Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2

Percentage Accurate: 82.9% → 94.0%

Time: 9.2s

Alternatives: 12

Speedup: 0.9×

• 25.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\frac{x\_m \cdot \frac{y\_m}{z}}{z \cdot z}\\ \mathbf{elif}\;t\_0 \leq 10^{-17}:\\ \;\;\;\;\frac{y\_m}{z \cdot \frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\left(z \cdot z\right) \cdot \frac{z}{y\_m}}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (* z z) (+ z 1.0))))
   (*
    x_s
    (*
     y_s
     (if (<= t_0 -5e+22)
       (/ (* x_m (/ y_m z)) (* z z))
       (if (<= t_0 1e-17)
         (/ y_m (* z (/ z x_m)))
         (/ x_m (* (* z z) (/ z y_m)))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = (z * z) * (z + 1.0);
	double tmp;
	if (t_0 <= -5e+22) {
		tmp = (x_m * (y_m / z)) / (z * z);
	} else if (t_0 <= 1e-17) {
		tmp = y_m / (z * (z / x_m));
	} else {
		tmp = x_m / ((z * z) * (z / y_m));
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (z * z) * (z + 1.0d0)
    if (t_0 <= (-5d+22)) then
        tmp = (x_m * (y_m / z)) / (z * z)
    else if (t_0 <= 1d-17) then
        tmp = y_m / (z * (z / x_m))
    else
        tmp = x_m / ((z * z) * (z / y_m))
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = (z * z) * (z + 1.0);
	double tmp;
	if (t_0 <= -5e+22) {
		tmp = (x_m * (y_m / z)) / (z * z);
	} else if (t_0 <= 1e-17) {
		tmp = y_m / (z * (z / x_m));
	} else {
		tmp = x_m / ((z * z) * (z / y_m));
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	t_0 = (z * z) * (z + 1.0)
	tmp = 0
	if t_0 <= -5e+22:
		tmp = (x_m * (y_m / z)) / (z * z)
	elif t_0 <= 1e-17:
		tmp = y_m / (z * (z / x_m))
	else:
		tmp = x_m / ((z * z) * (z / y_m))
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if (t_0 <= -5e+22)
		tmp = Float64(Float64(x_m * Float64(y_m / z)) / Float64(z * z));
	elseif (t_0 <= 1e-17)
		tmp = Float64(y_m / Float64(z * Float64(z / x_m)));
	else
		tmp = Float64(x_m / Float64(Float64(z * z) * Float64(z / y_m)));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	t_0 = (z * z) * (z + 1.0);
	tmp = 0.0;
	if (t_0 <= -5e+22)
		tmp = (x_m * (y_m / z)) / (z * z);
	elseif (t_0 <= 1e-17)
		tmp = y_m / (z * (z / x_m));
	else
		tmp = x_m / ((z * z) * (z / y_m));
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$0, -5e+22], N[(N[(x$95$m * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-17], N[(y$95$m / N[(z * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(N[(z * z), $MachinePrecision] * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -4.9999999999999996e22

   1. Initial program 91.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in z around inf

      \[\leadsto \frac{\frac{y}{z} \cdot x}{\color{blue}{{z}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 98.1

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

   7. Applied rewrites 98.1%

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

   if -4.9999999999999996e22 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 1.00000000000000007e-17

   1. Initial program 85.1%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{2}}} \]

      3. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{y}{{z}^{2}}} \]

      4. unpow2 N/A

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

      5. lower-*.f64 85.7

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

   5. Applied rewrites 85.7%

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

   7. Applied rewrites 84.2%

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

   9. Applied rewrites 90.3%

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

   if 1.00000000000000007e-17 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 92.9

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

   4. Applied rewrites 92.9%

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

   5. Taylor expanded in z around inf

      \[\leadsto \frac{\frac{y}{z} \cdot x}{\color{blue}{{z}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 90.2

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

   7. Applied rewrites 90.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. *-lft-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 90.2

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

   9. Applied rewrites 90.2%

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y} \cdot \left(z \cdot z\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z \cdot z}\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 10^{-17}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(z \cdot z\right) \cdot \frac{z}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := x\_m \cdot \frac{y\_m}{z \cdot \left(z \cdot z\right)}\\ t_1 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-310}:\\ \;\;\;\;x\_m \cdot \frac{\frac{y\_m}{z}}{z}\\ \mathbf{elif}\;t\_1 \leq 10^{-17}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* x_m (/ y_m (* z (* z z))))) (t_1 (* (* z z) (+ z 1.0))))
   (*
    x_s
    (*
     y_s
     (if (<= t_1 -5e+22)
       t_0
       (if (<= t_1 2e-310)
         (* x_m (/ (/ y_m z) z))
         (if (<= t_1 1e-17) (* y_m (/ x_m (* z z))) t_0)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = x_m * (y_m / (z * (z * z)));
	double t_1 = (z * z) * (z + 1.0);
	double tmp;
	if (t_1 <= -5e+22) {
		tmp = t_0;
	} else if (t_1 <= 2e-310) {
		tmp = x_m * ((y_m / z) / z);
	} else if (t_1 <= 1e-17) {
		tmp = y_m * (x_m / (z * z));
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x_m * (y_m / (z * (z * z)))
    t_1 = (z * z) * (z + 1.0d0)
    if (t_1 <= (-5d+22)) then
        tmp = t_0
    else if (t_1 <= 2d-310) then
        tmp = x_m * ((y_m / z) / z)
    else if (t_1 <= 1d-17) then
        tmp = y_m * (x_m / (z * z))
    else
        tmp = t_0
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = x_m * (y_m / (z * (z * z)));
	double t_1 = (z * z) * (z + 1.0);
	double tmp;
	if (t_1 <= -5e+22) {
		tmp = t_0;
	} else if (t_1 <= 2e-310) {
		tmp = x_m * ((y_m / z) / z);
	} else if (t_1 <= 1e-17) {
		tmp = y_m * (x_m / (z * z));
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	t_0 = x_m * (y_m / (z * (z * z)))
	t_1 = (z * z) * (z + 1.0)
	tmp = 0
	if t_1 <= -5e+22:
		tmp = t_0
	elif t_1 <= 2e-310:
		tmp = x_m * ((y_m / z) / z)
	elif t_1 <= 1e-17:
		tmp = y_m * (x_m / (z * z))
	else:
		tmp = t_0
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(x_m * Float64(y_m / Float64(z * Float64(z * z))))
	t_1 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if (t_1 <= -5e+22)
		tmp = t_0;
	elseif (t_1 <= 2e-310)
		tmp = Float64(x_m * Float64(Float64(y_m / z) / z));
	elseif (t_1 <= 1e-17)
		tmp = Float64(y_m * Float64(x_m / Float64(z * z)));
	else
		tmp = t_0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	t_0 = x_m * (y_m / (z * (z * z)));
	t_1 = (z * z) * (z + 1.0);
	tmp = 0.0;
	if (t_1 <= -5e+22)
		tmp = t_0;
	elseif (t_1 <= 2e-310)
		tmp = x_m * ((y_m / z) / z);
	elseif (t_1 <= 1e-17)
		tmp = y_m * (x_m / (z * z));
	else
		tmp = t_0;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(x$95$m * N[(y$95$m / N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$1, -5e+22], t$95$0, If[LessEqual[t$95$1, 2e-310], N[(x$95$m * N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-17], N[(y$95$m * N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -4.9999999999999996e22 or 1.00000000000000007e-17 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 87.4%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{3}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{3}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{3}}} \]

      3. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{y}{{z}^{3}}} \]

      4. cube-mult N/A

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

      5. unpow2 N/A

         \[\leadsto x \cdot \frac{y}{z \cdot \color{blue}{{z}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto x \cdot \frac{y}{\color{blue}{z \cdot {z}^{2}}} \]

      7. unpow2 N/A

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

      8. lower-*.f64 88.0

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

   5. Applied rewrites 88.0%

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

   if -4.9999999999999996e22 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 1.999999999999994e-310

   1. Initial program 78.4%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{2}}} \]

      3. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{y}{{z}^{2}}} \]

      4. unpow2 N/A

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

      5. lower-*.f64 79.6

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

   5. Applied rewrites 79.6%

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

   7. Applied rewrites 94.5%

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

   if 1.999999999999994e-310 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 1.00000000000000007e-17

   1. Initial program 93.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 88.3

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 88.3

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

   4. Applied rewrites 88.3%

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

   5. Taylor expanded in z around 0

      \[\leadsto \frac{x}{\color{blue}{{z}^{2}}} \cdot y \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 88.3

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

   7. Applied rewrites 88.3%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq -5 \cdot 10^{+22}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 2 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 10^{-17}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{x\_m}{\left(z \cdot z\right) \cdot \frac{z}{y\_m}}\\ t_1 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-17}:\\ \;\;\;\;\frac{y\_m}{z \cdot \frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (/ x_m (* (* z z) (/ z y_m)))) (t_1 (* (* z z) (+ z 1.0))))
   (*
    x_s
    (*
     y_s
     (if (<= t_1 -5e+22)
       t_0
       (if (<= t_1 1e-17) (/ y_m (* z (/ z x_m))) t_0))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = x_m / ((z * z) * (z / y_m));
	double t_1 = (z * z) * (z + 1.0);
	double tmp;
	if (t_1 <= -5e+22) {
		tmp = t_0;
	} else if (t_1 <= 1e-17) {
		tmp = y_m / (z * (z / x_m));
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x_m / ((z * z) * (z / y_m))
    t_1 = (z * z) * (z + 1.0d0)
    if (t_1 <= (-5d+22)) then
        tmp = t_0
    else if (t_1 <= 1d-17) then
        tmp = y_m / (z * (z / x_m))
    else
        tmp = t_0
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = x_m / ((z * z) * (z / y_m));
	double t_1 = (z * z) * (z + 1.0);
	double tmp;
	if (t_1 <= -5e+22) {
		tmp = t_0;
	} else if (t_1 <= 1e-17) {
		tmp = y_m / (z * (z / x_m));
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	t_0 = x_m / ((z * z) * (z / y_m))
	t_1 = (z * z) * (z + 1.0)
	tmp = 0
	if t_1 <= -5e+22:
		tmp = t_0
	elif t_1 <= 1e-17:
		tmp = y_m / (z * (z / x_m))
	else:
		tmp = t_0
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(x_m / Float64(Float64(z * z) * Float64(z / y_m)))
	t_1 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if (t_1 <= -5e+22)
		tmp = t_0;
	elseif (t_1 <= 1e-17)
		tmp = Float64(y_m / Float64(z * Float64(z / x_m)));
	else
		tmp = t_0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	t_0 = x_m / ((z * z) * (z / y_m));
	t_1 = (z * z) * (z + 1.0);
	tmp = 0.0;
	if (t_1 <= -5e+22)
		tmp = t_0;
	elseif (t_1 <= 1e-17)
		tmp = y_m / (z * (z / x_m));
	else
		tmp = t_0;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(x$95$m / N[(N[(z * z), $MachinePrecision] * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$1, -5e+22], t$95$0, If[LessEqual[t$95$1, 1e-17], N[(y$95$m / N[(z * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -4.9999999999999996e22 or 1.00000000000000007e-17 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 87.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 96.2

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

   4. Applied rewrites 96.2%

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

   5. Taylor expanded in z around inf

      \[\leadsto \frac{\frac{y}{z} \cdot x}{\color{blue}{{z}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 94.4

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

   7. Applied rewrites 94.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. *-lft-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 91.7

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

   9. Applied rewrites 91.7%

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

   if -4.9999999999999996e22 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 1.00000000000000007e-17

   1. Initial program 85.1%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{2}}} \]

      3. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{y}{{z}^{2}}} \]

      4. unpow2 N/A

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

      5. lower-*.f64 85.7

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

   5. Applied rewrites 85.7%

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

   7. Applied rewrites 84.2%

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

   9. Applied rewrites 90.3%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\frac{x}{\left(z \cdot z\right) \cdot \frac{z}{y}}\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 10^{-17}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(z \cdot z\right) \cdot \frac{z}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := x\_m \cdot \frac{y\_m}{z \cdot \left(z \cdot z\right)}\\ t_1 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-17}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* x_m (/ y_m (* z (* z z))))) (t_1 (* (* z z) (+ z 1.0))))
   (*
    x_s
    (*
     y_s
     (if (<= t_1 -5e+22)
       t_0
       (if (<= t_1 1e-17) (* y_m (/ x_m (* z z))) t_0))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = x_m * (y_m / (z * (z * z)));
	double t_1 = (z * z) * (z + 1.0);
	double tmp;
	if (t_1 <= -5e+22) {
		tmp = t_0;
	} else if (t_1 <= 1e-17) {
		tmp = y_m * (x_m / (z * z));
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x_m * (y_m / (z * (z * z)))
    t_1 = (z * z) * (z + 1.0d0)
    if (t_1 <= (-5d+22)) then
        tmp = t_0
    else if (t_1 <= 1d-17) then
        tmp = y_m * (x_m / (z * z))
    else
        tmp = t_0
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = x_m * (y_m / (z * (z * z)));
	double t_1 = (z * z) * (z + 1.0);
	double tmp;
	if (t_1 <= -5e+22) {
		tmp = t_0;
	} else if (t_1 <= 1e-17) {
		tmp = y_m * (x_m / (z * z));
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	t_0 = x_m * (y_m / (z * (z * z)))
	t_1 = (z * z) * (z + 1.0)
	tmp = 0
	if t_1 <= -5e+22:
		tmp = t_0
	elif t_1 <= 1e-17:
		tmp = y_m * (x_m / (z * z))
	else:
		tmp = t_0
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(x_m * Float64(y_m / Float64(z * Float64(z * z))))
	t_1 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if (t_1 <= -5e+22)
		tmp = t_0;
	elseif (t_1 <= 1e-17)
		tmp = Float64(y_m * Float64(x_m / Float64(z * z)));
	else
		tmp = t_0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	t_0 = x_m * (y_m / (z * (z * z)));
	t_1 = (z * z) * (z + 1.0);
	tmp = 0.0;
	if (t_1 <= -5e+22)
		tmp = t_0;
	elseif (t_1 <= 1e-17)
		tmp = y_m * (x_m / (z * z));
	else
		tmp = t_0;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(x$95$m * N[(y$95$m / N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$1, -5e+22], t$95$0, If[LessEqual[t$95$1, 1e-17], N[(y$95$m * N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -4.9999999999999996e22 or 1.00000000000000007e-17 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 87.4%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{3}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{3}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{3}}} \]

      3. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{y}{{z}^{3}}} \]

      4. cube-mult N/A

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

      5. unpow2 N/A

         \[\leadsto x \cdot \frac{y}{z \cdot \color{blue}{{z}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto x \cdot \frac{y}{\color{blue}{z \cdot {z}^{2}}} \]

      7. unpow2 N/A

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

      8. lower-*.f64 88.0

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

   5. Applied rewrites 88.0%

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

   if -4.9999999999999996e22 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 1.00000000000000007e-17

   1. Initial program 85.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 83.4

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 83.4

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

   4. Applied rewrites 83.4%

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

   5. Taylor expanded in z around 0

      \[\leadsto \frac{x}{\color{blue}{{z}^{2}}} \cdot y \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 83.4

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

   7. Applied rewrites 83.4%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq -5 \cdot 10^{+22}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 10^{-17}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := x\_m \cdot \frac{y\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-61}:\\ \;\;\;\;\frac{y\_m}{z \cdot \frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* x_m (/ y_m (* z (fma z z z))))))
   (*
    x_s
    (*
     y_s
     (if (<= z -2.4e-9)
       t_0
       (if (<= z 2.2e-61) (/ y_m (* z (/ z x_m))) t_0))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = x_m * (y_m / (z * fma(z, z, z)));
	double tmp;
	if (z <= -2.4e-9) {
		tmp = t_0;
	} else if (z <= 2.2e-61) {
		tmp = y_m / (z * (z / x_m));
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(x_m * Float64(y_m / Float64(z * fma(z, z, z))))
	tmp = 0.0
	if (z <= -2.4e-9)
		tmp = t_0;
	elseif (z <= 2.2e-61)
		tmp = Float64(y_m / Float64(z * Float64(z / x_m)));
	else
		tmp = t_0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(x$95$m * N[(y$95$m / N[(z * N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[z, -2.4e-9], t$95$0, If[LessEqual[z, 2.2e-61], N[(y$95$m / N[(z * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.4e-9 or 2.20000000000000009e-61 < z

   1. Initial program 87.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 90.6

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 90.6

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

   4. Applied rewrites 90.6%

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

   if -2.4e-9 < z < 2.20000000000000009e-61

   1. Initial program 84.6%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{2}}} \]

      3. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{y}{{z}^{2}}} \]

      4. unpow2 N/A

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

      5. lower-*.f64 84.4

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

   5. Applied rewrites 84.4%

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

   7. Applied rewrites 83.5%

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

   9. Applied rewrites 89.4%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-61}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.0% accurate, 0.7× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* x_m (/ y_m (* z (* z z))))))
   (*
    x_s
    (* y_s (if (<= z -1.0) t_0 (if (<= z 1.0) (/ y_m (* z (/ z x_m))) t_0))))))

C

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

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_m * (y_m / (z * (z * z)))
    if (z <= (-1.0d0)) then
        tmp = t_0
    else if (z <= 1.0d0) then
        tmp = y_m / (z * (z / x_m))
    else
        tmp = t_0
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = x_m * (y_m / (z * (z * z)));
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = y_m / (z * (z / x_m));
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	t_0 = x_m * (y_m / (z * (z * z)))
	tmp = 0
	if z <= -1.0:
		tmp = t_0
	elif z <= 1.0:
		tmp = y_m / (z * (z / x_m))
	else:
		tmp = t_0
	return x_s * (y_s * tmp)

Julia

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

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	t_0 = x_m * (y_m / (z * (z * z)));
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = y_m / (z * (z / x_m));
	else
		tmp = t_0;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(x$95$m * N[(y$95$m / N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[z, -1.0], t$95$0, If[LessEqual[z, 1.0], N[(y$95$m / N[(z * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 87.4%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{3}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{3}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{3}}} \]

      3. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{y}{{z}^{3}}} \]

      4. cube-mult N/A

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

      5. unpow2 N/A

         \[\leadsto x \cdot \frac{y}{z \cdot \color{blue}{{z}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto x \cdot \frac{y}{\color{blue}{z \cdot {z}^{2}}} \]

      7. unpow2 N/A

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

      8. lower-*.f64 88.0

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

   5. Applied rewrites 88.0%

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

   if -1 < z < 1

   1. Initial program 85.1%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{2}}} \]

      3. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{y}{{z}^{2}}} \]

      4. unpow2 N/A

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

      5. lower-*.f64 85.7

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

   5. Applied rewrites 85.7%

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

   7. Applied rewrites 84.2%

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

   9. Applied rewrites 90.3%

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

4. Final simplification 89.2%

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

Alternative 7: 90.7% accurate, 0.7× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* x_m (/ y_m (* z (* z z))))))
   (*
    x_s
    (* y_s (if (<= z -1.0) t_0 (if (<= z 1.0) (* (/ x_m z) (/ y_m z)) t_0))))))

C

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

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_m * (y_m / (z * (z * z)))
    if (z <= (-1.0d0)) then
        tmp = t_0
    else if (z <= 1.0d0) then
        tmp = (x_m / z) * (y_m / z)
    else
        tmp = t_0
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = x_m * (y_m / (z * (z * z)));
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = (x_m / z) * (y_m / z);
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	t_0 = x_m * (y_m / (z * (z * z)))
	tmp = 0
	if z <= -1.0:
		tmp = t_0
	elif z <= 1.0:
		tmp = (x_m / z) * (y_m / z)
	else:
		tmp = t_0
	return x_s * (y_s * tmp)

Julia

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

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	t_0 = x_m * (y_m / (z * (z * z)));
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = (x_m / z) * (y_m / z);
	else
		tmp = t_0;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(x$95$m * N[(y$95$m / N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[z, -1.0], t$95$0, If[LessEqual[z, 1.0], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 87.4%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{3}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{3}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{3}}} \]

      3. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{y}{{z}^{3}}} \]

      4. cube-mult N/A

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

      5. unpow2 N/A

         \[\leadsto x \cdot \frac{y}{z \cdot \color{blue}{{z}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto x \cdot \frac{y}{\color{blue}{z \cdot {z}^{2}}} \]

      7. unpow2 N/A

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

      8. lower-*.f64 88.0

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

   5. Applied rewrites 88.0%

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

   if -1 < z < 1

   1. Initial program 85.1%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{2}}} \]

      3. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{y}{{z}^{2}}} \]

      4. unpow2 N/A

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

      5. lower-*.f64 85.7

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

   5. Applied rewrites 85.7%

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

   7. Applied rewrites 97.5%

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

4. Final simplification 92.8%

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

Alternative 8: 95.2% accurate, 0.8× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (* x_s (* y_s (/ (/ x_m z) (/ (fma z z z) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * ((x_m / z) / (fma(z, z, z) / y_m)));
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	return Float64(x_s * Float64(y_s * Float64(Float64(x_m / z) / Float64(fma(z, z, z) / y_m))))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * N[(N[(x$95$m / z), $MachinePrecision] / N[(N[(z * z + z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.3%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. times-frac N/A

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

   8. associate-/r* N/A

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

   9. clear-num N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-+.f64 N/A

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

   15. distribute-lft1-in N/A

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

   16. lower-fma.f64 97.3

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

4. Applied rewrites 97.3%

   \[\leadsto \color{blue}{\frac{\frac{x}{z}}{\frac{\mathsf{fma}\left(z, z, z\right)}{y}}} \]
5. Add Preprocessing

Alternative 9: 94.3% accurate, 0.9× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (* x_s (* y_s (/ (* x_m (/ y_m (fma z z z))) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * ((x_m * (y_m / fma(z, z, z))) / z));
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	return Float64(x_s * Float64(y_s * Float64(Float64(x_m * Float64(y_m / fma(z, z, z))) / z)))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * N[(N[(x$95$m * N[(y$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.3%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. times-frac N/A

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

   8. associate-/r* N/A

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

   9. clear-num N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-+.f64 N/A

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

   15. distribute-lft1-in N/A

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

   16. lower-fma.f64 97.3

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

4. Applied rewrites 97.3%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f64 N/A

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

   5. clear-num N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 97.0

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

6. Applied rewrites 97.0%

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

7. Final simplification 97.0%

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

Alternative 10: 94.9% accurate, 0.9× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (* x_s (* y_s (* (/ x_m z) (/ y_m (fma z z z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * ((x_m / z) * (y_m / fma(z, z, z))));
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	return Float64(x_s * Float64(y_s * Float64(Float64(x_m / z) * Float64(y_m / fma(z, z, z)))))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. distribute-lft1-in N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-/.f64 97.1

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

4. Applied rewrites 97.1%

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

5. Final simplification 97.1%

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

Alternative 11: 74.5% accurate, 1.4× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (* x_s (* y_s (* y_m (/ x_m (* z z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * (y_m * (x_m / (z * z))));
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = x_s * (y_s * (y_m * (x_m / (z * z))))
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * (y_m * (x_m / (z * z))));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	return x_s * (y_s * (y_m * (x_m / (z * z))))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	return Float64(x_s * Float64(y_s * Float64(y_m * Float64(x_m / Float64(z * z)))))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(x_s, y_s, x_m, y_m, z)
	tmp = x_s * (y_s * (y_m * (x_m / (z * z))));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * N[(y$95$m * N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 88.0

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-+.f64 N/A

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

   14. distribute-lft1-in N/A

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

   15. lower-fma.f64 88.0

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

4. Applied rewrites 88.0%

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

5. Taylor expanded in z around 0

   \[\leadsto \frac{x}{\color{blue}{{z}^{2}}} \cdot y \]
6. Step-by-step derivation

   1. unpow2 N/A

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

   2. lower-*.f64 76.6

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

7. Applied rewrites 76.6%

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

8. Final simplification 76.6%

   \[\leadsto y \cdot \frac{x}{z \cdot z} \]
9. Add Preprocessing

Alternative 12: 68.9% accurate, 1.4× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (* x_s (* y_s (* x_m (/ y_m (* z z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * (x_m * (y_m / (z * z))));
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = x_s * (y_s * (x_m * (y_m / (z * z))))
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * (x_m * (y_m / (z * z))));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	return x_s * (y_s * (x_m * (y_m / (z * z))))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	return Float64(x_s * Float64(y_s * Float64(x_m * Float64(y_m / Float64(z * z)))))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(x_s, y_s, x_m, y_m, z)
	tmp = x_s * (y_s * (x_m * (y_m / (z * z))));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * N[(x$95$m * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.3%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{2}}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{2}}} \]

   3. lower-/.f64 N/A

      \[\leadsto x \cdot \color{blue}{\frac{y}{{z}^{2}}} \]

   4. unpow2 N/A

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

   5. lower-*.f64 77.0

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

5. Applied rewrites 77.0%

   \[\leadsto \color{blue}{x \cdot \frac{y}{z \cdot z}} \]
6. Add Preprocessing

Developer Target 1: 96.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< z 249.6182814532307)
   (/ (* y (/ x z)) (+ z (* z z)))
   (/ (* (/ (/ y z) (+ 1.0 z)) x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z < 249.6182814532307) {
		tmp = (y * (x / z)) / (z + (z * z));
	} else {
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z < 249.6182814532307d0) then
        tmp = (y * (x / z)) / (z + (z * z))
    else
        tmp = (((y / z) / (1.0d0 + z)) * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z < 249.6182814532307) {
		tmp = (y * (x / z)) / (z + (z * z));
	} else {
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z < 249.6182814532307:
		tmp = (y * (x / z)) / (z + (z * z))
	else:
		tmp = (((y / z) / (1.0 + z)) * x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z < 249.6182814532307)
		tmp = Float64(Float64(y * Float64(x / z)) / Float64(z + Float64(z * z)));
	else
		tmp = Float64(Float64(Float64(Float64(y / z) / Float64(1.0 + z)) * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z < 249.6182814532307)
		tmp = (y * (x / z)) / (z + (z * z));
	else
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[z, 249.6182814532307], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / N[(z + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y / z), $MachinePrecision] / N[(1.0 + z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision]]

Reproduce

herbie shell --seed 2024220 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z 2496182814532307/10000000000000) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1 z)) x) z)))

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