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

Percentage Accurate: 100.0% → 100.0%

Time: 3.2s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{1 - x}{y}\\ t_1 := \frac{-x}{y + -1}\\ \mathbf{if}\;y \leq -700:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.05 \cdot 10^{-10}:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{elif}\;y \leq 920:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (- 1.0 x) y))) (t_1 (/ (- x) (+ y -1.0))))
   (if (<= y -700.0)
     t_0
     (if (<= y 1.7e-98)
       t_1
       (if (<= y 3.05e-10) (/ y (+ y -1.0)) (if (<= y 920.0) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((1.0 - x) / y);
	double t_1 = -x / (y + -1.0);
	double tmp;
	if (y <= -700.0) {
		tmp = t_0;
	} else if (y <= 1.7e-98) {
		tmp = t_1;
	} else if (y <= 3.05e-10) {
		tmp = y / (y + -1.0);
	} else if (y <= 920.0) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + ((1.0d0 - x) / y)
    t_1 = -x / (y + (-1.0d0))
    if (y <= (-700.0d0)) then
        tmp = t_0
    else if (y <= 1.7d-98) then
        tmp = t_1
    else if (y <= 3.05d-10) then
        tmp = y / (y + (-1.0d0))
    else if (y <= 920.0d0) then
        tmp = t_1
    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 t_1 = -x / (y + -1.0);
	double tmp;
	if (y <= -700.0) {
		tmp = t_0;
	} else if (y <= 1.7e-98) {
		tmp = t_1;
	} else if (y <= 3.05e-10) {
		tmp = y / (y + -1.0);
	} else if (y <= 920.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((1.0 - x) / y)
	t_1 = -x / (y + -1.0)
	tmp = 0
	if y <= -700.0:
		tmp = t_0
	elif y <= 1.7e-98:
		tmp = t_1
	elif y <= 3.05e-10:
		tmp = y / (y + -1.0)
	elif y <= 920.0:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(1.0 - x) / y))
	t_1 = Float64(Float64(-x) / Float64(y + -1.0))
	tmp = 0.0
	if (y <= -700.0)
		tmp = t_0;
	elseif (y <= 1.7e-98)
		tmp = t_1;
	elseif (y <= 3.05e-10)
		tmp = Float64(y / Float64(y + -1.0));
	elseif (y <= 920.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + ((1.0 - x) / y);
	t_1 = -x / (y + -1.0);
	tmp = 0.0;
	if (y <= -700.0)
		tmp = t_0;
	elseif (y <= 1.7e-98)
		tmp = t_1;
	elseif (y <= 3.05e-10)
		tmp = y / (y + -1.0);
	elseif (y <= 920.0)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[((-x) / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -700.0], t$95$0, If[LessEqual[y, 1.7e-98], t$95$1, If[LessEqual[y, 3.05e-10], N[(y / N[(y + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 920.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -700 or 920 < y

   1. Initial program 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. associate-/r/ 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-out 100.0%

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

      9. remove-double-neg 100.0%

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

      10. sub-neg 100.0%

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

      11. metadata-eval 100.0%

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

      12. associate-/l* 100.0%

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

      13. associate-*l/ 100.0%

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

      14. /-rgt-identity 100.0%

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

      15. *-commutative 100.0%

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

      16. neg-mul-1 100.0%

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

      17. sub0-neg 100.0%

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

      18. associate--r- 100.0%

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

      19. metadata-eval 100.0%

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

      20. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. neg-sub0 99.6%

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

      3. associate-+l- 99.6%

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

      4. div-sub 99.6%

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

      5. neg-sub0 99.6%

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

      6. mul-1-neg 99.6%

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

      7. associate-*r/ 99.6%

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

      8. sub-neg 99.6%

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

      9. metadata-eval 99.6%

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

      10. distribute-lft-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. +-commutative 99.6%

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

      13. mul-1-neg 99.6%

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

      14. 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 -700 < y < 1.7000000000000001e-98 or 3.0499999999999998e-10 < y < 920

   1. Initial program 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. associate-/r/ 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-out 100.0%

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

      9. remove-double-neg 100.0%

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

      10. sub-neg 100.0%

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

      11. metadata-eval 100.0%

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

      12. associate-/l* 100.0%

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

      13. associate-*l/ 100.0%

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

      14. /-rgt-identity 100.0%

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

      15. *-commutative 100.0%

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

      16. neg-mul-1 100.0%

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

      17. sub0-neg 100.0%

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

      18. associate--r- 100.0%

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

      19. metadata-eval 100.0%

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

      20. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 86.7%

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

      1. associate-*r/ 86.7%

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

      2. mul-1-neg 86.7%

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

      3. sub-neg 86.7%

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

      4. metadata-eval 86.7%

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

      5. +-commutative 86.7%

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

   6. Simplified 86.7%

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

   if 1.7000000000000001e-98 < y < 3.0499999999999998e-10

   1. Initial program 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. associate-/r/ 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-out 100.0%

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

      9. remove-double-neg 100.0%

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

      10. sub-neg 100.0%

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

      11. metadata-eval 100.0%

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

      12. associate-/l* 100.0%

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

      13. associate-*l/ 100.0%

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

      14. /-rgt-identity 100.0%

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

      15. *-commutative 100.0%

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

      16. neg-mul-1 100.0%

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

      17. sub0-neg 100.0%

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

      18. associate--r- 100.0%

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

      19. metadata-eval 100.0%

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

      20. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 65.7%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -700:\\ \;\;\;\;1 + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-98}:\\ \;\;\;\;\frac{-x}{y + -1}\\ \mathbf{elif}\;y \leq 3.05 \cdot 10^{-10}:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{elif}\;y \leq 920:\\ \;\;\;\;\frac{-x}{y + -1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1 - x}{y}\\ \end{array} \]

Alternative 3: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ t_1 := \frac{-x}{y + -1}\\ \mathbf{if}\;y \leq -980:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.72 \cdot 10^{-9}:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{elif}\;y \leq 32000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))) (t_1 (/ (- x) (+ y -1.0))))
   (if (<= y -980.0)
     t_0
     (if (<= y 1.7e-98)
       t_1
       (if (<= y 1.72e-9) (/ y (+ y -1.0)) (if (<= y 32000000.0) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (x / y);
	double t_1 = -x / (y + -1.0);
	double tmp;
	if (y <= -980.0) {
		tmp = t_0;
	} else if (y <= 1.7e-98) {
		tmp = t_1;
	} else if (y <= 1.72e-9) {
		tmp = y / (y + -1.0);
	} else if (y <= 32000000.0) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (x / y)
    t_1 = -x / (y + (-1.0d0))
    if (y <= (-980.0d0)) then
        tmp = t_0
    else if (y <= 1.7d-98) then
        tmp = t_1
    else if (y <= 1.72d-9) then
        tmp = y / (y + (-1.0d0))
    else if (y <= 32000000.0d0) then
        tmp = t_1
    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 t_1 = -x / (y + -1.0);
	double tmp;
	if (y <= -980.0) {
		tmp = t_0;
	} else if (y <= 1.7e-98) {
		tmp = t_1;
	} else if (y <= 1.72e-9) {
		tmp = y / (y + -1.0);
	} else if (y <= 32000000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (x / y)
	t_1 = -x / (y + -1.0)
	tmp = 0
	if y <= -980.0:
		tmp = t_0
	elif y <= 1.7e-98:
		tmp = t_1
	elif y <= 1.72e-9:
		tmp = y / (y + -1.0)
	elif y <= 32000000.0:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(x / y))
	t_1 = Float64(Float64(-x) / Float64(y + -1.0))
	tmp = 0.0
	if (y <= -980.0)
		tmp = t_0;
	elseif (y <= 1.7e-98)
		tmp = t_1;
	elseif (y <= 1.72e-9)
		tmp = Float64(y / Float64(y + -1.0));
	elseif (y <= 32000000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (x / y);
	t_1 = -x / (y + -1.0);
	tmp = 0.0;
	if (y <= -980.0)
		tmp = t_0;
	elseif (y <= 1.7e-98)
		tmp = t_1;
	elseif (y <= 1.72e-9)
		tmp = y / (y + -1.0);
	elseif (y <= 32000000.0)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[((-x) / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -980.0], t$95$0, If[LessEqual[y, 1.7e-98], t$95$1, If[LessEqual[y, 1.72e-9], N[(y / N[(y + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 32000000.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -980 or 3.2e7 < y

   1. Initial program 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. associate-/r/ 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-out 100.0%

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

      9. remove-double-neg 100.0%

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

      10. sub-neg 100.0%

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

      11. metadata-eval 100.0%

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

      12. associate-/l* 100.0%

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

      13. associate-*l/ 100.0%

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

      14. /-rgt-identity 100.0%

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

      15. *-commutative 100.0%

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

      16. neg-mul-1 100.0%

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

      17. sub0-neg 100.0%

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

      18. associate--r- 100.0%

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

      19. metadata-eval 100.0%

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

      20. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. neg-sub0 99.6%

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

      3. associate-+l- 99.6%

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

      4. div-sub 99.6%

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

      5. neg-sub0 99.6%

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

      6. mul-1-neg 99.6%

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

      7. associate-*r/ 99.6%

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

      8. sub-neg 99.6%

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

      9. metadata-eval 99.6%

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

      10. distribute-lft-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. +-commutative 99.6%

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

      13. mul-1-neg 99.6%

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

      14. unsub-neg 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around inf 99.0%

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

      1. mul-1-neg 99.0%

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

      2. distribute-frac-neg 99.0%

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

   9. Simplified 99.0%

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

   if -980 < y < 1.7000000000000001e-98 or 1.72000000000000006e-9 < y < 3.2e7

   1. Initial program 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. associate-/r/ 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-out 100.0%

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

      9. remove-double-neg 100.0%

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

      10. sub-neg 100.0%

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

      11. metadata-eval 100.0%

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

      12. associate-/l* 100.0%

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

      13. associate-*l/ 100.0%

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

      14. /-rgt-identity 100.0%

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

      15. *-commutative 100.0%

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

      16. neg-mul-1 100.0%

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

      17. sub0-neg 100.0%

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

      18. associate--r- 100.0%

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

      19. metadata-eval 100.0%

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

      20. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 86.7%

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

      1. associate-*r/ 86.7%

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

      2. mul-1-neg 86.7%

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

      3. sub-neg 86.7%

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

      4. metadata-eval 86.7%

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

      5. +-commutative 86.7%

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

   6. Simplified 86.7%

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

   if 1.7000000000000001e-98 < y < 1.72000000000000006e-9

   1. Initial program 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. associate-/r/ 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-out 100.0%

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

      9. remove-double-neg 100.0%

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

      10. sub-neg 100.0%

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

      11. metadata-eval 100.0%

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

      12. associate-/l* 100.0%

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

      13. associate-*l/ 100.0%

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

      14. /-rgt-identity 100.0%

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

      15. *-commutative 100.0%

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

      16. neg-mul-1 100.0%

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

      17. sub0-neg 100.0%

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

      18. associate--r- 100.0%

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

      19. metadata-eval 100.0%

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

      20. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 65.7%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -980:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-98}:\\ \;\;\;\;\frac{-x}{y + -1}\\ \mathbf{elif}\;y \leq 1.72 \cdot 10^{-9}:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{elif}\;y \leq 32000000:\\ \;\;\;\;\frac{-x}{y + -1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]

Alternative 4: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ t_1 := \frac{y}{y + -1}\\ \mathbf{if}\;y \leq -840000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{-67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-99}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 26500000000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))) (t_1 (/ y (+ y -1.0))))
   (if (<= y -840000000.0)
     t_0
     (if (<= y -1.9e-67)
       t_1
       (if (<= y 6.5e-99) x (if (<= y 26500000000000.0) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (x / y);
	double t_1 = y / (y + -1.0);
	double tmp;
	if (y <= -840000000.0) {
		tmp = t_0;
	} else if (y <= -1.9e-67) {
		tmp = t_1;
	} else if (y <= 6.5e-99) {
		tmp = x;
	} else if (y <= 26500000000000.0) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (x / y)
    t_1 = y / (y + (-1.0d0))
    if (y <= (-840000000.0d0)) then
        tmp = t_0
    else if (y <= (-1.9d-67)) then
        tmp = t_1
    else if (y <= 6.5d-99) then
        tmp = x
    else if (y <= 26500000000000.0d0) then
        tmp = t_1
    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 t_1 = y / (y + -1.0);
	double tmp;
	if (y <= -840000000.0) {
		tmp = t_0;
	} else if (y <= -1.9e-67) {
		tmp = t_1;
	} else if (y <= 6.5e-99) {
		tmp = x;
	} else if (y <= 26500000000000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (x / y)
	t_1 = y / (y + -1.0)
	tmp = 0
	if y <= -840000000.0:
		tmp = t_0
	elif y <= -1.9e-67:
		tmp = t_1
	elif y <= 6.5e-99:
		tmp = x
	elif y <= 26500000000000.0:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(x / y))
	t_1 = Float64(y / Float64(y + -1.0))
	tmp = 0.0
	if (y <= -840000000.0)
		tmp = t_0;
	elseif (y <= -1.9e-67)
		tmp = t_1;
	elseif (y <= 6.5e-99)
		tmp = x;
	elseif (y <= 26500000000000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (x / y);
	t_1 = y / (y + -1.0);
	tmp = 0.0;
	if (y <= -840000000.0)
		tmp = t_0;
	elseif (y <= -1.9e-67)
		tmp = t_1;
	elseif (y <= 6.5e-99)
		tmp = x;
	elseif (y <= 26500000000000.0)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(y / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -840000000.0], t$95$0, If[LessEqual[y, -1.9e-67], t$95$1, If[LessEqual[y, 6.5e-99], x, If[LessEqual[y, 26500000000000.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.4e8 or 2.65e13 < y

   1. Initial program 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. associate-/r/ 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-out 100.0%

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

      9. remove-double-neg 100.0%

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

      10. sub-neg 100.0%

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

      11. metadata-eval 100.0%

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

      12. associate-/l* 100.0%

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

      13. associate-*l/ 100.0%

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

      14. /-rgt-identity 100.0%

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

      15. *-commutative 100.0%

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

      16. neg-mul-1 100.0%

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

      17. sub0-neg 100.0%

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

      18. associate--r- 100.0%

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

      19. metadata-eval 100.0%

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

      20. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. neg-sub0 99.9%

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

      3. associate-+l- 99.9%

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

      4. div-sub 99.9%

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

      5. neg-sub0 99.9%

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

      6. mul-1-neg 99.9%

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

      7. associate-*r/ 99.9%

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

      8. sub-neg 99.9%

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

      9. metadata-eval 99.9%

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

      10. distribute-lft-in 99.9%

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

      11. metadata-eval 99.9%

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

      12. +-commutative 99.9%

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

      13. mul-1-neg 99.9%

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

      14. unsub-neg 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

   9. Simplified 99.9%

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

   if -8.4e8 < y < -1.89999999999999994e-67 or 6.50000000000000033e-99 < y < 2.65e13

   1. Initial program 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. associate-/r/ 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-out 100.0%

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

      9. remove-double-neg 100.0%

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

      10. sub-neg 100.0%

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

      11. metadata-eval 100.0%

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

      12. associate-/l* 100.0%

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

      13. associate-*l/ 100.0%

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

      14. /-rgt-identity 100.0%

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

      15. *-commutative 100.0%

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

      16. neg-mul-1 100.0%

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

      17. sub0-neg 100.0%

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

      18. associate--r- 100.0%

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

      19. metadata-eval 100.0%

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

      20. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 65.1%

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

   if -1.89999999999999994e-67 < y < 6.50000000000000033e-99

   1. Initial program 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. associate-/r/ 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-out 100.0%

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

      9. remove-double-neg 100.0%

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

      10. sub-neg 100.0%

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

      11. metadata-eval 100.0%

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

      12. associate-/l* 100.0%

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

      13. associate-*l/ 100.0%

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

      14. /-rgt-identity 100.0%

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

      15. *-commutative 100.0%

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

      16. neg-mul-1 100.0%

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

      17. sub0-neg 100.0%

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

      18. associate--r- 100.0%

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

      19. metadata-eval 100.0%

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

      20. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 92.6%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -840000000:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{-67}:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-99}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 26500000000000:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]

Alternative 5: 72.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-67} \lor \neg \left(y \leq 1.7 \cdot 10^{-98}\right):\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.9e-67) (not (<= y 1.7e-98))) (/ y (+ y -1.0)) x))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.9e-67) || !(y <= 1.7e-98)) {
		tmp = y / (y + -1.0);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.9d-67)) .or. (.not. (y <= 1.7d-98))) then
        tmp = y / (y + (-1.0d0))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.9e-67) || !(y <= 1.7e-98)) {
		tmp = y / (y + -1.0);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.9e-67) or not (y <= 1.7e-98):
		tmp = y / (y + -1.0)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.9e-67) || !(y <= 1.7e-98))
		tmp = Float64(y / Float64(y + -1.0));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.9e-67) || ~((y <= 1.7e-98)))
		tmp = y / (y + -1.0);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.9e-67], N[Not[LessEqual[y, 1.7e-98]], $MachinePrecision]], N[(y / N[(y + -1.0), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -1.89999999999999994e-67 or 1.7000000000000001e-98 < y

   1. Initial program 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. associate-/r/ 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-out 100.0%

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

      9. remove-double-neg 100.0%

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

      10. sub-neg 100.0%

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

      11. metadata-eval 100.0%

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

      12. associate-/l* 100.0%

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

      13. associate-*l/ 100.0%

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

      14. /-rgt-identity 100.0%

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

      15. *-commutative 100.0%

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

      16. neg-mul-1 100.0%

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

      17. sub0-neg 100.0%

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

      18. associate--r- 100.0%

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

      19. metadata-eval 100.0%

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

      20. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 73.3%

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

   if -1.89999999999999994e-67 < y < 1.7000000000000001e-98

   1. Initial program 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. associate-/r/ 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-out 100.0%

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

      9. remove-double-neg 100.0%

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

      10. sub-neg 100.0%

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

      11. metadata-eval 100.0%

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

      12. associate-/l* 100.0%

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

      13. associate-*l/ 100.0%

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

      14. /-rgt-identity 100.0%

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

      15. *-commutative 100.0%

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

      16. neg-mul-1 100.0%

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

      17. sub0-neg 100.0%

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

      18. associate--r- 100.0%

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

      19. metadata-eval 100.0%

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

      20. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 92.6%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-67} \lor \neg \left(y \leq 1.7 \cdot 10^{-98}\right):\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 73.8% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (if (<= y -7.0) 1.0 (if (<= y 1.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.0) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-7.0d0)) then
        tmp = 1.0d0
    else if (y <= 1.0d0) then
        tmp = x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.0) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.0:
		tmp = 1.0
	elif y <= 1.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.0)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.0)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.0], 1.0, If[LessEqual[y, 1.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -7 or 1 < y

   1. Initial program 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. associate-/r/ 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-out 100.0%

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

      9. remove-double-neg 100.0%

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

      10. sub-neg 100.0%

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

      11. metadata-eval 100.0%

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

      12. associate-/l* 100.0%

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

      13. associate-*l/ 100.0%

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

      14. /-rgt-identity 100.0%

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

      15. *-commutative 100.0%

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

      16. neg-mul-1 100.0%

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

      17. sub0-neg 100.0%

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

      18. associate--r- 100.0%

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

      19. metadata-eval 100.0%

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

      20. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 74.7%

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

   if -7 < y < 1

   1. Initial program 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. associate-/r/ 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-out 100.0%

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

      9. remove-double-neg 100.0%

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

      10. sub-neg 100.0%

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

      11. metadata-eval 100.0%

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

      12. associate-/l* 100.0%

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

      13. associate-*l/ 100.0%

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

      14. /-rgt-identity 100.0%

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

      15. *-commutative 100.0%

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

      16. neg-mul-1 100.0%

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

      17. sub0-neg 100.0%

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

      18. associate--r- 100.0%

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

      19. metadata-eval 100.0%

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

      20. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 79.0%

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

4. Final simplification 76.7%

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

Alternative 7: 38.7% 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. *-rgt-identity 100.0%

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

   2. metadata-eval 100.0%

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

   3. distribute-rgt-neg-in 100.0%

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

   4. associate-/r/ 100.0%

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

   5. distribute-neg-frac 100.0%

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

   6. sub-neg 100.0%

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

   7. +-commutative 100.0%

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

   8. distribute-neg-out 100.0%

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

   9. remove-double-neg 100.0%

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

   10. sub-neg 100.0%

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

   11. metadata-eval 100.0%

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

   12. associate-/l* 100.0%

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

   13. associate-*l/ 100.0%

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

   14. /-rgt-identity 100.0%

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

   15. *-commutative 100.0%

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

   16. neg-mul-1 100.0%

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

   17. sub0-neg 100.0%

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

   18. associate--r- 100.0%

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

   19. metadata-eval 100.0%

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

   20. +-commutative 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in y around inf 40.4%

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

5. Final simplification 40.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023301 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))