Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 100.0%

Time: 3.5s

Alternatives: 5

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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x - y}\\ t_1 := \frac{y}{x - y}\\ \frac{t_0 \cdot t_0 - t_1 \cdot t_1}{t_0 - t_1} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (- x y))) (t_1 (/ y (- x y))))
   (/ (- (* t_0 t_0) (* t_1 t_1)) (- t_0 t_1))))

C

double code(double x, double y) {
	double t_0 = x / (x - y);
	double t_1 = y / (x - y);
	return ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    t_0 = x / (x - y)
    t_1 = y / (x - y)
    code = ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1)
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x - y);
	double t_1 = y / (x - y);
	return ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1);
}

Python

def code(x, y):
	t_0 = x / (x - y)
	t_1 = y / (x - y)
	return ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1)

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x - y))
	t_1 = Float64(y / Float64(x - y))
	return Float64(Float64(Float64(t_0 * t_0) - Float64(t_1 * t_1)) / Float64(t_0 - t_1))
end

MATLAB

function tmp = code(x, y)
	t_0 = x / (x - y);
	t_1 = y / (x - y);
	tmp = ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 99.9%

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

   1. add-cbrt-cube 99.9%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{x + y}{x - y} \cdot \frac{x + y}{x - y}\right) \cdot \frac{x + y}{x - y}}} \]

   2. pow3 100.0%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{x + y}{x - y}\right)}^{3}}} \]

3. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{x + y}{x - y}\right)}^{3}}} \]
4. Step-by-step derivation

   1. rem-cbrt-cube 99.9%

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

   2. *-un-lft-identity 99.9%

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

   3. associate-*l/ 99.7%

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

   4. distribute-lft-in 99.7%

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

   5. flip-+ 99.7%

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

   6. associate-*l/ 99.8%

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

   7. *-un-lft-identity 99.8%

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

   8. associate-*l/ 99.7%

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

   9. *-un-lft-identity 99.7%

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

   10. associate-*l/ 99.9%

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

   11. *-un-lft-identity 99.9%

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

   12. associate-*l/ 99.7%

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

   13. *-un-lft-identity 99.7%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

   \[\leadsto \frac{\frac{x}{x - y} \cdot \frac{x}{x - y} - \frac{y}{x - y} \cdot \frac{y}{x - y}}{\frac{x}{x - y} - \frac{y}{x - y}} \]

Alternative 2: 73.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-7} \lor \neg \left(x \leq 4.9 \cdot 10^{+58}\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -4.6e-7) (not (<= x 4.9e+58))) (+ 1.0 (* 2.0 (/ y x))) -1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -4.6e-7) || !(x <= 4.9e+58)) {
		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 <= (-4.6d-7)) .or. (.not. (x <= 4.9d+58))) 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 <= -4.6e-7) || !(x <= 4.9e+58)) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -4.6e-7) or not (x <= 4.9e+58):
		tmp = 1.0 + (2.0 * (y / x))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -4.6e-7) || !(x <= 4.9e+58))
		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 <= -4.6e-7) || ~((x <= 4.9e+58)))
		tmp = 1.0 + (2.0 * (y / x));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -4.6e-7], N[Not[LessEqual[x, 4.9e+58]], $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 < -4.5999999999999999e-7 or 4.90000000000000018e58 < x

   1. Initial program 99.9%

      \[\frac{x + y}{x - y} \]

   2. Taylor expanded in y around 0 78.9%

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

   if -4.5999999999999999e-7 < x < 4.90000000000000018e58

   1. Initial program 99.9%

      \[\frac{x + y}{x - y} \]

   2. Taylor expanded in x around 0 75.6%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-7} \lor \neg \left(x \leq 4.9 \cdot 10^{+58}\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3: 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 99.9%

   \[\frac{x + y}{x - y} \]

2. Final simplification 99.9%

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

Alternative 4: 72.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+33}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+59}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5e+33) 1.0 (if (<= x 1.4e+59) -1.0 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5e+33) {
		tmp = 1.0;
	} else if (x <= 1.4e+59) {
		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 <= (-5d+33)) then
        tmp = 1.0d0
    else if (x <= 1.4d+59) 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 <= -5e+33) {
		tmp = 1.0;
	} else if (x <= 1.4e+59) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5e+33:
		tmp = 1.0
	elif x <= 1.4e+59:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5e+33)
		tmp = 1.0;
	elseif (x <= 1.4e+59)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5e+33)
		tmp = 1.0;
	elseif (x <= 1.4e+59)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5e+33], 1.0, If[LessEqual[x, 1.4e+59], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -4.99999999999999973e33 or 1.3999999999999999e59 < x

   1. Initial program 99.9%

      \[\frac{x + y}{x - y} \]

   2. Taylor expanded in x around inf 80.1%

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

   if -4.99999999999999973e33 < x < 1.3999999999999999e59

   1. Initial program 99.9%

      \[\frac{x + y}{x - y} \]

   2. Taylor expanded in x around 0 74.3%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+33}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+59}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 50.0% 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 99.9%

   \[\frac{x + y}{x - y} \]

2. Taylor expanded in x around 0 53.8%

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

3. Final simplification 53.8%

   \[\leadsto -1 \]

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 2023238 
(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)))