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

Percentage Accurate: 100.0% → 100.0%

Time: 8.2s

Alternatives: 9

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x - y}{1 - y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - y}{1 - y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - y}{1 - y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{x - y}{1 - y} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 85.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t\_0 \leq 0.001:\\ \;\;\;\;x - y\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (<= t_0 0.001) (- x y) (if (<= t_0 2.0) 1.0 (- x (* x y))))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if (t_0 <= 0.001) {
		tmp = x - y;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x - (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / (1.0d0 - y)
    if (t_0 <= 0.001d0) then
        tmp = x - y
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = x - (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if (t_0 <= 0.001) {
		tmp = x - y;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / (1.0 - y)
	tmp = 0
	if t_0 <= 0.001:
		tmp = x - y
	elif t_0 <= 2.0:
		tmp = 1.0
	else:
		tmp = x - (x * y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(1.0 - y))
	tmp = 0.0
	if (t_0 <= 0.001)
		tmp = Float64(x - y);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(x - Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / (1.0 - y);
	tmp = 0.0;
	if (t_0 <= 0.001)
		tmp = x - y;
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = x - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.001], N[(x - y), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 1e-3

   1. Initial program 99.9%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 83.7%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 82.6

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

   8. Applied rewrites 82.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 83.2%

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. neg-mul-1 N/A

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

       4. unsub-neg N/A

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

       5. lower--.f64 83.2

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

   13. Applied rewrites 83.2%

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

   if 1e-3 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 2

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 96.5%

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

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

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 60.2

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

   8. Applied rewrites 60.2%

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

   10. Applied rewrites 62.1%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.001:\\ \;\;\;\;x - y\\ \mathbf{elif}\;\frac{x - y}{1 - y} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t\_0 \leq 0.001:\\ \;\;\;\;x - y\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (<= t_0 0.001) (- x y) (if (<= t_0 2.0) 1.0 (- x y)))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if (t_0 <= 0.001) {
		tmp = x - y;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / (1.0d0 - y)
    if (t_0 <= 0.001d0) then
        tmp = x - y
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = x - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if (t_0 <= 0.001) {
		tmp = x - y;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / (1.0 - y)
	tmp = 0
	if t_0 <= 0.001:
		tmp = x - y
	elif t_0 <= 2.0:
		tmp = 1.0
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(1.0 - y))
	tmp = 0.0
	if (t_0 <= 0.001)
		tmp = Float64(x - y);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(x - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / (1.0 - y);
	tmp = 0.0;
	if (t_0 <= 0.001)
		tmp = x - y;
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.001], N[(x - y), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, N[(x - y), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 1e-3 or 2 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 76.6

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

   8. Applied rewrites 76.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 77.3%

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. neg-mul-1 N/A

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

       4. unsub-neg N/A

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

       5. lower--.f64 77.3

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

   13. Applied rewrites 77.3%

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

   if 1e-3 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 2

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 96.5%

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

Alternative 4: 84.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+76}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -124000:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right) - y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.2e+76)
   1.0
   (if (<= y -124000.0) (/ x (- y)) (if (<= y 1.0) (- (fma y x x) y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.2e+76) {
		tmp = 1.0;
	} else if (y <= -124000.0) {
		tmp = x / -y;
	} else if (y <= 1.0) {
		tmp = fma(y, x, x) - y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.2e+76)
		tmp = 1.0;
	elseif (y <= -124000.0)
		tmp = Float64(x / Float64(-y));
	elseif (y <= 1.0)
		tmp = Float64(fma(y, x, x) - y);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.2e+76], 1.0, If[LessEqual[y, -124000.0], N[(x / (-y)), $MachinePrecision], If[LessEqual[y, 1.0], N[(N[(y * x + x), $MachinePrecision] - y), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.2e76 or 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 76.5%

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

   if -2.2e76 < y < -124000

   1. Initial program 99.8%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 67.0

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 64.9

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

   8. Applied rewrites 64.9%

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

   if -124000 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. associate--r+ N/A

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

      8. *-commutative N/A

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

      9. cancel-sign-sub N/A

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

      10. +-commutative N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. sub-neg N/A

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

      15. remove-double-neg N/A

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

      16. *-rgt-identity N/A

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

      17. distribute-lft-out N/A

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

      18. distribute-rgt-out N/A

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

      19. *-lft-identity N/A

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

      20. lower-fma.f64 97.7

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

   5. Applied rewrites 97.7%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+76}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -124000:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right) - y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 99.9%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. div-sub N/A

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

      5. lower-+.f64 N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 97.9

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

   5. Applied rewrites 97.9%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. mul-1-neg N/A

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

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

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-fma.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 98.7

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

   8. Applied rewrites 98.7%

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

4. Final simplification 99.0%

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

Alternative 6: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -0.85:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x + -1, \mathsf{fma}\left(y, y, y\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))))
   (if (<= y -0.85) t_0 (if (<= y 1.0) (fma (+ x -1.0) (fma y y y) x) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -0.85) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = fma((x + -1.0), fma(y, y, y), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(x / y))
	tmp = 0.0
	if (y <= -0.85)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = fma(Float64(x + -1.0), fma(y, y, y), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.85], t$95$0, If[LessEqual[y, 1.0], N[(N[(x + -1.0), $MachinePrecision] * N[(y * y + y), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.849999999999999978 or 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 97.7

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 97.8

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

   8. Applied rewrites 97.8%

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

   if -0.849999999999999978 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. mul-1-neg N/A

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

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

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-fma.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.85:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x + -1, \mathsf{fma}\left(y, y, y\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -0.82:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right) - y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))))
   (if (<= y -0.82) t_0 (if (<= y 1.0) (- (fma y x x) y) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -0.82) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = fma(y, x, x) - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(x / y))
	tmp = 0.0
	if (y <= -0.82)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = Float64(fma(y, x, x) - y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.82], t$95$0, If[LessEqual[y, 1.0], N[(N[(y * x + x), $MachinePrecision] - y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.819999999999999951 or 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 97.7

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 97.8

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

   8. Applied rewrites 97.8%

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

   if -0.819999999999999951 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. associate--r+ N/A

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

      8. *-commutative N/A

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

      9. cancel-sign-sub N/A

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

      10. +-commutative N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. sub-neg N/A

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

      15. remove-double-neg N/A

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

      16. *-rgt-identity N/A

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

      17. distribute-lft-out N/A

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

      18. distribute-rgt-out N/A

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

      19. *-lft-identity N/A

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

      20. lower-fma.f64 99.0

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

   5. Applied rewrites 99.0%

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

Alternative 8: 50.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 10^{-6}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 1e-6) (- y) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 1e-6) {
		tmp = -y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x - y) / (1.0d0 - y)) <= 1d-6) then
        tmp = -y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 1e-6) {
		tmp = -y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 1e-6:
		tmp = -y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 1e-6)
		tmp = Float64(-y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x - y) / (1.0 - y)) <= 1e-6)
		tmp = -y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 1e-6], (-y), 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 9.99999999999999955e-7

   1. Initial program 99.9%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. neg-sub0 N/A

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

      5. associate--r- N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 30.3

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

   5. Applied rewrites 30.3%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(y\right)} \]

      2. lower-neg.f64 29.5

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

   8. Applied rewrites 29.5%

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

   if 9.99999999999999955e-7 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 65.2%

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

Alternative 9: 38.8% accurate, 18.0× 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 100.0%

   \[\frac{x - y}{1 - y} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

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

5. Applied rewrites 36.1%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))