmp = 1.0;
	elseif (x <= 1e-36)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.8e+81], 1.0, If[LessEqual[x, -1.4e-34], -1.0, If[LessEqual[x, -5.4e-111], 1.0, If[LessEqual[x, 1e-36], -1.0, 1.0]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.80000000000000003e81 or -1.39999999999999998e-34 < x < -5.39999999999999977e-111 or 9.9999999999999994e-37 < x

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 75.1%

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

   if -1.80000000000000003e81 < x < -1.39999999999999998e-34 or -5.39999999999999977e-111 < x < 9.9999999999999994e-37

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 88.2%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+81}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-34}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-111}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 10^{-36}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \frac{x + y}{x - y} \]
4. Add Preprocessing

Alternative 6: 50.0% 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. Add Preprocessing

3. Taylor expanded in x around 0 52.4%

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

4. Final simplification 52.4%

   \[\leadsto -1 \]
5. Add Preprocessing

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 2024020 
(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)))