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

Percentage Accurate: 100.0% → 100.0%

Time: 4.2s

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. Final simplification 100.0%

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

Alternative 2: 98.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.25) (not (<= y 1.0))) (+ 1.0 (/ (- 1.0 x) y)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.25) || !(y <= 1.0)) {
		tmp = 1.0 + ((1.0 - x) / y);
	} 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) :: tmp
    if ((y <= (-1.25d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = 1.0d0 + ((1.0d0 - x) / y)
    else
        tmp = x - y
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if (y <= -1.25) or not (y <= 1.0):
		tmp = 1.0 + ((1.0 - x) / y)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.25) || !(y <= 1.0))
		tmp = Float64(1.0 + Float64(Float64(1.0 - x) / y));
	else
		tmp = Float64(x - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.25) || ~((y <= 1.0)))
		tmp = 1.0 + ((1.0 - x) / y);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.25], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(1.0 + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

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 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

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

      4. associate-+l- 100.0%

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

      5. sub0-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. neg-sub0 100.0%

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

      10. associate-+l- 100.0%

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

      11. sub0-neg 100.0%

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

      12. neg-mul-1 100.0%

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

      13. times-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. *-lft-identity 100.0%

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

      16. sub-neg 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 98.0%

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

      1. +-commutative 98.0%

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

      2. associate-+r+ 98.0%

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

      3. mul-1-neg 98.0%

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

      4. unsub-neg 98.0%

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

      5. div-sub 98.0%

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

      6. unsub-neg 98.0%

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

      7. mul-1-neg 98.0%

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

      8. +-commutative 98.0%

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

      9. metadata-eval 98.0%

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

      10. distribute-lft-in 98.0%

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

      11. metadata-eval 98.0%

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

      12. sub-neg 98.0%

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

      13. associate-*r/ 98.0%

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

      14. +-commutative 98.0%

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

      15. associate-*r/ 98.0%

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

      16. sub-neg 98.0%

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

      17. metadata-eval 98.0%

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

      18. distribute-lft-in 98.0%

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

      19. metadata-eval 98.0%

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

      20. +-commutative 98.0%

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

      21. mul-1-neg 98.0%

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

      22. unsub-neg 98.0%

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

   6. Simplified 98.0%

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

   if -1.25 < y < 1

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

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

      4. associate-+l- 100.0%

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

      5. sub0-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. neg-sub0 100.0%

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

      10. associate-+l- 100.0%

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

      11. sub0-neg 100.0%

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

      12. neg-mul-1 100.0%

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

      13. times-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. *-lft-identity 100.0%

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

      16. sub-neg 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 99.4%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(1 + -1 \cdot x\right)\right) + \left(-1 \cdot \left({y}^{2} \cdot \left(1 + -1 \cdot x\right)\right) + x\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-*r* 99.4%

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

      4. neg-mul-1 99.4%

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

      5. associate-*r* 99.4%

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

      6. distribute-rgt-out 99.4%

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

      7. mul-1-neg 99.4%

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

      8. unsub-neg 99.4%

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

      9. mul-1-neg 99.4%

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

      10. unsub-neg 99.4%

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

      11. unpow2 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in x around 0 98.7%

      \[\leadsto x + \color{blue}{-1 \cdot \left(y + {y}^{2}\right)} \]
   8. Step-by-step derivation

      1. unpow2 98.7%

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

      2. distribute-lft-in 98.7%

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

      3. mul-1-neg 98.7%

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

      4. distribute-lft-neg-in 98.7%

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

      5. mul-1-neg 98.7%

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

      6. distribute-rgt-in 98.7%

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

      7. mul-1-neg 98.7%

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

      8. sub-neg 98.7%

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

   9. Simplified 98.7%

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

   10. Taylor expanded in y around 0 97.8%

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

       1. neg-mul-1 97.8%

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

       2. +-commutative 97.8%

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

       3. unsub-neg 97.8%

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

   12. Simplified 97.8%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]

Alternative 3: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(-1 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0)))
   (+ 1.0 (/ (- 1.0 x) y))
   (+ x (* y (- -1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = 1.0 + ((1.0 - x) / y);
	} else {
		tmp = x + (y * (-1.0 - y));
	}
	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)) .or. (.not. (y <= 1.0d0))) then
        tmp = 1.0d0 + ((1.0d0 - x) / y)
    else
        tmp = x + (y * ((-1.0d0) - y))
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.0):
		tmp = 1.0 + ((1.0 - x) / y)
	else:
		tmp = x + (y * (-1.0 - y))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(1.0 + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

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 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

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

      4. associate-+l- 100.0%

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

      5. sub0-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. neg-sub0 100.0%

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

      10. associate-+l- 100.0%

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

      11. sub0-neg 100.0%

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

      12. neg-mul-1 100.0%

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

      13. times-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. *-lft-identity 100.0%

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

      16. sub-neg 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 98.0%

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

      1. +-commutative 98.0%

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

      2. associate-+r+ 98.0%

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

      3. mul-1-neg 98.0%

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

      4. unsub-neg 98.0%

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

      5. div-sub 98.0%

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

      6. unsub-neg 98.0%

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

      7. mul-1-neg 98.0%

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

      8. +-commutative 98.0%

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

      9. metadata-eval 98.0%

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

      10. distribute-lft-in 98.0%

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

      11. metadata-eval 98.0%

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

      12. sub-neg 98.0%

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

      13. associate-*r/ 98.0%

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

      14. +-commutative 98.0%

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

      15. associate-*r/ 98.0%

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

      16. sub-neg 98.0%

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

      17. metadata-eval 98.0%

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

      18. distribute-lft-in 98.0%

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

      19. metadata-eval 98.0%

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

      20. +-commutative 98.0%

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

      21. mul-1-neg 98.0%

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

      22. unsub-neg 98.0%

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

   6. Simplified 98.0%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

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

      4. associate-+l- 100.0%

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

      5. sub0-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. neg-sub0 100.0%

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

      10. associate-+l- 100.0%

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

      11. sub0-neg 100.0%

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

      12. neg-mul-1 100.0%

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

      13. times-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. *-lft-identity 100.0%

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

      16. sub-neg 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 99.4%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(1 + -1 \cdot x\right)\right) + \left(-1 \cdot \left({y}^{2} \cdot \left(1 + -1 \cdot x\right)\right) + x\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-*r* 99.4%

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

      4. neg-mul-1 99.4%

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

      5. associate-*r* 99.4%

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

      6. distribute-rgt-out 99.4%

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

      7. mul-1-neg 99.4%

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

      8. unsub-neg 99.4%

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

      9. mul-1-neg 99.4%

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

      10. unsub-neg 99.4%

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

      11. unpow2 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in x around 0 98.7%

      \[\leadsto x + \color{blue}{-1 \cdot \left(y + {y}^{2}\right)} \]
   8. Step-by-step derivation

      1. unpow2 98.7%

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

      2. distribute-lft-in 98.7%

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

      3. mul-1-neg 98.7%

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

      4. distribute-lft-neg-in 98.7%

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

      5. mul-1-neg 98.7%

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

      6. distribute-rgt-in 98.7%

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

      7. mul-1-neg 98.7%

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

      8. sub-neg 98.7%

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

   9. Simplified 98.7%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(-1 - y\right)\\ \end{array} \]

Alternative 4: 86.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-13} \lor \neg \left(y \leq 6 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -8.5e-13) (not (<= y 6e-10))) (/ y (+ y -1.0)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -8.5e-13) || !(y <= 6e-10)) {
		tmp = y / (y + -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) :: tmp
    if ((y <= (-8.5d-13)) .or. (.not. (y <= 6d-10))) then
        tmp = y / (y + (-1.0d0))
    else
        tmp = x - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -8.5e-13) || !(y <= 6e-10)) {
		tmp = y / (y + -1.0);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -8.5e-13) or not (y <= 6e-10):
		tmp = y / (y + -1.0)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -8.5e-13) || !(y <= 6e-10))
		tmp = Float64(y / Float64(y + -1.0));
	else
		tmp = Float64(x - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -8.5e-13) || ~((y <= 6e-10)))
		tmp = y / (y + -1.0);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -8.5e-13], N[Not[LessEqual[y, 6e-10]], $MachinePrecision]], N[(y / N[(y + -1.0), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.5000000000000001e-13 or 6e-10 < y

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

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

      4. associate-+l- 100.0%

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

      5. sub0-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. neg-sub0 100.0%

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

      10. associate-+l- 100.0%

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

      11. sub0-neg 100.0%

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

      12. neg-mul-1 100.0%

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

      13. times-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. *-lft-identity 100.0%

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

      16. sub-neg 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 75.4%

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

   if -8.5000000000000001e-13 < y < 6e-10

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

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

      4. associate-+l- 100.0%

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

      5. sub0-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. neg-sub0 100.0%

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

      10. associate-+l- 100.0%

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

      11. sub0-neg 100.0%

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

      12. neg-mul-1 100.0%

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

      13. times-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. *-lft-identity 100.0%

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

      16. sub-neg 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(1 + -1 \cdot x\right)\right) + \left(-1 \cdot \left({y}^{2} \cdot \left(1 + -1 \cdot x\right)\right) + x\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. neg-mul-1 100.0%

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

      5. associate-*r* 100.0%

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

      6. distribute-rgt-out 100.0%

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

      7. mul-1-neg 100.0%

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

      8. unsub-neg 100.0%

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

      9. mul-1-neg 100.0%

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

      10. unsub-neg 100.0%

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

      11. unpow2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 99.7%

      \[\leadsto x + \color{blue}{-1 \cdot \left(y + {y}^{2}\right)} \]
   8. Step-by-step derivation

      1. unpow2 99.7%

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

      2. distribute-lft-in 99.7%

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

      3. mul-1-neg 99.7%

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

      4. distribute-lft-neg-in 99.7%

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

      5. mul-1-neg 99.7%

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

      6. distribute-rgt-in 99.7%

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

      7. mul-1-neg 99.7%

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

      8. sub-neg 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in y around 0 99.5%

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

       1. neg-mul-1 99.5%

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

       2. +-commutative 99.5%

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

       3. unsub-neg 99.5%

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

   12. Simplified 99.5%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-13} \lor \neg \left(y \leq 6 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]

Alternative 5: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0))) (- 1.0 (/ x y)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = 1.0 - (x / y);
	} 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) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = x - y
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.0):
		tmp = 1.0 - (x / y)
	else:
		tmp = x - y
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

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 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

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

      4. associate-+l- 100.0%

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

      5. sub0-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. neg-sub0 100.0%

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

      10. associate-+l- 100.0%

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

      11. sub0-neg 100.0%

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

      12. neg-mul-1 100.0%

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

      13. times-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. *-lft-identity 100.0%

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

      16. sub-neg 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 98.0%

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

      1. +-commutative 98.0%

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

      2. associate-+r+ 98.0%

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

      3. mul-1-neg 98.0%

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

      4. unsub-neg 98.0%

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

      5. div-sub 98.0%

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

      6. unsub-neg 98.0%

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

      7. mul-1-neg 98.0%

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

      8. +-commutative 98.0%

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

      9. metadata-eval 98.0%

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

      10. distribute-lft-in 98.0%

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

      11. metadata-eval 98.0%

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

      12. sub-neg 98.0%

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

      13. associate-*r/ 98.0%

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

      14. +-commutative 98.0%

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

      15. associate-*r/ 98.0%

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

      16. sub-neg 98.0%

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

      17. metadata-eval 98.0%

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

      18. distribute-lft-in 98.0%

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

      19. metadata-eval 98.0%

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

      20. +-commutative 98.0%

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

      21. mul-1-neg 98.0%

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

      22. unsub-neg 98.0%

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

   6. Simplified 98.0%

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

   7. Taylor expanded in x around inf 97.3%

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

      1. neg-mul-1 97.3%

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

      2. distribute-neg-frac 97.3%

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

   9. Simplified 97.3%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

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

      4. associate-+l- 100.0%

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

      5. sub0-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. neg-sub0 100.0%

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

      10. associate-+l- 100.0%

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

      11. sub0-neg 100.0%

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

      12. neg-mul-1 100.0%

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

      13. times-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. *-lft-identity 100.0%

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

      16. sub-neg 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 99.4%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(1 + -1 \cdot x\right)\right) + \left(-1 \cdot \left({y}^{2} \cdot \left(1 + -1 \cdot x\right)\right) + x\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-*r* 99.4%

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

      4. neg-mul-1 99.4%

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

      5. associate-*r* 99.4%

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

      6. distribute-rgt-out 99.4%

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

      7. mul-1-neg 99.4%

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

      8. unsub-neg 99.4%

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

      9. mul-1-neg 99.4%

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

      10. unsub-neg 99.4%

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

      11. unpow2 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in x around 0 98.7%

      \[\leadsto x + \color{blue}{-1 \cdot \left(y + {y}^{2}\right)} \]
   8. Step-by-step derivation

      1. unpow2 98.7%

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

      2. distribute-lft-in 98.7%

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

      3. mul-1-neg 98.7%

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

      4. distribute-lft-neg-in 98.7%

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

      5. mul-1-neg 98.7%

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

      6. distribute-rgt-in 98.7%

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

      7. mul-1-neg 98.7%

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

      8. sub-neg 98.7%

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

   9. Simplified 98.7%

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

   10. Taylor expanded in y around 0 97.8%

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

       1. neg-mul-1 97.8%

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

       2. +-commutative 97.8%

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

       3. unsub-neg 97.8%

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

   12. Simplified 97.8%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]

Alternative 6: 86.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -800000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -800000000000.0) 1.0 (if (<= y 1.0) (- x y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -800000000000.0) {
		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 <= (-800000000000.0d0)) 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 <= -800000000000.0) {
		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 <= -800000000000.0:
		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 <= -800000000000.0)
		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 <= -800000000000.0)
		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, -800000000000.0], 1.0, If[LessEqual[y, 1.0], N[(x - y), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -8e11 or 1 < y

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

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

      4. associate-+l- 100.0%

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

      5. sub0-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. neg-sub0 100.0%

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

      10. associate-+l- 100.0%

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

      11. sub0-neg 100.0%

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

      12. neg-mul-1 100.0%

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

      13. times-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. *-lft-identity 100.0%

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

      16. sub-neg 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 75.6%

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

   if -8e11 < y < 1

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

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

      4. associate-+l- 100.0%

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

      5. sub0-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. neg-sub0 100.0%

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

      10. associate-+l- 100.0%

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

      11. sub0-neg 100.0%

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

      12. neg-mul-1 100.0%

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

      13. times-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. *-lft-identity 100.0%

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

      16. sub-neg 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 97.3%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(1 + -1 \cdot x\right)\right) + \left(-1 \cdot \left({y}^{2} \cdot \left(1 + -1 \cdot x\right)\right) + x\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 97.3%

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

      2. +-commutative 97.3%

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

      3. associate-*r* 97.3%

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

      4. neg-mul-1 97.3%

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

      5. associate-*r* 97.3%

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

      6. distribute-rgt-out 97.3%

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

      7. mul-1-neg 97.3%

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

      8. unsub-neg 97.3%

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

      9. mul-1-neg 97.3%

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

      10. unsub-neg 97.3%

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

      11. unpow2 97.3%

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

   6. Simplified 97.3%

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

   7. Taylor expanded in x around 0 96.7%

      \[\leadsto x + \color{blue}{-1 \cdot \left(y + {y}^{2}\right)} \]
   8. Step-by-step derivation

      1. unpow2 96.7%

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

      2. distribute-lft-in 96.7%

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

      3. mul-1-neg 96.7%

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

      4. distribute-lft-neg-in 96.7%

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

      5. mul-1-neg 96.7%

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

      6. distribute-rgt-in 96.7%

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

      7. mul-1-neg 96.7%

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

      8. sub-neg 96.7%

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

   9. Simplified 96.7%

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

   10. Taylor expanded in y around 0 95.8%

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

       1. neg-mul-1 95.8%

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

       2. +-commutative 95.8%

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

       3. unsub-neg 95.8%

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

   12. Simplified 95.8%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -800000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 74.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -800000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -800000000000.0) 1.0 (if (<= y 1.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -800000000000.0) {
		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 <= (-800000000000.0d0)) 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 <= -800000000000.0) {
		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 <= -800000000000.0:
		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 <= -800000000000.0)
		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 <= -800000000000.0)
		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, -800000000000.0], 1.0, If[LessEqual[y, 1.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -8e11 or 1 < y

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

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

      4. associate-+l- 100.0%

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

      5. sub0-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. neg-sub0 100.0%

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

      10. associate-+l- 100.0%

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

      11. sub0-neg 100.0%

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

      12. neg-mul-1 100.0%

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

      13. times-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. *-lft-identity 100.0%

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

      16. sub-neg 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 75.6%

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

   if -8e11 < y < 1

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

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

      4. associate-+l- 100.0%

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

      5. sub0-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. neg-sub0 100.0%

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

      10. associate-+l- 100.0%

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

      11. sub0-neg 100.0%

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

      12. neg-mul-1 100.0%

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

      13. times-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. *-lft-identity 100.0%

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

      16. sub-neg 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 70.0%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -800000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 38.6% 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 100.0%

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

   2. +-commutative 100.0%

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

   3. neg-sub0 100.0%

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

   4. associate-+l- 100.0%

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

   5. sub0-neg 100.0%

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

   6. neg-mul-1 100.0%

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

   7. sub-neg 100.0%

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

   8. +-commutative 100.0%

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

   9. neg-sub0 100.0%

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

   10. associate-+l- 100.0%

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

   11. sub0-neg 100.0%

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

   12. neg-mul-1 100.0%

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

   13. times-frac 100.0%

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

   14. metadata-eval 100.0%

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

   15. *-lft-identity 100.0%

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

   16. sub-neg 100.0%

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

   17. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in y around inf 38.7%

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

5. Final simplification 38.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023258 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))