Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, D

Percentage Accurate: 65.9% → 99.9%

Time: 11.2s

Alternatives: 12

Speedup: 2.1×

Specification

Math

\[\begin{array}{l} \\ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -11000:\\ \;\;\;\;x - \left(\frac{1 - x}{{y}^{2}} + \left(\frac{x + -1}{y} + \frac{x + -1}{{y}^{3}}\right)\right)\\ \mathbf{elif}\;y \leq 255000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{1 - {y}^{2}} \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + -1}{{y}^{2}} + \left(x + \frac{1 - x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -11000.0)
   (-
    x
    (+
     (/ (- 1.0 x) (pow y 2.0))
     (+ (/ (+ x -1.0) y) (/ (+ x -1.0) (pow y 3.0)))))
   (if (<= y 255000.0)
     (+ 1.0 (* (/ (* y (- 1.0 x)) (- 1.0 (pow y 2.0))) (+ y -1.0)))
     (+ (/ (+ x -1.0) (pow y 2.0)) (+ x (/ (- 1.0 x) y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -11000.0) {
		tmp = x - (((1.0 - x) / pow(y, 2.0)) + (((x + -1.0) / y) + ((x + -1.0) / pow(y, 3.0))));
	} else if (y <= 255000.0) {
		tmp = 1.0 + (((y * (1.0 - x)) / (1.0 - pow(y, 2.0))) * (y + -1.0));
	} else {
		tmp = ((x + -1.0) / pow(y, 2.0)) + (x + ((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 (y <= (-11000.0d0)) then
        tmp = x - (((1.0d0 - x) / (y ** 2.0d0)) + (((x + (-1.0d0)) / y) + ((x + (-1.0d0)) / (y ** 3.0d0))))
    else if (y <= 255000.0d0) then
        tmp = 1.0d0 + (((y * (1.0d0 - x)) / (1.0d0 - (y ** 2.0d0))) * (y + (-1.0d0)))
    else
        tmp = ((x + (-1.0d0)) / (y ** 2.0d0)) + (x + ((1.0d0 - x) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -11000.0) {
		tmp = x - (((1.0 - x) / Math.pow(y, 2.0)) + (((x + -1.0) / y) + ((x + -1.0) / Math.pow(y, 3.0))));
	} else if (y <= 255000.0) {
		tmp = 1.0 + (((y * (1.0 - x)) / (1.0 - Math.pow(y, 2.0))) * (y + -1.0));
	} else {
		tmp = ((x + -1.0) / Math.pow(y, 2.0)) + (x + ((1.0 - x) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -11000.0:
		tmp = x - (((1.0 - x) / math.pow(y, 2.0)) + (((x + -1.0) / y) + ((x + -1.0) / math.pow(y, 3.0))))
	elif y <= 255000.0:
		tmp = 1.0 + (((y * (1.0 - x)) / (1.0 - math.pow(y, 2.0))) * (y + -1.0))
	else:
		tmp = ((x + -1.0) / math.pow(y, 2.0)) + (x + ((1.0 - x) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -11000.0)
		tmp = Float64(x - Float64(Float64(Float64(1.0 - x) / (y ^ 2.0)) + Float64(Float64(Float64(x + -1.0) / y) + Float64(Float64(x + -1.0) / (y ^ 3.0)))));
	elseif (y <= 255000.0)
		tmp = Float64(1.0 + Float64(Float64(Float64(y * Float64(1.0 - x)) / Float64(1.0 - (y ^ 2.0))) * Float64(y + -1.0)));
	else
		tmp = Float64(Float64(Float64(x + -1.0) / (y ^ 2.0)) + Float64(x + Float64(Float64(1.0 - x) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -11000.0)
		tmp = x - (((1.0 - x) / (y ^ 2.0)) + (((x + -1.0) / y) + ((x + -1.0) / (y ^ 3.0))));
	elseif (y <= 255000.0)
		tmp = 1.0 + (((y * (1.0 - x)) / (1.0 - (y ^ 2.0))) * (y + -1.0));
	else
		tmp = ((x + -1.0) / (y ^ 2.0)) + (x + ((1.0 - x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -11000.0], N[(x - N[(N[(N[(1.0 - x), $MachinePrecision] / N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 255000.0], N[(1.0 + N[(N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + -1.0), $MachinePrecision] / N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -11000

   1. Initial program 28.8%

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

      1. *-commutative 28.8%

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

      2. associate-*l/ 51.2%

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

      3. +-commutative 51.2%

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

   3. Simplified 51.2%

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

   5. Taylor expanded in y around -inf 100.0%

      \[\leadsto \color{blue}{\left(x + \left(-1 \cdot \frac{x - 1}{y} + \left(-1 \cdot \frac{x - 1}{{y}^{3}} + \frac{x}{{y}^{2}}\right)\right)\right) - \frac{1}{{y}^{2}}} \]
   6. Step-by-step derivation

      1. associate--l+ 100.0%

         \[\leadsto \color{blue}{x + \left(\left(-1 \cdot \frac{x - 1}{y} + \left(-1 \cdot \frac{x - 1}{{y}^{3}} + \frac{x}{{y}^{2}}\right)\right) - \frac{1}{{y}^{2}}\right)} \]

      2. associate-+r+ 100.0%

         \[\leadsto x + \left(\color{blue}{\left(\left(-1 \cdot \frac{x - 1}{y} + -1 \cdot \frac{x - 1}{{y}^{3}}\right) + \frac{x}{{y}^{2}}\right)} - \frac{1}{{y}^{2}}\right) \]

      3. associate--l+ 100.0%

         \[\leadsto x + \color{blue}{\left(\left(-1 \cdot \frac{x - 1}{y} + -1 \cdot \frac{x - 1}{{y}^{3}}\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right)} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{x + \left(\left(\frac{1 - x}{y} + \frac{1 - x}{{y}^{3}}\right) + \frac{x + -1}{{y}^{2}}\right)} \]

   if -11000 < y < 255000

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. metadata-eval 100.0%

         \[\leadsto \color{blue}{\left(1 - 0\right)} + \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right) \]

      3. associate--r- 100.0%

         \[\leadsto \color{blue}{1 - \left(0 - \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)\right)} \]

      4. neg-sub0 100.0%

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

      5. remove-double-neg 100.0%

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

      6. associate-/l* 99.8%

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

      7. +-commutative 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{1 - \frac{1 - x}{\frac{1 + y}{y}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r/ 100.0%

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

      2. associate-*l/ 100.0%

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

      3. *-commutative 100.0%

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

      4. flip-+ 99.9%

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

      5. associate-/r/ 100.0%

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

      6. metadata-eval 100.0%

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

      7. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

   if 255000 < y

   1. Initial program 23.3%

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

      1. *-commutative 23.3%

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

      2. associate-*l/ 49.9%

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

      3. +-commutative 49.9%

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

   3. Simplified 49.9%

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

   5. Taylor expanded in y around -inf 100.0%

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

      1. associate-+r+ 100.0%

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

      2. associate--l+ 100.0%

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

      3. mul-1-neg 100.0%

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

      4. unsub-neg 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

      7. div-sub 100.0%

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

      8. sub-neg 100.0%

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

      9. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -11000:\\ \;\;\;\;x - \left(\frac{1 - x}{{y}^{2}} + \left(\frac{x + -1}{y} + \frac{x + -1}{{y}^{3}}\right)\right)\\ \mathbf{elif}\;y \leq 255000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{1 - {y}^{2}} \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + -1}{{y}^{2}} + \left(x + \frac{1 - x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -260000 \lor \neg \left(y \leq 255000\right):\\ \;\;\;\;\frac{x + -1}{{y}^{2}} + \left(x + \frac{1 - x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{1 - {y}^{2}} \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -260000.0) (not (<= y 255000.0)))
   (+ (/ (+ x -1.0) (pow y 2.0)) (+ x (/ (- 1.0 x) y)))
   (+ 1.0 (* (/ (* y (- 1.0 x)) (- 1.0 (pow y 2.0))) (+ y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -260000.0) || !(y <= 255000.0)) {
		tmp = ((x + -1.0) / pow(y, 2.0)) + (x + ((1.0 - x) / y));
	} else {
		tmp = 1.0 + (((y * (1.0 - x)) / (1.0 - pow(y, 2.0))) * (y + -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 <= (-260000.0d0)) .or. (.not. (y <= 255000.0d0))) then
        tmp = ((x + (-1.0d0)) / (y ** 2.0d0)) + (x + ((1.0d0 - x) / y))
    else
        tmp = 1.0d0 + (((y * (1.0d0 - x)) / (1.0d0 - (y ** 2.0d0))) * (y + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -260000.0) || !(y <= 255000.0)) {
		tmp = ((x + -1.0) / Math.pow(y, 2.0)) + (x + ((1.0 - x) / y));
	} else {
		tmp = 1.0 + (((y * (1.0 - x)) / (1.0 - Math.pow(y, 2.0))) * (y + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -260000.0) or not (y <= 255000.0):
		tmp = ((x + -1.0) / math.pow(y, 2.0)) + (x + ((1.0 - x) / y))
	else:
		tmp = 1.0 + (((y * (1.0 - x)) / (1.0 - math.pow(y, 2.0))) * (y + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -260000.0) || !(y <= 255000.0))
		tmp = Float64(Float64(Float64(x + -1.0) / (y ^ 2.0)) + Float64(x + Float64(Float64(1.0 - x) / y)));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(y * Float64(1.0 - x)) / Float64(1.0 - (y ^ 2.0))) * Float64(y + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -260000.0) || ~((y <= 255000.0)))
		tmp = ((x + -1.0) / (y ^ 2.0)) + (x + ((1.0 - x) / y));
	else
		tmp = 1.0 + (((y * (1.0 - x)) / (1.0 - (y ^ 2.0))) * (y + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -260000.0], N[Not[LessEqual[y, 255000.0]], $MachinePrecision]], N[(N[(N[(x + -1.0), $MachinePrecision] / N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.6e5 or 255000 < y

   1. Initial program 25.8%

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

      1. *-commutative 25.8%

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

      2. associate-*l/ 50.5%

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

      3. +-commutative 50.5%

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

   3. Simplified 50.5%

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

   5. Taylor expanded in y around -inf 99.8%

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

      1. associate-+r+ 99.8%

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

      2. associate--l+ 99.8%

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

      3. mul-1-neg 99.8%

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

      4. unsub-neg 99.8%

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

      5. sub-neg 99.8%

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

      6. metadata-eval 99.8%

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

      7. div-sub 99.8%

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

      8. sub-neg 99.8%

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

      9. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

   if -2.6e5 < y < 255000

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. metadata-eval 100.0%

         \[\leadsto \color{blue}{\left(1 - 0\right)} + \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right) \]

      3. associate--r- 100.0%

         \[\leadsto \color{blue}{1 - \left(0 - \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)\right)} \]

      4. neg-sub0 100.0%

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

      5. remove-double-neg 100.0%

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

      6. associate-/l* 99.8%

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

      7. +-commutative 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{1 - \frac{1 - x}{\frac{1 + y}{y}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r/ 100.0%

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

      2. associate-*l/ 100.0%

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

      3. *-commutative 100.0%

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

      4. flip-+ 99.9%

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

      5. associate-/r/ 100.0%

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

      6. metadata-eval 100.0%

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

      7. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -260000 \lor \neg \left(y \leq 255000\right):\\ \;\;\;\;\frac{x + -1}{{y}^{2}} + \left(x + \frac{1 - x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{1 - {y}^{2}} \cdot \left(y + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -300000 \lor \neg \left(y \leq 350000\right):\\ \;\;\;\;\frac{x + -1}{{y}^{2}} + \left(x + \frac{1 - x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y + 1}, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -300000.0) (not (<= y 350000.0)))
   (+ (/ (+ x -1.0) (pow y 2.0)) (+ x (/ (- 1.0 x) y)))
   (fma (/ (+ x -1.0) (+ y 1.0)) y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -300000.0) || !(y <= 350000.0)) {
		tmp = ((x + -1.0) / pow(y, 2.0)) + (x + ((1.0 - x) / y));
	} else {
		tmp = fma(((x + -1.0) / (y + 1.0)), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -300000.0) || !(y <= 350000.0))
		tmp = Float64(Float64(Float64(x + -1.0) / (y ^ 2.0)) + Float64(x + Float64(Float64(1.0 - x) / y)));
	else
		tmp = fma(Float64(Float64(x + -1.0) / Float64(y + 1.0)), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -300000.0], N[Not[LessEqual[y, 350000.0]], $MachinePrecision]], N[(N[(N[(x + -1.0), $MachinePrecision] / N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3e5 or 3.5e5 < y

   1. Initial program 25.8%

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

      1. *-commutative 25.8%

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

      2. associate-*l/ 50.5%

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

      3. +-commutative 50.5%

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

   3. Simplified 50.5%

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

   5. Taylor expanded in y around -inf 99.8%

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

      1. associate-+r+ 99.8%

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

      2. associate--l+ 99.8%

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

      3. mul-1-neg 99.8%

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

      4. unsub-neg 99.8%

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

      5. sub-neg 99.8%

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

      6. metadata-eval 99.8%

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

      7. div-sub 99.8%

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

      8. sub-neg 99.8%

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

      9. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

   if -3e5 < y < 3.5e5

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-/l* 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. associate-/r/ 100.0%

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

      6. fma-def 100.0%

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

      7. neg-sub0 100.0%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1}, y, 1\right) \]

      8. associate--r- 100.0%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1}, y, 1\right) \]

      9. metadata-eval 100.0%

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

      10. +-commutative 100.0%

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

      11. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x + -1}{1 + y}, y, 1\right)} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -300000 \lor \neg \left(y \leq 350000\right):\\ \;\;\;\;\frac{x + -1}{{y}^{2}} + \left(x + \frac{1 - x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y + 1}, y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -26000000000:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 210000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y + 1}, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -26000000000.0)
   (- x (/ -1.0 y))
   (if (<= y 210000000.0)
     (fma (/ (+ x -1.0) (+ y 1.0)) y 1.0)
     (+ x (/ (- 1.0 x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -26000000000.0) {
		tmp = x - (-1.0 / y);
	} else if (y <= 210000000.0) {
		tmp = fma(((x + -1.0) / (y + 1.0)), y, 1.0);
	} else {
		tmp = x + ((1.0 - x) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -26000000000.0)
		tmp = Float64(x - Float64(-1.0 / y));
	elseif (y <= 210000000.0)
		tmp = fma(Float64(Float64(x + -1.0) / Float64(y + 1.0)), y, 1.0);
	else
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -26000000000.0], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 210000000.0], N[(N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.6e10

   1. Initial program 26.9%

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

      1. *-commutative 26.9%

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

      2. associate-*l/ 49.9%

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

      3. +-commutative 49.9%

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

   3. Simplified 49.9%

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

   5. Taylor expanded in y around -inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0 99.9%

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

   if -2.6e10 < y < 2.1e8

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-/l* 99.6%

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

      4. distribute-neg-frac 99.6%

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

      5. associate-/r/ 99.7%

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

      6. fma-def 99.8%

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

      7. neg-sub0 99.8%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1}, y, 1\right) \]

      8. associate--r- 99.8%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1}, y, 1\right) \]

      9. metadata-eval 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x + -1}{1 + y}, y, 1\right)} \]
   4. Add Preprocessing

   if 2.1e8 < y

   1. Initial program 23.3%

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

      1. *-commutative 23.3%

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

      2. associate-*l/ 49.9%

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

      3. +-commutative 49.9%

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

   3. Simplified 49.9%

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

   5. Taylor expanded in y around -inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. sub-neg 99.8%

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

      4. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -26000000000:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 210000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y + 1}, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -26000000000:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 350000000:\\ \;\;\;\;1 + \frac{y}{y + 1} \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -26000000000.0)
   (- x (/ -1.0 y))
   (if (<= y 350000000.0)
     (+ 1.0 (* (/ y (+ y 1.0)) (+ x -1.0)))
     (+ x (/ (- 1.0 x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -26000000000.0) {
		tmp = x - (-1.0 / y);
	} else if (y <= 350000000.0) {
		tmp = 1.0 + ((y / (y + 1.0)) * (x + -1.0));
	} else {
		tmp = x + ((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 (y <= (-26000000000.0d0)) then
        tmp = x - ((-1.0d0) / y)
    else if (y <= 350000000.0d0) then
        tmp = 1.0d0 + ((y / (y + 1.0d0)) * (x + (-1.0d0)))
    else
        tmp = x + ((1.0d0 - x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -26000000000.0) {
		tmp = x - (-1.0 / y);
	} else if (y <= 350000000.0) {
		tmp = 1.0 + ((y / (y + 1.0)) * (x + -1.0));
	} else {
		tmp = x + ((1.0 - x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -26000000000.0:
		tmp = x - (-1.0 / y)
	elif y <= 350000000.0:
		tmp = 1.0 + ((y / (y + 1.0)) * (x + -1.0))
	else:
		tmp = x + ((1.0 - x) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -26000000000.0)
		tmp = Float64(x - Float64(-1.0 / y));
	elseif (y <= 350000000.0)
		tmp = Float64(1.0 + Float64(Float64(y / Float64(y + 1.0)) * Float64(x + -1.0)));
	else
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -26000000000.0)
		tmp = x - (-1.0 / y);
	elseif (y <= 350000000.0)
		tmp = 1.0 + ((y / (y + 1.0)) * (x + -1.0));
	else
		tmp = x + ((1.0 - x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -26000000000.0], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 350000000.0], N[(1.0 + N[(N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.6e10

   1. Initial program 26.9%

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

      1. *-commutative 26.9%

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

      2. associate-*l/ 49.9%

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

      3. +-commutative 49.9%

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

   3. Simplified 49.9%

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

   5. Taylor expanded in y around -inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0 99.9%

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

   if -2.6e10 < y < 3.5e8

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l/ 99.7%

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

      3. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   if 3.5e8 < y

   1. Initial program 23.3%

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

      1. *-commutative 23.3%

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

      2. associate-*l/ 49.9%

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

      3. +-commutative 49.9%

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

   3. Simplified 49.9%

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

   5. Taylor expanded in y around -inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. sub-neg 99.8%

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

      4. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -26000000000:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 350000000:\\ \;\;\;\;1 + \frac{y}{y + 1} \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.8\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.8)))
   (- x (/ -1.0 y))
   (+ 1.0 (* y (+ x -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.8)) {
		tmp = x - (-1.0 / y);
	} else {
		tmp = 1.0 + (y * (x + -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 <= (-1.0d0)) .or. (.not. (y <= 0.8d0))) then
        tmp = x - ((-1.0d0) / y)
    else
        tmp = 1.0d0 + (y * (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.8)) {
		tmp = x - (-1.0 / y);
	} else {
		tmp = 1.0 + (y * (x + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 0.8):
		tmp = x - (-1.0 / y)
	else:
		tmp = 1.0 + (y * (x + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.8))
		tmp = Float64(x - Float64(-1.0 / y));
	else
		tmp = Float64(1.0 + Float64(y * Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 0.8)))
		tmp = x - (-1.0 / y);
	else
		tmp = 1.0 + (y * (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.8]], $MachinePrecision]], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.80000000000000004 < y

   1. Initial program 26.9%

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

      1. *-commutative 26.9%

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

      2. associate-*l/ 51.2%

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

      3. +-commutative 51.2%

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

   3. Simplified 51.2%

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

   5. Taylor expanded in y around -inf 98.4%

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

      1. mul-1-neg 98.4%

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

      2. unsub-neg 98.4%

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

      3. sub-neg 98.4%

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

      4. metadata-eval 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in x around 0 97.9%

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

   if -1 < y < 0.80000000000000004

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

      3. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 98.2%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.8\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0)))
   (+ x (/ (- 1.0 x) y))
   (+ 1.0 (* y (+ x -1.0)))))

C

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

Java

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

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.0):
		tmp = x + ((1.0 - x) / y)
	else:
		tmp = 1.0 + (y * (x + -1.0))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 1.0)))
		tmp = x + ((1.0 - x) / y);
	else
		tmp = 1.0 + (y * (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 26.9%

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

      1. *-commutative 26.9%

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

      2. associate-*l/ 51.2%

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

      3. +-commutative 51.2%

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

   3. Simplified 51.2%

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

   5. Taylor expanded in y around -inf 98.4%

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

      1. mul-1-neg 98.4%

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

      2. unsub-neg 98.4%

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

      3. sub-neg 98.4%

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

      4. metadata-eval 98.4%

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

   7. Simplified 98.4%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

      3. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 98.2%

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

4. Final simplification 98.3%

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

Alternative 8: 85.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.0068\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.0068))) (- x (/ -1.0 y)) (- 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.0068)) {
		tmp = x - (-1.0 / y);
	} else {
		tmp = 1.0 - 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 <= (-1.0d0)) .or. (.not. (y <= 0.0068d0))) then
        tmp = x - ((-1.0d0) / y)
    else
        tmp = 1.0d0 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.0068)) {
		tmp = x - (-1.0 / y);
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 0.0068):
		tmp = x - (-1.0 / y)
	else:
		tmp = 1.0 - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.0068))
		tmp = Float64(x - Float64(-1.0 / y));
	else
		tmp = Float64(1.0 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 0.0068)))
		tmp = x - (-1.0 / y);
	else
		tmp = 1.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.0068]], $MachinePrecision]], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], N[(1.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.00679999999999999962 < y

   1. Initial program 27.4%

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

      1. *-commutative 27.4%

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

      2. associate-*l/ 51.5%

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

      3. +-commutative 51.5%

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

   3. Simplified 51.5%

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

   5. Taylor expanded in y around -inf 97.7%

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

      1. mul-1-neg 97.7%

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

      2. unsub-neg 97.7%

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

      3. sub-neg 97.7%

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

      4. metadata-eval 97.7%

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

   7. Simplified 97.7%

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

   8. Taylor expanded in x around 0 97.3%

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

   if -1 < y < 0.00679999999999999962

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

      3. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 75.4%

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

   6. Taylor expanded in y around 0 75.1%

      \[\leadsto \color{blue}{1 + -1 \cdot y} \]
   7. Step-by-step derivation

      1. neg-mul-1 75.1%

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

      2. sub-neg 75.1%

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

   8. Simplified 75.1%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.0068\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.0% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0))) (- x (/ -1.0 y)) (+ 1.0 (* y x))))

C

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

Java

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

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.0):
		tmp = x - (-1.0 / y)
	else:
		tmp = 1.0 + (y * x)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 1.0)))
		tmp = x - (-1.0 / y);
	else
		tmp = 1.0 + (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 26.9%

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

      1. *-commutative 26.9%

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

      2. associate-*l/ 51.2%

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

      3. +-commutative 51.2%

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

   3. Simplified 51.2%

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

   5. Taylor expanded in y around -inf 98.4%

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

      1. mul-1-neg 98.4%

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

      2. unsub-neg 98.4%

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

      3. sub-neg 98.4%

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

      4. metadata-eval 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in x around 0 97.9%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

      3. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 98.2%

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

   6. Taylor expanded in x around inf 97.3%

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

      1. mul-1-neg 97.3%

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

      2. distribute-lft-neg-out 97.3%

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

      3. *-commutative 97.3%

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

   8. Simplified 97.3%

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

4. Final simplification 97.6%

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

Alternative 10: 73.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.0068:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 0.0068) (- 1.0 y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.0068) {
		tmp = 1.0 - y;
	} else {
		tmp = 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 <= (-1.0d0)) then
        tmp = x
    else if (y <= 0.0068d0) then
        tmp = 1.0d0 - y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.0068) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 0.0068:
		tmp = 1.0 - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 0.0068)
		tmp = Float64(1.0 - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 0.0068)
		tmp = 1.0 - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 0.0068], N[(1.0 - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.00679999999999999962 < y

   1. Initial program 27.4%

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

      1. *-commutative 27.4%

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

      2. associate-*l/ 51.5%

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

      3. +-commutative 51.5%

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

   3. Simplified 51.5%

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

   5. Taylor expanded in y around inf 70.9%

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

   if -1 < y < 0.00679999999999999962

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

      3. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 75.4%

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

   6. Taylor expanded in y around 0 75.1%

      \[\leadsto \color{blue}{1 + -1 \cdot y} \]
   7. Step-by-step derivation

      1. neg-mul-1 75.1%

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

      2. sub-neg 75.1%

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

   8. Simplified 75.1%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.0068:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 17:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) x (if (<= y 17.0) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 17.0) {
		tmp = 1.0;
	} else {
		tmp = 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 <= (-1.0d0)) then
        tmp = x
    else if (y <= 17.0d0) then
        tmp = 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 17.0) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 17.0:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 17.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 17.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 17.0], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 17 < y

   1. Initial program 26.3%

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

      1. *-commutative 26.3%

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

      2. associate-*l/ 50.8%

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

      3. +-commutative 50.8%

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

   3. Simplified 50.8%

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

   5. Taylor expanded in y around inf 71.8%

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

   if -1 < y < 17

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

      3. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 73.2%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 17:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 38.8% accurate, 11.0× 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 61.4%

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

   1. *-commutative 61.4%

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

   2. associate-*l/ 74.2%

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

   3. +-commutative 74.2%

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

3. Simplified 74.2%

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

5. Taylor expanded in y around 0 36.7%

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

6. Final simplification 36.7%

   \[\leadsto 1 \]
7. Add Preprocessing

Developer target: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 y) (- (/ x y) x))))
   (if (< y -3693.8482788297247)
     t_0
     (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) t_0))))

C

double code(double x, double y) {
	double t_0 = (1.0 / y) - ((x / y) - x);
	double tmp;
	if (y < -3693.8482788297247) {
		tmp = t_0;
	} else if (y < 6799310503.41891) {
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	} 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 = (1.0d0 / y) - ((x / y) - x)
    if (y < (-3693.8482788297247d0)) then
        tmp = t_0
    else if (y < 6799310503.41891d0) then
        tmp = 1.0d0 - (((1.0d0 - x) * y) / (y + 1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.0 / y) - ((x / y) - x);
	double tmp;
	if (y < -3693.8482788297247) {
		tmp = t_0;
	} else if (y < 6799310503.41891) {
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 / y) - ((x / y) - x)
	tmp = 0
	if y < -3693.8482788297247:
		tmp = t_0
	elif y < 6799310503.41891:
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 / y) - Float64(Float64(x / y) - x))
	tmp = 0.0
	if (y < -3693.8482788297247)
		tmp = t_0;
	elseif (y < 6799310503.41891)
		tmp = Float64(1.0 - Float64(Float64(Float64(1.0 - x) * y) / Float64(y + 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 / y) - ((x / y) - x);
	tmp = 0.0;
	if (y < -3693.8482788297247)
		tmp = t_0;
	elseif (y < 6799310503.41891)
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 / y), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -3693.8482788297247], t$95$0, If[Less[y, 6799310503.41891], N[(1.0 - N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024010 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1.0 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) (- (/ 1.0 y) (- (/ x y) x))))

  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))