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

Percentage Accurate: 100.0% → 100.0%

Time: 9.3s

Alternatives: 15

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: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ t_1 := \frac{x}{1 - y}\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x + -1, \mathsf{fma}\left(y, y, y\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))) (t_1 (/ x (- 1.0 y))))
   (if (<= t_0 -10.0)
     t_1
     (if (<= t_0 4e-7)
       (fma (+ x -1.0) (fma y y y) x)
       (if (<= t_0 2.0) (/ y (+ y -1.0)) 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 <= -10.0) {
		tmp = t_1;
	} else if (t_0 <= 4e-7) {
		tmp = fma((x + -1.0), fma(y, y, y), x);
	} else if (t_0 <= 2.0) {
		tmp = y / (y + -1.0);
	} 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 <= -10.0)
		tmp = t_1;
	elseif (t_0 <= 4e-7)
		tmp = fma(Float64(x + -1.0), fma(y, y, y), x);
	elseif (t_0 <= 2.0)
		tmp = Float64(y / Float64(y + -1.0));
	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, -10.0], t$95$1, If[LessEqual[t$95$0, 4e-7], N[(N[(x + -1.0), $MachinePrecision] * N[(y * y + y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(y / N[(y + -1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < -10 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 98.8

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

   5. Applied rewrites 98.8%

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

   if -10 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 3.9999999999999998e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. mul-1-neg N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

         \[\leadsto \left(\color{blue}{{y}^{2}} \cdot \left(\mathsf{neg}\left(\left(1 + -1 \cdot x\right)\right)\right) + y \cdot \left(\mathsf{neg}\left(\left(1 + -1 \cdot x\right)\right)\right)\right) + x \]

      8. distribute-rgt-out N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(1 + -1 \cdot x\right)\right)\right) \cdot \left({y}^{2} + y\right)} + x \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\left(1 + -1 \cdot x\right)\right), {y}^{2} + y, x\right)} \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\left(1 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right), {y}^{2} + y, x\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)}, {y}^{2} + y, x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-1} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right), {y}^{2} + y, x\right) \]

      13. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(-1 + \color{blue}{x}, {y}^{2} + y, x\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-1 + x}, {y}^{2} + y, x\right) \]

      15. unpow2 N/A

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

      16. lower-fma.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 3.9999999999999998e-7 < (/.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. neg-sub0 N/A

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

      5. associate--r- N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 97.9

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

   5. Applied rewrites 97.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq -10:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{elif}\;\frac{x - y}{1 - y} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x + -1, \mathsf{fma}\left(y, y, y\right), x\right)\\ \mathbf{elif}\;\frac{x - y}{1 - y} \leq 2:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - y}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024223 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))