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

Percentage Accurate: 100.0% → 100.0%

Time: 4.4s

Alternatives: 8

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: 96.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ t_1 := \frac{x}{1 - y}\\ \mathbf{if}\;t\_0 \leq -30000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-77}:\\ \;\;\;\;\mathsf{fma}\left(-1, y, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{y}{y - 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))) (t_1 (/ x (- 1.0 y))))
   (if (<= t_0 -30000.0)
     t_1
     (if (<= t_0 1e-77)
       (fma -1.0 y x)
       (if (<= t_0 2.0) (/ y (- y 1.0)) t_1)))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double t_1 = x / (1.0 - y);
	double tmp;
	if (t_0 <= -30000.0) {
		tmp = t_1;
	} else if (t_0 <= 1e-77) {
		tmp = fma(-1.0, y, x);
	} else if (t_0 <= 2.0) {
		tmp = y / (y - 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(1.0 - y))
	t_1 = Float64(x / Float64(1.0 - y))
	tmp = 0.0
	if (t_0 <= -30000.0)
		tmp = t_1;
	elseif (t_0 <= 1e-77)
		tmp = fma(-1.0, y, x);
	elseif (t_0 <= 2.0)
		tmp = Float64(y / Float64(y - 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -30000.0], t$95$1, If[LessEqual[t$95$0, 1e-77], N[(-1.0 * y + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(y / N[(y - 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < -3e4 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 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 98.3

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

   5. Applied rewrites 98.3%

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

   if -3e4 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 9.9999999999999993e-78

   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. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.3%

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

   if 9.9999999999999993e-78 < (/.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 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. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 98.5

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

   5. Applied rewrites 98.5%

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

Alternative 3: 97.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ t_1 := \frac{x}{1 - y}\\ \mathbf{if}\;t\_0 \leq -30000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(-1, y, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))) (t_1 (/ x (- 1.0 y))))
   (if (<= t_0 -30000.0)
     t_1
     (if (<= t_0 0.02) (fma -1.0 y x) (if (<= t_0 2.0) 1.0 t_1)))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double t_1 = x / (1.0 - y);
	double tmp;
	if (t_0 <= -30000.0) {
		tmp = t_1;
	} else if (t_0 <= 0.02) {
		tmp = fma(-1.0, y, x);
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(1.0 - y))
	t_1 = Float64(x / Float64(1.0 - y))
	tmp = 0.0
	if (t_0 <= -30000.0)
		tmp = t_1;
	elseif (t_0 <= 0.02)
		tmp = fma(-1.0, y, x);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -30000.0], t$95$1, If[LessEqual[t$95$0, 0.02], N[(-1.0 * y + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < -3e4 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 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 98.3

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

   5. Applied rewrites 98.3%

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

   if -3e4 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.0200000000000000004

   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. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 96.8

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

   5. Applied rewrites 96.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 96.8%

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

   if 0.0200000000000000004 < (/.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 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 3.3

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

   5. Applied rewrites 3.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 2.8%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 97.1%

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

Alternative 4: 72.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(x, y, x\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-7}:\\ \;\;\;\;-y\\ \mathbf{elif}\;t\_0 \leq 40:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (<= t_0 -5e-15)
     (fma x y x)
     (if (<= t_0 4e-7) (- y) (if (<= t_0 40.0) 1.0 (fma x y x))))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if (t_0 <= -5e-15) {
		tmp = fma(x, y, x);
	} else if (t_0 <= 4e-7) {
		tmp = -y;
	} else if (t_0 <= 40.0) {
		tmp = 1.0;
	} else {
		tmp = fma(x, y, x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(1.0 - y))
	tmp = 0.0
	if (t_0 <= -5e-15)
		tmp = fma(x, y, x);
	elseif (t_0 <= 4e-7)
		tmp = Float64(-y);
	elseif (t_0 <= 40.0)
		tmp = 1.0;
	else
		tmp = fma(x, y, x);
	end
	return 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, -5e-15], N[(x * y + x), $MachinePrecision], If[LessEqual[t$95$0, 4e-7], (-y), If[LessEqual[t$95$0, 40.0], 1.0, N[(x * y + x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < -4.99999999999999999e-15 or 40 < (/.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 97.3

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 61.9%

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

   if -4.99999999999999999e-15 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 3.9999999999999998e-7

   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. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 66.4%

      \[\leadsto -y \]

   if 3.9999999999999998e-7 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 40

   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 4.7

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

   5. Applied rewrites 4.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 3.9%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 95.3%

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

Alternative 5: 50.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 4e-7) (- y) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 4e-7) {
		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)) <= 4d-7) 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)) <= 4e-7) {
		tmp = -y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 4e-7:
		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)) <= 4e-7)
		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)) <= 4e-7)
		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], 4e-7], (-y), 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 3.9999999999999998e-7

   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. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 82.9

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

   5. Applied rewrites 82.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 37.4%

      \[\leadsto -y \]

   if 3.9999999999999998e-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 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 39.8

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

   5. Applied rewrites 39.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 25.1%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 62.0%

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

Alternative 6: 86.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < 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 31.0

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

   5. Applied rewrites 31.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 2.5%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 69.2%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 96.1

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

   5. Applied rewrites 96.1%

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

Alternative 7: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(-1, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.1) 1.0 (if (<= y 1.0) (fma -1.0 y x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.1) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = fma(-1.0, y, x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.1)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = fma(-1.0, y, x);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.1], 1.0, If[LessEqual[y, 1.0], N[(-1.0 * y + x), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -3.10000000000000009 or 1 < 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 31.0

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

   5. Applied rewrites 31.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 2.5%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 69.2%

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

   if -3.10000000000000009 < 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. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 96.1

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

   5. Applied rewrites 96.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 95.7%

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

Alternative 8: 39.3% 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 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 50.6

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

5. Applied rewrites 50.6%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 35.9%

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

9. Taylor expanded in y around inf

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

11. Applied rewrites 35.1%

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

Reproduce

herbie shell --seed 2024327 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))