Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 100.0%

Time: 6.5s

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 100.0%

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

   1. clear-num 100.0%

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

   2. inv-pow 100.0%

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

4. Applied egg-rr 100.0%

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

   1. unpow-1 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 74.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-7} \lor \neg \left(y \leq 3.4 \cdot 10^{-39}\right) \land \left(y \leq 1.85 \cdot 10^{-22} \lor \neg \left(y \leq 125000000000\right)\right):\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.45e-7)
         (and (not (<= y 3.4e-39))
              (or (<= y 1.85e-22) (not (<= y 125000000000.0)))))
   (+ (* -2.0 (/ x y)) -1.0)
   (+ 1.0 (* 2.0 (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.45e-7) || (!(y <= 3.4e-39) && ((y <= 1.85e-22) || !(y <= 125000000000.0)))) {
		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 <= (-1.45d-7)) .or. (.not. (y <= 3.4d-39)) .and. (y <= 1.85d-22) .or. (.not. (y <= 125000000000.0d0))) 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 <= -1.45e-7) || (!(y <= 3.4e-39) && ((y <= 1.85e-22) || !(y <= 125000000000.0)))) {
		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 <= -1.45e-7) or (not (y <= 3.4e-39) and ((y <= 1.85e-22) or not (y <= 125000000000.0))):
		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 <= -1.45e-7) || (!(y <= 3.4e-39) && ((y <= 1.85e-22) || !(y <= 125000000000.0))))
		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 <= -1.45e-7) || (~((y <= 3.4e-39)) && ((y <= 1.85e-22) || ~((y <= 125000000000.0)))))
		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, -1.45e-7], And[N[Not[LessEqual[y, 3.4e-39]], $MachinePrecision], Or[LessEqual[y, 1.85e-22], N[Not[LessEqual[y, 125000000000.0]], $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 < -1.4499999999999999e-7 or 3.3999999999999999e-39 < y < 1.85e-22 or 1.25e11 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 79.8%

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

   if -1.4499999999999999e-7 < y < 3.3999999999999999e-39 or 1.85e-22 < y < 1.25e11

   1. Initial program 100.0%

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

   3. 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 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-7} \lor \neg \left(y \leq 3.4 \cdot 10^{-39}\right) \land \left(y \leq 1.85 \cdot 10^{-22} \lor \neg \left(y \leq 125000000000\right)\right):\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.000122:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-39} \lor \neg \left(y \leq 3.3 \cdot 10^{-24}\right) \land y \leq 42000000:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -0.000122)
   -1.0
   (if (or (<= y 1.85e-39) (and (not (<= y 3.3e-24)) (<= y 42000000.0)))
     (+ 1.0 (* 2.0 (/ y x)))
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.000122) {
		tmp = -1.0;
	} else if ((y <= 1.85e-39) || (!(y <= 3.3e-24) && (y <= 42000000.0))) {
		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 <= (-0.000122d0)) then
        tmp = -1.0d0
    else if ((y <= 1.85d-39) .or. (.not. (y <= 3.3d-24)) .and. (y <= 42000000.0d0)) 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 <= -0.000122) {
		tmp = -1.0;
	} else if ((y <= 1.85e-39) || (!(y <= 3.3e-24) && (y <= 42000000.0))) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -0.000122:
		tmp = -1.0
	elif (y <= 1.85e-39) or (not (y <= 3.3e-24) and (y <= 42000000.0)):
		tmp = 1.0 + (2.0 * (y / x))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -0.000122)
		tmp = -1.0;
	elseif ((y <= 1.85e-39) || (!(y <= 3.3e-24) && (y <= 42000000.0)))
		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 <= -0.000122)
		tmp = -1.0;
	elseif ((y <= 1.85e-39) || (~((y <= 3.3e-24)) && (y <= 42000000.0)))
		tmp = 1.0 + (2.0 * (y / x));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -0.000122], -1.0, If[Or[LessEqual[y, 1.85e-39], And[N[Not[LessEqual[y, 3.3e-24]], $MachinePrecision], LessEqual[y, 42000000.0]]], N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.21999999999999997e-4 or 1.85000000000000007e-39 < y < 3.29999999999999984e-24 or 4.2e7 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 79.0%

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

   if -1.21999999999999997e-4 < y < 1.85000000000000007e-39 or 3.29999999999999984e-24 < y < 4.2e7

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 81.8%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.000122:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-39} \lor \neg \left(y \leq 3.3 \cdot 10^{-24}\right) \land y \leq 42000000:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.52 \cdot 10^{-6}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{-39}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-27}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 5000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.52e-6)
   -1.0
   (if (<= y 1.02e-39)
     1.0
     (if (<= y 2e-27) -1.0 (if (<= y 5000000.0) 1.0 -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.52e-6) {
		tmp = -1.0;
	} else if (y <= 1.02e-39) {
		tmp = 1.0;
	} else if (y <= 2e-27) {
		tmp = -1.0;
	} else if (y <= 5000000.0) {
		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 <= (-1.52d-6)) then
        tmp = -1.0d0
    else if (y <= 1.02d-39) then
        tmp = 1.0d0
    else if (y <= 2d-27) then
        tmp = -1.0d0
    else if (y <= 5000000.0d0) 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 <= -1.52e-6) {
		tmp = -1.0;
	} else if (y <= 1.02e-39) {
		tmp = 1.0;
	} else if (y <= 2e-27) {
		tmp = -1.0;
	} else if (y <= 5000000.0) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.52e-6:
		tmp = -1.0
	elif y <= 1.02e-39:
		tmp = 1.0
	elif y <= 2e-27:
		tmp = -1.0
	elif y <= 5000000.0:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.52e-6)
		tmp = -1.0;
	elseif (y <= 1.02e-39)
		tmp = 1.0;
	elseif (y <= 2e-27)
		tmp = -1.0;
	elseif (y <= 5000000.0)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.52e-6)
		tmp = -1.0;
	elseif (y <= 1.02e-39)
		tmp = 1.0;
	elseif (y <= 2e-27)
		tmp = -1.0;
	elseif (y <= 5000000.0)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.52e-6], -1.0, If[LessEqual[y, 1.02e-39], 1.0, If[LessEqual[y, 2e-27], -1.0, If[LessEqual[y, 5000000.0], 1.0, -1.0]]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.52000000000000006e-6 or 1.02000000000000007e-39 < y < 2.0000000000000001e-27 or 5e6 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 79.0%

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

   if -1.52000000000000006e-6 < y < 1.02000000000000007e-39 or 2.0000000000000001e-27 < y < 5e6

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 80.8%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.52 \cdot 10^{-6}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{-39}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-27}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 5000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
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: 50.1% 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 51.2%

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

4. Final simplification 51.2%

   \[\leadsto -1 \]
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 2024044 
(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)))