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

Percentage Accurate: 65.8% → 99.9%

Time: 7.1s

Alternatives: 12

Speedup: 1.2×

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: 65.8% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{\left(x - \frac{x - 1}{y}\right) - 1}{y}\\ \mathbf{if}\;y \leq -9000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 14500:\\ \;\;\;\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{t\_0 - 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ (- (- x (/ (- x 1.0) y)) 1.0) y))))
   (if (<= y -9000000000.0)
     t_0
     (if (<= y 14500.0)
       (- 1.0 (/ (* (- x 1.0) y) (- -1.0 y)))
       (- x (/ (- t_0 1.0) y))))))

C

double code(double x, double y) {
	double t_0 = x - (((x - ((x - 1.0) / y)) - 1.0) / y);
	double tmp;
	if (y <= -9000000000.0) {
		tmp = t_0;
	} else if (y <= 14500.0) {
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	} else {
		tmp = x - ((t_0 - 1.0) / y);
	}
	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 = x - (((x - ((x - 1.0d0) / y)) - 1.0d0) / y)
    if (y <= (-9000000000.0d0)) then
        tmp = t_0
    else if (y <= 14500.0d0) then
        tmp = 1.0d0 - (((x - 1.0d0) * y) / ((-1.0d0) - y))
    else
        tmp = x - ((t_0 - 1.0d0) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x - (((x - ((x - 1.0) / y)) - 1.0) / y);
	double tmp;
	if (y <= -9000000000.0) {
		tmp = t_0;
	} else if (y <= 14500.0) {
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	} else {
		tmp = x - ((t_0 - 1.0) / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x - (((x - ((x - 1.0) / y)) - 1.0) / y)
	tmp = 0
	if y <= -9000000000.0:
		tmp = t_0
	elif y <= 14500.0:
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y))
	else:
		tmp = x - ((t_0 - 1.0) / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x - Float64(Float64(Float64(x - Float64(Float64(x - 1.0) / y)) - 1.0) / y))
	tmp = 0.0
	if (y <= -9000000000.0)
		tmp = t_0;
	elseif (y <= 14500.0)
		tmp = Float64(1.0 - Float64(Float64(Float64(x - 1.0) * y) / Float64(-1.0 - y)));
	else
		tmp = Float64(x - Float64(Float64(t_0 - 1.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x - (((x - ((x - 1.0) / y)) - 1.0) / y);
	tmp = 0.0;
	if (y <= -9000000000.0)
		tmp = t_0;
	elseif (y <= 14500.0)
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	else
		tmp = x - ((t_0 - 1.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(N[(N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9000000000.0], t$95$0, If[LessEqual[y, 14500.0], N[(1.0 - N[(N[(N[(x - 1.0), $MachinePrecision] * y), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(t$95$0 - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9e9

   1. Initial program 22.9%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. associate-*r/ N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. neg-sub0 N/A

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

      16. associate--r- N/A

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

      17. div-sub N/A

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

      18. neg-sub0 N/A

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

   5. Applied rewrites 100.0%

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

   if -9e9 < y < 14500

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   if 14500 < y

   1. Initial program 30.8%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9000000000:\\ \;\;\;\;x - \frac{\left(x - \frac{x - 1}{y}\right) - 1}{y}\\ \mathbf{elif}\;y \leq 14500:\\ \;\;\;\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(x - \frac{\left(x - \frac{x - 1}{y}\right) - 1}{y}\right) - 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{\left(x - 1\right) \cdot y}{-1 - y}\\ \mathbf{if}\;t\_0 \leq 0.95:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+93}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ (* (- x 1.0) y) (- -1.0 y)))))
   (if (<= t_0 0.95)
     (* 1.0 x)
     (if (<= t_0 2.0) (- 1.0 y) (if (<= t_0 2e+93) (* x y) (* 1.0 x))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	double tmp;
	if (t_0 <= 0.95) {
		tmp = 1.0 * x;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - y;
	} else if (t_0 <= 2e+93) {
		tmp = x * y;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (((x - 1.0d0) * y) / ((-1.0d0) - y))
    if (t_0 <= 0.95d0) then
        tmp = 1.0d0 * x
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0 - y
    else if (t_0 <= 2d+93) then
        tmp = x * y
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	double tmp;
	if (t_0 <= 0.95) {
		tmp = 1.0 * x;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - y;
	} else if (t_0 <= 2e+93) {
		tmp = x * y;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (((x - 1.0) * y) / (-1.0 - y))
	tmp = 0
	if t_0 <= 0.95:
		tmp = 1.0 * x
	elif t_0 <= 2.0:
		tmp = 1.0 - y
	elif t_0 <= 2e+93:
		tmp = x * y
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(Float64(Float64(x - 1.0) * y) / Float64(-1.0 - y)))
	tmp = 0.0
	if (t_0 <= 0.95)
		tmp = Float64(1.0 * x);
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 - y);
	elseif (t_0 <= 2e+93)
		tmp = Float64(x * y);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	tmp = 0.0;
	if (t_0 <= 0.95)
		tmp = 1.0 * x;
	elseif (t_0 <= 2.0)
		tmp = 1.0 - y;
	elseif (t_0 <= 2e+93)
		tmp = x * y;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(N[(N[(x - 1.0), $MachinePrecision] * y), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.95], N[(1.0 * x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(1.0 - y), $MachinePrecision], If[LessEqual[t$95$0, 2e+93], N[(x * y), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 0.94999999999999996 or 2.00000000000000009e93 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))))

   1. Initial program 32.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 75.6

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in y around inf

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 66.5%

      \[\leadsto 1 \cdot x \]

   if 0.94999999999999996 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 2

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.6%

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

   if 2 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 2.00000000000000009e93

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 68.4

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

   5. Applied rewrites 68.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 57.8%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y} \leq 0.95:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y} \leq 2:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y} \leq 2 \cdot 10^{+93}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x - 1\right) \cdot y}{-1 - y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+116}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;t\_0 \leq -20:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(y - 1, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x 1.0) y) (- -1.0 y))))
   (if (<= t_0 -2e+116)
     (* 1.0 x)
     (if (<= t_0 -20.0)
       (* x y)
       (if (<= t_0 0.01) (fma (- y 1.0) y 1.0) (* 1.0 x))))))

C

double code(double x, double y) {
	double t_0 = ((x - 1.0) * y) / (-1.0 - y);
	double tmp;
	if (t_0 <= -2e+116) {
		tmp = 1.0 * x;
	} else if (t_0 <= -20.0) {
		tmp = x * y;
	} else if (t_0 <= 0.01) {
		tmp = fma((y - 1.0), y, 1.0);
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - 1.0) * y) / Float64(-1.0 - y))
	tmp = 0.0
	if (t_0 <= -2e+116)
		tmp = Float64(1.0 * x);
	elseif (t_0 <= -20.0)
		tmp = Float64(x * y);
	elseif (t_0 <= 0.01)
		tmp = fma(Float64(y - 1.0), y, 1.0);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - 1.0), $MachinePrecision] * y), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+116], N[(1.0 * x), $MachinePrecision], If[LessEqual[t$95$0, -20.0], N[(x * y), $MachinePrecision], If[LessEqual[t$95$0, 0.01], N[(N[(y - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < -2.00000000000000003e116 or 0.0100000000000000002 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 32.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 75.6

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in y around inf

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 66.5%

      \[\leadsto 1 \cdot x \]

   if -2.00000000000000003e116 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < -20

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 68.4

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

   5. Applied rewrites 68.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 57.8%

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

   if -20 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 0.0100000000000000002

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(y - 1, y, 1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 98.2%

      \[\leadsto \mathsf{fma}\left(y - 1, y, 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 1\right) \cdot y}{-1 - y} \leq -2 \cdot 10^{+116}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;\frac{\left(x - 1\right) \cdot y}{-1 - y} \leq -20:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;\frac{\left(x - 1\right) \cdot y}{-1 - y} \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(y - 1, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{\left(x - 1\right) \cdot y}{-1 - y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-12}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ (* (- x 1.0) y) (- -1.0 y)))))
   (if (<= t_0 -1e-12) (* x y) (if (<= t_0 2.0) 1.0 (* x y)))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	double tmp;
	if (t_0 <= -1e-12) {
		tmp = x * y;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x * y;
	}
	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 - (((x - 1.0d0) * y) / ((-1.0d0) - y))
    if (t_0 <= (-1d-12)) then
        tmp = x * y
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	double tmp;
	if (t_0 <= -1e-12) {
		tmp = x * y;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (((x - 1.0) * y) / (-1.0 - y))
	tmp = 0
	if t_0 <= -1e-12:
		tmp = x * y
	elif t_0 <= 2.0:
		tmp = 1.0
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(Float64(Float64(x - 1.0) * y) / Float64(-1.0 - y)))
	tmp = 0.0
	if (t_0 <= -1e-12)
		tmp = Float64(x * y);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	tmp = 0.0;
	if (t_0 <= -1e-12)
		tmp = x * y;
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(N[(N[(x - 1.0), $MachinePrecision] * y), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-12], N[(x * y), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, N[(x * y), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < -9.9999999999999998e-13 or 2 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))))

   1. Initial program 63.4%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 27.7

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

   5. Applied rewrites 27.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 25.7%

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

   if -9.9999999999999998e-13 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 2

   1. Initial program 58.2%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 55.1%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y} \leq -1 \cdot 10^{-12}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{\left(x - \frac{x - 1}{y}\right) - 1}{y}\\ \mathbf{if}\;y \leq -9000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 330000:\\ \;\;\;\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ (- (- x (/ (- x 1.0) y)) 1.0) y))))
   (if (<= y -9000000000.0)
     t_0
     (if (<= y 330000.0) (- 1.0 (/ (* (- x 1.0) y) (- -1.0 y))) t_0))))

C

double code(double x, double y) {
	double t_0 = x - (((x - ((x - 1.0) / y)) - 1.0) / y);
	double tmp;
	if (y <= -9000000000.0) {
		tmp = t_0;
	} else if (y <= 330000.0) {
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - 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 = x - (((x - ((x - 1.0d0) / y)) - 1.0d0) / y)
    if (y <= (-9000000000.0d0)) then
        tmp = t_0
    else if (y <= 330000.0d0) then
        tmp = 1.0d0 - (((x - 1.0d0) * y) / ((-1.0d0) - y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x - (((x - ((x - 1.0) / y)) - 1.0) / y);
	double tmp;
	if (y <= -9000000000.0) {
		tmp = t_0;
	} else if (y <= 330000.0) {
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x - (((x - ((x - 1.0) / y)) - 1.0) / y)
	tmp = 0
	if y <= -9000000000.0:
		tmp = t_0
	elif y <= 330000.0:
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x - Float64(Float64(Float64(x - Float64(Float64(x - 1.0) / y)) - 1.0) / y))
	tmp = 0.0
	if (y <= -9000000000.0)
		tmp = t_0;
	elseif (y <= 330000.0)
		tmp = Float64(1.0 - Float64(Float64(Float64(x - 1.0) * y) / Float64(-1.0 - y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x - (((x - ((x - 1.0) / y)) - 1.0) / y);
	tmp = 0.0;
	if (y <= -9000000000.0)
		tmp = t_0;
	elseif (y <= 330000.0)
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(N[(N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9000000000.0], t$95$0, If[LessEqual[y, 330000.0], N[(1.0 - N[(N[(N[(x - 1.0), $MachinePrecision] * y), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -9e9 or 3.3e5 < y

   1. Initial program 26.8%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. associate-*r/ N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. neg-sub0 N/A

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

      16. associate--r- N/A

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

      17. div-sub N/A

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

      18. neg-sub0 N/A

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

   5. Applied rewrites 100.0%

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

   if -9e9 < y < 3.3e5

   1. Initial program 99.8%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9000000000:\\ \;\;\;\;x - \frac{\left(x - \frac{x - 1}{y}\right) - 1}{y}\\ \mathbf{elif}\;y \leq 330000:\\ \;\;\;\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(x - \frac{x - 1}{y}\right) - 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{x - 1}{y}\\ \mathbf{if}\;y \leq -9000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 190000000:\\ \;\;\;\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ (- x 1.0) y))))
   (if (<= y -9000000000.0)
     t_0
     (if (<= y 190000000.0) (- 1.0 (/ (* (- x 1.0) y) (- -1.0 y))) t_0))))

C

double code(double x, double y) {
	double t_0 = x - ((x - 1.0) / y);
	double tmp;
	if (y <= -9000000000.0) {
		tmp = t_0;
	} else if (y <= 190000000.0) {
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - 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 = x - ((x - 1.0d0) / y)
    if (y <= (-9000000000.0d0)) then
        tmp = t_0
    else if (y <= 190000000.0d0) then
        tmp = 1.0d0 - (((x - 1.0d0) * y) / ((-1.0d0) - y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x - ((x - 1.0) / y);
	double tmp;
	if (y <= -9000000000.0) {
		tmp = t_0;
	} else if (y <= 190000000.0) {
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x - ((x - 1.0) / y)
	tmp = 0
	if y <= -9000000000.0:
		tmp = t_0
	elif y <= 190000000.0:
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x - Float64(Float64(x - 1.0) / y))
	tmp = 0.0
	if (y <= -9000000000.0)
		tmp = t_0;
	elseif (y <= 190000000.0)
		tmp = Float64(1.0 - Float64(Float64(Float64(x - 1.0) * y) / Float64(-1.0 - y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x - ((x - 1.0) / y);
	tmp = 0.0;
	if (y <= -9000000000.0)
		tmp = t_0;
	elseif (y <= 190000000.0)
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9000000000.0], t$95$0, If[LessEqual[y, 190000000.0], N[(1.0 - N[(N[(N[(x - 1.0), $MachinePrecision] * y), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -9e9 or 1.9e8 < y

   1. Initial program 26.8%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -9e9 < y < 1.9e8

   1. Initial program 99.8%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9000000000:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{elif}\;y \leq 190000000:\\ \;\;\;\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{x - 1}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\left(y - 1\right) \cdot \left(1 - x\right), y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ (- x 1.0) y))))
   (if (<= y -1.0)
     t_0
     (if (<= y 1.0) (fma (* (- y 1.0) (- 1.0 x)) y 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = x - ((x - 1.0) / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = fma(((y - 1.0) * (1.0 - x)), y, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 28.7%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

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

4. Final simplification 98.9%

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

Alternative 8: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{x - 1}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\left(-x\right) \cdot \left(y - 1\right), y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ (- x 1.0) y))))
   (if (<= y -1.0) t_0 (if (<= y 1.0) (fma (* (- x) (- y 1.0)) y 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = x - ((x - 1.0) / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = fma((-x * (y - 1.0)), y, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 28.7%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\left(-1 \cdot x\right) \cdot \left(-1 + y\right), y, 1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 98.2%

      \[\leadsto \mathsf{fma}\left(\left(-x\right) \cdot \left(-1 + y\right), y, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

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

Alternative 9: 86.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{x}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 43000:\\ \;\;\;\;\mathsf{fma}\left(\left(-x\right) \cdot \left(y - 1\right), y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ x y))))
   (if (<= y -1.0)
     t_0
     (if (<= y 43000.0) (fma (* (- x) (- y 1.0)) y 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = x - (x / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 43000.0) {
		tmp = fma((-x * (y - 1.0)), y, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x - Float64(x / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 43000.0)
		tmp = fma(Float64(Float64(-x) * Float64(y - 1.0)), y, 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 43000.0], N[(N[((-x) * N[(y - 1.0), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 43000 < y

   1. Initial program 28.3%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 74.9

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 74.3%

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

   if -1 < y < 43000

   1. Initial program 99.8%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\left(-1 \cdot x\right) \cdot \left(-1 + y\right), y, 1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 97.4%

      \[\leadsto \mathsf{fma}\left(\left(-x\right) \cdot \left(-1 + y\right), y, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{elif}\;y \leq 43000:\\ \;\;\;\;\mathsf{fma}\left(\left(-x\right) \cdot \left(y - 1\right), y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{x}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.12:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ x y))))
   (if (<= y -1.0) t_0 (if (<= y 1.12) (fma (- x 1.0) y 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = x - (x / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.12) {
		tmp = fma((x - 1.0), y, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x - Float64(x / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 1.12)
		tmp = fma(Float64(x - 1.0), y, 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 1.12], N[(N[(x - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.1200000000000001 < y

   1. Initial program 28.7%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 74.4

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

   5. Applied rewrites 74.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 73.8%

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

   if -1 < y < 1.1200000000000001

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 97.6

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

   5. Applied rewrites 97.6%

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

Alternative 11: 86.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0 * x;
	} else if (y <= 1.0) {
		tmp = fma((x - 1.0), y, 1.0);
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 28.7%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 74.4

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

   5. Applied rewrites 74.4%

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

   6. Taylor expanded in y around inf

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 72.8%

      \[\leadsto 1 \cdot x \]

   if -1 < y < 1

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 97.6

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

   5. Applied rewrites 97.6%

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

Alternative 12: 38.8% accurate, 26.0× 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 60.1%

   \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

5. Applied rewrites 36.1%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.6× 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 2024240 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -36938482788297247/10000000000000) (- (/ 1 y) (- (/ x y) x)) (if (< y 679931050341891/100000) (- 1 (/ (* (- 1 x) y) (+ y 1))) (- (/ 1 y) (- (/ x y) x)))))

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