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

Percentage Accurate: 65.4% → 99.8%

Time: 6.7s

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: 65.4% 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: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -122000000:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{elif}\;y \leq 300000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x - 1}{y - -1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - x\right) + \frac{-1 + x}{y}}{y} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -122000000.0)
   (- x (/ (- x 1.0) y))
   (if (<= y 300000.0)
     (fma y (/ (- x 1.0) (- y -1.0)) 1.0)
     (+ (/ (+ (- 1.0 x) (/ (+ -1.0 x) y)) y) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -122000000.0) {
		tmp = x - ((x - 1.0) / y);
	} else if (y <= 300000.0) {
		tmp = fma(y, ((x - 1.0) / (y - -1.0)), 1.0);
	} else {
		tmp = (((1.0 - x) + ((-1.0 + x) / y)) / y) + x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -122000000.0)
		tmp = Float64(x - Float64(Float64(x - 1.0) / y));
	elseif (y <= 300000.0)
		tmp = fma(y, Float64(Float64(x - 1.0) / Float64(y - -1.0)), 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - x) + Float64(Float64(-1.0 + x) / y)) / y) + x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -122000000.0], N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 300000.0], N[(y * N[(N[(x - 1.0), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(N[(1.0 - x), $MachinePrecision] + N[(N[(-1.0 + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.22e8

   1. Initial program 33.7%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.22e8 < y < 3e5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 3.7

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

   5. Applied rewrites 3.7%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. distribute-rgt-out N/A

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

      9. metadata-eval N/A

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

      10. remove-double-neg N/A

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

      11. mul-1-neg N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. fp-cancel-sign-sub-inv N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

   10. Applied rewrites 100.0%

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

   if 3e5 < y

   1. Initial program 39.8%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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--l+ N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

   5. Applied rewrites 100.0%

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

Alternative 2: 73.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{if}\;t\_0 \leq -20000000:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- 1.0 x) y) (+ y 1.0))))
   (if (<= t_0 -20000000.0) x (if (<= t_0 0.01) (- 1.0 y) x))))

C

double code(double x, double y) {
	double t_0 = ((1.0 - x) * y) / (y + 1.0);
	double tmp;
	if (t_0 <= -20000000.0) {
		tmp = x;
	} else if (t_0 <= 0.01) {
		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) :: t_0
    real(8) :: tmp
    t_0 = ((1.0d0 - x) * y) / (y + 1.0d0)
    if (t_0 <= (-20000000.0d0)) then
        tmp = x
    else if (t_0 <= 0.01d0) then
        tmp = 1.0d0 - y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((1.0 - x) * y) / (y + 1.0);
	double tmp;
	if (t_0 <= -20000000.0) {
		tmp = x;
	} else if (t_0 <= 0.01) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((1.0 - x) * y) / (y + 1.0)
	tmp = 0
	if t_0 <= -20000000.0:
		tmp = x
	elif t_0 <= 0.01:
		tmp = 1.0 - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(1.0 - x) * y) / Float64(y + 1.0))
	tmp = 0.0
	if (t_0 <= -20000000.0)
		tmp = x;
	elseif (t_0 <= 0.01)
		tmp = Float64(1.0 - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((1.0 - x) * y) / (y + 1.0);
	tmp = 0.0;
	if (t_0 <= -20000000.0)
		tmp = x;
	elseif (t_0 <= 0.01)
		tmp = 1.0 - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -20000000.0], x, If[LessEqual[t$95$0, 0.01], N[(1.0 - y), $MachinePrecision], x]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < -2e7 or 0.0100000000000000002 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 53.3%

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

      1. lift--.f64 N/A

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

      2. *-lft-identity N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. distribute-lft-neg-in N/A

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

      5. *-lft-identity N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. distribute-lft-neg-in N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 66.2

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

   4. Applied rewrites 66.2%

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

   5. Taylor expanded in y around inf

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

      1. *-lft-identity N/A

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. metadata-eval N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. associate-+r+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \color{blue}{0} + x \]

      11. lower-+.f64 59.6

          \[\leadsto \color{blue}{0 + x} \]

   7. Applied rewrites 59.6%

      \[\leadsto \color{blue}{0 + x} \]
   8. Step-by-step derivation

   9. Applied rewrites 59.6%

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

   if -2e7 < (/.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

      \[\leadsto \color{blue}{1 + y \cdot \left(x - 1\right)} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.2%

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

Alternative 3: 73.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{if}\;t\_0 \leq -20000000:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- 1.0 x) y) (+ y 1.0))))
   (if (<= t_0 -20000000.0) x (if (<= t_0 0.01) 1.0 x))))

C

double code(double x, double y) {
	double t_0 = ((1.0 - x) * y) / (y + 1.0);
	double tmp;
	if (t_0 <= -20000000.0) {
		tmp = x;
	} else if (t_0 <= 0.01) {
		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) :: t_0
    real(8) :: tmp
    t_0 = ((1.0d0 - x) * y) / (y + 1.0d0)
    if (t_0 <= (-20000000.0d0)) then
        tmp = x
    else if (t_0 <= 0.01d0) then
        tmp = 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((1.0 - x) * y) / (y + 1.0);
	double tmp;
	if (t_0 <= -20000000.0) {
		tmp = x;
	} else if (t_0 <= 0.01) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((1.0 - x) * y) / (y + 1.0)
	tmp = 0
	if t_0 <= -20000000.0:
		tmp = x
	elif t_0 <= 0.01:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(1.0 - x) * y) / Float64(y + 1.0))
	tmp = 0.0
	if (t_0 <= -20000000.0)
		tmp = x;
	elseif (t_0 <= 0.01)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((1.0 - x) * y) / (y + 1.0);
	tmp = 0.0;
	if (t_0 <= -20000000.0)
		tmp = x;
	elseif (t_0 <= 0.01)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -20000000.0], x, If[LessEqual[t$95$0, 0.01], 1.0, x]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < -2e7 or 0.0100000000000000002 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 53.3%

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

      1. lift--.f64 N/A

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

      2. *-lft-identity N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. distribute-lft-neg-in N/A

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

      5. *-lft-identity N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. distribute-lft-neg-in N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 66.2

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

   4. Applied rewrites 66.2%

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

   5. Taylor expanded in y around inf

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

      1. *-lft-identity N/A

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. metadata-eval N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. associate-+r+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \color{blue}{0} + x \]

      11. lower-+.f64 59.6

          \[\leadsto \color{blue}{0 + x} \]

   7. Applied rewrites 59.6%

      \[\leadsto \color{blue}{0 + x} \]
   8. Step-by-step derivation

   9. Applied rewrites 59.6%

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

   if -2e7 < (/.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 inf

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

      1. lower--.f64 3.7

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

   5. Applied rewrites 3.7%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. distribute-rgt-out N/A

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

      9. metadata-eval N/A

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

      10. remove-double-neg N/A

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

      11. mul-1-neg N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. fp-cancel-sign-sub-inv N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 97.2%

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

Alternative 4: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -122000000 \lor \neg \left(y \leq 180000000\right):\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x - 1}{y - -1}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -122000000.0) (not (<= y 180000000.0)))
   (- x (/ (- x 1.0) y))
   (fma y (/ (- x 1.0) (- y -1.0)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -122000000.0) || !(y <= 180000000.0)) {
		tmp = x - ((x - 1.0) / y);
	} else {
		tmp = fma(y, ((x - 1.0) / (y - -1.0)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -122000000.0) || !(y <= 180000000.0))
		tmp = Float64(x - Float64(Float64(x - 1.0) / y));
	else
		tmp = fma(y, Float64(Float64(x - 1.0) / Float64(y - -1.0)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -122000000.0], N[Not[LessEqual[y, 180000000.0]], $MachinePrecision]], N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(x - 1.0), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.22e8 or 1.8e8 < y

   1. Initial program 36.4%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 -1.22e8 < y < 1.8e8

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 4.0

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

   5. Applied rewrites 4.0%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. distribute-rgt-out N/A

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

      9. metadata-eval N/A

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

      10. remove-double-neg N/A

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

      11. mul-1-neg N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. fp-cancel-sign-sub-inv N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

   10. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -122000000 \lor \neg \left(y \leq 180000000\right):\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x - 1}{y - -1}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.75 \lor \neg \left(y \leq 650000\right):\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{y + 1}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.75) (not (<= y 650000.0)))
   (- x (/ (- x 1.0) y))
   (fma y (/ x (+ y 1.0)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.75) || !(y <= 650000.0)) {
		tmp = x - ((x - 1.0) / y);
	} else {
		tmp = fma(y, (x / (y + 1.0)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.75) || !(y <= 650000.0))
		tmp = Float64(x - Float64(Float64(x - 1.0) / y));
	else
		tmp = fma(y, Float64(x / Float64(y + 1.0)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.75], N[Not[LessEqual[y, 650000.0]], $MachinePrecision]], N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.75 or 6.5e5 < y

   1. Initial program 36.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(x + \frac{1}{y}\right) - \frac{x}{y}} \]
   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 99.1

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

   5. Applied rewrites 99.1%

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

   if -1.75 < y < 6.5e5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 4.1

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

   5. Applied rewrites 4.1%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. distribute-rgt-out N/A

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

      9. metadata-eval N/A

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

      10. remove-double-neg N/A

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

      11. mul-1-neg N/A

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

      12. distribute-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. fp-cancel-sign-sub-inv N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.75 \lor \neg \left(y \leq 650000\right):\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{y + 1}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 38.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   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

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.6%

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

4. Final simplification 99.1%

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

Alternative 7: 98.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 38.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   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

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.6%

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

4. Final simplification 99.0%

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

Alternative 8: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.82\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - y\right) \cdot x, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.82)))
   (- x (/ -1.0 y))
   (fma (* (- 1.0 y) x) y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.82)) {
		tmp = x - (-1.0 / y);
	} else {
		tmp = fma(((1.0 - y) * x), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.82))
		tmp = Float64(x - Float64(-1.0 / y));
	else
		tmp = fma(Float64(Float64(1.0 - y) * x), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.82]], $MachinePrecision]], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - y), $MachinePrecision] * x), $MachinePrecision] * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.819999999999999951 < y

   1. Initial program 38.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.4%

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

   if -1 < y < 0.819999999999999951

   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

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.82\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - y\right) \cdot x, y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.8))) (- x (/ -1.0 y)) (fma (- x 1.0) y 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 = fma((x - 1.0), y, 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 = fma(Float64(x - 1.0), y, 1.0);
	end
	return 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[(N[(x - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.80000000000000004 < y

   1. Initial program 38.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.4%

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

   if -1 < y < 0.80000000000000004

   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 98.8

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

   5. Applied rewrites 98.8%

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

4. Final simplification 98.2%

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

Alternative 10: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.1\right):\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.1))) (- x (/ x y)) (fma (- x 1.0) y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.1)) {
		tmp = x - (x / y);
	} else {
		tmp = fma((x - 1.0), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.1))
		tmp = Float64(x - Float64(x / y));
	else
		tmp = fma(Float64(x - 1.0), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.1]], $MachinePrecision]], N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(x - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.1000000000000001 < y

   1. Initial program 38.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 78.9%

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

   if -1 < y < 1.1000000000000001

   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 98.8

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

   5. Applied rewrites 98.8%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.1\right):\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 38.0%

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

      1. lift--.f64 N/A

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

      2. *-lft-identity N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. distribute-lft-neg-in N/A

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

      5. *-lft-identity N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. distribute-lft-neg-in N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 55.2

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

   4. Applied rewrites 55.2%

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

   5. Taylor expanded in y around inf

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

      1. *-lft-identity N/A

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. metadata-eval N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. associate-+r+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \color{blue}{0} + x \]

      11. lower-+.f64 78.0

          \[\leadsto \color{blue}{0 + x} \]

   7. Applied rewrites 78.0%

      \[\leadsto \color{blue}{0 + x} \]
   8. Step-by-step derivation

   9. Applied rewrites 78.0%

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

   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 98.8

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

   5. Applied rewrites 98.8%

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

Alternative 12: 38.3% 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 72.6%

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

3. Taylor expanded in y around inf

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

   1. lower--.f64 25.2

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

5. Applied rewrites 25.2%

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

6. Taylor expanded in x around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. *-lft-identity N/A

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

   4. metadata-eval N/A

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

   5. fp-cancel-sign-sub-inv N/A

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

   6. +-commutative N/A

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

   7. associate-/l* N/A

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

   8. distribute-rgt-out N/A

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

   9. metadata-eval N/A

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

   10. remove-double-neg N/A

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

   11. mul-1-neg N/A

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

   12. distribute-neg-in N/A

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

   13. mul-1-neg N/A

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

   14. fp-cancel-sign-sub-inv N/A

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

   15. metadata-eval N/A

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

   16. *-lft-identity N/A

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

   17. lower-fma.f64 N/A

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

8. Applied rewrites 80.3%

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

10. Applied rewrites 80.2%

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

11. Taylor expanded in y around 0

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

13. Applied rewrites 42.3%

    \[\leadsto \color{blue}{1} \]
14. 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 2024339 
(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))))