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

Percentage Accurate: 100.0% → 100.0%

Time: 4.2s

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. Final simplification 100.0%

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

Alternative 2: 86.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -13000:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 320:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -13000.0)
   (- 1.0 (/ x y))
   (if (<= y 320.0) (/ x (- 1.0 y)) (+ 1.0 (/ (- 1.0 x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -13000.0) {
		tmp = 1.0 - (x / y);
	} else if (y <= 320.0) {
		tmp = x / (1.0 - 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 <= (-13000.0d0)) then
        tmp = 1.0d0 - (x / y)
    else if (y <= 320.0d0) then
        tmp = x / (1.0d0 - 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 <= -13000.0) {
		tmp = 1.0 - (x / y);
	} else if (y <= 320.0) {
		tmp = x / (1.0 - y);
	} else {
		tmp = 1.0 + ((1.0 - x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -13000.0:
		tmp = 1.0 - (x / y)
	elif y <= 320.0:
		tmp = x / (1.0 - y)
	else:
		tmp = 1.0 + ((1.0 - x) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -13000.0)
		tmp = Float64(1.0 - Float64(x / y));
	elseif (y <= 320.0)
		tmp = Float64(x / Float64(1.0 - 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 <= -13000.0)
		tmp = 1.0 - (x / y);
	elseif (y <= 320.0)
		tmp = x / (1.0 - y);
	else
		tmp = 1.0 + ((1.0 - x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -13000.0], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 320.0], N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -13000

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+l- 100.0%

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

      4. div-sub 100.0%

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

      5. neg-sub0 100.0%

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

      6. mul-1-neg 100.0%

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

      7. associate-*r/ 100.0%

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

      8. sub-neg 100.0%

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

      9. metadata-eval 100.0%

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

      10. distribute-lft-in 100.0%

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

      11. metadata-eval 100.0%

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

      12. +-commutative 100.0%

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

      13. mul-1-neg 100.0%

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

      14. unsub-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

   7. Simplified 100.0%

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

   if -13000 < y < 320

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 74.0%

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

   if 320 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 99.2%

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

      1. mul-1-neg 99.2%

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

      2. neg-sub0 99.2%

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

      3. associate-+l- 99.2%

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

      4. div-sub 99.2%

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

      5. neg-sub0 99.2%

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

      6. mul-1-neg 99.2%

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

      7. associate-*r/ 99.2%

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

      8. sub-neg 99.2%

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

      9. metadata-eval 99.2%

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

      10. distribute-lft-in 99.2%

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

      11. metadata-eval 99.2%

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

      12. +-commutative 99.2%

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

      13. mul-1-neg 99.2%

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

      14. unsub-neg 99.2%

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

   4. Simplified 99.2%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -13000:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 320:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1 - x}{y}\\ \end{array} \]

Alternative 3: 73.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{+60}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{+20}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq -0.54:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.22e+60)
   1.0
   (if (<= y -1.8e+20)
     (/ (- x) y)
     (if (<= y -0.54) 1.0 (if (<= y 1.0) x 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.22e+60) {
		tmp = 1.0;
	} else if (y <= -1.8e+20) {
		tmp = -x / y;
	} else if (y <= -0.54) {
		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 <= (-1.22d+60)) then
        tmp = 1.0d0
    else if (y <= (-1.8d+20)) then
        tmp = -x / y
    else if (y <= (-0.54d0)) 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 <= -1.22e+60) {
		tmp = 1.0;
	} else if (y <= -1.8e+20) {
		tmp = -x / y;
	} else if (y <= -0.54) {
		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 <= -1.22e+60:
		tmp = 1.0
	elif y <= -1.8e+20:
		tmp = -x / y
	elif y <= -0.54:
		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 <= -1.22e+60)
		tmp = 1.0;
	elseif (y <= -1.8e+20)
		tmp = Float64(Float64(-x) / y);
	elseif (y <= -0.54)
		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 <= -1.22e+60)
		tmp = 1.0;
	elseif (y <= -1.8e+20)
		tmp = -x / y;
	elseif (y <= -0.54)
		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, -1.22e+60], 1.0, If[LessEqual[y, -1.8e+20], N[((-x) / y), $MachinePrecision], If[LessEqual[y, -0.54], 1.0, If[LessEqual[y, 1.0], x, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.21999999999999995e60 or -1.8e20 < y < -0.54000000000000004 or 1 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 84.5%

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

   if -1.21999999999999995e60 < y < -1.8e20

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 71.7%

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

      1. div-inv 71.3%

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

      2. *-commutative 71.3%

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

      3. sub-neg 71.3%

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

      4. metadata-eval 71.3%

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

      5. distribute-neg-in 71.3%

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

      6. metadata-eval 71.3%

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

      7. frac-2neg 71.3%

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

   4. Applied egg-rr 71.3%

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

   5. Taylor expanded in y around inf 71.7%

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

      1. associate-*r/ 71.7%

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

      2. neg-mul-1 71.7%

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

   7. Simplified 71.7%

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

   if -0.54000000000000004 < y < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 72.2%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{+60}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{+20}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq -0.54:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 86.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -49000 \lor \neg \left(y \leq 16000\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -49000.0) (not (<= y 16000.0)))
   (- 1.0 (/ x y))
   (/ x (- 1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -49000.0) || !(y <= 16000.0)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / (1.0 - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-49000.0d0)) .or. (.not. (y <= 16000.0d0))) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = x / (1.0d0 - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -49000.0) || !(y <= 16000.0)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / (1.0 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -49000.0) or not (y <= 16000.0):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / (1.0 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -49000.0) || !(y <= 16000.0))
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(x / Float64(1.0 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -49000.0) || ~((y <= 16000.0)))
		tmp = 1.0 - (x / y);
	else
		tmp = x / (1.0 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -49000.0], N[Not[LessEqual[y, 16000.0]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -49000 or 16000 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. neg-sub0 99.7%

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

      3. associate-+l- 99.7%

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

      4. div-sub 99.7%

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

      5. neg-sub0 99.7%

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

      6. mul-1-neg 99.7%

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

      7. associate-*r/ 99.7%

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

      8. sub-neg 99.7%

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

      9. metadata-eval 99.7%

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

      10. distribute-lft-in 99.7%

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

      11. metadata-eval 99.7%

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

      12. +-commutative 99.7%

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

      13. mul-1-neg 99.7%

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

      14. unsub-neg 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in x around inf 99.1%

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

      1. mul-1-neg 99.1%

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

      2. distribute-neg-frac 99.1%

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

   7. Simplified 99.1%

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

   if -49000 < y < 16000

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 74.0%

      \[\leadsto \color{blue}{\frac{x}{1 - y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -49000 \lor \neg \left(y \leq 16000\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - y}\\ \end{array} \]

Alternative 5: 73.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+60}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 240000:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.15e+60) 1.0 (if (<= y 240000.0) (/ x (- 1.0 y)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.15e+60) {
		tmp = 1.0;
	} else if (y <= 240000.0) {
		tmp = x / (1.0 - y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.15d+60)) then
        tmp = 1.0d0
    else if (y <= 240000.0d0) then
        tmp = x / (1.0d0 - y)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.15e+60) {
		tmp = 1.0;
	} else if (y <= 240000.0) {
		tmp = x / (1.0 - y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.15e+60:
		tmp = 1.0
	elif y <= 240000.0:
		tmp = x / (1.0 - y)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.15e+60)
		tmp = 1.0;
	elseif (y <= 240000.0)
		tmp = Float64(x / Float64(1.0 - y));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.15e+60)
		tmp = 1.0;
	elseif (y <= 240000.0)
		tmp = x / (1.0 - y);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.15e+60], 1.0, If[LessEqual[y, 240000.0], N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.15000000000000008e60 or 2.4e5 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 85.0%

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

   if -1.15000000000000008e60 < y < 2.4e5

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 73.4%

      \[\leadsto \color{blue}{\frac{x}{1 - y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+60}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 240000:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 73.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+60}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{-57}:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.15e+60)
   1.0
   (if (<= y 1.22e-57) (/ x (- 1.0 y)) (/ y (+ y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.15e+60) {
		tmp = 1.0;
	} else if (y <= 1.22e-57) {
		tmp = x / (1.0 - y);
	} else {
		tmp = y / (y + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.15d+60)) then
        tmp = 1.0d0
    else if (y <= 1.22d-57) then
        tmp = x / (1.0d0 - y)
    else
        tmp = y / (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.15e+60) {
		tmp = 1.0;
	} else if (y <= 1.22e-57) {
		tmp = x / (1.0 - y);
	} else {
		tmp = y / (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.15e+60:
		tmp = 1.0
	elif y <= 1.22e-57:
		tmp = x / (1.0 - y)
	else:
		tmp = y / (y + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.15e+60)
		tmp = 1.0;
	elseif (y <= 1.22e-57)
		tmp = Float64(x / Float64(1.0 - y));
	else
		tmp = Float64(y / Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.15e+60)
		tmp = 1.0;
	elseif (y <= 1.22e-57)
		tmp = x / (1.0 - y);
	else
		tmp = y / (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.15e+60], 1.0, If[LessEqual[y, 1.22e-57], N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], N[(y / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.15000000000000008e60

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 84.3%

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

   if -1.15000000000000008e60 < y < 1.2200000000000001e-57

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 75.9%

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

   if 1.2200000000000001e-57 < y

   1. Initial program 100.0%

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

      1. div-inv 99.8%

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

      2. *-commutative 99.8%

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

      3. frac-2neg 99.8%

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

      4. metadata-eval 99.8%

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

      5. neg-sub0 99.8%

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

      6. associate--r- 99.8%

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

      7. metadata-eval 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 81.0%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+60}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{-57}:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + -1}\\ \end{array} \]

Alternative 7: 73.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.42:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -0.42) 1.0 (if (<= y 1.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.42) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-0.42d0)) then
        tmp = 1.0d0
    else if (y <= 1.0d0) then
        tmp = x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -0.42) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -0.42:
		tmp = 1.0
	elif y <= 1.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -0.42)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -0.42)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -0.42], 1.0, If[LessEqual[y, 1.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -0.419999999999999984 or 1 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 80.0%

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

   if -0.419999999999999984 < y < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 72.2%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.42:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 37.9% 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. Taylor expanded in y around inf 39.8%

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

3. Final simplification 39.8%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023326 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))