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

Percentage Accurate: 88.3% → 99.5%

Time: 6.6s

Alternatives: 5

Speedup: 0.7×

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: 88.3% 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: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m \cdot y}{y + 1}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{+233}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x\_m - \frac{x\_m}{y}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (let* ((t_0 (/ (* x_m y) (+ y 1.0))))
   (* x_s (if (<= t_0 1e+233) t_0 (- x_m (/ x_m y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double t_0 = (x_m * y) / (y + 1.0);
	double tmp;
	if (t_0 <= 1e+233) {
		tmp = t_0;
	} else {
		tmp = x_m - (x_m / y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_m * y) / (y + 1.0d0)
    if (t_0 <= 1d+233) then
        tmp = t_0
    else
        tmp = x_m - (x_m / y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y) {
	double t_0 = (x_m * y) / (y + 1.0);
	double tmp;
	if (t_0 <= 1e+233) {
		tmp = t_0;
	} else {
		tmp = x_m - (x_m / y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y):
	t_0 = (x_m * y) / (y + 1.0)
	tmp = 0
	if t_0 <= 1e+233:
		tmp = t_0
	else:
		tmp = x_m - (x_m / y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	t_0 = Float64(Float64(x_m * y) / Float64(y + 1.0))
	tmp = 0.0
	if (t_0 <= 1e+233)
		tmp = t_0;
	else
		tmp = Float64(x_m - Float64(x_m / y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y)
	t_0 = (x_m * y) / (y + 1.0);
	tmp = 0.0;
	if (t_0 <= 1e+233)
		tmp = t_0;
	else
		tmp = x_m - (x_m / y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := Block[{t$95$0 = N[(N[(x$95$m * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, 1e+233], t$95$0, N[(x$95$m - N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (+.f64 y #s(literal 1 binary64))) < 9.99999999999999974e232

   1. Initial program 91.2%

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

   if 9.99999999999999974e232 < (/.f64 (*.f64 x y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 20.8%

      \[\frac{x \cdot y}{y + 1} \]
   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. 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 84.2

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

   5. Simplified 84.2%

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

Alternative 2: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m - \frac{x\_m}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.2:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x\_m, y \cdot y - y, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (let* ((t_0 (- x_m (/ x_m y))))
   (*
    x_s
    (if (<= y -1.0)
      t_0
      (if (<= y 1.2) (* y (fma x_m (- (* y y) y) x_m)) t_0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double t_0 = x_m - (x_m / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.2) {
		tmp = y * fma(x_m, ((y * y) - y), x_m);
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	t_0 = Float64(x_m - Float64(x_m / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 1.2)
		tmp = Float64(y * fma(x_m, Float64(Float64(y * y) - y), x_m));
	else
		tmp = t_0;
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := Block[{t$95$0 = N[(x$95$m - N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 1.2], N[(y * N[(x$95$m * N[(N[(y * y), $MachinePrecision] - y), $MachinePrecision] + x$95$m), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.19999999999999996 < y

   1. Initial program 73.5%

      \[\frac{x \cdot y}{y + 1} \]
   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 98.6

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

   5. Simplified 98.6%

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

   if -1 < y < 1.19999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. remove-double-neg N/A

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

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

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

      12. distribute-neg-in N/A

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

      13. metadata-eval N/A

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

      14. distribute-lft1-in N/A

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

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

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

      16. unpow2 N/A

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

      17. distribute-neg-out N/A

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

      18. remove-double-neg N/A

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

      19. sub-neg N/A

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

      20. lower--.f64 N/A

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

      21. unpow2 N/A

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

      22. lower-*.f64 99.5

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

   5. Simplified 99.5%

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

Alternative 3: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m - \frac{x\_m}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y, -x\_m, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (let* ((t_0 (- x_m (/ x_m y))))
   (* x_s (if (<= y -1.0) t_0 (if (<= y 1.0) (* y (fma y (- x_m) x_m)) t_0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double t_0 = x_m - (x_m / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = y * fma(y, -x_m, x_m);
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	t_0 = Float64(x_m - Float64(x_m / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = Float64(y * fma(y, Float64(-x_m), x_m));
	else
		tmp = t_0;
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := Block[{t$95$0 = N[(x$95$m - N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 1.0], N[(y * N[(y * (-x$95$m) + x$95$m), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 73.5%

      \[\frac{x \cdot y}{y + 1} \]
   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 98.6

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

   5. Simplified 98.6%

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

   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. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-out N/A

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

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

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

      6. distribute-rgt1-in N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-in N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

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

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. distribute-lft1-in N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-neg.f64 99.0

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

   5. Simplified 99.0%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y, -y, y\right)} \]
   6. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-neg.f64 N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-*.f64 N/A

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

      11. *-lft-identity N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-fma.f64 N/A

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

      16. lower-neg.f64 99.0

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

   7. Applied egg-rr 99.0%

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

Alternative 4: 51.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y \cdot \mathsf{fma}\left(y, x\_m, x\_m\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y) :precision binary64 (* x_s (* y (fma y x_m x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	return x_s * (y * fma(y, x_m, x_m));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	return Float64(x_s * Float64(y * fma(y, x_m, x_m)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := N[(x$95$s * N[(y * N[(y * x$95$m + x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.0%

   \[\frac{x \cdot y}{y + 1} \]
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. distribute-lft-in N/A

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

   2. *-commutative N/A

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

   3. mul-1-neg N/A

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

   4. distribute-rgt-neg-out N/A

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

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

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

   6. distribute-rgt1-in N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. metadata-eval N/A

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

   10. distribute-neg-in N/A

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

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

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

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

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

   15. distribute-neg-in N/A

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

   16. metadata-eval N/A

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

   17. distribute-lft1-in N/A

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

   18. *-commutative N/A

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

   19. lower-fma.f64 N/A

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

   20. lower-neg.f64 47.3

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

5. Simplified 47.3%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y, -y, y\right)} \]
6. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lift-neg.f64 N/A

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

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

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

   7. lift-*.f64 N/A

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

   8. *-lft-identity N/A

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

   9. distribute-rgt-out N/A

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

   10. lower-*.f64 N/A

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

   11. *-lft-identity N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

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

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

   15. lower-fma.f64 N/A

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

   16. lower-neg.f64 47.3

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

7. Applied egg-rr 47.3%

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

9. Applied egg-rr 49.0%

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

10. Final simplification 49.0%

    \[\leadsto y \cdot \mathsf{fma}\left(y, x, x\right) \]
11. Add Preprocessing

Alternative 5: 50.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(x\_m \cdot y\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y) :precision binary64 (* x_s (* x_m y)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	return x_s * (x_m * y);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    code = x_s * (x_m * y)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y) {
	return x_s * (x_m * y);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y):
	return x_s * (x_m * y)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	return Float64(x_s * Float64(x_m * y))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y)
	tmp = x_s * (x_m * y);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := N[(x$95$s * N[(x$95$m * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.0%

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

3. Taylor expanded in y around 0

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

   1. lower-*.f64 48.4

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

5. Simplified 48.4%

   \[\leadsto \color{blue}{x \cdot y} \]
6. 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 2024212 
(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)))