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

Percentage Accurate: 100.0% → 100.0%

Time: 4.8s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 98.1% 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 99.9%

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

      1. *-lft-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-/r/ 99.9%

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

      4. associate-/l* 99.9%

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

      5. neg-mul-1 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-out 99.9%

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

      10. remove-double-neg 99.9%

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

      11. sub-neg 99.9%

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

      12. associate-/l/ 99.9%

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

      13. neg-mul-1 99.9%

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

      14. sub0-neg 99.9%

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

      15. associate--r- 99.9%

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

      16. metadata-eval 99.9%

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

      17. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 97.6%

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

      1. +-commutative 97.6%

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

      2. mul-1-neg 97.6%

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

      3. unsub-neg 97.6%

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

      4. div-sub 97.6%

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

   6. Simplified 97.6%

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

   if -1.25 < y < 1

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.7%

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

      4. associate-/l* 100.0%

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

      5. neg-mul-1 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-out 100.0%

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

      10. remove-double-neg 100.0%

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

      11. sub-neg 100.0%

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

      12. associate-/l/ 100.0%

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

      13. neg-mul-1 100.0%

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

      14. sub0-neg 100.0%

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

      15. associate--r- 100.0%

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

      16. metadata-eval 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around 0 99.0%

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

4. Final simplification 98.4%

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

Alternative 3: 74.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -18000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{-25}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -18000000000.0)
   1.0
   (if (<= y 5.2e-51) x (if (<= y 6.6e-25) (- y) (if (<= y 1.0) x 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -18000000000.0) {
		tmp = 1.0;
	} else if (y <= 5.2e-51) {
		tmp = x;
	} else if (y <= 6.6e-25) {
		tmp = -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 <= (-18000000000.0d0)) then
        tmp = 1.0d0
    else if (y <= 5.2d-51) then
        tmp = x
    else if (y <= 6.6d-25) then
        tmp = -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 <= -18000000000.0) {
		tmp = 1.0;
	} else if (y <= 5.2e-51) {
		tmp = x;
	} else if (y <= 6.6e-25) {
		tmp = -y;
	} else if (y <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -18000000000.0:
		tmp = 1.0
	elif y <= 5.2e-51:
		tmp = x
	elif y <= 6.6e-25:
		tmp = -y
	elif y <= 1.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -18000000000.0)
		tmp = 1.0;
	elseif (y <= 5.2e-51)
		tmp = x;
	elseif (y <= 6.6e-25)
		tmp = Float64(-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 <= -18000000000.0)
		tmp = 1.0;
	elseif (y <= 5.2e-51)
		tmp = x;
	elseif (y <= 6.6e-25)
		tmp = -y;
	elseif (y <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -18000000000.0], 1.0, If[LessEqual[y, 5.2e-51], x, If[LessEqual[y, 6.6e-25], (-y), If[LessEqual[y, 1.0], x, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.8e10 or 1 < y

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-/r/ 99.9%

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

      4. associate-/l* 99.9%

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

      5. neg-mul-1 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-out 99.9%

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

      10. remove-double-neg 99.9%

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

      11. sub-neg 99.9%

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

      12. associate-/l/ 99.9%

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

      13. neg-mul-1 99.9%

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

      14. sub0-neg 99.9%

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

      15. associate--r- 99.9%

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

      16. metadata-eval 99.9%

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

      17. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 76.0%

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

   if -1.8e10 < y < 5.2e-51 or 6.5999999999999997e-25 < y < 1

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.7%

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

      4. associate-/l* 100.0%

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

      5. neg-mul-1 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-out 100.0%

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

      10. remove-double-neg 100.0%

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

      11. sub-neg 100.0%

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

      12. associate-/l/ 100.0%

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

      13. neg-mul-1 100.0%

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

      14. sub0-neg 100.0%

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

      15. associate--r- 100.0%

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

      16. metadata-eval 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 82.9%

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

   if 5.2e-51 < y < 6.5999999999999997e-25

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.4%

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

      4. associate-/l* 100.0%

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

      5. neg-mul-1 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-out 100.0%

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

      10. remove-double-neg 100.0%

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

      11. sub-neg 100.0%

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

      12. associate-/l/ 100.0%

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

      13. neg-mul-1 100.0%

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

      14. sub0-neg 100.0%

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

      15. associate--r- 100.0%

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

      16. metadata-eval 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

   5. Taylor expanded in y around 0 100.0%

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

      1. neg-mul-1 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -18000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{-25}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 86.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -18000000000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -18000000000.0) (not (<= y 1.0))) (+ 1.0 (/ 1.0 y)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -18000000000.0) || !(y <= 1.0)) {
		tmp = 1.0 + (1.0 / 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 <= (-18000000000.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = 1.0d0 + (1.0d0 / y)
    else
        tmp = x - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -18000000000.0) || !(y <= 1.0)) {
		tmp = 1.0 + (1.0 / y);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -18000000000.0) or not (y <= 1.0):
		tmp = 1.0 + (1.0 / y)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -18000000000.0) || !(y <= 1.0))
		tmp = Float64(1.0 + Float64(1.0 / y));
	else
		tmp = Float64(x - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -18000000000.0) || ~((y <= 1.0)))
		tmp = 1.0 + (1.0 / y);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -18000000000.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(1.0 + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8e10 or 1 < y

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-/r/ 99.9%

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

      4. associate-/l* 99.9%

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

      5. neg-mul-1 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-out 99.9%

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

      10. remove-double-neg 99.9%

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

      11. sub-neg 99.9%

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

      12. associate-/l/ 99.9%

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

      13. neg-mul-1 99.9%

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

      14. sub0-neg 99.9%

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

      15. associate--r- 99.9%

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

      16. metadata-eval 99.9%

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

      17. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 76.2%

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

   5. Taylor expanded in y around inf 76.3%

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

   if -1.8e10 < y < 1

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.7%

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

      4. associate-/l* 100.0%

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

      5. neg-mul-1 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-out 100.0%

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

      10. remove-double-neg 100.0%

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

      11. sub-neg 100.0%

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

      12. associate-/l/ 100.0%

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

      13. neg-mul-1 100.0%

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

      14. sub0-neg 100.0%

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

      15. associate--r- 100.0%

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

      16. metadata-eval 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around 0 97.1%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -18000000000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]

Alternative 5: 97.8% 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 99.9%

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

      1. *-lft-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-/r/ 99.9%

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

      4. associate-/l* 99.9%

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

      5. neg-mul-1 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-out 99.9%

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

      10. remove-double-neg 99.9%

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

      11. sub-neg 99.9%

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

      12. associate-/l/ 99.9%

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

      13. neg-mul-1 99.9%

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

      14. sub0-neg 99.9%

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

      15. associate--r- 99.9%

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

      16. metadata-eval 99.9%

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

      17. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 97.6%

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

      1. +-commutative 97.6%

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

      2. mul-1-neg 97.6%

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

      3. unsub-neg 97.6%

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

      4. div-sub 97.6%

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

   6. Simplified 97.6%

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

   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. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.7%

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

      4. associate-/l* 100.0%

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

      5. neg-mul-1 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-out 100.0%

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

      10. remove-double-neg 100.0%

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

      11. sub-neg 100.0%

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

      12. associate-/l/ 100.0%

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

      13. neg-mul-1 100.0%

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

      14. sub0-neg 100.0%

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

      15. associate--r- 100.0%

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

      16. metadata-eval 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around 0 99.0%

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

4. Final simplification 98.2%

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

Alternative 6: 86.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -18000000000:\\ \;\;\;\;1 + \frac{1}{y}\\ \mathbf{elif}\;y \leq 1.38 \cdot 10^{-14}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -18000000000.0)
   (+ 1.0 (/ 1.0 y))
   (if (<= y 1.38e-14) (- x y) (/ y (+ y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -18000000000.0) {
		tmp = 1.0 + (1.0 / y);
	} else if (y <= 1.38e-14) {
		tmp = x - y;
	} else {
		tmp = y / (y + -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 <= (-18000000000.0d0)) then
        tmp = 1.0d0 + (1.0d0 / y)
    else if (y <= 1.38d-14) then
        tmp = x - y
    else
        tmp = y / (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -18000000000.0) {
		tmp = 1.0 + (1.0 / y);
	} else if (y <= 1.38e-14) {
		tmp = x - y;
	} else {
		tmp = y / (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -18000000000.0:
		tmp = 1.0 + (1.0 / y)
	elif y <= 1.38e-14:
		tmp = x - y
	else:
		tmp = y / (y + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -18000000000.0)
		tmp = Float64(1.0 + Float64(1.0 / y));
	elseif (y <= 1.38e-14)
		tmp = Float64(x - y);
	else
		tmp = Float64(y / Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -18000000000.0)
		tmp = 1.0 + (1.0 / y);
	elseif (y <= 1.38e-14)
		tmp = x - y;
	else
		tmp = y / (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -18000000000.0], N[(1.0 + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.38e-14], N[(x - y), $MachinePrecision], N[(y / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.8e10

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-/r/ 99.9%

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

      4. associate-/l* 99.9%

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

      5. neg-mul-1 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-out 99.9%

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

      10. remove-double-neg 99.9%

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

      11. sub-neg 99.9%

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

      12. associate-/l/ 99.9%

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

      13. neg-mul-1 99.9%

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

      14. sub0-neg 99.9%

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

      15. associate--r- 99.9%

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

      16. metadata-eval 99.9%

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

      17. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 75.2%

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

   5. Taylor expanded in y around inf 75.2%

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

   if -1.8e10 < y < 1.38000000000000002e-14

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.7%

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

      4. associate-/l* 100.0%

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

      5. neg-mul-1 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-out 100.0%

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

      10. remove-double-neg 100.0%

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

      11. sub-neg 100.0%

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

      12. associate-/l/ 100.0%

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

      13. neg-mul-1 100.0%

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

      14. sub0-neg 100.0%

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

      15. associate--r- 100.0%

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

      16. metadata-eval 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around 0 97.8%

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

   if 1.38000000000000002e-14 < y

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-/r/ 99.9%

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

      4. associate-/l* 99.9%

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

      5. neg-mul-1 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-out 99.9%

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

      10. remove-double-neg 99.9%

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

      11. sub-neg 99.9%

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

      12. associate-/l/ 99.9%

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

      13. neg-mul-1 99.9%

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

      14. sub0-neg 99.9%

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

      15. associate--r- 99.9%

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

      16. metadata-eval 99.9%

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

      17. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 76.8%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -18000000000:\\ \;\;\;\;1 + \frac{1}{y}\\ \mathbf{elif}\;y \leq 1.38 \cdot 10^{-14}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + -1}\\ \end{array} \]

Alternative 7: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -18000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -18000000000.0) 1.0 (if (<= y 1.0) (- x y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -18000000000.0) {
		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 <= (-18000000000.0d0)) 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 <= -18000000000.0) {
		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 <= -18000000000.0:
		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 <= -18000000000.0)
		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 <= -18000000000.0)
		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, -18000000000.0], 1.0, If[LessEqual[y, 1.0], N[(x - y), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8e10 or 1 < y

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-/r/ 99.9%

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

      4. associate-/l* 99.9%

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

      5. neg-mul-1 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-out 99.9%

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

      10. remove-double-neg 99.9%

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

      11. sub-neg 99.9%

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

      12. associate-/l/ 99.9%

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

      13. neg-mul-1 99.9%

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

      14. sub0-neg 99.9%

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

      15. associate--r- 99.9%

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

      16. metadata-eval 99.9%

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

      17. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 76.0%

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

   if -1.8e10 < y < 1

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.7%

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

      4. associate-/l* 100.0%

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

      5. neg-mul-1 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-out 100.0%

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

      10. remove-double-neg 100.0%

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

      11. sub-neg 100.0%

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

      12. associate-/l/ 100.0%

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

      13. neg-mul-1 100.0%

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

      14. sub0-neg 100.0%

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

      15. associate--r- 100.0%

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

      16. metadata-eval 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around 0 97.1%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -18000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 74.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -18000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -18000000000.0) 1.0 (if (<= y 1.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -18000000000.0) {
		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 <= (-18000000000.0d0)) 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 <= -18000000000.0) {
		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 <= -18000000000.0:
		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 <= -18000000000.0)
		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 <= -18000000000.0)
		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, -18000000000.0], 1.0, If[LessEqual[y, 1.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8e10 or 1 < y

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-/r/ 99.9%

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

      4. associate-/l* 99.9%

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

      5. neg-mul-1 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-out 99.9%

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

      10. remove-double-neg 99.9%

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

      11. sub-neg 99.9%

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

      12. associate-/l/ 99.9%

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

      13. neg-mul-1 99.9%

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

      14. sub0-neg 99.9%

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

      15. associate--r- 99.9%

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

      16. metadata-eval 99.9%

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

      17. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 76.0%

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

   if -1.8e10 < y < 1

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.7%

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

      4. associate-/l* 100.0%

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

      5. neg-mul-1 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-out 100.0%

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

      10. remove-double-neg 100.0%

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

      11. sub-neg 100.0%

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

      12. associate-/l/ 100.0%

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

      13. neg-mul-1 100.0%

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

      14. sub0-neg 100.0%

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

      15. associate--r- 100.0%

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

      16. metadata-eval 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 80.0%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -18000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 39.6% 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. *-lft-identity 100.0%

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

   2. metadata-eval 100.0%

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

   3. associate-/r/ 99.8%

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

   4. associate-/l* 100.0%

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

   5. neg-mul-1 100.0%

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

   6. distribute-frac-neg 100.0%

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

   7. sub-neg 100.0%

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

   8. +-commutative 100.0%

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

   9. distribute-neg-out 100.0%

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

   10. remove-double-neg 100.0%

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

   11. sub-neg 100.0%

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

   12. associate-/l/ 100.0%

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

   13. neg-mul-1 100.0%

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

   14. sub0-neg 100.0%

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

   15. associate--r- 100.0%

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

   16. metadata-eval 100.0%

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

   17. +-commutative 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in y around inf 37.7%

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

5. Final simplification 37.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023310 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))