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

Percentage Accurate: 100.0% → 100.0%

Time: 3.8s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 85.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -30000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - y\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+57}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+82}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -30000.0)
   1.0
   (if (<= y 1.0)
     (- x y)
     (if (<= y 9e+57) 1.0 (if (<= y 4.7e+82) (/ (- x) y) 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -30000.0) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = x - y;
	} else if (y <= 9e+57) {
		tmp = 1.0;
	} else if (y <= 4.7e+82) {
		tmp = -x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-30000.0d0)) then
        tmp = 1.0d0
    else if (y <= 1.0d0) then
        tmp = x - y
    else if (y <= 9d+57) then
        tmp = 1.0d0
    else if (y <= 4.7d+82) then
        tmp = -x / y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -30000.0) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = x - y;
	} else if (y <= 9e+57) {
		tmp = 1.0;
	} else if (y <= 4.7e+82) {
		tmp = -x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -30000.0:
		tmp = 1.0
	elif y <= 1.0:
		tmp = x - y
	elif y <= 9e+57:
		tmp = 1.0
	elif y <= 4.7e+82:
		tmp = -x / y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -30000.0)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = Float64(x - y);
	elseif (y <= 9e+57)
		tmp = 1.0;
	elseif (y <= 4.7e+82)
		tmp = Float64(Float64(-x) / y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -30000.0)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = x - y;
	elseif (y <= 9e+57)
		tmp = 1.0;
	elseif (y <= 4.7e+82)
		tmp = -x / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -30000.0], 1.0, If[LessEqual[y, 1.0], N[(x - y), $MachinePrecision], If[LessEqual[y, 9e+57], 1.0, If[LessEqual[y, 4.7e+82], N[((-x) / y), $MachinePrecision], 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3e4 or 1 < y < 8.99999999999999991e57 or 4.7e82 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 78.9%

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

   if -3e4 < y < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.1%

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

      1. mul-1-neg 99.1%

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

      2. *-commutative 99.1%

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

      3. distribute-lft-neg-in 99.1%

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

      4. cancel-sign-sub 99.1%

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

      5. +-commutative 99.1%

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

      6. metadata-eval 99.1%

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

      7. distribute-lft-in 99.1%

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

      8. metadata-eval 99.1%

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

      9. sub-neg 99.1%

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

      10. mul-1-neg 99.1%

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

      11. remove-double-neg 99.1%

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

      12. sub-neg 99.1%

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

      13. metadata-eval 99.1%

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

      14. +-commutative 99.1%

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

      15. distribute-neg-in 99.1%

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

      16. metadata-eval 99.1%

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

      17. mul-1-neg 99.1%

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

      18. *-commutative 99.1%

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

      19. mul-1-neg 99.1%

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

      20. unsub-neg 99.1%

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

   4. Simplified 99.1%

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

   5. Taylor expanded in x around 0 99.0%

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

   if 8.99999999999999991e57 < y < 4.7e82

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 84.9%

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

   3. Taylor expanded in y around inf 84.9%

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

      1. associate-*r/ 84.9%

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

      2. neg-mul-1 84.9%

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

   5. Simplified 84.9%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -30000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - y\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+57}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+82}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 86.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{-1}{y}\\ \mathbf{if}\;y \leq -30000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - y\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+82}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ -1.0 y))))
   (if (<= y -30000.0)
     t_0
     (if (<= y 1.0)
       (- x y)
       (if (<= y 8.6e+57) t_0 (if (<= y 4.7e+82) (/ (- x) y) 1.0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (-1.0 / y);
	double tmp;
	if (y <= -30000.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = x - y;
	} else if (y <= 8.6e+57) {
		tmp = t_0;
	} else if (y <= 4.7e+82) {
		tmp = -x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((-1.0d0) / y)
    if (y <= (-30000.0d0)) then
        tmp = t_0
    else if (y <= 1.0d0) then
        tmp = x - y
    else if (y <= 8.6d+57) then
        tmp = t_0
    else if (y <= 4.7d+82) then
        tmp = -x / y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (-1.0 / y);
	double tmp;
	if (y <= -30000.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = x - y;
	} else if (y <= 8.6e+57) {
		tmp = t_0;
	} else if (y <= 4.7e+82) {
		tmp = -x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (-1.0 / y)
	tmp = 0
	if y <= -30000.0:
		tmp = t_0
	elif y <= 1.0:
		tmp = x - y
	elif y <= 8.6e+57:
		tmp = t_0
	elif y <= 4.7e+82:
		tmp = -x / y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(-1.0 / y))
	tmp = 0.0
	if (y <= -30000.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = Float64(x - y);
	elseif (y <= 8.6e+57)
		tmp = t_0;
	elseif (y <= 4.7e+82)
		tmp = Float64(Float64(-x) / y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (-1.0 / y);
	tmp = 0.0;
	if (y <= -30000.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = x - y;
	elseif (y <= 8.6e+57)
		tmp = t_0;
	elseif (y <= 4.7e+82)
		tmp = -x / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -30000.0], t$95$0, If[LessEqual[y, 1.0], N[(x - y), $MachinePrecision], If[LessEqual[y, 8.6e+57], t$95$0, If[LessEqual[y, 4.7e+82], N[((-x) / y), $MachinePrecision], 1.0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3e4 or 1 < y < 8.60000000000000066e57

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 99.6%

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

      1. +-commutative 99.6%

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

      2. mul-1-neg 99.6%

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

      3. unsub-neg 99.6%

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

      4. div-sub 99.6%

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

      5. unsub-neg 99.6%

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

      6. mul-1-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. metadata-eval 99.6%

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

      9. distribute-lft-in 99.6%

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

      10. metadata-eval 99.6%

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

      11. sub-neg 99.6%

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

      12. associate-*r/ 99.6%

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

      13. mul-1-neg 99.6%

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

      14. unsub-neg 99.6%

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

      15. sub-neg 99.6%

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

      16. metadata-eval 99.6%

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

      17. +-commutative 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in x around 0 79.6%

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

   if -3e4 < y < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.1%

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

      1. mul-1-neg 99.1%

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

      2. *-commutative 99.1%

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

      3. distribute-lft-neg-in 99.1%

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

      4. cancel-sign-sub 99.1%

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

      5. +-commutative 99.1%

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

      6. metadata-eval 99.1%

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

      7. distribute-lft-in 99.1%

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

      8. metadata-eval 99.1%

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

      9. sub-neg 99.1%

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

      10. mul-1-neg 99.1%

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

      11. remove-double-neg 99.1%

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

      12. sub-neg 99.1%

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

      13. metadata-eval 99.1%

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

      14. +-commutative 99.1%

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

      15. distribute-neg-in 99.1%

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

      16. metadata-eval 99.1%

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

      17. mul-1-neg 99.1%

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

      18. *-commutative 99.1%

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

      19. mul-1-neg 99.1%

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

      20. unsub-neg 99.1%

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

   4. Simplified 99.1%

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

   5. Taylor expanded in x around 0 99.0%

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

   if 8.60000000000000066e57 < y < 4.7e82

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 84.9%

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

   3. Taylor expanded in y around inf 84.9%

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

      1. associate-*r/ 84.9%

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

      2. neg-mul-1 84.9%

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

   5. Simplified 84.9%

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

   if 4.7e82 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 78.8%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -30000:\\ \;\;\;\;1 - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - y\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+57}:\\ \;\;\;\;1 - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+82}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.25d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = 1.0d0 + ((1.0d0 - x) / y)
    else
        tmp = x - y
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1.25 or 1 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 99.4%

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

      1. +-commutative 99.4%

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

      2. mul-1-neg 99.4%

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

      3. unsub-neg 99.4%

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

      4. div-sub 99.4%

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

      5. unsub-neg 99.4%

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

      6. mul-1-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. metadata-eval 99.4%

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

      9. distribute-lft-in 99.4%

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

      10. metadata-eval 99.4%

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

      11. sub-neg 99.4%

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

      12. associate-*r/ 99.4%

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

      13. mul-1-neg 99.4%

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

      14. unsub-neg 99.4%

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

      15. sub-neg 99.4%

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

      16. metadata-eval 99.4%

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

      17. +-commutative 99.4%

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

   4. Simplified 99.4%

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

   if -1.25 < y < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. cancel-sign-sub 100.0%

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

      5. +-commutative 100.0%

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

      6. metadata-eval 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

      10. mul-1-neg 100.0%

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

      11. remove-double-neg 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

      15. distribute-neg-in 100.0%

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

      16. metadata-eval 100.0%

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

      17. mul-1-neg 100.0%

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

      18. *-commutative 100.0%

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

      19. mul-1-neg 100.0%

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

      20. unsub-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 99.7%

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

4. Final simplification 99.5%

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

Alternative 5: 98.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 99.4%

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

      1. +-commutative 99.4%

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

      2. mul-1-neg 99.4%

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

      3. unsub-neg 99.4%

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

      4. div-sub 99.4%

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

      5. unsub-neg 99.4%

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

      6. mul-1-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. metadata-eval 99.4%

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

      9. distribute-lft-in 99.4%

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

      10. metadata-eval 99.4%

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

      11. sub-neg 99.4%

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

      12. associate-*r/ 99.4%

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

      13. mul-1-neg 99.4%

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

      14. unsub-neg 99.4%

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

      15. sub-neg 99.4%

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

      16. metadata-eval 99.4%

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

      17. +-commutative 99.4%

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

   4. Simplified 99.4%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. cancel-sign-sub 100.0%

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

      5. +-commutative 100.0%

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

      6. metadata-eval 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

      10. mul-1-neg 100.0%

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

      11. remove-double-neg 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

      15. distribute-neg-in 100.0%

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

      16. metadata-eval 100.0%

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

      17. mul-1-neg 100.0%

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

      18. *-commutative 100.0%

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

      19. mul-1-neg 100.0%

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

      20. unsub-neg 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 99.6%

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

Alternative 6: 98.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 99.4%

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

      1. +-commutative 99.4%

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

      2. mul-1-neg 99.4%

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

      3. unsub-neg 99.4%

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

      4. div-sub 99.4%

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

      5. unsub-neg 99.4%

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

      6. mul-1-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. metadata-eval 99.4%

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

      9. distribute-lft-in 99.4%

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

      10. metadata-eval 99.4%

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

      11. sub-neg 99.4%

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

      12. associate-*r/ 99.4%

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

      13. mul-1-neg 99.4%

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

      14. unsub-neg 99.4%

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

      15. sub-neg 99.4%

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

      16. metadata-eval 99.4%

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

      17. +-commutative 99.4%

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

   4. Simplified 99.4%

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

   5. Taylor expanded in x around inf 99.0%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. cancel-sign-sub 100.0%

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

      5. +-commutative 100.0%

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

      6. metadata-eval 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

      10. mul-1-neg 100.0%

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

      11. remove-double-neg 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

      15. distribute-neg-in 100.0%

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

      16. metadata-eval 100.0%

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

      17. mul-1-neg 100.0%

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

      18. *-commutative 100.0%

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

      19. mul-1-neg 100.0%

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

      20. unsub-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 99.7%

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

4. Final simplification 99.3%

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

Alternative 7: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -30000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -30000.0) 1.0 (if (<= y 1.0) (- x y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -30000.0) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = x - y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-30000.0d0)) then
        tmp = 1.0d0
    else if (y <= 1.0d0) then
        tmp = x - y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -30000.0) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = x - y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -30000.0:
		tmp = 1.0
	elif y <= 1.0:
		tmp = x - y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -30000.0)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = Float64(x - y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -30000.0)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = x - y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -30000.0], 1.0, If[LessEqual[y, 1.0], N[(x - y), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -3e4 or 1 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 73.4%

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

   if -3e4 < y < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.1%

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

      1. mul-1-neg 99.1%

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

      2. *-commutative 99.1%

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

      3. distribute-lft-neg-in 99.1%

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

      4. cancel-sign-sub 99.1%

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

      5. +-commutative 99.1%

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

      6. metadata-eval 99.1%

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

      7. distribute-lft-in 99.1%

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

      8. metadata-eval 99.1%

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

      9. sub-neg 99.1%

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

      10. mul-1-neg 99.1%

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

      11. remove-double-neg 99.1%

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

      12. sub-neg 99.1%

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

      13. metadata-eval 99.1%

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

      14. +-commutative 99.1%

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

      15. distribute-neg-in 99.1%

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

      16. metadata-eval 99.1%

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

      17. mul-1-neg 99.1%

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

      18. *-commutative 99.1%

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

      19. mul-1-neg 99.1%

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

      20. unsub-neg 99.1%

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

   4. Simplified 99.1%

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

   5. Taylor expanded in x around 0 99.0%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -30000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 74.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -30000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -30000.0) 1.0 (if (<= y 1.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -30000.0) {
		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 <= (-30000.0d0)) 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 <= -30000.0) {
		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 <= -30000.0:
		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 <= -30000.0)
		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 <= -30000.0)
		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, -30000.0], 1.0, If[LessEqual[y, 1.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -3e4 or 1 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 73.4%

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

   if -3e4 < y < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 77.2%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -30000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 39.3% 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. Taylor expanded in y around inf 41.8%

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

3. Final simplification 41.8%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023320 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))