Linear.Projection:perspective from linear-1.19.1.3, B

Percentage Accurate: 77.4% → 98.2%

Time: 3.0s

Alternatives: 6

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: 77.4% 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: 98.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-13} \lor \neg \left(y \leq 8 \cdot 10^{-161}\right):\\ \;\;\;\;\frac{x \cdot 2}{\frac{x}{y} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\frac{y}{x}, -0.5, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1e-13) (not (<= y 8e-161)))
   (/ (* x 2.0) (+ (/ x y) -1.0))
   (/ y (fma (/ y x) -0.5 0.5))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1e-13) || !(y <= 8e-161)) {
		tmp = (x * 2.0) / ((x / y) + -1.0);
	} else {
		tmp = y / fma((y / x), -0.5, 0.5);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1e-13) || !(y <= 8e-161))
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(x / y) + -1.0));
	else
		tmp = Float64(y / fma(Float64(y / x), -0.5, 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1e-13], N[Not[LessEqual[y, 8e-161]], $MachinePrecision]], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(y / N[(N[(y / x), $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1e-13 or 8.00000000000000022e-161 < y

   1. Initial program 82.6%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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 0 100.0%

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

   if -1e-13 < y < 8.00000000000000022e-161

   1. Initial program 67.7%

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

      1. *-commutative 67.7%

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

      2. associate-/l* 99.9%

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

      3. div-sub 99.9%

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

      4. sub-neg 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-neg-frac 99.9%

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

      7. neg-mul-1 99.9%

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

      8. *-commutative 99.9%

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

      9. times-frac 99.9%

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

      10. metadata-eval 99.9%

          \[\leadsto \frac{y}{\frac{y}{x} \cdot \color{blue}{-0.5} + \frac{x}{x \cdot 2}} \]

      11. metadata-eval 99.9%

          \[\leadsto \frac{y}{\frac{y}{x} \cdot \color{blue}{\left(-0.5\right)} + \frac{x}{x \cdot 2}} \]

      12. metadata-eval 99.9%

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

      13. *-inverses 99.9%

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

      14. associate-/r* 99.9%

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

      15. *-commutative 99.9%

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

      16. associate-/r* 99.9%

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

      17. *-inverses 99.9%

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

      18. *-inverses 99.9%

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

      19. associate-/r* 99.9%

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

      20. *-commutative 99.9%

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

      21. fma-def 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(\frac{y}{x}, -0.5, 0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-13} \lor \neg \left(y \leq 8 \cdot 10^{-161}\right):\\ \;\;\;\;\frac{x \cdot 2}{\frac{x}{y} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\frac{y}{x}, -0.5, 0.5\right)}\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-32} \lor \neg \left(x \leq 5 \cdot 10^{-32}\right):\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{\frac{x - y}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2e-32) (not (<= x 5e-32)))
   (* y (/ (* x 2.0) (- x y)))
   (* x (/ 2.0 (/ (- x y) y)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2e-32) || !(x <= 5e-32)) {
		tmp = y * ((x * 2.0) / (x - y));
	} else {
		tmp = x * (2.0 / ((x - y) / 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 <= (-2d-32)) .or. (.not. (x <= 5d-32))) then
        tmp = y * ((x * 2.0d0) / (x - y))
    else
        tmp = x * (2.0d0 / ((x - y) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2e-32) || !(x <= 5e-32)) {
		tmp = y * ((x * 2.0) / (x - y));
	} else {
		tmp = x * (2.0 / ((x - y) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2e-32) or not (x <= 5e-32):
		tmp = y * ((x * 2.0) / (x - y))
	else:
		tmp = x * (2.0 / ((x - y) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2e-32) || !(x <= 5e-32))
		tmp = Float64(y * Float64(Float64(x * 2.0) / Float64(x - y)));
	else
		tmp = Float64(x * Float64(2.0 / Float64(Float64(x - y) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2e-32) || ~((x <= 5e-32)))
		tmp = y * ((x * 2.0) / (x - y));
	else
		tmp = x * (2.0 / ((x - y) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2e-32], N[Not[LessEqual[x, 5e-32]], $MachinePrecision]], N[(y * N[(N[(x * 2.0), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(2.0 / N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.00000000000000011e-32 or 5e-32 < x

   1. Initial program 81.4%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   if -2.00000000000000011e-32 < x < 5e-32

   1. Initial program 72.5%

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

      1. associate-/l* 99.9%

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

      2. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-32} \lor \neg \left(x \leq 5 \cdot 10^{-32}\right):\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{\frac{x - y}{y}}\\ \end{array} \]

Alternative 3: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-38} \lor \neg \left(y \leq 8 \cdot 10^{-161}\right):\\ \;\;\;\;\frac{x \cdot 2}{\frac{x}{y} + -1}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4e-38) (not (<= y 8e-161)))
   (/ (* x 2.0) (+ (/ x y) -1.0))
   (* y (/ (* x 2.0) (- x y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4e-38) || !(y <= 8e-161)) {
		tmp = (x * 2.0) / ((x / y) + -1.0);
	} else {
		tmp = y * ((x * 2.0) / (x - y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-4d-38)) .or. (.not. (y <= 8d-161))) then
        tmp = (x * 2.0d0) / ((x / y) + (-1.0d0))
    else
        tmp = y * ((x * 2.0d0) / (x - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4e-38) || !(y <= 8e-161)) {
		tmp = (x * 2.0) / ((x / y) + -1.0);
	} else {
		tmp = y * ((x * 2.0) / (x - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4e-38) or not (y <= 8e-161):
		tmp = (x * 2.0) / ((x / y) + -1.0)
	else:
		tmp = y * ((x * 2.0) / (x - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4e-38) || !(y <= 8e-161))
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(x / y) + -1.0));
	else
		tmp = Float64(y * Float64(Float64(x * 2.0) / Float64(x - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4e-38) || ~((y <= 8e-161)))
		tmp = (x * 2.0) / ((x / y) + -1.0);
	else
		tmp = y * ((x * 2.0) / (x - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4e-38], N[Not[LessEqual[y, 8e-161]], $MachinePrecision]], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(x * 2.0), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.9999999999999998e-38 or 8.00000000000000022e-161 < y

   1. Initial program 83.5%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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 0 100.0%

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

   if -3.9999999999999998e-38 < y < 8.00000000000000022e-161

   1. Initial program 64.1%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-38} \lor \neg \left(y \leq 8 \cdot 10^{-161}\right):\\ \;\;\;\;\frac{x \cdot 2}{\frac{x}{y} + -1}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \end{array} \]

Alternative 4: 93.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+170}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{+109}:\\ \;\;\;\;x \cdot \frac{2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7e+170)
   (* y 2.0)
   (if (<= x 4.7e+109) (* x (/ 2.0 (/ (- x y) y))) (* y 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7e+170) {
		tmp = y * 2.0;
	} else if (x <= 4.7e+109) {
		tmp = x * (2.0 / ((x - y) / y));
	} 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 <= (-7d+170)) then
        tmp = y * 2.0d0
    else if (x <= 4.7d+109) then
        tmp = x * (2.0d0 / ((x - y) / y))
    else
        tmp = y * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -7e+170) {
		tmp = y * 2.0;
	} else if (x <= 4.7e+109) {
		tmp = x * (2.0 / ((x - y) / y));
	} else {
		tmp = y * 2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7e+170:
		tmp = y * 2.0
	elif x <= 4.7e+109:
		tmp = x * (2.0 / ((x - y) / y))
	else:
		tmp = y * 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7e+170)
		tmp = Float64(y * 2.0);
	elseif (x <= 4.7e+109)
		tmp = Float64(x * Float64(2.0 / Float64(Float64(x - y) / y)));
	else
		tmp = Float64(y * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7e+170)
		tmp = y * 2.0;
	elseif (x <= 4.7e+109)
		tmp = x * (2.0 / ((x - y) / y));
	else
		tmp = y * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7e+170], N[(y * 2.0), $MachinePrecision], If[LessEqual[x, 4.7e+109], N[(x * N[(2.0 / N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.00000000000000011e170 or 4.69999999999999998e109 < x

   1. Initial program 73.5%

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

      1. associate-/l* 61.8%

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

      2. associate-*r/ 61.6%

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

   3. Simplified 61.6%

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

   4. Taylor expanded in x around inf 90.7%

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

   if -7.00000000000000011e170 < x < 4.69999999999999998e109

   1. Initial program 78.3%

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

      1. associate-/l* 98.6%

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

      2. associate-*r/ 98.4%

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

   3. Simplified 98.4%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+170}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{+109}:\\ \;\;\;\;x \cdot \frac{2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]

Alternative 5: 75.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-38}:\\ \;\;\;\;x \cdot -2\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{-6}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.65e-38) (* x -2.0) (if (<= y 2.55e-6) (* y 2.0) (* x -2.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.65e-38) {
		tmp = x * -2.0;
	} else if (y <= 2.55e-6) {
		tmp = y * 2.0;
	} 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 (y <= (-1.65d-38)) then
        tmp = x * (-2.0d0)
    else if (y <= 2.55d-6) then
        tmp = y * 2.0d0
    else
        tmp = x * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.65e-38) {
		tmp = x * -2.0;
	} else if (y <= 2.55e-6) {
		tmp = y * 2.0;
	} else {
		tmp = x * -2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.65e-38:
		tmp = x * -2.0
	elif y <= 2.55e-6:
		tmp = y * 2.0
	else:
		tmp = x * -2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.65e-38)
		tmp = Float64(x * -2.0);
	elseif (y <= 2.55e-6)
		tmp = Float64(y * 2.0);
	else
		tmp = Float64(x * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.65e-38)
		tmp = x * -2.0;
	elseif (y <= 2.55e-6)
		tmp = y * 2.0;
	else
		tmp = x * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.65e-38], N[(x * -2.0), $MachinePrecision], If[LessEqual[y, 2.55e-6], N[(y * 2.0), $MachinePrecision], N[(x * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6500000000000001e-38 or 2.5500000000000001e-6 < y

   1. Initial program 81.4%

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

      1. associate-/l* 99.9%

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

      2. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 80.2%

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

   if -1.6500000000000001e-38 < y < 2.5500000000000001e-6

   1. Initial program 72.4%

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

      1. associate-/l* 80.0%

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

      2. associate-*r/ 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in x around inf 78.9%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-38}:\\ \;\;\;\;x \cdot -2\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{-6}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \]

Alternative 6: 50.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ y \cdot 2 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.3%

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

   1. associate-/l* 90.8%

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

   2. associate-*r/ 90.7%

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

3. Simplified 90.7%

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

4. Taylor expanded in x around inf 47.8%

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

5. Final simplification 47.8%

   \[\leadsto y \cdot 2 \]

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 2023194 
(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)))