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: 73.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{y}\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{+71}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.02 \cdot 10^{-63}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+50}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x) y)))
   (if (<= y -2.4e+71)
     1.0
     (if (<= y -5.2e+52)
       t_0
       (if (<= y -1.0)
         1.0
         (if (<= y -1.02e-63)
           (- y)
           (if (<= y 1.0) x (if (<= y 1.65e+50) t_0 1.0))))))))

C

double code(double x, double y) {
	double t_0 = -x / y;
	double tmp;
	if (y <= -2.4e+71) {
		tmp = 1.0;
	} else if (y <= -5.2e+52) {
		tmp = t_0;
	} else if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= -1.02e-63) {
		tmp = -y;
	} else if (y <= 1.0) {
		tmp = x;
	} else if (y <= 1.65e+50) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = -x / y
    if (y <= (-2.4d+71)) then
        tmp = 1.0d0
    else if (y <= (-5.2d+52)) then
        tmp = t_0
    else if (y <= (-1.0d0)) then
        tmp = 1.0d0
    else if (y <= (-1.02d-63)) then
        tmp = -y
    else if (y <= 1.0d0) then
        tmp = x
    else if (y <= 1.65d+50) then
        tmp = t_0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = -x / y;
	double tmp;
	if (y <= -2.4e+71) {
		tmp = 1.0;
	} else if (y <= -5.2e+52) {
		tmp = t_0;
	} else if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= -1.02e-63) {
		tmp = -y;
	} else if (y <= 1.0) {
		tmp = x;
	} else if (y <= 1.65e+50) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -x / y
	tmp = 0
	if y <= -2.4e+71:
		tmp = 1.0
	elif y <= -5.2e+52:
		tmp = t_0
	elif y <= -1.0:
		tmp = 1.0
	elif y <= -1.02e-63:
		tmp = -y
	elif y <= 1.0:
		tmp = x
	elif y <= 1.65e+50:
		tmp = t_0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(-x) / y)
	tmp = 0.0
	if (y <= -2.4e+71)
		tmp = 1.0;
	elseif (y <= -5.2e+52)
		tmp = t_0;
	elseif (y <= -1.0)
		tmp = 1.0;
	elseif (y <= -1.02e-63)
		tmp = Float64(-y);
	elseif (y <= 1.0)
		tmp = x;
	elseif (y <= 1.65e+50)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = -x / y;
	tmp = 0.0;
	if (y <= -2.4e+71)
		tmp = 1.0;
	elseif (y <= -5.2e+52)
		tmp = t_0;
	elseif (y <= -1.0)
		tmp = 1.0;
	elseif (y <= -1.02e-63)
		tmp = -y;
	elseif (y <= 1.0)
		tmp = x;
	elseif (y <= 1.65e+50)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[((-x) / y), $MachinePrecision]}, If[LessEqual[y, -2.4e+71], 1.0, If[LessEqual[y, -5.2e+52], t$95$0, If[LessEqual[y, -1.0], 1.0, If[LessEqual[y, -1.02e-63], (-y), If[LessEqual[y, 1.0], x, If[LessEqual[y, 1.65e+50], t$95$0, 1.0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.39999999999999981e71 or -5.2e52 < y < -1 or 1.65e50 < 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 83.6%

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

   if -2.39999999999999981e71 < y < -5.2e52 or 1 < y < 1.65e50

   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 84.6%

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

      1. sub-neg 84.6%

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

      2. metadata-eval 84.6%

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

      3. neg-mul-1 84.6%

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

      4. distribute-neg-frac 84.6%

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

      5. +-commutative 84.6%

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

   6. Simplified 84.6%

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

      1. frac-2neg 84.6%

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

      2. +-commutative 84.6%

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

      3. metadata-eval 84.6%

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

      4. sub-neg 84.6%

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

      5. div-inv 84.4%

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

      6. remove-double-neg 84.4%

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

      7. sub-neg 84.4%

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

      8. metadata-eval 84.4%

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

      9. distribute-neg-in 84.4%

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

      10. neg-mul-1 84.4%

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

      11. *-commutative 84.4%

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

      12. metadata-eval 84.4%

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

      13. fma-def 84.4%

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

   8. Applied egg-rr 84.4%

      \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(y, -1, 1\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ 84.6%

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

      2. *-rgt-identity 84.6%

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

      3. fma-udef 84.6%

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

      4. *-commutative 84.6%

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

      5. neg-mul-1 84.6%

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

      6. +-commutative 84.6%

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

      7. unsub-neg 84.6%

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

   10. Simplified 84.6%

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

   11. Taylor expanded in y around inf 79.3%

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

       1. associate-*r/ 79.3%

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

       2. neg-mul-1 79.3%

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

   13. Simplified 79.3%

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

   if -1 < y < -1.01999999999999997e-63

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. neg-sub0 99.8%

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

      4. associate-+l- 99.8%

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

      5. sub0-neg 99.8%

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

      6. neg-mul-1 99.8%

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

      7. sub-neg 99.8%

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

      8. +-commutative 99.8%

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

      9. neg-sub0 99.8%

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

      10. associate-+l- 99.8%

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

      11. sub0-neg 99.8%

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

      12. neg-mul-1 99.8%

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

      13. times-frac 99.8%

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

      14. metadata-eval 99.8%

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

      15. *-lft-identity 99.8%

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

      16. sub-neg 99.8%

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

      17. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 62.4%

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

   5. Taylor expanded in y around 0 54.3%

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

      1. neg-mul-1 54.3%

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

   7. Simplified 54.3%

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

   if -1.01999999999999997e-63 < 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 82.1%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+71}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{+52}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.02 \cdot 10^{-63}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+50}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 74.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+71}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2.85 \cdot 10^{+51}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq -1 \cdot 10^{-63} \lor \neg \left(y \leq 1.8 \cdot 10^{+50}\right):\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4e+71)
   1.0
   (if (<= y -2.85e+51)
     (/ (- x) y)
     (if (or (<= y -1e-63) (not (<= y 1.8e+50)))
       (/ y (+ y -1.0))
       (/ x (- 1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4e+71) {
		tmp = 1.0;
	} else if (y <= -2.85e+51) {
		tmp = -x / y;
	} else if ((y <= -1e-63) || !(y <= 1.8e+50)) {
		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 <= (-4d+71)) then
        tmp = 1.0d0
    else if (y <= (-2.85d+51)) then
        tmp = -x / y
    else if ((y <= (-1d-63)) .or. (.not. (y <= 1.8d+50))) 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 <= -4e+71) {
		tmp = 1.0;
	} else if (y <= -2.85e+51) {
		tmp = -x / y;
	} else if ((y <= -1e-63) || !(y <= 1.8e+50)) {
		tmp = y / (y + -1.0);
	} else {
		tmp = x / (1.0 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4e+71:
		tmp = 1.0
	elif y <= -2.85e+51:
		tmp = -x / y
	elif (y <= -1e-63) or not (y <= 1.8e+50):
		tmp = y / (y + -1.0)
	else:
		tmp = x / (1.0 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4e+71)
		tmp = 1.0;
	elseif (y <= -2.85e+51)
		tmp = Float64(Float64(-x) / y);
	elseif ((y <= -1e-63) || !(y <= 1.8e+50))
		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 <= -4e+71)
		tmp = 1.0;
	elseif (y <= -2.85e+51)
		tmp = -x / y;
	elseif ((y <= -1e-63) || ~((y <= 1.8e+50)))
		tmp = y / (y + -1.0);
	else
		tmp = x / (1.0 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4e+71], 1.0, If[LessEqual[y, -2.85e+51], N[((-x) / y), $MachinePrecision], If[Or[LessEqual[y, -1e-63], N[Not[LessEqual[y, 1.8e+50]], $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 < -4.0000000000000002e71

   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 y around inf 82.2%

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

   if -4.0000000000000002e71 < y < -2.8500000000000001e51

   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 90.4%

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

      1. sub-neg 90.4%

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

      2. metadata-eval 90.4%

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

      3. neg-mul-1 90.4%

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

      4. distribute-neg-frac 90.4%

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

      5. +-commutative 90.4%

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

   6. Simplified 90.4%

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

      1. frac-2neg 90.4%

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

      2. +-commutative 90.4%

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

      3. metadata-eval 90.4%

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

      4. sub-neg 90.4%

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

      5. div-inv 90.4%

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

      6. remove-double-neg 90.4%

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

      7. sub-neg 90.4%

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

      8. metadata-eval 90.4%

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

      9. distribute-neg-in 90.4%

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

      10. neg-mul-1 90.4%

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

      11. *-commutative 90.4%

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

      12. metadata-eval 90.4%

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

      13. fma-def 90.4%

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

   8. Applied egg-rr 90.4%

      \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(y, -1, 1\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ 90.4%

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

      2. *-rgt-identity 90.4%

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

      3. fma-udef 90.4%

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

      4. *-commutative 90.4%

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

      5. neg-mul-1 90.4%

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

      6. +-commutative 90.4%

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

      7. unsub-neg 90.4%

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

   10. Simplified 90.4%

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

   11. Taylor expanded in y around inf 90.4%

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

       1. associate-*r/ 90.4%

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

       2. neg-mul-1 90.4%

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

   13. Simplified 90.4%

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

   if -2.8500000000000001e51 < y < -1.00000000000000007e-63 or 1.79999999999999993e50 < 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 80.2%

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

   if -1.00000000000000007e-63 < y < 1.79999999999999993e50

   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 83.1%

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

      1. sub-neg 83.1%

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

      2. metadata-eval 83.1%

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

      3. neg-mul-1 83.1%

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

      4. distribute-neg-frac 83.1%

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

      5. +-commutative 83.1%

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

   6. Simplified 83.1%

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

      1. frac-2neg 83.1%

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

      2. +-commutative 83.1%

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

      3. metadata-eval 83.1%

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

      4. sub-neg 83.1%

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

      5. div-inv 83.0%

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

      6. remove-double-neg 83.0%

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

      7. sub-neg 83.0%

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

      8. metadata-eval 83.0%

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

      9. distribute-neg-in 83.0%

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

      10. neg-mul-1 83.0%

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

      11. *-commutative 83.0%

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

      12. metadata-eval 83.0%

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

      13. fma-def 83.0%

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

   8. Applied egg-rr 83.0%

      \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(y, -1, 1\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ 83.1%

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

      2. *-rgt-identity 83.1%

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

      3. fma-udef 83.1%

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

      4. *-commutative 83.1%

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

      5. neg-mul-1 83.1%

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

      6. +-commutative 83.1%

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

      7. unsub-neg 83.1%

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

   10. Simplified 83.1%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+71}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2.85 \cdot 10^{+51}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq -1 \cdot 10^{-63} \lor \neg \left(y \leq 1.8 \cdot 10^{+50}\right):\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - y}\\ \end{array} \]

Alternative 4: 74.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+72}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -3 \cdot 10^{+53}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq -0.0023:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+50}:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.7e+72)
   1.0
   (if (<= y -3e+53)
     (/ (- x) y)
     (if (<= y -0.0023) 1.0 (if (<= y 2.2e+50) (/ x (- 1.0 y)) 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.7e+72) {
		tmp = 1.0;
	} else if (y <= -3e+53) {
		tmp = -x / y;
	} else if (y <= -0.0023) {
		tmp = 1.0;
	} else if (y <= 2.2e+50) {
		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 <= (-2.7d+72)) then
        tmp = 1.0d0
    else if (y <= (-3d+53)) then
        tmp = -x / y
    else if (y <= (-0.0023d0)) then
        tmp = 1.0d0
    else if (y <= 2.2d+50) 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 <= -2.7e+72) {
		tmp = 1.0;
	} else if (y <= -3e+53) {
		tmp = -x / y;
	} else if (y <= -0.0023) {
		tmp = 1.0;
	} else if (y <= 2.2e+50) {
		tmp = x / (1.0 - y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.7e+72:
		tmp = 1.0
	elif y <= -3e+53:
		tmp = -x / y
	elif y <= -0.0023:
		tmp = 1.0
	elif y <= 2.2e+50:
		tmp = x / (1.0 - y)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.7e+72)
		tmp = 1.0;
	elseif (y <= -3e+53)
		tmp = Float64(Float64(-x) / y);
	elseif (y <= -0.0023)
		tmp = 1.0;
	elseif (y <= 2.2e+50)
		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 <= -2.7e+72)
		tmp = 1.0;
	elseif (y <= -3e+53)
		tmp = -x / y;
	elseif (y <= -0.0023)
		tmp = 1.0;
	elseif (y <= 2.2e+50)
		tmp = x / (1.0 - y);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.7e+72], 1.0, If[LessEqual[y, -3e+53], N[((-x) / y), $MachinePrecision], If[LessEqual[y, -0.0023], 1.0, If[LessEqual[y, 2.2e+50], N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.7000000000000001e72 or -2.99999999999999998e53 < y < -0.0023 or 2.20000000000000017e50 < 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 y around inf 82.4%

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

   if -2.7000000000000001e72 < y < -2.99999999999999998e53

   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 90.4%

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

      1. sub-neg 90.4%

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

      2. metadata-eval 90.4%

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

      3. neg-mul-1 90.4%

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

      4. distribute-neg-frac 90.4%

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

      5. +-commutative 90.4%

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

   6. Simplified 90.4%

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

      1. frac-2neg 90.4%

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

      2. +-commutative 90.4%

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

      3. metadata-eval 90.4%

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

      4. sub-neg 90.4%

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

      5. div-inv 90.4%

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

      6. remove-double-neg 90.4%

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

      7. sub-neg 90.4%

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

      8. metadata-eval 90.4%

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

      9. distribute-neg-in 90.4%

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

      10. neg-mul-1 90.4%

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

      11. *-commutative 90.4%

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

      12. metadata-eval 90.4%

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

      13. fma-def 90.4%

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

   8. Applied egg-rr 90.4%

      \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(y, -1, 1\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ 90.4%

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

      2. *-rgt-identity 90.4%

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

      3. fma-udef 90.4%

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

      4. *-commutative 90.4%

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

      5. neg-mul-1 90.4%

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

      6. +-commutative 90.4%

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

      7. unsub-neg 90.4%

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

   10. Simplified 90.4%

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

   11. Taylor expanded in y around inf 90.4%

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

       1. associate-*r/ 90.4%

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

       2. neg-mul-1 90.4%

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

   13. Simplified 90.4%

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

   if -0.0023 < y < 2.20000000000000017e50

   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 78.1%

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

      1. sub-neg 78.1%

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

      2. metadata-eval 78.1%

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

      3. neg-mul-1 78.1%

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

      4. distribute-neg-frac 78.1%

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

      5. +-commutative 78.1%

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

   6. Simplified 78.1%

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

      1. frac-2neg 78.1%

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

      2. +-commutative 78.1%

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

      3. metadata-eval 78.1%

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

      4. sub-neg 78.1%

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

      5. div-inv 78.0%

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

      6. remove-double-neg 78.0%

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

      7. sub-neg 78.0%

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

      8. metadata-eval 78.0%

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

      9. distribute-neg-in 78.0%

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

      10. neg-mul-1 78.0%

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

      11. *-commutative 78.0%

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

      12. metadata-eval 78.0%

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

      13. fma-def 78.0%

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

   8. Applied egg-rr 78.0%

      \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(y, -1, 1\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ 78.1%

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

      2. *-rgt-identity 78.1%

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

      3. fma-udef 78.1%

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

      4. *-commutative 78.1%

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

      5. neg-mul-1 78.1%

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

      6. +-commutative 78.1%

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

      7. unsub-neg 78.1%

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

   10. Simplified 78.1%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+72}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -3 \cdot 10^{+53}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq -0.0023:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+50}:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -11000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{-63}:\\ \;\;\;\;y \cdot \frac{1}{y + -1}\\ \mathbf{elif}\;y \leq 10000000000000:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (- 1.0 x) y))))
   (if (<= y -11000.0)
     t_0
     (if (<= y -1.05e-63)
       (* y (/ 1.0 (+ y -1.0)))
       (if (<= y 10000000000000.0) (/ x (- 1.0 y)) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((1.0 - x) / y);
	double tmp;
	if (y <= -11000.0) {
		tmp = t_0;
	} else if (y <= -1.05e-63) {
		tmp = y * (1.0 / (y + -1.0));
	} else if (y <= 10000000000000.0) {
		tmp = x / (1.0 - y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((1.0d0 - x) / y)
    if (y <= (-11000.0d0)) then
        tmp = t_0
    else if (y <= (-1.05d-63)) then
        tmp = y * (1.0d0 / (y + (-1.0d0)))
    else if (y <= 10000000000000.0d0) then
        tmp = x / (1.0d0 - y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + ((1.0 - x) / y);
	double tmp;
	if (y <= -11000.0) {
		tmp = t_0;
	} else if (y <= -1.05e-63) {
		tmp = y * (1.0 / (y + -1.0));
	} else if (y <= 10000000000000.0) {
		tmp = x / (1.0 - y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((1.0 - x) / y)
	tmp = 0
	if y <= -11000.0:
		tmp = t_0
	elif y <= -1.05e-63:
		tmp = y * (1.0 / (y + -1.0))
	elif y <= 10000000000000.0:
		tmp = x / (1.0 - y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(1.0 - x) / y))
	tmp = 0.0
	if (y <= -11000.0)
		tmp = t_0;
	elseif (y <= -1.05e-63)
		tmp = Float64(y * Float64(1.0 / Float64(y + -1.0)));
	elseif (y <= 10000000000000.0)
		tmp = Float64(x / Float64(1.0 - y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + ((1.0 - x) / y);
	tmp = 0.0;
	if (y <= -11000.0)
		tmp = t_0;
	elseif (y <= -1.05e-63)
		tmp = y * (1.0 / (y + -1.0));
	elseif (y <= 10000000000000.0)
		tmp = x / (1.0 - y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -11000.0], t$95$0, If[LessEqual[y, -1.05e-63], N[(y * N[(1.0 / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 10000000000000.0], N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -11000 or 1e13 < 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 99.6%

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

      1. +-commutative 99.6%

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

      2. associate-+r+ 99.6%

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

      3. mul-1-neg 99.6%

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

      4. unsub-neg 99.6%

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

      5. div-sub 99.6%

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

      6. unsub-neg 99.6%

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

      7. mul-1-neg 99.6%

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

      8. +-commutative 99.6%

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

      9. metadata-eval 99.6%

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

      10. distribute-lft-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. sub-neg 99.6%

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

      13. associate-*r/ 99.6%

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

      14. +-commutative 99.6%

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

      15. associate-*r/ 99.6%

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

      16. sub-neg 99.6%

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

      17. metadata-eval 99.6%

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

      18. distribute-lft-in 99.6%

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

      19. metadata-eval 99.6%

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

      20. +-commutative 99.6%

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

      21. mul-1-neg 99.6%

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

      22. unsub-neg 99.6%

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

   6. Simplified 99.6%

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

   if -11000 < y < -1.05e-63

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. neg-sub0 99.8%

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

      4. associate-+l- 99.8%

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

      5. sub0-neg 99.8%

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

      6. neg-mul-1 99.8%

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

      7. sub-neg 99.8%

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

      8. +-commutative 99.8%

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

      9. neg-sub0 99.8%

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

      10. associate-+l- 99.8%

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

      11. sub0-neg 99.8%

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

      12. neg-mul-1 99.8%

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

      13. times-frac 99.8%

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

      14. metadata-eval 99.8%

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

      15. *-lft-identity 99.8%

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

      16. sub-neg 99.8%

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

      17. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 62.4%

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

      1. clear-num 62.3%

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

      2. associate-/r/ 62.5%

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

      3. sub-neg 62.5%

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

      4. metadata-eval 62.5%

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

   6. Applied egg-rr 62.5%

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

   if -1.05e-63 < y < 1e13

   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 83.5%

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

      1. sub-neg 83.5%

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

      2. metadata-eval 83.5%

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

      3. neg-mul-1 83.5%

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

      4. distribute-neg-frac 83.5%

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

      5. +-commutative 83.5%

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

   6. Simplified 83.5%

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

      1. frac-2neg 83.5%

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

      2. +-commutative 83.5%

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

      3. metadata-eval 83.5%

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

      4. sub-neg 83.5%

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

      5. div-inv 83.5%

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

      6. remove-double-neg 83.5%

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

      7. sub-neg 83.5%

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

      8. metadata-eval 83.5%

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

      9. distribute-neg-in 83.5%

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

      10. neg-mul-1 83.5%

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

      11. *-commutative 83.5%

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

      12. metadata-eval 83.5%

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

      13. fma-def 83.5%

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

   8. Applied egg-rr 83.5%

      \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(y, -1, 1\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ 83.5%

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

      2. *-rgt-identity 83.5%

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

      3. fma-udef 83.5%

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

      4. *-commutative 83.5%

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

      5. neg-mul-1 83.5%

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

      6. +-commutative 83.5%

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

      7. unsub-neg 83.5%

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

   10. Simplified 83.5%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -11000:\\ \;\;\;\;1 + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{-63}:\\ \;\;\;\;y \cdot \frac{1}{y + -1}\\ \mathbf{elif}\;y \leq 10000000000000:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1 - x}{y}\\ \end{array} \]

Alternative 6: 85.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -15600000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-64}:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{elif}\;y \leq 10000000000000:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))))
   (if (<= y -15600000000.0)
     t_0
     (if (<= y -6e-64)
       (/ y (+ y -1.0))
       (if (<= y 10000000000000.0) (/ x (- 1.0 y)) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -15600000000.0) {
		tmp = t_0;
	} else if (y <= -6e-64) {
		tmp = y / (y + -1.0);
	} else if (y <= 10000000000000.0) {
		tmp = x / (1.0 - y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x / y)
    if (y <= (-15600000000.0d0)) then
        tmp = t_0
    else if (y <= (-6d-64)) then
        tmp = y / (y + (-1.0d0))
    else if (y <= 10000000000000.0d0) then
        tmp = x / (1.0d0 - y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -15600000000.0) {
		tmp = t_0;
	} else if (y <= -6e-64) {
		tmp = y / (y + -1.0);
	} else if (y <= 10000000000000.0) {
		tmp = x / (1.0 - y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (x / y)
	tmp = 0
	if y <= -15600000000.0:
		tmp = t_0
	elif y <= -6e-64:
		tmp = y / (y + -1.0)
	elif y <= 10000000000000.0:
		tmp = x / (1.0 - y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(x / y))
	tmp = 0.0
	if (y <= -15600000000.0)
		tmp = t_0;
	elseif (y <= -6e-64)
		tmp = Float64(y / Float64(y + -1.0));
	elseif (y <= 10000000000000.0)
		tmp = Float64(x / Float64(1.0 - y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (x / y);
	tmp = 0.0;
	if (y <= -15600000000.0)
		tmp = t_0;
	elseif (y <= -6e-64)
		tmp = y / (y + -1.0);
	elseif (y <= 10000000000000.0)
		tmp = x / (1.0 - y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -15600000000.0], t$95$0, If[LessEqual[y, -6e-64], N[(y / N[(y + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 10000000000000.0], N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.56e10 or 1e13 < 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%

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

   7. Taylor expanded in x around inf 99.9%

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

      1. neg-mul-1 99.9%

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

      2. distribute-neg-frac 99.9%

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

   9. Simplified 99.9%

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

   if -1.56e10 < y < -6.0000000000000001e-64

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. neg-sub0 99.8%

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

      4. associate-+l- 99.8%

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

      5. sub0-neg 99.8%

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

      6. neg-mul-1 99.8%

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

      7. sub-neg 99.8%

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

      8. +-commutative 99.8%

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

      9. neg-sub0 99.8%

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

      10. associate-+l- 99.8%

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

      11. sub0-neg 99.8%

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

      12. neg-mul-1 99.8%

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

      13. times-frac 99.8%

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

      14. metadata-eval 99.8%

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

      15. *-lft-identity 99.8%

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

      16. sub-neg 99.8%

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

      17. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 62.8%

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

   if -6.0000000000000001e-64 < y < 1e13

   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 83.5%

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

      1. sub-neg 83.5%

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

      2. metadata-eval 83.5%

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

      3. neg-mul-1 83.5%

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

      4. distribute-neg-frac 83.5%

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

      5. +-commutative 83.5%

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

   6. Simplified 83.5%

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

      1. frac-2neg 83.5%

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

      2. +-commutative 83.5%

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

      3. metadata-eval 83.5%

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

      4. sub-neg 83.5%

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

      5. div-inv 83.5%

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

      6. remove-double-neg 83.5%

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

      7. sub-neg 83.5%

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

      8. metadata-eval 83.5%

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

      9. distribute-neg-in 83.5%

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

      10. neg-mul-1 83.5%

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

      11. *-commutative 83.5%

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

      12. metadata-eval 83.5%

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

      13. fma-def 83.5%

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

   8. Applied egg-rr 83.5%

      \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(y, -1, 1\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ 83.5%

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

      2. *-rgt-identity 83.5%

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

      3. fma-udef 83.5%

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

      4. *-commutative 83.5%

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

      5. neg-mul-1 83.5%

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

      6. +-commutative 83.5%

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

      7. unsub-neg 83.5%

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

   10. Simplified 83.5%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -15600000000:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-64}:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{elif}\;y \leq 10000000000000:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]

Alternative 7: 73.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-64}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y -5e-64) (- y) (if (<= y 1.0) x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= -5e-64) {
		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 <= (-1.0d0)) then
        tmp = 1.0d0
    else if (y <= (-5d-64)) 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 <= -1.0) {
		tmp = 1.0;
	} else if (y <= -5e-64) {
		tmp = -y;
	} else if (y <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= -5e-64:
		tmp = -y
	elif y <= 1.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = 1.0;
	elseif (y <= -5e-64)
		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 <= -1.0)
		tmp = 1.0;
	elseif (y <= -5e-64)
		tmp = -y;
	elseif (y <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], 1.0, If[LessEqual[y, -5e-64], (-y), If[LessEqual[y, 1.0], x, 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 1 < y

   1. Initial program 100.0%

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

      1. sub-neg 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 74.1%

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

   if -1 < y < -5.00000000000000033e-64

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. neg-sub0 99.8%

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

      4. associate-+l- 99.8%

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

      5. sub0-neg 99.8%

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

      6. neg-mul-1 99.8%

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

      7. sub-neg 99.8%

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

      8. +-commutative 99.8%

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

      9. neg-sub0 99.8%

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

      10. associate-+l- 99.8%

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

      11. sub0-neg 99.8%

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

      12. neg-mul-1 99.8%

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

      13. times-frac 99.8%

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

      14. metadata-eval 99.8%

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

      15. *-lft-identity 99.8%

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

      16. sub-neg 99.8%

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

      17. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 62.4%

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

   5. Taylor expanded in y around 0 54.3%

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

      1. neg-mul-1 54.3%

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

   7. Simplified 54.3%

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

   if -5.00000000000000033e-64 < 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 82.1%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-64}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 74.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0023:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -0.0023) 1.0 (if (<= y 1.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.0023) {
		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 <= (-0.0023d0)) 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 <= -0.0023) {
		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 <= -0.0023:
		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 <= -0.0023)
		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 <= -0.0023)
		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, -0.0023], 1.0, If[LessEqual[y, 1.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0023 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 73.3%

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

   if -0.0023 < 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 75.6%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0023:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 38.6% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

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

   1. sub-neg 100.0%

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

   2. +-commutative 100.0%

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

   3. neg-sub0 100.0%

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

   4. associate-+l- 100.0%

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

   5. sub0-neg 100.0%

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

   6. neg-mul-1 100.0%

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

   7. sub-neg 100.0%

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

   8. +-commutative 100.0%

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

   9. neg-sub0 100.0%

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

   10. associate-+l- 100.0%

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

   11. sub0-neg 100.0%

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

   12. neg-mul-1 100.0%

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

   13. times-frac 100.0%

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

   14. metadata-eval 100.0%

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

   15. *-lft-identity 100.0%

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

   16. sub-neg 100.0%

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

   17. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in y around inf 40.3%

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

5. Final simplification 40.3%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023252 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))