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

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

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: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;x - y\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (<= t_0 2e-6) (- x y) (if (<= t_0 2.0) 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 <= 2e-6) {
		tmp = x - y;
	} else if (t_0 <= 2.0) {
		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 <= 2e-6)
		tmp = Float64(x - y);
	elseif (t_0 <= 2.0)
		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, 2e-6], N[(x - y), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 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)) < 1.99999999999999991e-6

   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. --lowering--.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. accelerator-lowering-fma.f64 83.9

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

   5. Simplified 83.9%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 83.8%

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

   if 1.99999999999999991e-6 < (/.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. Simplified 97.1%

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

   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. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 100.0

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

   5. Simplified 100.0%

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

   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. accelerator-lowering-fma.f64 68.7

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

   8. Simplified 68.7%

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

Alternative 3: 85.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;x - y\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (<= t_0 2e-6) (- 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 <= 2e-6) {
		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 <= 2d-6) 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 <= 2e-6) {
		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 <= 2e-6:
		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 <= 2e-6)
		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 <= 2e-6)
		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, 2e-6], 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)) < 1.99999999999999991e-6 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

      \[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot \left(1 + -1 \cdot x\right)\right)} \]
   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. --lowering--.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. accelerator-lowering-fma.f64 79.0

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

   5. Simplified 79.0%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 78.9%

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

   if 1.99999999999999991e-6 < (/.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. Simplified 97.1%

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

Alternative 4: 74.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (<= t_0 2e-6) x (if (<= t_0 2.0) 1.0 x))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if (t_0 <= 2e-6) {
		tmp = x;
	} else if (t_0 <= 2.0) {
		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 = (x - y) / (1.0d0 - y)
    if (t_0 <= 2d-6) then
        tmp = x
    else if (t_0 <= 2.0d0) 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 = (x - y) / (1.0 - y);
	double tmp;
	if (t_0 <= 2e-6) {
		tmp = x;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / (1.0 - y)
	tmp = 0
	if t_0 <= 2e-6:
		tmp = x
	elif t_0 <= 2.0:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(1.0 - y))
	tmp = 0.0
	if (t_0 <= 2e-6)
		tmp = x;
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / (1.0 - y);
	tmp = 0.0;
	if (t_0 <= 2e-6)
		tmp = x;
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = x;
	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, 2e-6], x, If[LessEqual[t$95$0, 2.0], 1.0, x]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 1.99999999999999991e-6 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

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

   5. Simplified 61.3%

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

   if 1.99999999999999991e-6 < (/.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. Simplified 97.1%

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

Alternative 5: 98.8% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (- 1.0 x) y))))
   (if (<= y -1.0) 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 + ((1.0 - x) / y);
	double tmp;
	if (y <= -1.0) {
		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(Float64(1.0 - x) / y))
	tmp = 0.0
	if (y <= -1.0)
		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[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], 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 < -1 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1 + \left(-1 \cdot \frac{x}{y} + \frac{1}{y}\right)} \]
   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. +-lowering-+.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. /-lowering-/.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. --lowering--.f64 99.1

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

   5. Simplified 99.1%

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

   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. accelerator-lowering-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. +-lowering-+.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. accelerator-lowering-fma.f64 99.4

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

   5. Simplified 99.4%

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

4. Final simplification 99.2%

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

Alternative 6: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -0.86:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x + -1, \mathsf{fma}\left(y, y, y\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))))
   (if (<= y -0.86) 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 <= -0.86) {
		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 <= -0.86)
		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, -0.86], 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 < -0.859999999999999987 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1 + \left(-1 \cdot \frac{x}{y} + \frac{1}{y}\right)} \]
   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. +-lowering-+.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. /-lowering-/.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. --lowering--.f64 99.1

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

   5. Simplified 99.1%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. neg-lowering-neg.f64 98.5

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

   8. Simplified 98.5%

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

      1. distribute-frac-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. --lowering--.f64 N/A

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

      4. /-lowering-/.f64 98.5

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

   10. Applied egg-rr 98.5%

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

   if -0.859999999999999987 < 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. accelerator-lowering-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. +-lowering-+.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. accelerator-lowering-fma.f64 99.4

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

   5. Simplified 99.4%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.86:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x + -1, \mathsf{fma}\left(y, y, y\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]
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

      \[\leadsto \color{blue}{1 + \left(-1 \cdot \frac{x}{y} + \frac{1}{y}\right)} \]
   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. +-lowering-+.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. /-lowering-/.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. --lowering--.f64 99.1

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

   5. Simplified 99.1%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. neg-lowering-neg.f64 98.5

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

   8. Simplified 98.5%

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

      1. distribute-frac-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. --lowering--.f64 N/A

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

      4. /-lowering-/.f64 98.5

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

   10. Applied egg-rr 98.5%

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

   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. --lowering--.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. accelerator-lowering-fma.f64 98.8

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

   5. Simplified 98.8%

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

Alternative 8: 86.1% accurate, 0.8× speedup

Math

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

FPCore

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

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

   5. Simplified 73.9%

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

   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. 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. --lowering--.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. accelerator-lowering-fma.f64 98.8

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

   5. Simplified 98.8%

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

Alternative 9: 39.2% 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. Simplified 39.0%

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

Reproduce

herbie shell --seed 2024204 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))