Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 100.0%

Time: 9.9s

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, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. clear-num 99.9%

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

   2. associate-/r/ 99.7%

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

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{1}{x - y} \cdot \left(x + y\right)} \]
5. Step-by-step derivation

   1. +-commutative 99.7%

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

   2. distribute-rgt-in 99.7%

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

   3. un-div-inv 99.8%

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

   4. un-div-inv 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{y}{x - y} + \frac{x}{x - y}} \]
7. Step-by-step derivation

   1. clear-num 99.9%

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

   2. inv-pow 99.9%

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

8. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{{\left(\frac{x - y}{y}\right)}^{-1}} + \frac{x}{x - y} \]
9. Step-by-step derivation

   1. unpow-1 99.9%

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

   2. div-sub 100.0%

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

   3. sub-neg 100.0%

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

   4. *-inverses 100.0%

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

   5. metadata-eval 100.0%

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

10. Simplified 100.0%

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

11. Final simplification 100.0%

    \[\leadsto \frac{1}{\frac{x}{y} + -1} + \frac{x}{x - y} \]
12. Add Preprocessing

Alternative 2: 72.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+133}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.9e-9) 1.0 (if (<= x 1.85e+133) (+ -1.0 (* (/ x y) -2.0)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.9e-9) {
		tmp = 1.0;
	} else if (x <= 1.85e+133) {
		tmp = -1.0 + ((x / y) * -2.0);
	} 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 <= (-5.9d-9)) then
        tmp = 1.0d0
    else if (x <= 1.85d+133) then
        tmp = (-1.0d0) + ((x / y) * (-2.0d0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.9e-9) {
		tmp = 1.0;
	} else if (x <= 1.85e+133) {
		tmp = -1.0 + ((x / y) * -2.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.9e-9:
		tmp = 1.0
	elif x <= 1.85e+133:
		tmp = -1.0 + ((x / y) * -2.0)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.9e-9)
		tmp = 1.0;
	elseif (x <= 1.85e+133)
		tmp = Float64(-1.0 + Float64(Float64(x / y) * -2.0));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.9e-9)
		tmp = 1.0;
	elseif (x <= 1.85e+133)
		tmp = -1.0 + ((x / y) * -2.0);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.9e-9], 1.0, If[LessEqual[x, 1.85e+133], N[(-1.0 + N[(N[(x / y), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -5.8999999999999999e-9 or 1.85000000000000012e133 < x

   1. Initial program 99.9%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 77.0%

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

   if -5.8999999999999999e-9 < x < 1.85000000000000012e133

   1. Initial program 99.9%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 75.5%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+133}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+133}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.45e-9) 1.0 (if (<= x 1.8e+133) -1.0 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.45e-9) {
		tmp = 1.0;
	} else if (x <= 1.8e+133) {
		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 <= (-1.45d-9)) then
        tmp = 1.0d0
    else if (x <= 1.8d+133) 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 <= -1.45e-9) {
		tmp = 1.0;
	} else if (x <= 1.8e+133) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.45e-9:
		tmp = 1.0
	elif x <= 1.8e+133:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.45e-9)
		tmp = 1.0;
	elseif (x <= 1.8e+133)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.45e-9)
		tmp = 1.0;
	elseif (x <= 1.8e+133)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.45e-9], 1.0, If[LessEqual[x, 1.8e+133], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1.44999999999999996e-9 or 1.79999999999999989e133 < x

   1. Initial program 99.9%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 77.0%

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

   if -1.44999999999999996e-9 < x < 1.79999999999999989e133

   1. Initial program 99.9%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 73.9%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+133}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 100.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. clear-num 99.9%

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

   2. associate-/r/ 99.7%

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

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{1}{x - y} \cdot \left(x + y\right)} \]
5. Step-by-step derivation

   1. +-commutative 99.7%

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

   2. distribute-rgt-in 99.7%

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

   3. un-div-inv 99.8%

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

   4. un-div-inv 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

   \[\leadsto \frac{x}{x - y} + \frac{y}{x - y} \]
8. Add Preprocessing

Alternative 5: 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. Add Preprocessing

3. Final simplification 99.9%

   \[\leadsto \frac{x + y}{x - y} \]
4. Add Preprocessing

Alternative 6: 49.1% 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. Add Preprocessing

3. Taylor expanded in x around 0 52.1%

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

4. Final simplification 52.1%

   \[\leadsto -1 \]
5. Add Preprocessing

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 2024067 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (/ 1.0 (- (/ x (+ x y)) (/ y (+ x y))))

  (/ (+ x y) (- x y)))