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, 0.6× 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. Add Preprocessing
3. Step-by-step derivation

   1. clear-num 99.9%

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

   2. inv-pow 99.9%

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

4. Applied egg-rr 99.9%

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

   1. unpow-1 99.9%

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

   2. associate-/r/ 99.8%

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

   3. +-commutative 99.8%

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

   4. distribute-rgt-in 99.8%

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

   5. un-div-inv 99.9%

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

   6. un-div-inv 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{y}{x - y} + \frac{x}{x - y}} \]
7. Add Preprocessing

Alternative 2: 72.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{+171} \lor \neg \left(x \leq -8.6 \cdot 10^{+122}\right) \land \left(x \leq -7.6 \cdot 10^{-28} \lor \neg \left(x \leq 8.5 \cdot 10^{+64}\right)\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.85e+171)
         (and (not (<= x -8.6e+122))
              (or (<= x -7.6e-28) (not (<= x 8.5e+64)))))
   (+ 1.0 (* 2.0 (/ y x)))
   (+ (* -2.0 (/ x y)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.85e+171) || (!(x <= -8.6e+122) && ((x <= -7.6e-28) || !(x <= 8.5e+64)))) {
		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 <= (-2.85d+171)) .or. (.not. (x <= (-8.6d+122))) .and. (x <= (-7.6d-28)) .or. (.not. (x <= 8.5d+64))) 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 <= -2.85e+171) || (!(x <= -8.6e+122) && ((x <= -7.6e-28) || !(x <= 8.5e+64)))) {
		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 <= -2.85e+171) or (not (x <= -8.6e+122) and ((x <= -7.6e-28) or not (x <= 8.5e+64))):
		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 <= -2.85e+171) || (!(x <= -8.6e+122) && ((x <= -7.6e-28) || !(x <= 8.5e+64))))
		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 <= -2.85e+171) || (~((x <= -8.6e+122)) && ((x <= -7.6e-28) || ~((x <= 8.5e+64)))))
		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, -2.85e+171], And[N[Not[LessEqual[x, -8.6e+122]], $MachinePrecision], Or[LessEqual[x, -7.6e-28], N[Not[LessEqual[x, 8.5e+64]], $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 < -2.85e171 or -8.59999999999999943e122 < x < -7.60000000000000018e-28 or 8.4999999999999998e64 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 85.9%

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

   if -2.85e171 < x < -8.59999999999999943e122 or -7.60000000000000018e-28 < x < 8.4999999999999998e64

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 74.8%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{+171} \lor \neg \left(x \leq -8.6 \cdot 10^{+122}\right) \land \left(x \leq -7.6 \cdot 10^{-28} \lor \neg \left(x \leq 8.5 \cdot 10^{+64}\right)\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+166} \lor \neg \left(x \leq -8.6 \cdot 10^{+122} \lor \neg \left(x \leq -4 \cdot 10^{-28}\right) \land x \leq 2.3 \cdot 10^{+64}\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -7.5e+166)
         (not (or (<= x -8.6e+122) (and (not (<= x -4e-28)) (<= x 2.3e+64)))))
   (+ 1.0 (* 2.0 (/ y x)))
   -1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -7.5e+166) || !((x <= -8.6e+122) || (!(x <= -4e-28) && (x <= 2.3e+64)))) {
		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 <= (-7.5d+166)) .or. (.not. (x <= (-8.6d+122)) .or. (.not. (x <= (-4d-28))) .and. (x <= 2.3d+64))) 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 <= -7.5e+166) || !((x <= -8.6e+122) || (!(x <= -4e-28) && (x <= 2.3e+64)))) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -7.5e+166) or not ((x <= -8.6e+122) or (not (x <= -4e-28) and (x <= 2.3e+64))):
		tmp = 1.0 + (2.0 * (y / x))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -7.5e+166) || !((x <= -8.6e+122) || (!(x <= -4e-28) && (x <= 2.3e+64))))
		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 <= -7.5e+166) || ~(((x <= -8.6e+122) || (~((x <= -4e-28)) && (x <= 2.3e+64)))))
		tmp = 1.0 + (2.0 * (y / x));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -7.5e+166], N[Not[Or[LessEqual[x, -8.6e+122], And[N[Not[LessEqual[x, -4e-28]], $MachinePrecision], LessEqual[x, 2.3e+64]]]], $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 < -7.50000000000000029e166 or -8.59999999999999943e122 < x < -3.99999999999999988e-28 or 2.3e64 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 85.4%

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

   if -7.50000000000000029e166 < x < -8.59999999999999943e122 or -3.99999999999999988e-28 < x < 2.3e64

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 73.4%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+166} \lor \neg \left(x \leq -8.6 \cdot 10^{+122} \lor \neg \left(x \leq -4 \cdot 10^{-28}\right) \land x \leq 2.3 \cdot 10^{+64}\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{+171}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -8.6 \cdot 10^{+122}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{-28}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+64}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.85e+171)
   1.0
   (if (<= x -8.6e+122)
     -1.0
     (if (<= x -3.7e-28) 1.0 (if (<= x 4e+64) -1.0 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.85e+171) {
		tmp = 1.0;
	} else if (x <= -8.6e+122) {
		tmp = -1.0;
	} else if (x <= -3.7e-28) {
		tmp = 1.0;
	} else if (x <= 4e+64) {
		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 <= (-2.85d+171)) then
        tmp = 1.0d0
    else if (x <= (-8.6d+122)) then
        tmp = -1.0d0
    else if (x <= (-3.7d-28)) then
        tmp = 1.0d0
    else if (x <= 4d+64) 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 <= -2.85e+171) {
		tmp = 1.0;
	} else if (x <= -8.6e+122) {
		tmp = -1.0;
	} else if (x <= -3.7e-28) {
		tmp = 1.0;
	} else if (x <= 4e+64) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.85e+171:
		tmp = 1.0
	elif x <= -8.6e+122:
		tmp = -1.0
	elif x <= -3.7e-28:
		tmp = 1.0
	elif x <= 4e+64:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.85e+171)
		tmp = 1.0;
	elseif (x <= -8.6e+122)
		tmp = -1.0;
	elseif (x <= -3.7e-28)
		tmp = 1.0;
	elseif (x <= 4e+64)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.85e+171)
		tmp = 1.0;
	elseif (x <= -8.6e+122)
		tmp = -1.0;
	elseif (x <= -3.7e-28)
		tmp = 1.0;
	elseif (x <= 4e+64)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.85e+171], 1.0, If[LessEqual[x, -8.6e+122], -1.0, If[LessEqual[x, -3.7e-28], 1.0, If[LessEqual[x, 4e+64], -1.0, 1.0]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.85e171 or -8.59999999999999943e122 < x < -3.7000000000000002e-28 or 4.00000000000000009e64 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 84.9%

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

   if -2.85e171 < x < -8.59999999999999943e122 or -3.7000000000000002e-28 < x < 4.00000000000000009e64

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 73.1%

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

Alternative 5: 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. Add Preprocessing

3. Final simplification 100.0%

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

Alternative 6: 49.2% 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 100.0%

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

3. Taylor expanded in x around 0 48.7%

   \[\leadsto \color{blue}{-1} \]
4. 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 2024103 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (/ 1.0 (- (/ x (+ x y)) (/ y (+ x y))))

  (/ (+ x y) (- x y)))