Linear.Projection:perspective from linear-1.19.1.3, B

Percentage Accurate: 77.5% → 99.6%

Time: 5.7s

Alternatives: 6

Speedup: 0.5×

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: 77.5% 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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-37} \lor \neg \left(y \leq 6 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{x \cdot -2}{1 - \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4e-37) (not (<= y 6e-55)))
   (/ (* x -2.0) (- 1.0 (/ x y)))
   (* y (/ (* x 2.0) (- x y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4e-37) || !(y <= 6e-55)) {
		tmp = (x * -2.0) / (1.0 - (x / y));
	} 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-37)) .or. (.not. (y <= 6d-55))) then
        tmp = (x * (-2.0d0)) / (1.0d0 - (x / y))
    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-37) || !(y <= 6e-55)) {
		tmp = (x * -2.0) / (1.0 - (x / y));
	} else {
		tmp = y * ((x * 2.0) / (x - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4e-37) or not (y <= 6e-55):
		tmp = (x * -2.0) / (1.0 - (x / y))
	else:
		tmp = y * ((x * 2.0) / (x - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4e-37) || !(y <= 6e-55))
		tmp = Float64(Float64(x * -2.0) / Float64(1.0 - Float64(x / y)));
	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-37) || ~((y <= 6e-55)))
		tmp = (x * -2.0) / (1.0 - (x / y));
	else
		tmp = y * ((x * 2.0) / (x - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4e-37], N[Not[LessEqual[y, 6e-55]], $MachinePrecision]], N[(N[(x * -2.0), $MachinePrecision] / N[(1.0 - N[(x / y), $MachinePrecision]), $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 < -4.00000000000000027e-37 or 6.00000000000000033e-55 < y

   1. Initial program 80.7%

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

      1. *-lft-identity 80.7%

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

      2. metadata-eval 80.7%

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

      3. times-frac 80.7%

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

      4. neg-mul-1 80.7%

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

      5. sub-neg 80.7%

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

      6. +-commutative 80.7%

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

      7. distribute-neg-out 80.7%

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

      8. remove-double-neg 80.7%

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

      9. sub-neg 80.7%

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

      10. associate-*r* 80.7%

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

      11. neg-mul-1 80.7%

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

      12. distribute-lft-neg-out 80.7%

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

      13. associate-/l* 99.9%

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

      14. distribute-lft-neg-out 99.9%

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

      15. distribute-rgt-neg-in 99.9%

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

      16. metadata-eval 99.9%

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

      17. div-sub 99.9%

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

      18. *-inverses 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{1 - \frac{x}{y}}} \]
   4. Add Preprocessing

   if -4.00000000000000027e-37 < y < 6.00000000000000033e-55

   1. Initial program 73.3%

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

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-37} \lor \neg \left(y \leq 6 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{x \cdot -2}{1 - \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+77} \lor \neg \left(y \leq 2.2 \cdot 10^{+67}\right):\\ \;\;\;\;\left(x \cdot 2\right) \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.4e+77) (not (<= y 2.2e+67)))
   (* (* x 2.0) (- -1.0 (/ x y)))
   (* y 2.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.4e+77) || !(y <= 2.2e+67)) {
		tmp = (x * 2.0) * (-1.0 - (x / 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 ((y <= (-4.4d+77)) .or. (.not. (y <= 2.2d+67))) then
        tmp = (x * 2.0d0) * ((-1.0d0) - (x / y))
    else
        tmp = y * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4.4e+77) || !(y <= 2.2e+67)) {
		tmp = (x * 2.0) * (-1.0 - (x / y));
	} else {
		tmp = y * 2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.4e+77) or not (y <= 2.2e+67):
		tmp = (x * 2.0) * (-1.0 - (x / y))
	else:
		tmp = y * 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.4e+77) || !(y <= 2.2e+67))
		tmp = Float64(Float64(x * 2.0) * Float64(-1.0 - Float64(x / y)));
	else
		tmp = Float64(y * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4.4e+77) || ~((y <= 2.2e+67)))
		tmp = (x * 2.0) * (-1.0 - (x / y));
	else
		tmp = y * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.4e+77], N[Not[LessEqual[y, 2.2e+67]], $MachinePrecision]], N[(N[(x * 2.0), $MachinePrecision] * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.4000000000000001e77 or 2.2e67 < y

   1. Initial program 74.1%

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

      1. associate-*l/ 64.9%

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

      2. div-inv 64.9%

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

      3. associate-*l* 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around 0 85.3%

      \[\leadsto \left(x \cdot 2\right) \cdot \color{blue}{\left(-1 \cdot \frac{x}{y} - 1\right)} \]
   6. Step-by-step derivation

      1. sub-neg 85.3%

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

      2. metadata-eval 85.3%

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

      3. +-commutative 85.3%

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

      4. mul-1-neg 85.3%

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

      5. unsub-neg 85.3%

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

   7. Simplified 85.3%

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

   if -4.4000000000000001e77 < y < 2.2e67

   1. Initial program 80.8%

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

      1. associate-*l/ 99.1%

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

   3. Simplified 99.1%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{x - y} \cdot y} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 76.4%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+77} \lor \neg \left(y \leq 2.2 \cdot 10^{+67}\right):\\ \;\;\;\;\left(x \cdot 2\right) \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-145} \lor \neg \left(x \leq 5 \cdot 10^{-162}\right):\\ \;\;\;\;y \cdot \frac{2}{1 - \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2\right) \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.55e-145) (not (<= x 5e-162)))
   (* y (/ 2.0 (- 1.0 (/ y x))))
   (* (* x 2.0) (- -1.0 (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.55e-145) || !(x <= 5e-162)) {
		tmp = y * (2.0 / (1.0 - (y / x)));
	} else {
		tmp = (x * 2.0) * (-1.0 - (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 ((x <= (-1.55d-145)) .or. (.not. (x <= 5d-162))) then
        tmp = y * (2.0d0 / (1.0d0 - (y / x)))
    else
        tmp = (x * 2.0d0) * ((-1.0d0) - (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.55e-145) || !(x <= 5e-162)) {
		tmp = y * (2.0 / (1.0 - (y / x)));
	} else {
		tmp = (x * 2.0) * (-1.0 - (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.55e-145) or not (x <= 5e-162):
		tmp = y * (2.0 / (1.0 - (y / x)))
	else:
		tmp = (x * 2.0) * (-1.0 - (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.55e-145) || !(x <= 5e-162))
		tmp = Float64(y * Float64(2.0 / Float64(1.0 - Float64(y / x))));
	else
		tmp = Float64(Float64(x * 2.0) * Float64(-1.0 - Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.55e-145) || ~((x <= 5e-162)))
		tmp = y * (2.0 / (1.0 - (y / x)));
	else
		tmp = (x * 2.0) * (-1.0 - (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.55e-145], N[Not[LessEqual[x, 5e-162]], $MachinePrecision]], N[(y * N[(2.0 / N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55e-145 or 5.00000000000000014e-162 < x

   1. Initial program 79.0%

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

      1. associate-*l/ 96.3%

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

   3. Simplified 96.3%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{x - y} \cdot y} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. expm1-log1p-u 95.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot 2}{x - y}\right)\right)} \cdot y \]

      2. expm1-udef 65.5%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x \cdot 2}{x - y}\right)} - 1\right)} \cdot y \]

      3. *-commutative 65.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{2 \cdot x}}{x - y}\right)} - 1\right) \cdot y \]

      4. *-un-lft-identity 65.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{2 \cdot x}{\color{blue}{1 \cdot \left(x - y\right)}}\right)} - 1\right) \cdot y \]

      5. times-frac 65.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{2}{1} \cdot \frac{x}{x - y}}\right)} - 1\right) \cdot y \]

      6. metadata-eval 65.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{2} \cdot \frac{x}{x - y}\right)} - 1\right) \cdot y \]

   6. Applied egg-rr 65.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \frac{x}{x - y}\right)} - 1\right)} \cdot y \]
   7. Step-by-step derivation

      1. expm1-def 95.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \frac{x}{x - y}\right)\right)} \cdot y \]

      2. expm1-log1p 96.3%

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

      3. associate-*r/ 96.3%

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

      4. associate-/l* 95.8%

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

      5. div-sub 95.9%

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

      6. *-inverses 95.9%

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

   8. Simplified 95.9%

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

   if -1.55e-145 < x < 5.00000000000000014e-162

   1. Initial program 74.4%

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

      1. associate-*l/ 46.7%

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

      2. div-inv 46.6%

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

      3. associate-*l* 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around 0 93.9%

      \[\leadsto \left(x \cdot 2\right) \cdot \color{blue}{\left(-1 \cdot \frac{x}{y} - 1\right)} \]
   6. Step-by-step derivation

      1. sub-neg 93.9%

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

      2. metadata-eval 93.9%

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

      3. +-commutative 93.9%

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

      4. mul-1-neg 93.9%

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

      5. unsub-neg 93.9%

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

   7. Simplified 93.9%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-145} \lor \neg \left(x \leq 5 \cdot 10^{-162}\right):\\ \;\;\;\;y \cdot \frac{2}{1 - \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2\right) \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{-145} \lor \neg \left(x \leq 2.65 \cdot 10^{-163}\right):\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2\right) \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -5.6e-145) (not (<= x 2.65e-163)))
   (* y (/ (* x 2.0) (- x y)))
   (* (* x 2.0) (- -1.0 (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -5.6e-145) || !(x <= 2.65e-163)) {
		tmp = y * ((x * 2.0) / (x - y));
	} else {
		tmp = (x * 2.0) * (-1.0 - (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 ((x <= (-5.6d-145)) .or. (.not. (x <= 2.65d-163))) then
        tmp = y * ((x * 2.0d0) / (x - y))
    else
        tmp = (x * 2.0d0) * ((-1.0d0) - (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -5.6e-145) || !(x <= 2.65e-163)) {
		tmp = y * ((x * 2.0) / (x - y));
	} else {
		tmp = (x * 2.0) * (-1.0 - (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -5.6e-145) or not (x <= 2.65e-163):
		tmp = y * ((x * 2.0) / (x - y))
	else:
		tmp = (x * 2.0) * (-1.0 - (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -5.6e-145) || !(x <= 2.65e-163))
		tmp = Float64(y * Float64(Float64(x * 2.0) / Float64(x - y)));
	else
		tmp = Float64(Float64(x * 2.0) * Float64(-1.0 - Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -5.6e-145) || ~((x <= 2.65e-163)))
		tmp = y * ((x * 2.0) / (x - y));
	else
		tmp = (x * 2.0) * (-1.0 - (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -5.6e-145], N[Not[LessEqual[x, 2.65e-163]], $MachinePrecision]], N[(y * N[(N[(x * 2.0), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.6000000000000002e-145 or 2.65000000000000008e-163 < x

   1. Initial program 79.0%

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

      1. associate-*l/ 96.3%

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

   3. Simplified 96.3%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{x - y} \cdot y} \]
   4. Add Preprocessing

   if -5.6000000000000002e-145 < x < 2.65000000000000008e-163

   1. Initial program 74.4%

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

      1. associate-*l/ 46.7%

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

      2. div-inv 46.6%

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

      3. associate-*l* 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around 0 93.9%

      \[\leadsto \left(x \cdot 2\right) \cdot \color{blue}{\left(-1 \cdot \frac{x}{y} - 1\right)} \]
   6. Step-by-step derivation

      1. sub-neg 93.9%

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

      2. metadata-eval 93.9%

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

      3. +-commutative 93.9%

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

      4. mul-1-neg 93.9%

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

      5. unsub-neg 93.9%

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

   7. Simplified 93.9%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{-145} \lor \neg \left(x \leq 2.65 \cdot 10^{-163}\right):\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2\right) \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -29500000 \lor \neg \left(y \leq 5.2 \cdot 10^{+67}\right):\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -29500000.0) (not (<= y 5.2e+67))) (* x -2.0) (* y 2.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -29500000.0) || !(y <= 5.2e+67)) {
		tmp = x * -2.0;
	} 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 ((y <= (-29500000.0d0)) .or. (.not. (y <= 5.2d+67))) then
        tmp = x * (-2.0d0)
    else
        tmp = y * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -29500000.0) || !(y <= 5.2e+67)) {
		tmp = x * -2.0;
	} else {
		tmp = y * 2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -29500000.0) or not (y <= 5.2e+67):
		tmp = x * -2.0
	else:
		tmp = y * 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -29500000.0) || !(y <= 5.2e+67))
		tmp = Float64(x * -2.0);
	else
		tmp = Float64(y * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -29500000.0) || ~((y <= 5.2e+67)))
		tmp = x * -2.0;
	else
		tmp = y * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -29500000.0], N[Not[LessEqual[y, 5.2e+67]], $MachinePrecision]], N[(x * -2.0), $MachinePrecision], N[(y * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.95e7 or 5.2000000000000001e67 < y

   1. Initial program 77.0%

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

      1. associate-*l/ 68.0%

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

   3. Simplified 68.0%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{x - y} \cdot y} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 80.9%

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

   if -2.95e7 < y < 5.2000000000000001e67

   1. Initial program 78.8%

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

      \[\leadsto \color{blue}{\frac{x \cdot 2}{x - y} \cdot y} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 79.1%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -29500000 \lor \neg \left(y \leq 5.2 \cdot 10^{+67}\right):\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ x \cdot -2 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

   1. associate-*l/ 84.3%

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

3. Simplified 84.3%

   \[\leadsto \color{blue}{\frac{x \cdot 2}{x - y} \cdot y} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 50.9%

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

6. Final simplification 50.9%

   \[\leadsto x \cdot -2 \]
7. Add Preprocessing

Developer target: 99.5% accurate, 0.5× 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 2024036 
(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)))