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

Percentage Accurate: 100.0% → 100.0%

Time: 5.9s

Alternatives: 7

Speedup: 1.0×

• could not determine a ground truth

  1. x = -8.268636701890469e-126
  2. y = 2.2168813064779523e-125

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. Final simplification 100.0%

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

Alternative 2: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{1 - y}\\ \mathbf{if}\;y \leq -0.0033:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-111}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.31:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{elif}\;y \leq 290:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (- 1.0 y))))
   (if (<= y -0.0033)
     (- 1.0 (/ x y))
     (if (<= y 1.1e-111)
       t_0
       (if (<= y 0.31)
         (/ y (+ y -1.0))
         (if (<= y 290.0) t_0 (+ 1.0 (/ (- 1.0 x) y))))))))

C

double code(double x, double y) {
	double t_0 = x / (1.0 - y);
	double tmp;
	if (y <= -0.0033) {
		tmp = 1.0 - (x / y);
	} else if (y <= 1.1e-111) {
		tmp = t_0;
	} else if (y <= 0.31) {
		tmp = y / (y + -1.0);
	} else if (y <= 290.0) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = x / (1.0d0 - y)
    if (y <= (-0.0033d0)) then
        tmp = 1.0d0 - (x / y)
    else if (y <= 1.1d-111) then
        tmp = t_0
    else if (y <= 0.31d0) then
        tmp = y / (y + (-1.0d0))
    else if (y <= 290.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0 + ((1.0d0 - x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (1.0 - y);
	double tmp;
	if (y <= -0.0033) {
		tmp = 1.0 - (x / y);
	} else if (y <= 1.1e-111) {
		tmp = t_0;
	} else if (y <= 0.31) {
		tmp = y / (y + -1.0);
	} else if (y <= 290.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 + ((1.0 - x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (1.0 - y)
	tmp = 0
	if y <= -0.0033:
		tmp = 1.0 - (x / y)
	elif y <= 1.1e-111:
		tmp = t_0
	elif y <= 0.31:
		tmp = y / (y + -1.0)
	elif y <= 290.0:
		tmp = t_0
	else:
		tmp = 1.0 + ((1.0 - x) / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(1.0 - y))
	tmp = 0.0
	if (y <= -0.0033)
		tmp = Float64(1.0 - Float64(x / y));
	elseif (y <= 1.1e-111)
		tmp = t_0;
	elseif (y <= 0.31)
		tmp = Float64(y / Float64(y + -1.0));
	elseif (y <= 290.0)
		tmp = t_0;
	else
		tmp = Float64(1.0 + Float64(Float64(1.0 - x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (1.0 - y);
	tmp = 0.0;
	if (y <= -0.0033)
		tmp = 1.0 - (x / y);
	elseif (y <= 1.1e-111)
		tmp = t_0;
	elseif (y <= 0.31)
		tmp = y / (y + -1.0);
	elseif (y <= 290.0)
		tmp = t_0;
	else
		tmp = 1.0 + ((1.0 - x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0033], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e-111], t$95$0, If[LessEqual[y, 0.31], N[(y / N[(y + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 290.0], t$95$0, N[(1.0 + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -0.0033

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. div-inv 99.9%

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

      3. fma-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around -inf 97.0%

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

      1. mul-1-neg 97.0%

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

      2. unsub-neg 97.0%

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

      3. sub-neg 97.0%

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

      4. metadata-eval 97.0%

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

      5. +-commutative 97.0%

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

   7. Simplified 97.0%

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

   8. Taylor expanded in x around inf 97.2%

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

   if -0.0033 < y < 1.1e-111 or 0.309999999999999998 < y < 290

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 79.5%

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

   if 1.1e-111 < y < 0.309999999999999998

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 57.3%

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

      1. metadata-eval 57.3%

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

      2. times-frac 57.3%

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

      3. *-lft-identity 57.3%

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

      4. neg-mul-1 57.3%

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

      5. neg-sub0 57.3%

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

      6. associate--r- 57.3%

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

      7. metadata-eval 57.3%

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

   5. Simplified 57.3%

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

   if 290 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. div-sub 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0033:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-111}:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{elif}\;y \leq 0.31:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{elif}\;y \leq 290:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1 - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ t_1 := \frac{x}{1 - y}\\ \mathbf{if}\;y \leq -0.0033:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-112}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.92:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{elif}\;y \leq 3350000:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))) (t_1 (/ x (- 1.0 y))))
   (if (<= y -0.0033)
     t_0
     (if (<= y 5.2e-112)
       t_1
       (if (<= y 0.92) (/ y (+ y -1.0)) (if (<= y 3350000.0) 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 <= -0.0033) {
		tmp = t_0;
	} else if (y <= 5.2e-112) {
		tmp = t_1;
	} else if (y <= 0.92) {
		tmp = y / (y + -1.0);
	} else if (y <= 3350000.0) {
		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 <= (-0.0033d0)) then
        tmp = t_0
    else if (y <= 5.2d-112) then
        tmp = t_1
    else if (y <= 0.92d0) then
        tmp = y / (y + (-1.0d0))
    else if (y <= 3350000.0d0) 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 <= -0.0033) {
		tmp = t_0;
	} else if (y <= 5.2e-112) {
		tmp = t_1;
	} else if (y <= 0.92) {
		tmp = y / (y + -1.0);
	} else if (y <= 3350000.0) {
		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 <= -0.0033:
		tmp = t_0
	elif y <= 5.2e-112:
		tmp = t_1
	elif y <= 0.92:
		tmp = y / (y + -1.0)
	elif y <= 3350000.0:
		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 <= -0.0033)
		tmp = t_0;
	elseif (y <= 5.2e-112)
		tmp = t_1;
	elseif (y <= 0.92)
		tmp = Float64(y / Float64(y + -1.0));
	elseif (y <= 3350000.0)
		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 <= -0.0033)
		tmp = t_0;
	elseif (y <= 5.2e-112)
		tmp = t_1;
	elseif (y <= 0.92)
		tmp = y / (y + -1.0);
	elseif (y <= 3350000.0)
		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, -0.0033], t$95$0, If[LessEqual[y, 5.2e-112], t$95$1, If[LessEqual[y, 0.92], N[(y / N[(y + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3350000.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0033 or 3.35e6 < y

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. div-inv 99.9%

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

      3. fma-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around -inf 98.6%

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

      1. mul-1-neg 98.6%

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

      2. unsub-neg 98.6%

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

      3. sub-neg 98.6%

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

      4. metadata-eval 98.6%

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

      5. +-commutative 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in x around inf 98.6%

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

   if -0.0033 < y < 5.19999999999999983e-112 or 0.92000000000000004 < y < 3.35e6

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 79.5%

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

   if 5.19999999999999983e-112 < y < 0.92000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 57.3%

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

      1. metadata-eval 57.3%

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

      2. times-frac 57.3%

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

      3. *-lft-identity 57.3%

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

      4. neg-mul-1 57.3%

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

      5. neg-sub0 57.3%

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

      6. associate--r- 57.3%

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

      7. metadata-eval 57.3%

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

   5. Simplified 57.3%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0033:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{elif}\;y \leq 0.92:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{elif}\;y \leq 3350000:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0025 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -0.0025) (not (<= y 1.0))) (- 1.0 (/ x y)) x))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -0.0025) || !(y <= 1.0)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-0.0025d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -0.0025) || !(y <= 1.0)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -0.0025) or not (y <= 1.0):
		tmp = 1.0 - (x / y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -0.0025) || !(y <= 1.0))
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -0.0025) || ~((y <= 1.0)))
		tmp = 1.0 - (x / y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -0.0025], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -0.00250000000000000005 or 1 < y

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. div-inv 99.9%

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

      3. fma-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around -inf 97.3%

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

      1. mul-1-neg 97.3%

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

      2. unsub-neg 97.3%

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

      3. sub-neg 97.3%

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

      4. metadata-eval 97.3%

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

      5. +-commutative 97.3%

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

   7. Simplified 97.3%

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

   8. Taylor expanded in x around inf 97.3%

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

   if -0.00250000000000000005 < 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. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0025 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0033 \lor \neg \left(y \leq 12200\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -0.0033) (not (<= y 12200.0))) (- 1.0 (/ x y)) (/ x (- 1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -0.0033) || !(y <= 12200.0)) {
		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 <= (-0.0033d0)) .or. (.not. (y <= 12200.0d0))) 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 <= -0.0033) || !(y <= 12200.0)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / (1.0 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -0.0033) or not (y <= 12200.0):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / (1.0 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -0.0033) || !(y <= 12200.0))
		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 <= -0.0033) || ~((y <= 12200.0)))
		tmp = 1.0 - (x / y);
	else
		tmp = x / (1.0 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -0.0033], N[Not[LessEqual[y, 12200.0]], $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 < -0.0033 or 12200 < y

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. div-inv 99.9%

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

      3. fma-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around -inf 98.6%

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

      1. mul-1-neg 98.6%

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

      2. unsub-neg 98.6%

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

      3. sub-neg 98.6%

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

      4. metadata-eval 98.6%

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

      5. +-commutative 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in x around inf 98.6%

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

   if -0.0033 < y < 12200

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 71.8%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0033 \lor \neg \left(y \leq 12200\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0021:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -0.0021) 1.0 (if (<= y 1.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.0021) {
		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 <= (-0.0021d0)) 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 <= -0.0021) {
		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 <= -0.0021:
		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 <= -0.0021)
		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 <= -0.0021)
		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, -0.0021], 1.0, If[LessEqual[y, 1.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -0.00209999999999999987 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 79.1%

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

   if -0.00209999999999999987 < 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. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0021:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 39.4% 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 41.4%

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

4. Final simplification 41.4%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))