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

Percentage Accurate: 65.0% → 99.9%

Time: 8.7s

Alternatives: 15

Speedup: 2.0×

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: 65.0% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{1 - \mathsf{fma}\left(\frac{x + -1}{y}, -1 + \frac{1}{y}, x\right)}{y}\\ \mathbf{if}\;y \leq -12500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 14200:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ (- 1.0 (fma (/ (+ x -1.0) y) (+ -1.0 (/ 1.0 y)) x)) y))))
   (if (<= y -12500.0)
     t_0
     (if (<= y 14200.0) (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y))) t_0))))

C

double code(double x, double y) {
	double t_0 = x + ((1.0 - fma(((x + -1.0) / y), (-1.0 + (1.0 / y)), x)) / y);
	double tmp;
	if (y <= -12500.0) {
		tmp = t_0;
	} else if (y <= 14200.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x + Float64(Float64(1.0 - fma(Float64(Float64(x + -1.0) / y), Float64(-1.0 + Float64(1.0 / y)), x)) / y))
	tmp = 0.0
	if (y <= -12500.0)
		tmp = t_0;
	elseif (y <= 14200.0)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(N[(1.0 - N[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] * N[(-1.0 + N[(1.0 / y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -12500.0], t$95$0, If[LessEqual[y, 14200.0], N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -12500 or 14200 < y

   1. Initial program 30.4%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 99.9%

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

   if -12500 < y < 14200

   1. Initial program 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -12500:\\ \;\;\;\;x + \frac{1 - \mathsf{fma}\left(\frac{x + -1}{y}, -1 + \frac{1}{y}, x\right)}{y}\\ \mathbf{elif}\;y \leq 14200:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - \mathsf{fma}\left(\frac{x + -1}{y}, -1 + \frac{1}{y}, x\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+206}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{+16}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;t\_0 \leq 10^{-15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* y (- 1.0 x)) (+ y 1.0))))
   (if (<= t_0 -5e+206)
     x
     (if (<= t_0 -1e+16) (* y x) (if (<= t_0 1e-15) 1.0 x)))))

C

double code(double x, double y) {
	double t_0 = (y * (1.0 - x)) / (y + 1.0);
	double tmp;
	if (t_0 <= -5e+206) {
		tmp = x;
	} else if (t_0 <= -1e+16) {
		tmp = y * x;
	} else if (t_0 <= 1e-15) {
		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 = (y * (1.0d0 - x)) / (y + 1.0d0)
    if (t_0 <= (-5d+206)) then
        tmp = x
    else if (t_0 <= (-1d+16)) then
        tmp = y * x
    else if (t_0 <= 1d-15) 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 = (y * (1.0 - x)) / (y + 1.0);
	double tmp;
	if (t_0 <= -5e+206) {
		tmp = x;
	} else if (t_0 <= -1e+16) {
		tmp = y * x;
	} else if (t_0 <= 1e-15) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * (1.0 - x)) / (y + 1.0)
	tmp = 0
	if t_0 <= -5e+206:
		tmp = x
	elif t_0 <= -1e+16:
		tmp = y * x
	elif t_0 <= 1e-15:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * Float64(1.0 - x)) / Float64(y + 1.0))
	tmp = 0.0
	if (t_0 <= -5e+206)
		tmp = x;
	elseif (t_0 <= -1e+16)
		tmp = Float64(y * x);
	elseif (t_0 <= 1e-15)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * (1.0 - x)) / (y + 1.0);
	tmp = 0.0;
	if (t_0 <= -5e+206)
		tmp = x;
	elseif (t_0 <= -1e+16)
		tmp = y * x;
	elseif (t_0 <= 1e-15)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+206], x, If[LessEqual[t$95$0, -1e+16], N[(y * x), $MachinePrecision], If[LessEqual[t$95$0, 1e-15], 1.0, x]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < -5.0000000000000002e206 or 1.0000000000000001e-15 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 33.9%

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

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

      7. associate-/r/ N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-rgt-identity N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   4. Applied rewrites 30.9%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-in N/A

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

      3. metadata-eval N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. associate-+r+ N/A

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

      7. metadata-eval N/A

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

      8. +-lft-identity 59.5

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

   7. Applied rewrites 59.5%

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

   if -5.0000000000000002e206 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < -1e16

   1. Initial program 99.9%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 65.0%

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

   if -1e16 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 1.0000000000000001e-15

   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

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

   5. Applied rewrites 97.5%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(1 - x\right)}{y + 1} \leq -5 \cdot 10^{+206}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{y \cdot \left(1 - x\right)}{y + 1} \leq -1 \cdot 10^{+16}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;\frac{y \cdot \left(1 - x\right)}{y + 1} \leq 10^{-15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3050000:\\ \;\;\;\;x - \frac{-1 - \frac{-1 + \frac{1}{y}}{y}}{y}\\ \mathbf{elif}\;y \leq 320000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\mathsf{fma}\left(y, y, -1\right)}, \left(x + -1\right) \cdot \left(y - y \cdot y\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(\frac{x + -1}{y} - -1\right) - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3050000.0)
   (- x (/ (- -1.0 (/ (+ -1.0 (/ 1.0 y)) y)) y))
   (if (<= y 320000.0)
     (fma (/ -1.0 (fma y y -1.0)) (* (+ x -1.0) (- y (* y y))) 1.0)
     (+ x (/ (- (- (/ (+ x -1.0) y) -1.0) x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3050000.0) {
		tmp = x - ((-1.0 - ((-1.0 + (1.0 / y)) / y)) / y);
	} else if (y <= 320000.0) {
		tmp = fma((-1.0 / fma(y, y, -1.0)), ((x + -1.0) * (y - (y * y))), 1.0);
	} else {
		tmp = x + (((((x + -1.0) / y) - -1.0) - x) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3050000.0)
		tmp = Float64(x - Float64(Float64(-1.0 - Float64(Float64(-1.0 + Float64(1.0 / y)) / y)) / y));
	elseif (y <= 320000.0)
		tmp = fma(Float64(-1.0 / fma(y, y, -1.0)), Float64(Float64(x + -1.0) * Float64(y - Float64(y * y))), 1.0);
	else
		tmp = Float64(x + Float64(Float64(Float64(Float64(Float64(x + -1.0) / y) - -1.0) - x) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3050000.0], N[(x - N[(N[(-1.0 - N[(N[(-1.0 + N[(1.0 / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 320000.0], N[(N[(-1.0 / N[(y * y + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(x + -1.0), $MachinePrecision] * N[(y - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(x + N[(N[(N[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] - -1.0), $MachinePrecision] - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.05e6

   1. Initial program 34.1%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 99.8%

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

   if -3.05e6 < y < 3.2e5

   1. Initial program 99.8%

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

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. lift-+.f64 N/A

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

      9. flip-+ N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r/ N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   if 3.2e5 < y

   1. Initial program 26.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 4.2%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3050000:\\ \;\;\;\;x - \frac{-1 - \frac{-1 + \frac{1}{y}}{y}}{y}\\ \mathbf{elif}\;y \leq 320000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\mathsf{fma}\left(y, y, -1\right)}, \left(x + -1\right) \cdot \left(y - y \cdot y\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(\frac{x + -1}{y} - -1\right) - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{\left(\frac{x + -1}{y} - -1\right) - x}{y}\\ \mathbf{if}\;y \leq -300000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 320000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\mathsf{fma}\left(y, y, -1\right)}, \left(x + -1\right) \cdot \left(y - y \cdot y\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ (- (- (/ (+ x -1.0) y) -1.0) x) y))))
   (if (<= y -300000.0)
     t_0
     (if (<= y 320000.0)
       (fma (/ -1.0 (fma y y -1.0)) (* (+ x -1.0) (- y (* y y))) 1.0)
       t_0))))

C

double code(double x, double y) {
	double t_0 = x + (((((x + -1.0) / y) - -1.0) - x) / y);
	double tmp;
	if (y <= -300000.0) {
		tmp = t_0;
	} else if (y <= 320000.0) {
		tmp = fma((-1.0 / fma(y, y, -1.0)), ((x + -1.0) * (y - (y * y))), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x + Float64(Float64(Float64(Float64(Float64(x + -1.0) / y) - -1.0) - x) / y))
	tmp = 0.0
	if (y <= -300000.0)
		tmp = t_0;
	elseif (y <= 320000.0)
		tmp = fma(Float64(-1.0 / fma(y, y, -1.0)), Float64(Float64(x + -1.0) * Float64(y - Float64(y * y))), 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(N[(N[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] - -1.0), $MachinePrecision] - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -300000.0], t$95$0, If[LessEqual[y, 320000.0], N[(N[(-1.0 / N[(y * y + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(x + -1.0), $MachinePrecision] * N[(y - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3e5 or 3.2e5 < y

   1. Initial program 30.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 3.7%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 99.9%

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

   if -3e5 < y < 3.2e5

   1. Initial program 99.8%

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

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. lift-+.f64 N/A

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

      9. flip-+ N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r/ N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -300000:\\ \;\;\;\;x + \frac{\left(\frac{x + -1}{y} - -1\right) - x}{y}\\ \mathbf{elif}\;y \leq 320000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\mathsf{fma}\left(y, y, -1\right)}, \left(x + -1\right) \cdot \left(y - y \cdot y\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(\frac{x + -1}{y} - -1\right) - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{\left(\frac{x + -1}{y} - -1\right) - x}{y}\\ \mathbf{if}\;y \leq -300000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 240000:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(y, -x, y\right)}{\mathsf{fma}\left(y, y, -1\right)}, 1 - y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ (- (- (/ (+ x -1.0) y) -1.0) x) y))))
   (if (<= y -300000.0)
     t_0
     (if (<= y 240000.0)
       (fma (/ (fma y (- x) y) (fma y y -1.0)) (- 1.0 y) 1.0)
       t_0))))

C

double code(double x, double y) {
	double t_0 = x + (((((x + -1.0) / y) - -1.0) - x) / y);
	double tmp;
	if (y <= -300000.0) {
		tmp = t_0;
	} else if (y <= 240000.0) {
		tmp = fma((fma(y, -x, y) / fma(y, y, -1.0)), (1.0 - y), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x + Float64(Float64(Float64(Float64(Float64(x + -1.0) / y) - -1.0) - x) / y))
	tmp = 0.0
	if (y <= -300000.0)
		tmp = t_0;
	elseif (y <= 240000.0)
		tmp = fma(Float64(fma(y, Float64(-x), y) / fma(y, y, -1.0)), Float64(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[(N[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] - -1.0), $MachinePrecision] - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -300000.0], t$95$0, If[LessEqual[y, 240000.0], N[(N[(N[(y * (-x) + y), $MachinePrecision] / N[(y * y + -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 - y), $MachinePrecision] + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3e5 or 2.4e5 < y

   1. Initial program 30.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 3.7%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 99.9%

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

   if -3e5 < y < 2.4e5

   1. Initial program 99.8%

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

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

      7. associate-/r/ N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-rgt-identity N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -300000:\\ \;\;\;\;x + \frac{\left(\frac{x + -1}{y} - -1\right) - x}{y}\\ \mathbf{elif}\;y \leq 240000:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(y, -x, y\right)}{\mathsf{fma}\left(y, y, -1\right)}, 1 - y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(\frac{x + -1}{y} - -1\right) - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{\left(\frac{x + -1}{y} - -1\right) - x}{y}\\ \mathbf{if}\;y \leq -270000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 360000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ (- (- (/ (+ x -1.0) y) -1.0) x) y))))
   (if (<= y -270000.0)
     t_0
     (if (<= y 360000.0) (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y))) t_0))))

C

double code(double x, double y) {
	double t_0 = x + (((((x + -1.0) / y) - -1.0) - x) / y);
	double tmp;
	if (y <= -270000.0) {
		tmp = t_0;
	} else if (y <= 360000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-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) - (-1.0d0)) - x) / y)
    if (y <= (-270000.0d0)) then
        tmp = t_0
    else if (y <= 360000.0d0) then
        tmp = 1.0d0 + ((y * (1.0d0 - x)) / ((-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) - -1.0) - x) / y);
	double tmp;
	if (y <= -270000.0) {
		tmp = t_0;
	} else if (y <= 360000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (((((x + -1.0) / y) - -1.0) - x) / y)
	tmp = 0
	if y <= -270000.0:
		tmp = t_0
	elif y <= 360000.0:
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(Float64(Float64(Float64(Float64(x + -1.0) / y) - -1.0) - x) / y))
	tmp = 0.0
	if (y <= -270000.0)
		tmp = t_0;
	elseif (y <= 360000.0)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / 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) - -1.0) - x) / y);
	tmp = 0.0;
	if (y <= -270000.0)
		tmp = t_0;
	elseif (y <= 360000.0)
		tmp = 1.0 + ((y * (1.0 - x)) / (-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[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] - -1.0), $MachinePrecision] - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -270000.0], t$95$0, If[LessEqual[y, 360000.0], N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.7e5 or 3.6e5 < y

   1. Initial program 30.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 3.7%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 99.9%

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

   if -2.7e5 < y < 3.6e5

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -270000:\\ \;\;\;\;x + \frac{\left(\frac{x + -1}{y} - -1\right) - x}{y}\\ \mathbf{elif}\;y \leq 360000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(\frac{x + -1}{y} - -1\right) - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -120000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 22000000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ (- 1.0 x) y))))
   (if (<= y -120000000.0)
     t_0
     (if (<= y 22000000.0) (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y))) t_0))))

C

double code(double x, double y) {
	double t_0 = x + ((1.0 - x) / y);
	double tmp;
	if (y <= -120000000.0) {
		tmp = t_0;
	} else if (y <= 22000000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-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 + ((1.0d0 - x) / y)
    if (y <= (-120000000.0d0)) then
        tmp = t_0
    else if (y <= 22000000.0d0) then
        tmp = 1.0d0 + ((y * (1.0d0 - x)) / ((-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 + ((1.0 - x) / y);
	double tmp;
	if (y <= -120000000.0) {
		tmp = t_0;
	} else if (y <= 22000000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + ((1.0 - x) / y)
	tmp = 0
	if y <= -120000000.0:
		tmp = t_0
	elif y <= 22000000.0:
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(Float64(1.0 - x) / y))
	tmp = 0.0
	if (y <= -120000000.0)
		tmp = t_0;
	elseif (y <= 22000000.0)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + ((1.0 - x) / y);
	tmp = 0.0;
	if (y <= -120000000.0)
		tmp = t_0;
	elseif (y <= 22000000.0)
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -120000000.0], t$95$0, If[LessEqual[y, 22000000.0], N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.2e8 or 2.2e7 < y

   1. Initial program 29.7%

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

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -1.2e8 < y < 2.2e7

   1. Initial program 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -120000000:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 22000000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 30.9%

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

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 98.5

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

   5. Applied rewrites 98.5%

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

   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

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l- N/A

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

      4. unsub-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-in N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft-out N/A

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

      11. lower-*.f64 N/A

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

      12. distribute-lft-out-- N/A

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

      13. *-commutative N/A

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

      14. *-rgt-identity N/A

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

      15. lower--.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 9: 98.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 30.9%

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

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 98.5

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

   5. Applied rewrites 98.5%

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

   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

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

   5. Applied rewrites 73.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

   8. Applied rewrites 100.0%

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

Alternative 10: 98.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 30.9%

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

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 98.5

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

   5. Applied rewrites 98.5%

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

   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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

Alternative 11: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{1}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(y, x + -1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ 1.0 y))))
   (if (<= y -1.0) t_0 (if (<= y 0.8) (fma y (+ x -1.0) 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = x + (1.0 / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 0.8) {
		tmp = fma(y, (x + -1.0), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x + Float64(1.0 / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 0.8)
		tmp = fma(y, Float64(x + -1.0), 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 0.8], N[(y * N[(x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.80000000000000004 < y

   1. Initial program 30.9%

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

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.1%

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

   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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

Alternative 12: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{x}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.05:\\ \;\;\;\;\mathsf{fma}\left(y, x + -1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ x y))))
   (if (<= y -1.0) t_0 (if (<= y 1.05) (fma y (+ x -1.0) 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 <= 1.05) {
		tmp = fma(y, (x + -1.0), 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 <= 1.05)
		tmp = fma(y, Float64(x + -1.0), 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, 1.05], N[(y * N[(x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.05000000000000004 < y

   1. Initial program 30.9%

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

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 68.9%

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

   if -1 < y < 1.05000000000000004

   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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

Alternative 13: 86.1% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 1.0) (fma y (+ x -1.0) 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(y, (x + -1.0), 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(y, Float64(x + -1.0), 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[(y * N[(x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 30.9%

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

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

      7. associate-/r/ N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-rgt-identity N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   4. Applied rewrites 24.9%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-in N/A

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

      3. metadata-eval N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. associate-+r+ N/A

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

      7. metadata-eval N/A

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

      8. +-lft-identity 68.5

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

   7. Applied rewrites 68.5%

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

   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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

Alternative 14: 73.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) x (if (<= y 7e-15) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 7e-15) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 7d-15) then
        tmp = 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 7e-15) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 7e-15:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 7e-15)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 7e-15)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 7e-15], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 7.0000000000000001e-15 < y

   1. Initial program 31.9%

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

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

      7. associate-/r/ N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-rgt-identity N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   4. Applied rewrites 26.0%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-in N/A

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

      3. metadata-eval N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. associate-+r+ N/A

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

      7. metadata-eval N/A

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

      8. +-lft-identity 67.6

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

   7. Applied rewrites 67.6%

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

   if -1 < y < 7.0000000000000001e-15

   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

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

   5. Applied rewrites 75.1%

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

Alternative 15: 39.1% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 64.1%

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

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. flip-+ N/A

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

   7. associate-/r/ N/A

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

   8. distribute-rgt-neg-in N/A

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-in N/A

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

   12. metadata-eval N/A

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

   13. +-commutative N/A

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

   14. *-rgt-identity N/A

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

   15. sub-neg N/A

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

   16. metadata-eval N/A

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

   17. lower-fma.f64 N/A

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

4. Applied rewrites 61.0%

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

5. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-in N/A

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

   3. metadata-eval N/A

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

   4. mul-1-neg N/A

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

   5. remove-double-neg N/A

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

   6. associate-+r+ N/A

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

   7. metadata-eval N/A

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

   8. +-lft-identity 37.3

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

7. Applied rewrites 37.3%

   \[\leadsto \color{blue}{x} \]
8. Add Preprocessing

Developer Target 1: 99.6% 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 2024233 
(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))))