Linear.Projection:perspective from linear-1.19.1.3, B

Percentage Accurate: 77.4% → 99.7%

Time: 3.4s

Alternatives: 3

Speedup: 1.0×

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: 77.4% 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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-2}{\frac{-1}{y} + \frac{1}{x}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ -2.0 (+ (/ -1.0 y) (/ 1.0 x))))

C

double code(double x, double y) {
	return -2.0 / ((-1.0 / y) + (1.0 / x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (-2.0d0) / (((-1.0d0) / y) + (1.0d0 / x))
end function

Java

public static double code(double x, double y) {
	return -2.0 / ((-1.0 / y) + (1.0 / x));
}

Python

def code(x, y):
	return -2.0 / ((-1.0 / y) + (1.0 / x))

Julia

function code(x, y)
	return Float64(-2.0 / Float64(Float64(-1.0 / y) + Float64(1.0 / x)))
end

MATLAB

function tmp = code(x, y)
	tmp = -2.0 / ((-1.0 / y) + (1.0 / x));
end

Wolfram

code[x_, y_] := N[(-2.0 / N[(N[(-1.0 / y), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.1%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \frac{-2}{\frac{-1}{y} + \frac{1}{x}} \]

Alternative 2: 75.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+17} \lor \neg \left(x \leq 17\right):\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3e+17) (not (<= x 17.0))) (* y 2.0) (* -2.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3e+17) || !(x <= 17.0)) {
		tmp = y * 2.0;
	} else {
		tmp = -2.0 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-3d+17)) .or. (.not. (x <= 17.0d0))) then
        tmp = y * 2.0d0
    else
        tmp = (-2.0d0) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -3e+17) || !(x <= 17.0)) {
		tmp = y * 2.0;
	} else {
		tmp = -2.0 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3e+17) or not (x <= 17.0):
		tmp = y * 2.0
	else:
		tmp = -2.0 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3e+17) || !(x <= 17.0))
		tmp = Float64(y * 2.0);
	else
		tmp = Float64(-2.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -3e+17) || ~((x <= 17.0)))
		tmp = y * 2.0;
	else
		tmp = -2.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3e+17], N[Not[LessEqual[x, 17.0]], $MachinePrecision]], N[(y * 2.0), $MachinePrecision], N[(-2.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3e17 or 17 < x

   1. Initial program 76.3%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 78.7%

      \[\leadsto \color{blue}{2 \cdot y} \]

   if -3e17 < x < 17

   1. Initial program 77.9%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around inf 79.6%

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

      1. *-commutative 79.6%

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

   6. Simplified 79.6%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+17} \lor \neg \left(x \leq 17\right):\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot x\\ \end{array} \]

Alternative 3: 50.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ y \cdot 2 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

3. Simplified 99.7%

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

4. Taylor expanded in y around 0 50.7%

   \[\leadsto \color{blue}{2 \cdot y} \]

5. Final simplification 50.7%

   \[\leadsto y \cdot 2 \]

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 2023318 
(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)))