Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 100.0%

Time: 7.6s

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, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. log1p-expm1-u 99.9%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x + y}{x - y}\right)\right)} \]

3. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x + y}{x - y}\right)\right)} \]
4. Step-by-step derivation

   1. log1p-expm1 99.9%

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

   2. div-inv 99.7%

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

   3. *-commutative 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. +-commutative 99.7%

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

   6. associate-*l/ 99.8%

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

   7. *-un-lft-identity 99.8%

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

   8. associate-*l/ 100.0%

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

   9. *-un-lft-identity 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 74.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+76} \lor \neg \left(y \leq 8 \cdot 10^{-35}\right):\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot 2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.4e+76) (not (<= y 8e-35)))
   (+ (* -2.0 (/ x y)) -1.0)
   (+ 1.0 (/ (* y 2.0) x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.4e+76) || !(y <= 8e-35)) {
		tmp = (-2.0 * (x / y)) + -1.0;
	} else {
		tmp = 1.0 + ((y * 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 ((y <= (-3.4d+76)) .or. (.not. (y <= 8d-35))) then
        tmp = ((-2.0d0) * (x / y)) + (-1.0d0)
    else
        tmp = 1.0d0 + ((y * 2.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -3.4e+76) || !(y <= 8e-35)) {
		tmp = (-2.0 * (x / y)) + -1.0;
	} else {
		tmp = 1.0 + ((y * 2.0) / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.4e+76) or not (y <= 8e-35):
		tmp = (-2.0 * (x / y)) + -1.0
	else:
		tmp = 1.0 + ((y * 2.0) / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.4e+76) || !(y <= 8e-35))
		tmp = Float64(Float64(-2.0 * Float64(x / y)) + -1.0);
	else
		tmp = Float64(1.0 + Float64(Float64(y * 2.0) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -3.4e+76) || ~((y <= 8e-35)))
		tmp = (-2.0 * (x / y)) + -1.0;
	else
		tmp = 1.0 + ((y * 2.0) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.4e+76], N[Not[LessEqual[y, 8e-35]], $MachinePrecision]], N[(N[(-2.0 * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 + N[(N[(y * 2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.3999999999999997e76 or 8.00000000000000006e-35 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 82.8%

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

   if -3.3999999999999997e76 < y < 8.00000000000000006e-35

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 82.7%

      \[\leadsto \color{blue}{1 + 2 \cdot \frac{y}{x}} \]
   3. Step-by-step derivation

      1. associate-*r/ 82.7%

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

   4. Simplified 82.7%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+76} \lor \neg \left(y \leq 8 \cdot 10^{-35}\right):\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot 2}{x}\\ \end{array} \]

Alternative 3: 73.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+76}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-36}:\\ \;\;\;\;1 + \frac{y \cdot 2}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.5e+76) -1.0 (if (<= y 1.55e-36) (+ 1.0 (/ (* y 2.0) x)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.5e+76) {
		tmp = -1.0;
	} else if (y <= 1.55e-36) {
		tmp = 1.0 + ((y * 2.0) / 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 <= (-6.5d+76)) then
        tmp = -1.0d0
    else if (y <= 1.55d-36) then
        tmp = 1.0d0 + ((y * 2.0d0) / x)
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -6.5e+76) {
		tmp = -1.0;
	} else if (y <= 1.55e-36) {
		tmp = 1.0 + ((y * 2.0) / x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6.5e+76:
		tmp = -1.0
	elif y <= 1.55e-36:
		tmp = 1.0 + ((y * 2.0) / x)
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6.5e+76)
		tmp = -1.0;
	elseif (y <= 1.55e-36)
		tmp = Float64(1.0 + Float64(Float64(y * 2.0) / x));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -6.5e+76)
		tmp = -1.0;
	elseif (y <= 1.55e-36)
		tmp = 1.0 + ((y * 2.0) / x);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6.5e+76], -1.0, If[LessEqual[y, 1.55e-36], N[(1.0 + N[(N[(y * 2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -6.5000000000000005e76 or 1.5499999999999999e-36 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 81.6%

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

   if -6.5000000000000005e76 < y < 1.5499999999999999e-36

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 82.7%

      \[\leadsto \color{blue}{1 + 2 \cdot \frac{y}{x}} \]
   3. Step-by-step derivation

      1. associate-*r/ 82.7%

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

   4. Simplified 82.7%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+76}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-36}:\\ \;\;\;\;1 + \frac{y \cdot 2}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 5: 73.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+76}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{-35}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.5e+76) -1.0 (if (<= y 4.7e-35) 1.0 -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.5e+76) {
		tmp = -1.0;
	} else if (y <= 4.7e-35) {
		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 <= (-1.5d+76)) then
        tmp = -1.0d0
    else if (y <= 4.7d-35) 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 <= -1.5e+76) {
		tmp = -1.0;
	} else if (y <= 4.7e-35) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.5e+76:
		tmp = -1.0
	elif y <= 4.7e-35:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.5e+76)
		tmp = -1.0;
	elseif (y <= 4.7e-35)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.5e+76)
		tmp = -1.0;
	elseif (y <= 4.7e-35)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.5e+76], -1.0, If[LessEqual[y, 4.7e-35], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4999999999999999e76 or 4.7e-35 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 81.6%

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

   if -1.4999999999999999e76 < y < 4.7e-35

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 81.2%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+76}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{-35}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 6: 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 47.0%

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

3. Final simplification 47.0%

   \[\leadsto -1 \]

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 2023301 
(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)))