Linear.Projection:perspective from linear-1.19.1.3, B

Percentage Accurate: 77.3% → 99.0%

Time: 2.7s

Alternatives: 4

Speedup: 0.7×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot 2\right) \cdot y}{x - y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (* x 2.0) y) (- x y)))

C

double code(double x, double y) {
	return ((x * 2.0) * y) / (x - y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * 2.0d0) * y) / (x - y)
end function

Java

public static double code(double x, double y) {
	return ((x * 2.0) * y) / (x - y);
}

Python

def code(x, y):
	return ((x * 2.0) * y) / (x - y)

Julia

function code(x, y)
	return Float64(Float64(Float64(x * 2.0) * y) / Float64(x - y))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * 2.0) * y) / (x - y);
end

Wolfram

code[x_, y_] := N[(N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]

Initial Program: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot 2\right) \cdot y}{x - y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (* x 2.0) y) (- x y)))

C

double code(double x, double y) {
	return ((x * 2.0) * y) / (x - y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * 2.0d0) * y) / (x - y)
end function

Java

public static double code(double x, double y) {
	return ((x * 2.0) * y) / (x - y);
}

Python

def code(x, y):
	return ((x * 2.0) * y) / (x - y)

Julia

function code(x, y)
	return Float64(Float64(Float64(x * 2.0) * y) / Float64(x - y))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * 2.0) * y) / (x - y);
end

Wolfram

code[x_, y_] := N[(N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{+106} \lor \neg \left(x \leq 1.36 \cdot 10^{-50}\right):\\ \;\;\;\;\frac{y}{1 - \frac{y}{x}} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -6.6e+106) (not (<= x 1.36e-50)))
   (* (/ y (- 1.0 (/ y x))) 2.0)
   (/ (* x 2.0) (/ (- x y) y))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -6.6e+106) || !(x <= 1.36e-50)) {
		tmp = (y / (1.0 - (y / x))) * 2.0;
	} else {
		tmp = (x * 2.0) / ((x - y) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-6.6d+106)) .or. (.not. (x <= 1.36d-50))) then
        tmp = (y / (1.0d0 - (y / x))) * 2.0d0
    else
        tmp = (x * 2.0d0) / ((x - y) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -6.6e+106) || !(x <= 1.36e-50)) {
		tmp = (y / (1.0 - (y / x))) * 2.0;
	} else {
		tmp = (x * 2.0) / ((x - y) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -6.6e+106) or not (x <= 1.36e-50):
		tmp = (y / (1.0 - (y / x))) * 2.0
	else:
		tmp = (x * 2.0) / ((x - y) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -6.6e+106) || !(x <= 1.36e-50))
		tmp = Float64(Float64(y / Float64(1.0 - Float64(y / x))) * 2.0);
	else
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(x - y) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -6.6e+106) || ~((x <= 1.36e-50)))
		tmp = (y / (1.0 - (y / x))) * 2.0;
	else
		tmp = (x * 2.0) / ((x - y) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -6.6e+106], N[Not[LessEqual[x, 1.36e-50]], $MachinePrecision]], N[(N[(y / N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.60000000000000015e106 or 1.36000000000000001e-50 < x

   1. Initial program 76.1%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Step-by-step derivation

      1. *-commutative 76.1%

         \[\leadsto \frac{\color{blue}{y \cdot \left(x \cdot 2\right)}}{x - y} \]

      2. associate-/l* 99.9%

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

      3. associate-/r* 99.9%

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

      4. associate-/r/ 99.9%

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

      5. div-sub 99.9%

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

      6. *-inverses 99.9%

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

   3. Simplified 99.9%

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

   if -6.60000000000000015e106 < x < 1.36000000000000001e-50

   1. Initial program 82.8%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Step-by-step derivation

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{+106} \lor \neg \left(x \leq 1.36 \cdot 10^{-50}\right):\\ \;\;\;\;\frac{y}{1 - \frac{y}{x}} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \end{array} \]

Alternative 2: 92.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-212} \lor \neg \left(x \leq 1.6 \cdot 10^{-228}\right):\\ \;\;\;\;\frac{y}{1 - \frac{y}{x}} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.5e-212) (not (<= x 1.6e-228)))
   (* (/ y (- 1.0 (/ y x))) 2.0)
   (* x -2.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.5e-212) || !(x <= 1.6e-228)) {
		tmp = (y / (1.0 - (y / x))) * 2.0;
	} else {
		tmp = x * -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.5d-212)) .or. (.not. (x <= 1.6d-228))) then
        tmp = (y / (1.0d0 - (y / x))) * 2.0d0
    else
        tmp = x * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.5e-212) || !(x <= 1.6e-228)) {
		tmp = (y / (1.0 - (y / x))) * 2.0;
	} else {
		tmp = x * -2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.5e-212) or not (x <= 1.6e-228):
		tmp = (y / (1.0 - (y / x))) * 2.0
	else:
		tmp = x * -2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.5e-212) || !(x <= 1.6e-228))
		tmp = Float64(Float64(y / Float64(1.0 - Float64(y / x))) * 2.0);
	else
		tmp = Float64(x * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.5e-212) || ~((x <= 1.6e-228)))
		tmp = (y / (1.0 - (y / x))) * 2.0;
	else
		tmp = x * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.5e-212], N[Not[LessEqual[x, 1.6e-228]], $MachinePrecision]], N[(N[(y / N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(x * -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5000000000000001e-212 or 1.60000000000000011e-228 < x

   1. Initial program 80.0%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Step-by-step derivation

      1. *-commutative 80.0%

         \[\leadsto \frac{\color{blue}{y \cdot \left(x \cdot 2\right)}}{x - y} \]

      2. associate-/l* 94.8%

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

      3. associate-/r* 94.8%

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

      4. associate-/r/ 94.8%

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

      5. div-sub 94.8%

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

      6. *-inverses 94.8%

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

   3. Simplified 94.8%

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

   if -1.5000000000000001e-212 < x < 1.60000000000000011e-228

   1. Initial program 77.2%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-2 \cdot x} \]
   5. Step-by-step derivation

      1. *-commutative 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-212} \lor \neg \left(x \leq 1.6 \cdot 10^{-228}\right):\\ \;\;\;\;\frac{y}{1 - \frac{y}{x}} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \]

Alternative 3: 74.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-80}:\\ \;\;\;\;x \cdot -2\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{+72}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.05e-80) (* x -2.0) (if (<= y 2.45e+72) (* y 2.0) (* x -2.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.05e-80) {
		tmp = x * -2.0;
	} else if (y <= 2.45e+72) {
		tmp = y * 2.0;
	} else {
		tmp = x * -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.05d-80)) then
        tmp = x * (-2.0d0)
    else if (y <= 2.45d+72) then
        tmp = y * 2.0d0
    else
        tmp = x * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.05e-80) {
		tmp = x * -2.0;
	} else if (y <= 2.45e+72) {
		tmp = y * 2.0;
	} else {
		tmp = x * -2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.05e-80:
		tmp = x * -2.0
	elif y <= 2.45e+72:
		tmp = y * 2.0
	else:
		tmp = x * -2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.05e-80)
		tmp = Float64(x * -2.0);
	elseif (y <= 2.45e+72)
		tmp = Float64(y * 2.0);
	else
		tmp = Float64(x * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.05e-80)
		tmp = x * -2.0;
	elseif (y <= 2.45e+72)
		tmp = y * 2.0;
	else
		tmp = x * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.05e-80], N[(x * -2.0), $MachinePrecision], If[LessEqual[y, 2.45e+72], N[(y * 2.0), $MachinePrecision], N[(x * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.05000000000000001e-80 or 2.45000000000000003e72 < y

   1. Initial program 76.1%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Step-by-step derivation

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 76.8%

      \[\leadsto \color{blue}{-2 \cdot x} \]
   5. Step-by-step derivation

      1. *-commutative 76.8%

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

   6. Simplified 76.8%

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

   if -1.05000000000000001e-80 < y < 2.45000000000000003e72

   1. Initial program 82.9%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Step-by-step derivation

      1. *-commutative 82.9%

         \[\leadsto \frac{\color{blue}{y \cdot \left(x \cdot 2\right)}}{x - y} \]

      2. associate-/l* 99.9%

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

      3. associate-/r* 99.9%

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

      4. associate-/r/ 99.9%

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

      5. div-sub 99.9%

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

      6. *-inverses 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 80.7%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-80}:\\ \;\;\;\;x \cdot -2\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{+72}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \]

Alternative 4: 50.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ x \cdot -2 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x -2.0))

C

double code(double x, double y) {
	return x * -2.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (-2.0d0)
end function

Java

public static double code(double x, double y) {
	return x * -2.0;
}

Python

def code(x, y):
	return x * -2.0

Julia

function code(x, y)
	return Float64(x * -2.0)
end

MATLAB

function tmp = code(x, y)
	tmp = x * -2.0;
end

Wolfram

code[x_, y_] := N[(x * -2.0), $MachinePrecision]

Derivation

1. Initial program 79.7%

   \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
2. Step-by-step derivation

   1. associate-/l* 86.8%

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

3. Simplified 86.8%

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

4. Taylor expanded in x around 0 47.8%

   \[\leadsto \color{blue}{-2 \cdot x} \]
5. Step-by-step derivation

   1. *-commutative 47.8%

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

6. Simplified 47.8%

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

7. Final simplification 47.8%

   \[\leadsto x \cdot -2 \]

Developer target: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2 \cdot x}{x - y} \cdot y\\ \mathbf{if}\;x < -1.7210442634149447 \cdot 10^{+81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x < 83645045635564430:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ (* 2.0 x) (- x y)) y)))
   (if (< x -1.7210442634149447e+81)
     t_0
     (if (< x 83645045635564430.0) (/ (* x 2.0) (/ (- x y) y)) t_0))))

C

double code(double x, double y) {
	double t_0 = ((2.0 * x) / (x - y)) * y;
	double tmp;
	if (x < -1.7210442634149447e+81) {
		tmp = t_0;
	} else if (x < 83645045635564430.0) {
		tmp = (x * 2.0) / ((x - y) / y);
	} 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 = ((2.0d0 * x) / (x - y)) * y
    if (x < (-1.7210442634149447d+81)) then
        tmp = t_0
    else if (x < 83645045635564430.0d0) then
        tmp = (x * 2.0d0) / ((x - y) / y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((2.0 * x) / (x - y)) * y;
	double tmp;
	if (x < -1.7210442634149447e+81) {
		tmp = t_0;
	} else if (x < 83645045635564430.0) {
		tmp = (x * 2.0) / ((x - y) / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((2.0 * x) / (x - y)) * y
	tmp = 0
	if x < -1.7210442634149447e+81:
		tmp = t_0
	elif x < 83645045635564430.0:
		tmp = (x * 2.0) / ((x - y) / y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(2.0 * x) / Float64(x - y)) * y)
	tmp = 0.0
	if (x < -1.7210442634149447e+81)
		tmp = t_0;
	elseif (x < 83645045635564430.0)
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(x - y) / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((2.0 * x) / (x - y)) * y;
	tmp = 0.0;
	if (x < -1.7210442634149447e+81)
		tmp = t_0;
	elseif (x < 83645045635564430.0)
		tmp = (x * 2.0) / ((x - y) / y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(2.0 * x), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[Less[x, -1.7210442634149447e+81], t$95$0, If[Less[x, 83645045635564430.0], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2023290 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, B"
  :precision binary64

  :herbie-target
  (if (< x -1.7210442634149447e+81) (* (/ (* 2.0 x) (- x y)) y) (if (< x 83645045635564430.0) (/ (* x 2.0) (/ (- x y) y)) (* (/ (* 2.0 x) (- x y)) y)))

  (/ (* (* x 2.0) y) (- x y)))