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

Percentage Accurate: 100.0% → 100.0%

Time: 5.2s

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. Add Preprocessing
3. Add Preprocessing

Alternative 2: 98.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ t_1 := \frac{x}{1 - y}\\ \mathbf{if}\;t\_0 \leq -400:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(-1 + x, y, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{y}{-1 + y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))) (t_1 (/ x (- 1.0 y))))
   (if (<= t_0 -400.0)
     t_1
     (if (<= t_0 5e-10)
       (fma (+ -1.0 x) y x)
       (if (<= t_0 2.0) (/ y (+ -1.0 y)) t_1)))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double t_1 = x / (1.0 - y);
	double tmp;
	if (t_0 <= -400.0) {
		tmp = t_1;
	} else if (t_0 <= 5e-10) {
		tmp = fma((-1.0 + x), y, x);
	} else if (t_0 <= 2.0) {
		tmp = y / (-1.0 + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(1.0 - y))
	t_1 = Float64(x / Float64(1.0 - y))
	tmp = 0.0
	if (t_0 <= -400.0)
		tmp = t_1;
	elseif (t_0 <= 5e-10)
		tmp = fma(Float64(-1.0 + x), y, x);
	elseif (t_0 <= 2.0)
		tmp = Float64(y / Float64(-1.0 + y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -400.0], t$95$1, If[LessEqual[t$95$0, 5e-10], N[(N[(-1.0 + x), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(y / N[(-1.0 + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < -400 or 2 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -400 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 5.00000000000000031e-10

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lower-fma.f64 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-+.f64 98.4

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

   5. Applied rewrites 98.4%

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

   if 5.00000000000000031e-10 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower-+.f64 98.9

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

   5. Applied rewrites 98.9%

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

Alternative 3: 63.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t\_0 \leq -400 \lor \neg \left(t\_0 \leq 0.9999999999997695\right):\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 + x, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (or (<= t_0 -400.0) (not (<= t_0 0.9999999999997695)))
     (/ x (- 1.0 y))
     (fma (+ -1.0 x) y x))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if ((t_0 <= -400.0) || !(t_0 <= 0.9999999999997695)) {
		tmp = x / (1.0 - y);
	} else {
		tmp = fma((-1.0 + x), y, x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(1.0 - y))
	tmp = 0.0
	if ((t_0 <= -400.0) || !(t_0 <= 0.9999999999997695))
		tmp = Float64(x / Float64(1.0 - y));
	else
		tmp = fma(Float64(-1.0 + x), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -400.0], N[Not[LessEqual[t$95$0, 0.9999999999997695]], $MachinePrecision]], N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 + x), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < -400 or 0.99999999999976952 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 49.0

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

   5. Applied rewrites 49.0%

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

   if -400 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.99999999999976952

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lower-fma.f64 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-+.f64 93.4

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

   5. Applied rewrites 93.4%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq -400 \lor \neg \left(\frac{x - y}{1 - y} \leq 0.9999999999997695\right):\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 + x, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.02 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 + x, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.02) (not (<= y 1.0)))
   (- 1.0 (/ (- x 1.0) y))
   (fma (+ -1.0 x) y x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.02) || !(y <= 1.0)) {
		tmp = 1.0 - ((x - 1.0) / y);
	} else {
		tmp = fma((-1.0 + x), y, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.02) || !(y <= 1.0))
		tmp = Float64(1.0 - Float64(Float64(x - 1.0) / y));
	else
		tmp = fma(Float64(-1.0 + x), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.02], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(1.0 - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 + x), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.02 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. associate-+r+ N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if -1.02 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lower-fma.f64 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-+.f64 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.02 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 + x, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -11800000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 + x, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -11800000.0) (not (<= y 1.0)))
   (/ (- x) y)
   (fma (+ -1.0 x) y x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -11800000.0) || !(y <= 1.0)) {
		tmp = -x / y;
	} else {
		tmp = fma((-1.0 + x), y, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -11800000.0) || !(y <= 1.0))
		tmp = Float64(Float64(-x) / y);
	else
		tmp = fma(Float64(-1.0 + x), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -11800000.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[((-x) / y), $MachinePrecision], N[(N[(-1.0 + x), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.18e7 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 26.7

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

   5. Applied rewrites 26.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 26.1%

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

   if -1.18e7 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lower-fma.f64 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-+.f64 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -11800000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 + x, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 44.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-127} \lor \neg \left(x \leq 7 \cdot 10^{-72}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -6.2e-127) (not (<= x 7e-72))) (fma y x x) (- y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -6.2e-127) || !(x <= 7e-72)) {
		tmp = fma(y, x, x);
	} else {
		tmp = -y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -6.2e-127) || !(x <= 7e-72))
		tmp = fma(y, x, x);
	else
		tmp = Float64(-y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -6.2e-127], N[Not[LessEqual[x, 7e-72]], $MachinePrecision]], N[(y * x + x), $MachinePrecision], (-y)]

Derivation

1. Split input into 2 regimes

2. if x < -6.2e-127 or 7.00000000000000001e-72 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lower-fma.f64 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-+.f64 49.2

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

   5. Applied rewrites 49.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 45.6%

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

   if -6.2e-127 < x < 7.00000000000000001e-72

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lower-fma.f64 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-+.f64 54.4

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

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 40.1%

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

   9. Taylor expanded in x around inf

      \[\leadsto x \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 4.2%

       \[\leadsto x \cdot y \]

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 40.1%

       \[\leadsto -y \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-127} \lor \neg \left(x \leq 7 \cdot 10^{-72}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-1, y, x\right) \end{array} \]

FPCore

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

C

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

Julia

function code(x, y)
	return fma(-1.0, y, x)
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-neg-in N/A

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

   5. lower-fma.f64 N/A

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

   6. distribute-neg-in N/A

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

   7. metadata-eval N/A

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

   8. mul-1-neg N/A

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

   9. remove-double-neg N/A

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

   10. lower-+.f64 51.2

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

5. Applied rewrites 51.2%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(-1, y, x\right) \]
7. Step-by-step derivation

8. Applied rewrites 51.7%

   \[\leadsto \mathsf{fma}\left(-1, y, x\right) \]
9. Add Preprocessing

Alternative 8: 14.8% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ -y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -y

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := (-y)

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-neg-in N/A

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

   5. lower-fma.f64 N/A

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

   6. distribute-neg-in N/A

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

   7. metadata-eval N/A

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

   8. mul-1-neg N/A

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

   9. remove-double-neg N/A

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

   10. lower-+.f64 51.2

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

5. Applied rewrites 51.2%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 19.0%

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

9. Taylor expanded in x around inf

   \[\leadsto x \cdot y \]
10. Step-by-step derivation

11. Applied rewrites 3.2%

    \[\leadsto x \cdot y \]

12. Taylor expanded in x around 0

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

14. Applied rewrites 19.4%

    \[\leadsto -y \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024337 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))