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

Percentage Accurate: 88.2% → 100.0%

Time: 7.0s

Alternatives: 7

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: 88.2% 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

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

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(x * Float64(y / Float64(1.0 + y)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.4%

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 93.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot y}{y + 1}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{+213}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x y) (+ y 1.0)))) (if (<= t_0 4e+213) t_0 (- x (/ x y)))))

C

double code(double x, double y) {
	double t_0 = (x * y) / (y + 1.0);
	double tmp;
	if (t_0 <= 4e+213) {
		tmp = t_0;
	} else {
		tmp = x - (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 * y) / (y + 1.0d0)
    if (t_0 <= 4d+213) then
        tmp = t_0
    else
        tmp = x - (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * y) / (y + 1.0);
	double tmp;
	if (t_0 <= 4e+213) {
		tmp = t_0;
	} else {
		tmp = x - (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * y) / (y + 1.0)
	tmp = 0
	if t_0 <= 4e+213:
		tmp = t_0
	else:
		tmp = x - (x / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * y) / Float64(y + 1.0))
	tmp = 0.0
	if (t_0 <= 4e+213)
		tmp = t_0;
	else
		tmp = Float64(x - Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * y) / (y + 1.0);
	tmp = 0.0;
	if (t_0 <= 4e+213)
		tmp = t_0;
	else
		tmp = x - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e+213], t$95$0, N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (+.f64 y #s(literal 1 binary64))) < 3.99999999999999994e213

   1. Initial program 93.5%

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

   if 3.99999999999999994e213 < (/.f64 (*.f64 x y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 15.2%

      \[\frac{x \cdot y}{y + 1} \]
   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. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 95.0

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

   5. Applied rewrites 95.0%

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

Alternative 3: 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 74.8%

      \[\frac{x \cdot y}{y + 1} \]
   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. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 99.4

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

   5. Applied rewrites 99.4%

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

   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

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

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

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

      5. *-commutative 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. fp-cancel-sub-sign-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. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      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 98.6

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

   5. Applied rewrites 98.6%

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

4. Final simplification 99.0%

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

Alternative 4: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.76\right):\\ \;\;\;\;\left(-x\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, y, x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.76))) (* (- x) -1.0) (* (fma (- x) y x) y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.76)) {
		tmp = -x * -1.0;
	} else {
		tmp = fma(-x, y, x) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.76))
		tmp = Float64(Float64(-x) * -1.0);
	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, 0.76]], $MachinePrecision]], N[((-x) * -1.0), $MachinePrecision], N[(N[((-x) * y + x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.76000000000000001 < y

   1. Initial program 74.8%

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 44.9

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

   6. Applied rewrites 44.9%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 98.3%

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

   if -1 < y < 0.76000000000000001

   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

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

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

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

      5. *-commutative 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. fp-cancel-sub-sign-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. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      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 98.6

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

   5. Applied rewrites 98.6%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.76\right):\\ \;\;\;\;\left(-x\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, y, x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.76\right):\\ \;\;\;\;\left(-x\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - y\right) \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.76))) (* (- x) -1.0) (* (* (- 1.0 y) x) y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.76)) {
		tmp = -x * -1.0;
	} else {
		tmp = ((1.0 - y) * 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 <= (-1.0d0)) .or. (.not. (y <= 0.76d0))) then
        tmp = -x * (-1.0d0)
    else
        tmp = ((1.0d0 - y) * x) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.76)) {
		tmp = -x * -1.0;
	} else {
		tmp = ((1.0 - y) * x) * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 0.76):
		tmp = -x * -1.0
	else:
		tmp = ((1.0 - y) * x) * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.76))
		tmp = Float64(Float64(-x) * -1.0);
	else
		tmp = Float64(Float64(Float64(1.0 - y) * x) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 0.76)))
		tmp = -x * -1.0;
	else
		tmp = ((1.0 - y) * x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.76]], $MachinePrecision]], N[((-x) * -1.0), $MachinePrecision], N[(N[(N[(1.0 - y), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.76000000000000001 < y

   1. Initial program 74.8%

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 44.9

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

   6. Applied rewrites 44.9%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 98.3%

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

   if -1 < y < 0.76000000000000001

   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

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

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

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

      5. *-commutative 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. fp-cancel-sub-sign-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. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      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 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.5%

      \[\leadsto \left(\left(1 - y\right) \cdot x\right) \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.76\right):\\ \;\;\;\;\left(-x\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - y\right) \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = -x * -1.0;
	} else {
		tmp = y * 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 <= (-1.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = -x * (-1.0d0)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 74.8%

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 44.9

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

   6. Applied rewrites 44.9%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 98.3%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 97.7

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

   7. Applied rewrites 97.7%

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

4. Final simplification 98.0%

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

Alternative 7: 51.4% 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 87.4%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 50.7

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

7. Applied rewrites 50.7%

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

8. Final simplification 50.7%

   \[\leadsto y \cdot x \]
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 2024332 
(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)))