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

Percentage Accurate: 100.0% → 100.0%

Time: 6.3s

Alternatives: 8

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. Add Preprocessing
3. Add Preprocessing

Alternative 2: 73.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-12}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-59}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9.6:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5e-12) 1.0 (if (<= y 1.1e-59) x (if (<= y 9.6) (- y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5e-12) {
		tmp = 1.0;
	} else if (y <= 1.1e-59) {
		tmp = x;
	} else if (y <= 9.6) {
		tmp = -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 <= (-5d-12)) then
        tmp = 1.0d0
    else if (y <= 1.1d-59) then
        tmp = x
    else if (y <= 9.6d0) then
        tmp = -y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -5e-12) {
		tmp = 1.0;
	} else if (y <= 1.1e-59) {
		tmp = x;
	} else if (y <= 9.6) {
		tmp = -y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5e-12:
		tmp = 1.0
	elif y <= 1.1e-59:
		tmp = x
	elif y <= 9.6:
		tmp = -y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5e-12)
		tmp = 1.0;
	elseif (y <= 1.1e-59)
		tmp = x;
	elseif (y <= 9.6)
		tmp = Float64(-y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5e-12)
		tmp = 1.0;
	elseif (y <= 1.1e-59)
		tmp = x;
	elseif (y <= 9.6)
		tmp = -y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5e-12], 1.0, If[LessEqual[y, 1.1e-59], x, If[LessEqual[y, 9.6], (-y), 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.9999999999999997e-12 or 9.59999999999999964 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 76.7%

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

   if -4.9999999999999997e-12 < y < 1.0999999999999999e-59

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 75.2%

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

   if 1.0999999999999999e-59 < y < 9.59999999999999964

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 62.5%

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

      1. neg-mul-1 62.5%

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

      2. distribute-neg-frac2 62.5%

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

      3. neg-sub0 62.5%

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

      4. associate--r- 62.5%

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

      5. metadata-eval 62.5%

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

   5. Simplified 62.5%

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

   6. Taylor expanded in y around 0 62.7%

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

      1. sub-neg 62.7%

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

      2. neg-mul-1 62.7%

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

      3. metadata-eval 62.7%

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

      4. +-commutative 62.7%

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

      5. sub-neg 62.7%

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

   8. Simplified 62.7%

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

   9. Taylor expanded in y around 0 62.3%

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

   10. Taylor expanded in y around 0 62.3%

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

       1. neg-mul-1 62.3%

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

   12. Simplified 62.3%

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

Alternative 3: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.82:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x + y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -0.82)
   (- 1.0 (/ x y))
   (if (<= y 1.0) (+ x (* y (+ x -1.0))) (+ 1.0 (/ (- 1.0 x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.82) {
		tmp = 1.0 - (x / y);
	} else if (y <= 1.0) {
		tmp = x + (y * (x + -1.0));
	} else {
		tmp = 1.0 + ((1.0 - 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 <= (-0.82d0)) then
        tmp = 1.0d0 - (x / y)
    else if (y <= 1.0d0) then
        tmp = x + (y * (x + (-1.0d0)))
    else
        tmp = 1.0d0 + ((1.0d0 - x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -0.82) {
		tmp = 1.0 - (x / y);
	} else if (y <= 1.0) {
		tmp = x + (y * (x + -1.0));
	} else {
		tmp = 1.0 + ((1.0 - x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -0.82:
		tmp = 1.0 - (x / y)
	elif y <= 1.0:
		tmp = x + (y * (x + -1.0))
	else:
		tmp = 1.0 + ((1.0 - x) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -0.82)
		tmp = Float64(1.0 - Float64(x / y));
	elseif (y <= 1.0)
		tmp = Float64(x + Float64(y * Float64(x + -1.0)));
	else
		tmp = Float64(1.0 + Float64(Float64(1.0 - x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -0.82)
		tmp = 1.0 - (x / y);
	elseif (y <= 1.0)
		tmp = x + (y * (x + -1.0));
	else
		tmp = 1.0 + ((1.0 - x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -0.82], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(x + N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.819999999999999951

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 100.0%

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

   4. Taylor expanded in y around inf 99.4%

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

      1. neg-mul-1 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in x around 0 99.4%

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

      1. neg-mul-1 99.4%

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

      2. unsub-neg 99.4%

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

   9. Simplified 99.4%

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

   if -0.819999999999999951 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. unsub-neg 99.5%

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

      3. mul-1-neg 99.5%

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

      4. sub-neg 99.5%

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

   5. Simplified 99.5%

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

   if 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 97.9%

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

      1. +-commutative 97.9%

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

      2. mul-1-neg 97.9%

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

      3. sub-neg 97.9%

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

      4. div-sub 97.9%

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

   5. Simplified 97.9%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.82:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x + y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1 - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 100.0%

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

   4. Taylor expanded in y around inf 99.4%

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

      1. neg-mul-1 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in x around 0 99.4%

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

      1. neg-mul-1 99.4%

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

      2. unsub-neg 99.4%

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

   9. Simplified 99.4%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. unsub-neg 99.5%

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

      3. mul-1-neg 99.5%

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

      4. sub-neg 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 99.2%

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

   if 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 97.9%

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

      1. +-commutative 97.9%

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

      2. mul-1-neg 97.9%

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

      3. sub-neg 97.9%

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

      4. div-sub 97.9%

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

   5. Simplified 97.9%

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

Alternative 5: 97.9% accurate, 0.5× 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. Add Preprocessing

   3. Taylor expanded in y around inf 100.0%

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

   4. Taylor expanded in y around inf 98.3%

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

      1. neg-mul-1 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in x around 0 98.4%

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

      1. neg-mul-1 98.4%

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

      2. unsub-neg 98.4%

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

   9. Simplified 98.4%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. unsub-neg 99.5%

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

      3. mul-1-neg 99.5%

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

      4. sub-neg 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 99.2%

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

4. Final simplification 98.8%

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

Alternative 6: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -880000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -880000000.0) 1.0 (if (<= y 1.0) (- x y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -880000000.0) {
		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 <= (-880000000.0d0)) 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 <= -880000000.0) {
		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 <= -880000000.0:
		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 <= -880000000.0)
		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 <= -880000000.0)
		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, -880000000.0], 1.0, If[LessEqual[y, 1.0], N[(x - y), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -8.8e8 or 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 77.7%

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

   if -8.8e8 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 98.7%

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

      1. mul-1-neg 98.7%

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

      2. unsub-neg 98.7%

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

      3. mul-1-neg 98.7%

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

      4. sub-neg 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in x around 0 98.5%

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

Alternative 7: 73.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-12}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -5e-12) 1.0 (if (<= y 1.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5e-12) {
		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 <= (-5d-12)) 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 <= -5e-12) {
		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 <= -5e-12:
		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 <= -5e-12)
		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 <= -5e-12)
		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, -5e-12], 1.0, If[LessEqual[y, 1.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -4.9999999999999997e-12 or 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 76.2%

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

   if -4.9999999999999997e-12 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 70.6%

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

Alternative 8: 38.4% 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. Add Preprocessing

3. Taylor expanded in y around inf 43.8%

   \[\leadsto \color{blue}{1} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024133 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))