Linear.Projection:perspective from linear-1.19.1.3, B

Percentage Accurate: 77.7% → 99.8%

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: 77.7% 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: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+37} \lor \neg \left(y \leq 500000\right):\\ \;\;\;\;\frac{x \cdot 2}{\frac{x}{y} + -1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{y}{1 - \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.5e+37) (not (<= y 500000.0)))
   (/ (* x 2.0) (+ (/ x y) -1.0))
   (* 2.0 (/ y (- 1.0 (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.5e+37) || !(y <= 500000.0)) {
		tmp = (x * 2.0) / ((x / y) + -1.0);
	} else {
		tmp = 2.0 * (y / (1.0 - (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 <= (-3.5d+37)) .or. (.not. (y <= 500000.0d0))) then
        tmp = (x * 2.0d0) / ((x / y) + (-1.0d0))
    else
        tmp = 2.0d0 * (y / (1.0d0 - (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -3.5e+37) || !(y <= 500000.0)) {
		tmp = (x * 2.0) / ((x / y) + -1.0);
	} else {
		tmp = 2.0 * (y / (1.0 - (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.5e+37) or not (y <= 500000.0):
		tmp = (x * 2.0) / ((x / y) + -1.0)
	else:
		tmp = 2.0 * (y / (1.0 - (y / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.5e+37) || !(y <= 500000.0))
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(x / y) + -1.0));
	else
		tmp = Float64(2.0 * Float64(y / Float64(1.0 - Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -3.5e+37) || ~((y <= 500000.0)))
		tmp = (x * 2.0) / ((x / y) + -1.0);
	else
		tmp = 2.0 * (y / (1.0 - (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.5e+37], N[Not[LessEqual[y, 500000.0]], $MachinePrecision]], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(y / N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.5e37 or 5e5 < y

   1. Initial program 77.7%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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 99.9%

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

   if -3.5e37 < y < 5e5

   1. Initial program 79.3%

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

      1. *-commutative 79.3%

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

      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 100.0%

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

      6. *-inverses 100.0%

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

   3. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+37} \lor \neg \left(y \leq 500000\right):\\ \;\;\;\;\frac{x \cdot 2}{\frac{x}{y} + -1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{y}{1 - \frac{y}{x}}\\ \end{array} \]

Alternative 2: 93.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.05 \cdot 10^{-170} \lor \neg \left(x \leq 1.9 \cdot 10^{-107}\right):\\ \;\;\;\;2 \cdot \frac{y}{1 - \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.05e-170) (not (<= x 1.9e-107)))
   (* 2.0 (/ y (- 1.0 (/ y x))))
   (* x -2.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.05e-170) || !(x <= 1.9e-107)) {
		tmp = 2.0 * (y / (1.0 - (y / x)));
	} 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 <= (-3.05d-170)) .or. (.not. (x <= 1.9d-107))) then
        tmp = 2.0d0 * (y / (1.0d0 - (y / x)))
    else
        tmp = x * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -3.05e-170) || !(x <= 1.9e-107)) {
		tmp = 2.0 * (y / (1.0 - (y / x)));
	} else {
		tmp = x * -2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3.05e-170) or not (x <= 1.9e-107):
		tmp = 2.0 * (y / (1.0 - (y / x)))
	else:
		tmp = x * -2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.05e-170) || !(x <= 1.9e-107))
		tmp = Float64(2.0 * Float64(y / Float64(1.0 - Float64(y / x))));
	else
		tmp = Float64(x * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -3.05e-170) || ~((x <= 1.9e-107)))
		tmp = 2.0 * (y / (1.0 - (y / x)));
	else
		tmp = x * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.05e-170], N[Not[LessEqual[x, 1.9e-107]], $MachinePrecision]], N[(2.0 * N[(y / N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.05e-170 or 1.9000000000000001e-107 < x

   1. Initial program 82.8%

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

      1. *-commutative 82.8%

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

      2. associate-/l* 98.3%

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

      3. associate-/r* 98.3%

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

      4. associate-/r/ 98.3%

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

      5. div-sub 98.3%

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

      6. *-inverses 98.3%

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

   3. Simplified 98.3%

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

   if -3.05e-170 < x < 1.9000000000000001e-107

   1. Initial program 69.4%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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 88.8%

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

      1. *-commutative 88.8%

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

   6. Simplified 88.8%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.05 \cdot 10^{-170} \lor \neg \left(x \leq 1.9 \cdot 10^{-107}\right):\\ \;\;\;\;2 \cdot \frac{y}{1 - \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \]

Alternative 3: 73.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-58}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+92}:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.1e-58) (* y 2.0) (if (<= x 2.5e+92) (* x -2.0) (* y 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.1e-58) {
		tmp = y * 2.0;
	} else if (x <= 2.5e+92) {
		tmp = x * -2.0;
	} else {
		tmp = y * 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 <= (-2.1d-58)) then
        tmp = y * 2.0d0
    else if (x <= 2.5d+92) then
        tmp = x * (-2.0d0)
    else
        tmp = y * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.1e-58) {
		tmp = y * 2.0;
	} else if (x <= 2.5e+92) {
		tmp = x * -2.0;
	} else {
		tmp = y * 2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.1e-58:
		tmp = y * 2.0
	elif x <= 2.5e+92:
		tmp = x * -2.0
	else:
		tmp = y * 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.1e-58)
		tmp = Float64(y * 2.0);
	elseif (x <= 2.5e+92)
		tmp = Float64(x * -2.0);
	else
		tmp = Float64(y * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.1e-58)
		tmp = y * 2.0;
	elseif (x <= 2.5e+92)
		tmp = x * -2.0;
	else
		tmp = y * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.1e-58], N[(y * 2.0), $MachinePrecision], If[LessEqual[x, 2.5e+92], N[(x * -2.0), $MachinePrecision], N[(y * 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.09999999999999988e-58 or 2.50000000000000011e92 < x

   1. Initial program 80.3%

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

      1. *-commutative 80.3%

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

      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 100.0%

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

      6. *-inverses 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 84.0%

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

   if -2.09999999999999988e-58 < x < 2.50000000000000011e92

   1. Initial program 77.1%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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 78.4%

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

      1. *-commutative 78.4%

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

   6. Simplified 78.4%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-58}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+92}:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]

Alternative 4: 50.1% 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 78.6%

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

   1. associate-/l* 85.6%

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

3. Simplified 85.6%

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

4. Taylor expanded in x around 0 49.1%

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

   1. *-commutative 49.1%

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

6. Simplified 49.1%

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

7. Final simplification 49.1%

   \[\leadsto x \cdot -2 \]

Developer target: 99.5% 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 2023268 
(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)))