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

Percentage Accurate: 100.0% → 100.0%

Time: 5.2s

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: 74.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+95}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{+68}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq -16200000000 \lor \neg \left(y \leq 500000000000\right):\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.2e+95)
   1.0
   (if (<= y -8.2e+68)
     (/ (- x) y)
     (if (or (<= y -16200000000.0) (not (<= y 500000000000.0)))
       (/ y (+ y -1.0))
       (/ x (- 1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.2e+95) {
		tmp = 1.0;
	} else if (y <= -8.2e+68) {
		tmp = -x / y;
	} else if ((y <= -16200000000.0) || !(y <= 500000000000.0)) {
		tmp = y / (y + -1.0);
	} else {
		tmp = x / (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 <= (-7.2d+95)) then
        tmp = 1.0d0
    else if (y <= (-8.2d+68)) then
        tmp = -x / y
    else if ((y <= (-16200000000.0d0)) .or. (.not. (y <= 500000000000.0d0))) then
        tmp = y / (y + (-1.0d0))
    else
        tmp = x / (1.0d0 - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.2e+95) {
		tmp = 1.0;
	} else if (y <= -8.2e+68) {
		tmp = -x / y;
	} else if ((y <= -16200000000.0) || !(y <= 500000000000.0)) {
		tmp = y / (y + -1.0);
	} else {
		tmp = x / (1.0 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.2e+95:
		tmp = 1.0
	elif y <= -8.2e+68:
		tmp = -x / y
	elif (y <= -16200000000.0) or not (y <= 500000000000.0):
		tmp = y / (y + -1.0)
	else:
		tmp = x / (1.0 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.2e+95)
		tmp = 1.0;
	elseif (y <= -8.2e+68)
		tmp = Float64(Float64(-x) / y);
	elseif ((y <= -16200000000.0) || !(y <= 500000000000.0))
		tmp = Float64(y / Float64(y + -1.0));
	else
		tmp = Float64(x / Float64(1.0 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.2e+95)
		tmp = 1.0;
	elseif (y <= -8.2e+68)
		tmp = -x / y;
	elseif ((y <= -16200000000.0) || ~((y <= 500000000000.0)))
		tmp = y / (y + -1.0);
	else
		tmp = x / (1.0 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.2e+95], 1.0, If[LessEqual[y, -8.2e+68], N[((-x) / y), $MachinePrecision], If[Or[LessEqual[y, -16200000000.0], N[Not[LessEqual[y, 500000000000.0]], $MachinePrecision]], N[(y / N[(y + -1.0), $MachinePrecision]), $MachinePrecision], N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -7.19999999999999955e95

   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 93.9%

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

   if -7.19999999999999955e95 < y < -8.1999999999999998e68

   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 inf 100.0%

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

      1. sub-neg 100.0%

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

      2. metadata-eval 100.0%

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

      3. neg-mul-1 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around inf 100.0%

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

      1. neg-mul-1 100.0%

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

      2. distribute-neg-frac 100.0%

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

   9. Simplified 100.0%

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

   if -8.1999999999999998e68 < y < -1.62e10 or 5e11 < y

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. neg-sub0 99.9%

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

      4. associate-+l- 99.9%

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

      5. sub0-neg 99.9%

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

      6. neg-mul-1 99.9%

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

      7. sub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. neg-sub0 99.9%

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

      10. associate-+l- 99.9%

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

      11. sub0-neg 99.9%

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

      12. neg-mul-1 99.9%

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

      13. times-frac 99.9%

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

      14. metadata-eval 99.9%

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

      15. *-lft-identity 99.9%

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

      16. sub-neg 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 84.1%

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

   if -1.62e10 < y < 5e11

   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. Step-by-step derivation

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

      2. frac-2neg 99.8%

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

      3. metadata-eval 99.8%

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

      4. metadata-eval 99.8%

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

      5. sub-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. add-sqr-sqrt 54.3%

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

      8. sqrt-prod 88.9%

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

      9. sqr-neg 88.9%

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

      10. sqrt-unprod 38.2%

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

      11. add-sqr-sqrt 80.1%

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

      12. add-sqr-sqrt 35.7%

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

      13. sqrt-unprod 20.1%

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

      14. sqr-neg 20.1%

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

      15. sqrt-unprod 0.7%

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

      16. add-sqr-sqrt 1.4%

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

      17. distribute-neg-in 1.4%

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

      18. sub-neg 1.4%

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

      19. distribute-frac-neg 1.4%

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

   7. Applied egg-rr 80.1%

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

   8. Taylor expanded in x around inf 80.5%

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

   9. Taylor expanded in x around 0 80.6%

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

       1. metadata-eval 80.6%

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

       2. sub-neg 80.6%

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

       3. metadata-eval 80.6%

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

       4. +-commutative 80.6%

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

       5. times-frac 80.6%

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

       6. *-lft-identity 80.6%

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

       7. neg-mul-1 80.6%

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

       8. distribute-neg-in 80.6%

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

       9. metadata-eval 80.6%

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

       10. unsub-neg 80.6%

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

   11. Simplified 80.6%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+95}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{+68}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq -16200000000 \lor \neg \left(y \leq 500000000000\right):\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - y}\\ \end{array} \]

Alternative 3: 73.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+96}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{+69}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq -0.6:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.8e+96)
   1.0
   (if (<= y -2.2e+69)
     (/ (- x) y)
     (if (<= y -0.6) 1.0 (if (<= y 1.0) (+ x (* x y)) 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.8e+96) {
		tmp = 1.0;
	} else if (y <= -2.2e+69) {
		tmp = -x / y;
	} else if (y <= -0.6) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = x + (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 <= (-2.8d+96)) then
        tmp = 1.0d0
    else if (y <= (-2.2d+69)) then
        tmp = -x / y
    else if (y <= (-0.6d0)) then
        tmp = 1.0d0
    else if (y <= 1.0d0) then
        tmp = x + (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 <= -2.8e+96) {
		tmp = 1.0;
	} else if (y <= -2.2e+69) {
		tmp = -x / y;
	} else if (y <= -0.6) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = x + (x * y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.8e+96:
		tmp = 1.0
	elif y <= -2.2e+69:
		tmp = -x / y
	elif y <= -0.6:
		tmp = 1.0
	elif y <= 1.0:
		tmp = x + (x * y)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.8e+96)
		tmp = 1.0;
	elseif (y <= -2.2e+69)
		tmp = Float64(Float64(-x) / y);
	elseif (y <= -0.6)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = Float64(x + Float64(x * y));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.8e+96)
		tmp = 1.0;
	elseif (y <= -2.2e+69)
		tmp = -x / y;
	elseif (y <= -0.6)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = x + (x * y);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.8e+96], 1.0, If[LessEqual[y, -2.2e+69], N[((-x) / y), $MachinePrecision], If[LessEqual[y, -0.6], 1.0, If[LessEqual[y, 1.0], N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision], 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.8e96 or -2.2000000000000002e69 < y < -0.599999999999999978 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 83.4%

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

   if -2.8e96 < y < -2.2000000000000002e69

   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 inf 100.0%

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

      1. sub-neg 100.0%

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

      2. metadata-eval 100.0%

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

      3. neg-mul-1 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around inf 100.0%

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

      1. neg-mul-1 100.0%

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

      2. distribute-neg-frac 100.0%

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

   9. Simplified 100.0%

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

   if -0.599999999999999978 < 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 x around inf 79.6%

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

      1. sub-neg 79.6%

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

      2. metadata-eval 79.6%

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

      3. neg-mul-1 79.6%

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

      4. distribute-neg-frac 79.6%

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

      5. +-commutative 79.6%

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

   6. Simplified 79.6%

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

   7. Taylor expanded in y around 0 78.6%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+96}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{+69}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq -0.6:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 74.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+99}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq -21000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3e+99)
   1.0
   (if (<= y -2.6e+69)
     (/ (- x) y)
     (if (<= y -21000000000.0) 1.0 (if (<= y 5.1e+16) (/ x (- 1.0 y)) 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3e+99) {
		tmp = 1.0;
	} else if (y <= -2.6e+69) {
		tmp = -x / y;
	} else if (y <= -21000000000.0) {
		tmp = 1.0;
	} else if (y <= 5.1e+16) {
		tmp = x / (1.0 - 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 <= (-3d+99)) then
        tmp = 1.0d0
    else if (y <= (-2.6d+69)) then
        tmp = -x / y
    else if (y <= (-21000000000.0d0)) then
        tmp = 1.0d0
    else if (y <= 5.1d+16) then
        tmp = x / (1.0d0 - y)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3e+99) {
		tmp = 1.0;
	} else if (y <= -2.6e+69) {
		tmp = -x / y;
	} else if (y <= -21000000000.0) {
		tmp = 1.0;
	} else if (y <= 5.1e+16) {
		tmp = x / (1.0 - y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3e+99:
		tmp = 1.0
	elif y <= -2.6e+69:
		tmp = -x / y
	elif y <= -21000000000.0:
		tmp = 1.0
	elif y <= 5.1e+16:
		tmp = x / (1.0 - y)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3e+99)
		tmp = 1.0;
	elseif (y <= -2.6e+69)
		tmp = Float64(Float64(-x) / y);
	elseif (y <= -21000000000.0)
		tmp = 1.0;
	elseif (y <= 5.1e+16)
		tmp = Float64(x / Float64(1.0 - y));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3e+99)
		tmp = 1.0;
	elseif (y <= -2.6e+69)
		tmp = -x / y;
	elseif (y <= -21000000000.0)
		tmp = 1.0;
	elseif (y <= 5.1e+16)
		tmp = x / (1.0 - y);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3e+99], 1.0, If[LessEqual[y, -2.6e+69], N[((-x) / y), $MachinePrecision], If[LessEqual[y, -21000000000.0], 1.0, If[LessEqual[y, 5.1e+16], N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.00000000000000014e99 or -2.6000000000000002e69 < y < -2.1e10 or 5.1e16 < 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 88.0%

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

   if -3.00000000000000014e99 < y < -2.6000000000000002e69

   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 inf 100.0%

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

      1. sub-neg 100.0%

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

      2. metadata-eval 100.0%

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

      3. neg-mul-1 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around inf 100.0%

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

      1. neg-mul-1 100.0%

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

      2. distribute-neg-frac 100.0%

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

   9. Simplified 100.0%

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

   if -2.1e10 < y < 5.1e16

   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. Step-by-step derivation

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

      2. frac-2neg 99.8%

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

      3. metadata-eval 99.8%

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

      4. metadata-eval 99.8%

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

      5. sub-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. add-sqr-sqrt 55.0%

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

      8. sqrt-prod 89.1%

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

      9. sqr-neg 89.1%

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

      10. sqrt-unprod 37.6%

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

      11. add-sqr-sqrt 79.6%

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

      12. add-sqr-sqrt 35.2%

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

      13. sqrt-unprod 19.8%

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

      14. sqr-neg 19.8%

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

      15. sqrt-unprod 0.7%

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

      16. add-sqr-sqrt 1.4%

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

      17. distribute-neg-in 1.4%

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

      18. sub-neg 1.4%

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

      19. distribute-frac-neg 1.4%

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

   7. Applied egg-rr 79.6%

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

   8. Taylor expanded in x around inf 80.0%

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

   9. Taylor expanded in x around 0 80.2%

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

       1. metadata-eval 80.2%

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

       2. sub-neg 80.2%

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

       3. metadata-eval 80.2%

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

       4. +-commutative 80.2%

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

       5. times-frac 80.2%

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

       6. *-lft-identity 80.2%

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

       7. neg-mul-1 80.2%

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

       8. distribute-neg-in 80.2%

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

       9. metadata-eval 80.2%

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

       10. unsub-neg 80.2%

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

   11. Simplified 80.2%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+99}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq -21000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 86.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -15500000000 \lor \neg \left(y \leq 460000000000\right):\\ \;\;\;\;1 + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -15500000000.0) (not (<= y 460000000000.0)))
   (+ 1.0 (/ (- 1.0 x) y))
   (/ x (- 1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -15500000000.0) || !(y <= 460000000000.0)) {
		tmp = 1.0 + ((1.0 - x) / y);
	} else {
		tmp = x / (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 <= (-15500000000.0d0)) .or. (.not. (y <= 460000000000.0d0))) then
        tmp = 1.0d0 + ((1.0d0 - x) / y)
    else
        tmp = x / (1.0d0 - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -15500000000.0) || !(y <= 460000000000.0)) {
		tmp = 1.0 + ((1.0 - x) / y);
	} else {
		tmp = x / (1.0 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -15500000000.0) or not (y <= 460000000000.0):
		tmp = 1.0 + ((1.0 - x) / y)
	else:
		tmp = x / (1.0 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -15500000000.0) || !(y <= 460000000000.0))
		tmp = Float64(1.0 + Float64(Float64(1.0 - x) / y));
	else
		tmp = Float64(x / Float64(1.0 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -15500000000.0) || ~((y <= 460000000000.0)))
		tmp = 1.0 + ((1.0 - x) / y);
	else
		tmp = x / (1.0 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -15500000000.0], N[Not[LessEqual[y, 460000000000.0]], $MachinePrecision]], N[(1.0 + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.55e10 or 4.6e11 < 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 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+r+ 100.0%

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

      3. mul-1-neg 100.0%

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

      4. unsub-neg 100.0%

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

      5. div-sub 100.0%

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

      6. unsub-neg 100.0%

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

      7. mul-1-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. metadata-eval 100.0%

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

      10. distribute-lft-in 100.0%

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

      11. metadata-eval 100.0%

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

      12. sub-neg 100.0%

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

      13. associate-*r/ 100.0%

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

      14. +-commutative 100.0%

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

      15. associate-*r/ 100.0%

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

      16. sub-neg 100.0%

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

      17. metadata-eval 100.0%

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

      18. distribute-lft-in 100.0%

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

      19. metadata-eval 100.0%

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

      20. +-commutative 100.0%

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

      21. mul-1-neg 100.0%

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

      22. unsub-neg 100.0%

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

   6. Simplified 100.0%

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

   if -1.55e10 < y < 4.6e11

   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. Step-by-step derivation

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

      2. frac-2neg 99.8%

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

      3. metadata-eval 99.8%

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

      4. metadata-eval 99.8%

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

      5. sub-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. add-sqr-sqrt 54.3%

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

      8. sqrt-prod 88.9%

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

      9. sqr-neg 88.9%

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

      10. sqrt-unprod 38.2%

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

      11. add-sqr-sqrt 80.1%

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

      12. add-sqr-sqrt 35.7%

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

      13. sqrt-unprod 20.1%

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

      14. sqr-neg 20.1%

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

      15. sqrt-unprod 0.7%

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

      16. add-sqr-sqrt 1.4%

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

      17. distribute-neg-in 1.4%

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

      18. sub-neg 1.4%

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

      19. distribute-frac-neg 1.4%

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

   7. Applied egg-rr 80.1%

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

   8. Taylor expanded in x around inf 80.5%

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

   9. Taylor expanded in x around 0 80.6%

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

       1. metadata-eval 80.6%

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

       2. sub-neg 80.6%

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

       3. metadata-eval 80.6%

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

       4. +-commutative 80.6%

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

       5. times-frac 80.6%

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

       6. *-lft-identity 80.6%

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

       7. neg-mul-1 80.6%

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

       8. distribute-neg-in 80.6%

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

       9. metadata-eval 80.6%

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

       10. unsub-neg 80.6%

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

   11. Simplified 80.6%

       \[\leadsto \color{blue}{\frac{x}{1 - y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -15500000000 \lor \neg \left(y \leq 460000000000\right):\\ \;\;\;\;1 + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - y}\\ \end{array} \]

Alternative 6: 73.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+95}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{+69}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq -15500000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.2e+95)
   1.0
   (if (<= y -2.8e+69)
     (/ (- x) y)
     (if (<= y -15500000000.0) 1.0 (if (<= y 1.0) x 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.2e+95) {
		tmp = 1.0;
	} else if (y <= -2.8e+69) {
		tmp = -x / y;
	} else if (y <= -15500000000.0) {
		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 <= (-7.2d+95)) then
        tmp = 1.0d0
    else if (y <= (-2.8d+69)) then
        tmp = -x / y
    else if (y <= (-15500000000.0d0)) 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 <= -7.2e+95) {
		tmp = 1.0;
	} else if (y <= -2.8e+69) {
		tmp = -x / y;
	} else if (y <= -15500000000.0) {
		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 <= -7.2e+95:
		tmp = 1.0
	elif y <= -2.8e+69:
		tmp = -x / y
	elif y <= -15500000000.0:
		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 <= -7.2e+95)
		tmp = 1.0;
	elseif (y <= -2.8e+69)
		tmp = Float64(Float64(-x) / y);
	elseif (y <= -15500000000.0)
		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 <= -7.2e+95)
		tmp = 1.0;
	elseif (y <= -2.8e+69)
		tmp = -x / y;
	elseif (y <= -15500000000.0)
		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, -7.2e+95], 1.0, If[LessEqual[y, -2.8e+69], N[((-x) / y), $MachinePrecision], If[LessEqual[y, -15500000000.0], 1.0, If[LessEqual[y, 1.0], x, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.19999999999999955e95 or -2.79999999999999982e69 < y < -1.55e10 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 84.7%

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

   if -7.19999999999999955e95 < y < -2.79999999999999982e69

   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 inf 100.0%

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

      1. sub-neg 100.0%

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

      2. metadata-eval 100.0%

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

      3. neg-mul-1 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around inf 100.0%

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

      1. neg-mul-1 100.0%

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

      2. distribute-neg-frac 100.0%

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

   9. Simplified 100.0%

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

   if -1.55e10 < 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 76.9%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+95}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{+69}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq -15500000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 74.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -15500000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -15500000000.0) 1.0 (if (<= y 1.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -15500000000.0) {
		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 <= (-15500000000.0d0)) 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 <= -15500000000.0) {
		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 <= -15500000000.0:
		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 <= -15500000000.0)
		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 <= -15500000000.0)
		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, -15500000000.0], 1.0, If[LessEqual[y, 1.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.55e10 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 81.7%

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

   if -1.55e10 < 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 76.9%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -15500000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 2.9% 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. Step-by-step derivation

   1. clear-num 99.9%

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

   2. inv-pow 99.9%

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

5. Applied egg-rr 99.9%

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

   1. unpow-1 99.9%

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

   2. frac-2neg 99.9%

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

   3. metadata-eval 99.9%

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

   4. metadata-eval 99.9%

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

   5. sub-neg 99.9%

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

   6. sub-neg 99.9%

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

   7. add-sqr-sqrt 50.8%

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

   8. sqrt-prod 60.1%

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

   9. sqr-neg 60.1%

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

   10. sqrt-unprod 22.5%

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

   11. add-sqr-sqrt 46.8%

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

   12. add-sqr-sqrt 20.1%

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

   13. sqrt-unprod 11.3%

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

   14. sqr-neg 11.3%

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

   15. sqrt-unprod 0.7%

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

   16. add-sqr-sqrt 1.5%

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

   17. distribute-neg-in 1.5%

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

   18. sub-neg 1.5%

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

   19. distribute-frac-neg 1.5%

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

7. Applied egg-rr 46.8%

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

8. Taylor expanded in y around inf 2.8%

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

9. Final simplification 2.8%

   \[\leadsto -1 \]

Alternative 9: 39.1% 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 45.2%

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

5. Final simplification 45.2%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023175 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))