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

Percentage Accurate: 100.0% → 100.0%

Time: 6.2s

Alternatives: 7

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: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -1.25:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (- 1.0 x) y))))
   (if (<= y -1.25) t_0 (if (<= y 1.0) (- x y) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((1.0 - x) / y);
	double tmp;
	if (y <= -1.25) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = x - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((1.0d0 - x) / y)
    if (y <= (-1.25d0)) then
        tmp = t_0
    else if (y <= 1.0d0) then
        tmp = x - y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + ((1.0 - x) / y);
	double tmp;
	if (y <= -1.25) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = x - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((1.0 - x) / y)
	tmp = 0
	if y <= -1.25:
		tmp = t_0
	elif y <= 1.0:
		tmp = x - y
	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 <= -1.25)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = Float64(x - y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + ((1.0 - x) / y);
	tmp = 0.0;
	if (y <= -1.25)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = x - y;
	else
		tmp = t_0;
	end
	tmp_2 = 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, -1.25], t$95$0, If[LessEqual[y, 1.0], N[(x - y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.25 or 1 < y

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. sub0-neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-neg-frac N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-out N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{\left(y - 1\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(\color{blue}{y} - 1\right)\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(y + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(y + -1\right)\right) \]

      17. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{+.f64}\left(y, \color{blue}{-1}\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{y - x}{y + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. div-sub N/A

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

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1 - x}{y}\right)}\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1 + \left(\mathsf{neg}\left(x\right)\right)}{y}\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1 + -1 \cdot x}{y}\right)\right) \]

      8. /-lowering-/.f64 N/A

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

      9. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right), y\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(1 - x\right), y\right)\right) \]

      11. --lowering--.f64 99.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), y\right)\right) \]

   7. Simplified 99.6%

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

   if -1.25 < y < 1

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. sub0-neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-neg-frac N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-out N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{\left(y - 1\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(\color{blue}{y} - 1\right)\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(y + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(y + -1\right)\right) \]

      17. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{+.f64}\left(y, \color{blue}{-1}\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{y - x}{y + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \color{blue}{-1}\right) \]
   6. Step-by-step derivation

   7. Simplified 99.0%

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

   8. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 99.0%

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{y}\right) \]

   10. Simplified 99.0%

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

Alternative 3: 73.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-72}:\\ \;\;\;\;0 - y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y -1.8e-72) (- 0.0 y) (if (<= y 1.0) x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= -1.8e-72) {
		tmp = 0.0 - y;
	} else if (y <= 1.0) {
		tmp = x;
	} 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 (y <= (-1.0d0)) then
        tmp = 1.0d0
    else if (y <= (-1.8d-72)) then
        tmp = 0.0d0 - y
    else if (y <= 1.0d0) then
        tmp = x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= -1.8e-72) {
		tmp = 0.0 - y;
	} else if (y <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= -1.8e-72:
		tmp = 0.0 - y
	elif y <= 1.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = 1.0;
	elseif (y <= -1.8e-72)
		tmp = Float64(0.0 - y);
	elseif (y <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = 1.0;
	elseif (y <= -1.8e-72)
		tmp = 0.0 - y;
	elseif (y <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], 1.0, If[LessEqual[y, -1.8e-72], N[(0.0 - y), $MachinePrecision], If[LessEqual[y, 1.0], x, 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 1 < y

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. sub0-neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-neg-frac N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-out N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{\left(y - 1\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(\color{blue}{y} - 1\right)\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(y + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(y + -1\right)\right) \]

      17. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{+.f64}\left(y, \color{blue}{-1}\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{y - x}{y + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

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

   7. Simplified 69.6%

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

   if -1 < y < -1.8e-72

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. sub0-neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-neg-frac N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-out N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{\left(y - 1\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(\color{blue}{y} - 1\right)\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(y + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(y + -1\right)\right) \]

      17. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{+.f64}\left(y, \color{blue}{-1}\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{y - x}{y + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \color{blue}{-1}\right) \]
   6. Step-by-step derivation

   7. Simplified 96.8%

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

   8. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{y}, -1\right) \]
   9. Step-by-step derivation

   10. Simplified 64.6%

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

       1. frac-2neg N/A

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

       2. metadata-eval N/A

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

       3. /-rgt-identity N/A

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

       4. neg-lowering-neg.f64 64.6%

          \[\leadsto \mathsf{neg.f64}\left(y\right) \]

   12. Applied egg-rr 64.6%

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

   if -1.8e-72 < y < 1

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. sub0-neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-neg-frac N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-out N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{\left(y - 1\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(\color{blue}{y} - 1\right)\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(y + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(y + -1\right)\right) \]

      17. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{+.f64}\left(y, \color{blue}{-1}\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{y - x}{y + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

   7. Simplified 79.9%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-72}:\\ \;\;\;\;0 - y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	t_0 = 1.0 - (x / y)
	tmp = 0
	if y <= -1.0:
		tmp = t_0
	elif y <= 1.0:
		tmp = 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 <= -1.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = Float64(x - y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

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

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 1.0], N[(x - y), $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. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. sub0-neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-neg-frac N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-out N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{\left(y - 1\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(\color{blue}{y} - 1\right)\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(y + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(y + -1\right)\right) \]

      17. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{+.f64}\left(y, \color{blue}{-1}\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{y - x}{y + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \color{blue}{y}\right) \]
   6. Step-by-step derivation

   7. Simplified 98.9%

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. metadata-eval N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(-1\right)\right), \color{blue}{\left(\frac{x}{y}\right)}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\frac{\color{blue}{x}}{y}\right)\right) \]

      6. /-lowering-/.f64 98.9%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right) \]

   9. Applied egg-rr 98.9%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. sub0-neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-neg-frac N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-out N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{\left(y - 1\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(\color{blue}{y} - 1\right)\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(y + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(y + -1\right)\right) \]

      17. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{+.f64}\left(y, \color{blue}{-1}\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{y - x}{y + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \color{blue}{-1}\right) \]
   6. Step-by-step derivation

   7. Simplified 99.0%

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

   8. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 99.0%

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{y}\right) \]

   10. Simplified 99.0%

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

Alternative 5: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5400:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5400.0) 1.0 (if (<= y 1.0) (- x y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5400.0) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = x - 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 (y <= (-5400.0d0)) then
        tmp = 1.0d0
    else if (y <= 1.0d0) then
        tmp = x - y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -5400.0) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = x - y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5400.0:
		tmp = 1.0
	elif y <= 1.0:
		tmp = x - y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5400.0)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = Float64(x - y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5400.0)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = x - y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5400.0], 1.0, If[LessEqual[y, 1.0], N[(x - y), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -5400 or 1 < y

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. sub0-neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-neg-frac N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-out N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{\left(y - 1\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(\color{blue}{y} - 1\right)\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(y + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(y + -1\right)\right) \]

      17. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{+.f64}\left(y, \color{blue}{-1}\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{y - x}{y + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

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

   7. Simplified 69.6%

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

   if -5400 < y < 1

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. sub0-neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-neg-frac N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-out N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{\left(y - 1\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(\color{blue}{y} - 1\right)\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(y + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(y + -1\right)\right) \]

      17. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{+.f64}\left(y, \color{blue}{-1}\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{y - x}{y + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \color{blue}{-1}\right) \]
   6. Step-by-step derivation

   7. Simplified 99.0%

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

   8. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 99.0%

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{y}\right) \]

   10. Simplified 99.0%

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

Alternative 6: 73.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.4e-7) 1.0 (if (<= y 1.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.4e-7) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = x;
	} 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 (y <= (-3.4d-7)) then
        tmp = 1.0d0
    else if (y <= 1.0d0) then
        tmp = x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.4e-7) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.4e-7:
		tmp = 1.0
	elif y <= 1.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.4e-7)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.4e-7)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.4e-7], 1.0, If[LessEqual[y, 1.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -3.39999999999999974e-7 or 1 < y

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. sub0-neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-neg-frac N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-out N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{\left(y - 1\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(\color{blue}{y} - 1\right)\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(y + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(y + -1\right)\right) \]

      17. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{+.f64}\left(y, \color{blue}{-1}\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{y - x}{y + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

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

   7. Simplified 69.2%

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

   if -3.39999999999999974e-7 < y < 1

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. sub0-neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-neg-frac N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-out N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{\left(y - 1\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(\color{blue}{y} - 1\right)\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(y + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(y + -1\right)\right) \]

      17. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{+.f64}\left(y, \color{blue}{-1}\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{y - x}{y + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

   7. Simplified 75.0%

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

Alternative 7: 38.2% accurate, 7.0× 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. Step-by-step derivation

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. neg-sub0 N/A

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

   4. associate-+l- N/A

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

   5. sub0-neg N/A

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

   6. distribute-frac-neg2 N/A

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

   7. distribute-neg-frac N/A

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

   8. sub-neg N/A

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

   9. +-commutative N/A

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

   10. distribute-neg-out N/A

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

   11. remove-double-neg N/A

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

   12. sub-neg N/A

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

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{\left(y - 1\right)}\right) \]

   14. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(\color{blue}{y} - 1\right)\right) \]

   15. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(y + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(y + -1\right)\right) \]

   17. +-lowering-+.f64 100.0%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{+.f64}\left(y, \color{blue}{-1}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{y - x}{y + -1}} \]
4. Add Preprocessing

5. Taylor expanded in y around inf

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

7. Simplified 37.5%

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

Reproduce

herbie shell --seed 2024150 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))