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

Percentage Accurate: 100.0% → 100.0%

Time: 7.1s

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, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. div-sub 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \frac{x}{1 - y} + \frac{y}{y + -1} \]
6. Add Preprocessing

Alternative 2: 86.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ t_1 := \frac{x}{1 - y}\\ \mathbf{if}\;y \leq -235000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-32}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 27000000000:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))) (t_1 (/ x (- 1.0 y))))
   (if (<= y -235000000.0)
     t_0
     (if (<= y 7.5e-49)
       t_1
       (if (<= y 5.2e-32) (- y) (if (<= y 27000000000.0) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (x / y);
	double t_1 = x / (1.0 - y);
	double tmp;
	if (y <= -235000000.0) {
		tmp = t_0;
	} else if (y <= 7.5e-49) {
		tmp = t_1;
	} else if (y <= 5.2e-32) {
		tmp = -y;
	} else if (y <= 27000000000.0) {
		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 / (1.0d0 - y)
    if (y <= (-235000000.0d0)) then
        tmp = t_0
    else if (y <= 7.5d-49) then
        tmp = t_1
    else if (y <= 5.2d-32) then
        tmp = -y
    else if (y <= 27000000000.0d0) 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 / (1.0 - y);
	double tmp;
	if (y <= -235000000.0) {
		tmp = t_0;
	} else if (y <= 7.5e-49) {
		tmp = t_1;
	} else if (y <= 5.2e-32) {
		tmp = -y;
	} else if (y <= 27000000000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (x / y)
	t_1 = x / (1.0 - y)
	tmp = 0
	if y <= -235000000.0:
		tmp = t_0
	elif y <= 7.5e-49:
		tmp = t_1
	elif y <= 5.2e-32:
		tmp = -y
	elif y <= 27000000000.0:
		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(x / Float64(1.0 - y))
	tmp = 0.0
	if (y <= -235000000.0)
		tmp = t_0;
	elseif (y <= 7.5e-49)
		tmp = t_1;
	elseif (y <= 5.2e-32)
		tmp = Float64(-y);
	elseif (y <= 27000000000.0)
		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 / (1.0 - y);
	tmp = 0.0;
	if (y <= -235000000.0)
		tmp = t_0;
	elseif (y <= 7.5e-49)
		tmp = t_1;
	elseif (y <= 5.2e-32)
		tmp = -y;
	elseif (y <= 27000000000.0)
		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[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -235000000.0], t$95$0, If[LessEqual[y, 7.5e-49], t$95$1, If[LessEqual[y, 5.2e-32], (-y), If[LessEqual[y, 27000000000.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.35e8 or 2.7e10 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 100.0%

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

      1. neg-mul-1 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   8. Simplified 100.0%

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

   if -2.35e8 < y < 7.4999999999999998e-49 or 5.1999999999999995e-32 < y < 2.7e10

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 75.6%

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

   if 7.4999999999999998e-49 < y < 5.1999999999999995e-32

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. neg-sub0 100.0%

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

      4. associate--r- 100.0%

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

      5. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 100.0%

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

      1. neg-mul-1 100.0%

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

   8. Simplified 100.0%

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

Alternative 3: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ t_1 := x + x \cdot y\\ \mathbf{if}\;y \leq -0.17:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.9 \cdot 10^{-49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-31}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))) (t_1 (+ x (* x y))))
   (if (<= y -0.17)
     t_0
     (if (<= y 5.9e-49)
       t_1
       (if (<= y 1.6e-31) (- y) (if (<= y 1.0) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (x / y);
	double t_1 = x + (x * y);
	double tmp;
	if (y <= -0.17) {
		tmp = t_0;
	} else if (y <= 5.9e-49) {
		tmp = t_1;
	} else if (y <= 1.6e-31) {
		tmp = -y;
	} else if (y <= 1.0) {
		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 + (x * y)
    if (y <= (-0.17d0)) then
        tmp = t_0
    else if (y <= 5.9d-49) then
        tmp = t_1
    else if (y <= 1.6d-31) then
        tmp = -y
    else if (y <= 1.0d0) 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 + (x * y);
	double tmp;
	if (y <= -0.17) {
		tmp = t_0;
	} else if (y <= 5.9e-49) {
		tmp = t_1;
	} else if (y <= 1.6e-31) {
		tmp = -y;
	} else if (y <= 1.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (x / y)
	t_1 = x + (x * y)
	tmp = 0
	if y <= -0.17:
		tmp = t_0
	elif y <= 5.9e-49:
		tmp = t_1
	elif y <= 1.6e-31:
		tmp = -y
	elif y <= 1.0:
		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(x + Float64(x * y))
	tmp = 0.0
	if (y <= -0.17)
		tmp = t_0;
	elseif (y <= 5.9e-49)
		tmp = t_1;
	elseif (y <= 1.6e-31)
		tmp = Float64(-y);
	elseif (y <= 1.0)
		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 + (x * y);
	tmp = 0.0;
	if (y <= -0.17)
		tmp = t_0;
	elseif (y <= 5.9e-49)
		tmp = t_1;
	elseif (y <= 1.6e-31)
		tmp = -y;
	elseif (y <= 1.0)
		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[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.17], t$95$0, If[LessEqual[y, 5.9e-49], t$95$1, If[LessEqual[y, 1.6e-31], (-y), If[LessEqual[y, 1.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.170000000000000012 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 96.7%

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

      1. neg-mul-1 96.7%

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

   5. Simplified 96.7%

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

   6. Taylor expanded in x around 0 96.7%

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

      1. neg-mul-1 96.7%

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

      2. unsub-neg 96.7%

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

   8. Simplified 96.7%

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

   if -0.170000000000000012 < y < 5.90000000000000037e-49 or 1.60000000000000009e-31 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 75.5%

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

   4. Taylor expanded in y around 0 75.3%

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

      1. *-commutative 75.3%

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

   6. Simplified 75.3%

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

   if 5.90000000000000037e-49 < y < 1.60000000000000009e-31

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. neg-sub0 100.0%

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

      4. associate--r- 100.0%

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

      5. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 100.0%

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

      1. neg-mul-1 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.17:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 5.9 \cdot 10^{-49}:\\ \;\;\;\;x + x \cdot y\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-31}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + x \cdot y\\ \mathbf{if}\;y \leq -0.75:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-49}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-31}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (* x y))))
   (if (<= y -0.75)
     1.0
     (if (<= y 6.8e-49)
       t_0
       (if (<= y 1.65e-31) (- y) (if (<= y 1.0) t_0 1.0))))))

C

double code(double x, double y) {
	double t_0 = x + (x * y);
	double tmp;
	if (y <= -0.75) {
		tmp = 1.0;
	} else if (y <= 6.8e-49) {
		tmp = t_0;
	} else if (y <= 1.65e-31) {
		tmp = -y;
	} else if (y <= 1.0) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (x * y)
    if (y <= (-0.75d0)) then
        tmp = 1.0d0
    else if (y <= 6.8d-49) then
        tmp = t_0
    else if (y <= 1.65d-31) then
        tmp = -y
    else if (y <= 1.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + (x * y);
	double tmp;
	if (y <= -0.75) {
		tmp = 1.0;
	} else if (y <= 6.8e-49) {
		tmp = t_0;
	} else if (y <= 1.65e-31) {
		tmp = -y;
	} else if (y <= 1.0) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (x * y)
	tmp = 0
	if y <= -0.75:
		tmp = 1.0
	elif y <= 6.8e-49:
		tmp = t_0
	elif y <= 1.65e-31:
		tmp = -y
	elif y <= 1.0:
		tmp = t_0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(x * y))
	tmp = 0.0
	if (y <= -0.75)
		tmp = 1.0;
	elseif (y <= 6.8e-49)
		tmp = t_0;
	elseif (y <= 1.65e-31)
		tmp = Float64(-y);
	elseif (y <= 1.0)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (x * y);
	tmp = 0.0;
	if (y <= -0.75)
		tmp = 1.0;
	elseif (y <= 6.8e-49)
		tmp = t_0;
	elseif (y <= 1.65e-31)
		tmp = -y;
	elseif (y <= 1.0)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.75], 1.0, If[LessEqual[y, 6.8e-49], t$95$0, If[LessEqual[y, 1.65e-31], (-y), If[LessEqual[y, 1.0], t$95$0, 1.0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.75 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 79.0%

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

   if -0.75 < y < 6.8000000000000001e-49 or 1.65e-31 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 75.5%

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

   4. Taylor expanded in y around 0 75.3%

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

      1. *-commutative 75.3%

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

   6. Simplified 75.3%

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

   if 6.8000000000000001e-49 < y < 1.65e-31

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. neg-sub0 100.0%

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

      4. associate--r- 100.0%

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

      5. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 100.0%

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

      1. neg-mul-1 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.75:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-49}:\\ \;\;\;\;x + x \cdot y\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-31}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -66000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-49}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-30}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -66000.0)
   1.0
   (if (<= y 5.8e-49) x (if (<= y 1.3e-30) (- y) (if (<= y 1.0) x 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -66000.0) {
		tmp = 1.0;
	} else if (y <= 5.8e-49) {
		tmp = x;
	} else if (y <= 1.3e-30) {
		tmp = -y;
	} else if (y <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-66000.0d0)) then
        tmp = 1.0d0
    else if (y <= 5.8d-49) then
        tmp = x
    else if (y <= 1.3d-30) then
        tmp = -y
    else if (y <= 1.0d0) then
        tmp = x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -66000.0) {
		tmp = 1.0;
	} else if (y <= 5.8e-49) {
		tmp = x;
	} else if (y <= 1.3e-30) {
		tmp = -y;
	} else if (y <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -66000.0:
		tmp = 1.0
	elif y <= 5.8e-49:
		tmp = x
	elif y <= 1.3e-30:
		tmp = -y
	elif y <= 1.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -66000.0)
		tmp = 1.0;
	elseif (y <= 5.8e-49)
		tmp = x;
	elseif (y <= 1.3e-30)
		tmp = Float64(-y);
	elseif (y <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -66000.0)
		tmp = 1.0;
	elseif (y <= 5.8e-49)
		tmp = x;
	elseif (y <= 1.3e-30)
		tmp = -y;
	elseif (y <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -66000.0], 1.0, If[LessEqual[y, 5.8e-49], x, If[LessEqual[y, 1.3e-30], (-y), If[LessEqual[y, 1.0], x, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -66000 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 80.2%

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

   if -66000 < y < 5.8e-49 or 1.29999999999999993e-30 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 73.6%

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

   if 5.8e-49 < y < 1.29999999999999993e-30

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. neg-sub0 100.0%

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

      4. associate--r- 100.0%

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

      5. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 100.0%

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

      1. neg-mul-1 100.0%

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

   8. Simplified 100.0%

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

Alternative 6: 98.3% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -0.82) (not (<= y 1.0)))
   (- 1.0 (/ x y))
   (+ x (* y (+ x -1.0)))))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-0.82d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = x + (y * (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if (y <= -0.82) or not (y <= 1.0):
		tmp = 1.0 - (x / y)
	else:
		tmp = x + (y * (x + -1.0))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -0.82], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.819999999999999951 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 96.7%

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

      1. neg-mul-1 96.7%

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

   5. Simplified 96.7%

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

   6. Taylor expanded in x around 0 96.7%

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

      1. neg-mul-1 96.7%

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

      2. unsub-neg 96.7%

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

   8. Simplified 96.7%

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

   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 99.4%

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

      1. mul-1-neg 99.4%

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

      2. unsub-neg 99.4%

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

      3. mul-1-neg 99.4%

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

      4. sub-neg 99.4%

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

   5. Simplified 99.4%

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

4. Final simplification 98.1%

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

Alternative 7: 73.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -66000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -66000.0) 1.0 (if (<= y 1.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -66000.0) {
		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 <= (-66000.0d0)) 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 <= -66000.0) {
		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 <= -66000.0:
		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 <= -66000.0)
		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 <= -66000.0)
		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, -66000.0], 1.0, If[LessEqual[y, 1.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -66000 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 80.2%

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

   if -66000 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 71.1%

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

Alternative 8: 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 9: 38.9% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around inf 39.3%

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

Reproduce

herbie shell --seed 2024111 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))