Linear.Projection:inversePerspective from linear-1.19.1.3, B

Percentage Accurate: 77.3% → 100.0%

Time: 10.4s

Alternatives: 5

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x - y}{\left(x \cdot 2\right) \cdot y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - y}{\left(x \cdot 2\right) \cdot y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{y} - \frac{0.5}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (/ 0.5 y) (/ 0.5 x)))

C

double code(double x, double y) {
	return (0.5 / y) - (0.5 / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (0.5d0 / y) - (0.5d0 / x)
end function

Java

public static double code(double x, double y) {
	return (0.5 / y) - (0.5 / x);
}

Python

def code(x, y):
	return (0.5 / y) - (0.5 / x)

Julia

function code(x, y)
	return Float64(Float64(0.5 / y) - Float64(0.5 / x))
end

MATLAB

function tmp = code(x, y)
	tmp = (0.5 / y) - (0.5 / x);
end

Wolfram

code[x_, y_] := N[(N[(0.5 / y), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.2%

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

   1. associate-/l/ N/A

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

   2. div-sub N/A

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

   3. *-inverses N/A

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

   4. sub-div N/A

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

   5. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{x}{\left(x \cdot 2\right) \cdot y}} - \frac{1}{x \cdot 2} \]

   6. --lowering--.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{\left(x \cdot 2\right) \cdot y} - \frac{1}{x \cdot 2}} \]

   7. associate-/r* N/A

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

   8. /-lowering-/.f64 N/A

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

   9. associate-/r* N/A

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

   10. *-inverses N/A

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

   11. metadata-eval N/A

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

   12. *-commutative N/A

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

   13. associate-/r* N/A

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

   14. *-inverses N/A

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

   15. associate-/r* N/A

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

   16. /-lowering-/.f64 N/A

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

   17. associate-/r* N/A

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

   18. *-inverses N/A

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

   19. metadata-eval 100.0

       \[\leadsto \frac{0.5}{y} - \frac{\color{blue}{0.5}}{x} \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{0.5}{y} - \frac{0.5}{x}} \]
5. Add Preprocessing

Alternative 2: 86.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{y \cdot \left(x \cdot 2\right)}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{-78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+298}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (* y (* x 2.0)))))
   (if (<= t_0 (- INFINITY))
     (/ 0.5 y)
     (if (<= t_0 -1e-78)
       t_0
       (if (<= t_0 0.0) (/ 0.5 y) (if (<= t_0 4e+298) t_0 (/ -0.5 x)))))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (y * (x * 2.0));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 0.5 / y;
	} else if (t_0 <= -1e-78) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = 0.5 / y;
	} else if (t_0 <= 4e+298) {
		tmp = t_0;
	} else {
		tmp = -0.5 / x;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (y * (x * 2.0));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = 0.5 / y;
	} else if (t_0 <= -1e-78) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = 0.5 / y;
	} else if (t_0 <= 4e+298) {
		tmp = t_0;
	} else {
		tmp = -0.5 / x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / (y * (x * 2.0))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = 0.5 / y
	elif t_0 <= -1e-78:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = 0.5 / y
	elif t_0 <= 4e+298:
		tmp = t_0
	else:
		tmp = -0.5 / x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(y * Float64(x * 2.0)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(0.5 / y);
	elseif (t_0 <= -1e-78)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(0.5 / y);
	elseif (t_0 <= 4e+298)
		tmp = t_0;
	else
		tmp = Float64(-0.5 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / (y * (x * 2.0));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = 0.5 / y;
	elseif (t_0 <= -1e-78)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = 0.5 / y;
	elseif (t_0 <= 4e+298)
		tmp = t_0;
	else
		tmp = -0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(y * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(0.5 / y), $MachinePrecision], If[LessEqual[t$95$0, -1e-78], t$95$0, If[LessEqual[t$95$0, 0.0], N[(0.5 / y), $MachinePrecision], If[LessEqual[t$95$0, 4e+298], t$95$0, N[(-0.5 / x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < -inf.0 or -9.99999999999999999e-79 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < 0.0

   1. Initial program 5.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{y}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 59.5

         \[\leadsto \color{blue}{\frac{0.5}{y}} \]

   5. Simplified 59.5%

      \[\leadsto \color{blue}{\frac{0.5}{y}} \]

   if -inf.0 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < -9.99999999999999999e-79 or 0.0 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < 3.9999999999999998e298

   1. Initial program 99.3%

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

   if 3.9999999999999998e298 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y))

   1. Initial program 7.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{x}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 51.7

         \[\leadsto \color{blue}{\frac{-0.5}{x}} \]

   5. Simplified 51.7%

      \[\leadsto \color{blue}{\frac{-0.5}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{y \cdot \left(x \cdot 2\right)} \leq -\infty:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;\frac{x - y}{y \cdot \left(x \cdot 2\right)} \leq -1 \cdot 10^{-78}:\\ \;\;\;\;\frac{x - y}{y \cdot \left(x \cdot 2\right)}\\ \mathbf{elif}\;\frac{x - y}{y \cdot \left(x \cdot 2\right)} \leq 0:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;\frac{x - y}{y \cdot \left(x \cdot 2\right)} \leq 4 \cdot 10^{+298}:\\ \;\;\;\;\frac{x - y}{y \cdot \left(x \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - y\right) \cdot \frac{0.5}{y \cdot x}\\ t_1 := \frac{x - y}{y \cdot \left(x \cdot 2\right)}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+305}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+298}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- x y) (/ 0.5 (* y x)))) (t_1 (/ (- x y) (* y (* x 2.0)))))
   (if (<= t_1 -1e+305)
     (/ 0.5 y)
     (if (<= t_1 -1e-78)
       t_0
       (if (<= t_1 0.0) (/ 0.5 y) (if (<= t_1 4e+298) t_0 (/ -0.5 x)))))))

C

double code(double x, double y) {
	double t_0 = (x - y) * (0.5 / (y * x));
	double t_1 = (x - y) / (y * (x * 2.0));
	double tmp;
	if (t_1 <= -1e+305) {
		tmp = 0.5 / y;
	} else if (t_1 <= -1e-78) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = 0.5 / y;
	} else if (t_1 <= 4e+298) {
		tmp = t_0;
	} else {
		tmp = -0.5 / x;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (x - y) * (0.5d0 / (y * x))
    t_1 = (x - y) / (y * (x * 2.0d0))
    if (t_1 <= (-1d+305)) then
        tmp = 0.5d0 / y
    else if (t_1 <= (-1d-78)) then
        tmp = t_0
    else if (t_1 <= 0.0d0) then
        tmp = 0.5d0 / y
    else if (t_1 <= 4d+298) then
        tmp = t_0
    else
        tmp = (-0.5d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) * (0.5 / (y * x));
	double t_1 = (x - y) / (y * (x * 2.0));
	double tmp;
	if (t_1 <= -1e+305) {
		tmp = 0.5 / y;
	} else if (t_1 <= -1e-78) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = 0.5 / y;
	} else if (t_1 <= 4e+298) {
		tmp = t_0;
	} else {
		tmp = -0.5 / x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) * (0.5 / (y * x))
	t_1 = (x - y) / (y * (x * 2.0))
	tmp = 0
	if t_1 <= -1e+305:
		tmp = 0.5 / y
	elif t_1 <= -1e-78:
		tmp = t_0
	elif t_1 <= 0.0:
		tmp = 0.5 / y
	elif t_1 <= 4e+298:
		tmp = t_0
	else:
		tmp = -0.5 / x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) * Float64(0.5 / Float64(y * x)))
	t_1 = Float64(Float64(x - y) / Float64(y * Float64(x * 2.0)))
	tmp = 0.0
	if (t_1 <= -1e+305)
		tmp = Float64(0.5 / y);
	elseif (t_1 <= -1e-78)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(0.5 / y);
	elseif (t_1 <= 4e+298)
		tmp = t_0;
	else
		tmp = Float64(-0.5 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) * (0.5 / (y * x));
	t_1 = (x - y) / (y * (x * 2.0));
	tmp = 0.0;
	if (t_1 <= -1e+305)
		tmp = 0.5 / y;
	elseif (t_1 <= -1e-78)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = 0.5 / y;
	elseif (t_1 <= 4e+298)
		tmp = t_0;
	else
		tmp = -0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] * N[(0.5 / N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(y * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+305], N[(0.5 / y), $MachinePrecision], If[LessEqual[t$95$1, -1e-78], t$95$0, If[LessEqual[t$95$1, 0.0], N[(0.5 / y), $MachinePrecision], If[LessEqual[t$95$1, 4e+298], t$95$0, N[(-0.5 / x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < -9.9999999999999994e304 or -9.99999999999999999e-79 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < 0.0

   1. Initial program 8.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{y}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 60.4

         \[\leadsto \color{blue}{\frac{0.5}{y}} \]

   5. Simplified 60.4%

      \[\leadsto \color{blue}{\frac{0.5}{y}} \]

   if -9.9999999999999994e304 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < -9.99999999999999999e-79 or 0.0 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < 3.9999999999999998e298

   1. Initial program 99.3%

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

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(x \cdot 2\right) \cdot y}{x - y}}} \]

      2. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{1}{\left(x \cdot 2\right) \cdot y} \cdot \left(x - y\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\left(x \cdot 2\right) \cdot y} \cdot \left(x - y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot x\right)} \cdot y} \cdot \left(x - y\right) \]

      5. associate-*l* N/A

         \[\leadsto \frac{1}{\color{blue}{2 \cdot \left(x \cdot y\right)}} \cdot \left(x - y\right) \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{x \cdot y}} \cdot \left(x - y\right) \]

      7. *-inverses N/A

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

      8. associate-/r* N/A

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

      9. /-lowering-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. *-inverses N/A

          \[\leadsto \frac{\frac{\color{blue}{1}}{2}}{x \cdot y} \cdot \left(x - y\right) \]

      12. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{2}}}{x \cdot y} \cdot \left(x - y\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{1}{2}}{\color{blue}{x \cdot y}} \cdot \left(x - y\right) \]

      14. --lowering--.f64 99.0

          \[\leadsto \frac{0.5}{x \cdot y} \cdot \color{blue}{\left(x - y\right)} \]

   4. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\frac{0.5}{x \cdot y} \cdot \left(x - y\right)} \]

   if 3.9999999999999998e298 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y))

   1. Initial program 7.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{x}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 51.7

         \[\leadsto \color{blue}{\frac{-0.5}{x}} \]

   5. Simplified 51.7%

      \[\leadsto \color{blue}{\frac{-0.5}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{y \cdot \left(x \cdot 2\right)} \leq -1 \cdot 10^{+305}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;\frac{x - y}{y \cdot \left(x \cdot 2\right)} \leq -1 \cdot 10^{-78}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{0.5}{y \cdot x}\\ \mathbf{elif}\;\frac{x - y}{y \cdot \left(x \cdot 2\right)} \leq 0:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;\frac{x - y}{y \cdot \left(x \cdot 2\right)} \leq 4 \cdot 10^{+298}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{0.5}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+84}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;x \leq 3.15 \cdot 10^{+29}:\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7e+84) (/ 0.5 y) (if (<= x 3.15e+29) (/ -0.5 x) (/ 0.5 y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7e+84) {
		tmp = 0.5 / y;
	} else if (x <= 3.15e+29) {
		tmp = -0.5 / x;
	} else {
		tmp = 0.5 / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7d+84)) then
        tmp = 0.5d0 / y
    else if (x <= 3.15d+29) then
        tmp = (-0.5d0) / x
    else
        tmp = 0.5d0 / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -7e+84) {
		tmp = 0.5 / y;
	} else if (x <= 3.15e+29) {
		tmp = -0.5 / x;
	} else {
		tmp = 0.5 / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7e+84:
		tmp = 0.5 / y
	elif x <= 3.15e+29:
		tmp = -0.5 / x
	else:
		tmp = 0.5 / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7e+84)
		tmp = Float64(0.5 / y);
	elseif (x <= 3.15e+29)
		tmp = Float64(-0.5 / x);
	else
		tmp = Float64(0.5 / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7e+84)
		tmp = 0.5 / y;
	elseif (x <= 3.15e+29)
		tmp = -0.5 / x;
	else
		tmp = 0.5 / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7e+84], N[(0.5 / y), $MachinePrecision], If[LessEqual[x, 3.15e+29], N[(-0.5 / x), $MachinePrecision], N[(0.5 / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.9999999999999998e84 or 3.1499999999999999e29 < x

   1. Initial program 71.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{y}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 86.9

         \[\leadsto \color{blue}{\frac{0.5}{y}} \]

   5. Simplified 86.9%

      \[\leadsto \color{blue}{\frac{0.5}{y}} \]

   if -6.9999999999999998e84 < x < 3.1499999999999999e29

   1. Initial program 82.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{x}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 72.2

         \[\leadsto \color{blue}{\frac{-0.5}{x}} \]

   5. Simplified 72.2%

      \[\leadsto \color{blue}{\frac{-0.5}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 50.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ -0.5 x))

C

double code(double x, double y) {
	return -0.5 / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (-0.5d0) / x
end function

Java

public static double code(double x, double y) {
	return -0.5 / x;
}

Python

def code(x, y):
	return -0.5 / x

Julia

function code(x, y)
	return Float64(-0.5 / x)
end

MATLAB

function tmp = code(x, y)
	tmp = -0.5 / x;
end

Wolfram

code[x_, y_] := N[(-0.5 / x), $MachinePrecision]

Derivation

1. Initial program 78.2%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{x}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 49.2

      \[\leadsto \color{blue}{\frac{-0.5}{x}} \]

5. Simplified 49.2%

   \[\leadsto \color{blue}{\frac{-0.5}{x}} \]
6. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{y} - \frac{0.5}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (/ 0.5 y) (/ 0.5 x)))

C

double code(double x, double y) {
	return (0.5 / y) - (0.5 / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (0.5d0 / y) - (0.5d0 / x)
end function

Java

public static double code(double x, double y) {
	return (0.5 / y) - (0.5 / x);
}

Python

def code(x, y):
	return (0.5 / y) - (0.5 / x)

Julia

function code(x, y)
	return Float64(Float64(0.5 / y) - Float64(0.5 / x))
end

MATLAB

function tmp = code(x, y)
	tmp = (0.5 / y) - (0.5 / x);
end

Wolfram

code[x_, y_] := N[(N[(0.5 / y), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024195 
(FPCore (x y)
  :name "Linear.Projection:inversePerspective from linear-1.19.1.3, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (/ 1/2 y) (/ 1/2 x)))

  (/ (- x y) (* (* x 2.0) y)))