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

Percentage Accurate: 88.0% → 100.0%

Time: 3.7s

Alternatives: 3

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 100.0

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{y}{1 + y} \cdot x} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 79.2%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. lower-*.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 99.3%

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

Alternative 3: 51.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ y \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return y * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.6%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. remove-double-neg N/A

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

   4. mul-1-neg N/A

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

   5. sub-neg N/A

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

   6. lower-*.f64 N/A

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

   7. cancel-sign-sub-inv N/A

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

   8. metadata-eval N/A

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

   9. *-lft-identity N/A

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

   10. associate-*r* N/A

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

   11. lower-fma.f64 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f64 45.4

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

5. Applied rewrites 45.4%

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

6. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 46.6

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

8. Applied rewrites 46.6%

   \[\leadsto \color{blue}{y \cdot x} \]
9. Add Preprocessing

Developer Target 1: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot y} - \left(\frac{x}{y} - x\right)\\ \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;\frac{x \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (/ x (* y y)) (- (/ x y) x))))
   (if (< y -3693.8482788297247)
     t_0
     (if (< y 6799310503.41891) (/ (* x y) (+ y 1.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = (x / (y * y)) - ((x / y) - x);
	double tmp;
	if (y < -3693.8482788297247) {
		tmp = t_0;
	} else if (y < 6799310503.41891) {
		tmp = (x * y) / (y + 1.0);
	} 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) :: tmp
    t_0 = (x / (y * y)) - ((x / y) - x)
    if (y < (-3693.8482788297247d0)) then
        tmp = t_0
    else if (y < 6799310503.41891d0) then
        tmp = (x * y) / (y + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x / (y * y)) - ((x / y) - x);
	double tmp;
	if (y < -3693.8482788297247) {
		tmp = t_0;
	} else if (y < 6799310503.41891) {
		tmp = (x * y) / (y + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / (y * y)) - ((x / y) - x)
	tmp = 0
	if y < -3693.8482788297247:
		tmp = t_0
	elif y < 6799310503.41891:
		tmp = (x * y) / (y + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / Float64(y * y)) - Float64(Float64(x / y) - x))
	tmp = 0.0
	if (y < -3693.8482788297247)
		tmp = t_0;
	elseif (y < 6799310503.41891)
		tmp = Float64(Float64(x * y) / Float64(y + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / (y * y)) - ((x / y) - x);
	tmp = 0.0;
	if (y < -3693.8482788297247)
		tmp = t_0;
	elseif (y < 6799310503.41891)
		tmp = (x * y) / (y + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -3693.8482788297247], t$95$0, If[Less[y, 6799310503.41891], N[(N[(x * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024323 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -36938482788297247/10000000000000) (- (/ x (* y y)) (- (/ x y) x)) (if (< y 679931050341891/100000) (/ (* x y) (+ y 1)) (- (/ x (* y y)) (- (/ x y) x)))))

  (/ (* x y) (+ y 1.0)))