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

Percentage Accurate: 83.5% → 96.9%

Time: 10.0s

Alternatives: 7

Speedup: 1.0×

• 26.0% 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: 83.5% 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: 96.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{\frac{x\_m}{\frac{z}{\frac{y\_m}{z}}}}{z}\\ t_1 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -40000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{y\_m}{\frac{z}{x\_m}}}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (/ (/ x_m (/ z (/ y_m z))) z)) (t_1 (* (+ z 1.0) (* z z))))
   (*
    y_s
    (*
     x_s
     (if (<= t_1 -40000000000.0)
       t_0
       (if (<= t_1 2e-11) (/ (/ y_m (/ z x_m)) z) t_0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (x_m / (z / (y_m / z))) / z;
	double t_1 = (z + 1.0) * (z * z);
	double tmp;
	if (t_1 <= -40000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 2e-11) {
		tmp = (y_m / (z / x_m)) / z;
	} else {
		tmp = t_0;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_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 / (y_m / z))) / z
    t_1 = (z + 1.0d0) * (z * z)
    if (t_1 <= (-40000000000.0d0)) then
        tmp = t_0
    else if (t_1 <= 2d-11) then
        tmp = (y_m / (z / x_m)) / z
    else
        tmp = t_0
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (x_m / (z / (y_m / z))) / z;
	double t_1 = (z + 1.0) * (z * z);
	double tmp;
	if (t_1 <= -40000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 2e-11) {
		tmp = (y_m / (z / x_m)) / z;
	} else {
		tmp = t_0;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	t_0 = (x_m / (z / (y_m / z))) / z
	t_1 = (z + 1.0) * (z * z)
	tmp = 0
	if t_1 <= -40000000000.0:
		tmp = t_0
	elif t_1 <= 2e-11:
		tmp = (y_m / (z / x_m)) / z
	else:
		tmp = t_0
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(Float64(x_m / Float64(z / Float64(y_m / z))) / z)
	t_1 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_1 <= -40000000000.0)
		tmp = t_0;
	elseif (t_1 <= 2e-11)
		tmp = Float64(Float64(y_m / Float64(z / x_m)) / z);
	else
		tmp = t_0;
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = (x_m / (z / (y_m / z))) / z;
	t_1 = (z + 1.0) * (z * z);
	tmp = 0.0;
	if (t_1 <= -40000000000.0)
		tmp = t_0;
	elseif (t_1 <= 2e-11)
		tmp = (y_m / (z / x_m)) / z;
	else
		tmp = t_0;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, 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[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x$95$m / N[(z / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$1, -40000000000.0], t$95$0, If[LessEqual[t$95$1, 2e-11], N[(N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $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))) < -4e10 or 1.99999999999999988e-11 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 83.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(z \cdot z\right) \cdot \left(z + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(z \cdot z\right)} \cdot \left(z + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{\left(z \cdot \left(z + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(z + 1\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(z + 1\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 83.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(z, \color{blue}{1}\right)\right)\right)\right) \]

   3. Simplified 83.7%

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

   5. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left({z}^{3}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left({\color{blue}{z}}^{3}\right)\right) \]

      3. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot {z}^{\color{blue}{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{\left({z}^{2}\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(z \cdot \color{blue}{z}\right)\right)\right) \]

      7. *-lowering-*.f64 82.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   7. Simplified 82.4%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{z \cdot z}\right), \color{blue}{\left(\frac{z}{y}\right)}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(z \cdot z\right)\right), \left(\frac{\color{blue}{z}}{y}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, z\right)\right), \left(\frac{z}{y}\right)\right) \]

      8. /-lowering-/.f64 91.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, z\right)\right), \mathsf{/.f64}\left(z, \color{blue}{y}\right)\right) \]

   9. Applied egg-rr 91.9%

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

   11. Applied egg-rr 94.0%

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

   if -4e10 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 1.99999999999999988e-11

   1. Initial program 84.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(z \cdot z\right) \cdot \left(z + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(z \cdot z\right)} \cdot \left(z + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{\left(z \cdot \left(z + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(z + 1\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(z + 1\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 84.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(z, \color{blue}{1}\right)\right)\right)\right) \]

   3. Simplified 84.0%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \color{blue}{\left({z}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{z}\right)\right) \]

      2. *-lowering-*.f64 83.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right) \]

   7. Simplified 83.1%

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

      1. times-frac N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{z \cdot z}{x}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z\right), \color{blue}{x}\right)\right) \]

      8. *-lowering-*.f64 82.3%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), x\right)\right) \]

   9. Applied egg-rr 82.3%

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

       1. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(y, \left(\frac{z}{x} \cdot \color{blue}{z}\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\frac{z}{x}\right), \color{blue}{z}\right)\right) \]

       3. /-lowering-/.f64 88.1%

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, x\right), z\right)\right) \]

   11. Applied egg-rr 88.1%

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

       1. associate-/l/ N/A

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

       2. clear-num N/A

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

       3. associate-/l/ N/A

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

       4. associate-/r/ N/A

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

       5. frac-2neg N/A

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

       6. associate-/r/ N/A

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

       7. inv-pow N/A

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

       8. metadata-eval N/A

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

       9. metadata-eval N/A

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

       10. pow-pow N/A

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

       11. pow2 N/A

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

       12. sqr-neg N/A

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

       13. pow-prod-down N/A

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

       14. sqr-pow N/A

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

       15. inv-pow N/A

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

       16. associate-/r/ N/A

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

       17. clear-num N/A

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

       18. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{x}{\frac{z}{y}}\right)\right), \color{blue}{z}\right) \]

   13. Applied egg-rr 95.6%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -40000000000:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{\frac{y}{z}}}}{z}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{\frac{y}{z}}}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 83.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(z \cdot z\right) \cdot \left(z + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(z \cdot z\right)} \cdot \left(z + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{\left(z \cdot \left(z + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(z + 1\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(z + 1\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 83.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(z, \color{blue}{1}\right)\right)\right)\right) \]

   3. Simplified 83.6%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. clear-num N/A

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

      5. associate-/r* N/A

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

      6. frac-times N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

      9. clear-num N/A

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

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{z}\right), \color{blue}{\left(\frac{z + 1}{y} \cdot z\right)}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \left(\color{blue}{\frac{z + 1}{y}} \cdot z\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{*.f64}\left(\left(\frac{z + 1}{y}\right), \color{blue}{z}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(z + 1\right), y\right), z\right)\right) \]

      14. +-lowering-+.f64 95.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(z, 1\right), y\right), z\right)\right) \]

   6. Applied egg-rr 95.4%

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

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{*.f64}\left(\color{blue}{\left(\frac{z}{y}\right)}, z\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 94.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, y\right), z\right)\right) \]

   9. Simplified 94.0%

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

   if -1 < z < 1

   1. Initial program 84.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(z \cdot z\right) \cdot \left(z + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(z \cdot z\right)} \cdot \left(z + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{\left(z \cdot \left(z + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(z + 1\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(z + 1\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 84.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(z, \color{blue}{1}\right)\right)\right)\right) \]

   3. Simplified 84.1%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \color{blue}{\left({z}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{z}\right)\right) \]

      2. *-lowering-*.f64 83.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right) \]

   7. Simplified 83.2%

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

      1. times-frac N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{z \cdot z}{x}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z\right), \color{blue}{x}\right)\right) \]

      8. *-lowering-*.f64 82.4%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), x\right)\right) \]

   9. Applied egg-rr 82.4%

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

       1. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(y, \left(\frac{z}{x} \cdot \color{blue}{z}\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\frac{z}{x}\right), \color{blue}{z}\right)\right) \]

       3. /-lowering-/.f64 88.2%

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, x\right), z\right)\right) \]

   11. Applied egg-rr 88.2%

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

       1. associate-/l/ N/A

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

       2. clear-num N/A

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

       3. associate-/l/ N/A

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

       4. associate-/r/ N/A

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

       5. frac-2neg N/A

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

       6. associate-/r/ N/A

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

       7. inv-pow N/A

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

       8. metadata-eval N/A

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

       9. metadata-eval N/A

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

       10. pow-pow N/A

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

       11. pow2 N/A

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

       12. sqr-neg N/A

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

       13. pow-prod-down N/A

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

       14. sqr-pow N/A

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

       15. inv-pow N/A

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

       16. associate-/r/ N/A

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

       17. clear-num N/A

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

       18. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{x}{\frac{z}{y}}\right)\right), \color{blue}{z}\right) \]

   13. Applied egg-rr 95.6%

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

4. Final simplification 94.8%

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

Alternative 3: 95.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 83.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(z \cdot z\right) \cdot \left(z + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(z \cdot z\right)} \cdot \left(z + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{\left(z \cdot \left(z + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(z + 1\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(z + 1\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 83.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(z, \color{blue}{1}\right)\right)\right)\right) \]

   3. Simplified 83.6%

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

   5. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left({z}^{3}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left({\color{blue}{z}}^{3}\right)\right) \]

      3. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot {z}^{\color{blue}{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{\left({z}^{2}\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(z \cdot \color{blue}{z}\right)\right)\right) \]

      7. *-lowering-*.f64 82.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   7. Simplified 82.2%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{z \cdot z}\right), \color{blue}{\left(\frac{z}{y}\right)}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(z \cdot z\right)\right), \left(\frac{\color{blue}{z}}{y}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, z\right)\right), \left(\frac{z}{y}\right)\right) \]

      8. /-lowering-/.f64 91.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, z\right)\right), \mathsf{/.f64}\left(z, \color{blue}{y}\right)\right) \]

   9. Applied egg-rr 91.8%

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

       1. associate-/r* N/A

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

       2. div-inv N/A

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

       3. associate-/l* N/A

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

       4. frac-2neg N/A

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

       5. distribute-frac-neg2 N/A

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

       6. inv-pow N/A

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

       7. metadata-eval N/A

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

       8. metadata-eval N/A

          \[\leadsto \frac{x}{z} \cdot \frac{{\left(\mathsf{neg}\left(z\right)\right)}^{\left(2 \cdot \frac{-1}{2}\right)}}{\mathsf{neg}\left(\frac{z}{y}\right)} \]

       9. pow-pow N/A

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

       10. pow2 N/A

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

       11. sqr-neg N/A

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

       12. pow-prod-down N/A

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

       13. sqr-pow N/A

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

       14. inv-pow N/A

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

       15. associate-/l* N/A

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

       16. div-inv N/A

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

       17. associate-/r* N/A

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

       18. associate-/l/ N/A

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

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

           \[\leadsto \frac{x}{\mathsf{neg}\left(\frac{z}{y} \cdot \left(z \cdot z\right)\right)} \]

       20. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{z}{y} \cdot \left(z \cdot z\right)\right)\right)}\right) \]

       21. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(x, \left(\mathsf{neg}\left(\left(z \cdot z\right) \cdot \frac{z}{y}\right)\right)\right) \]

   11. Applied egg-rr 84.7%

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

   if -1 < z < 1

   1. Initial program 84.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(z \cdot z\right) \cdot \left(z + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(z \cdot z\right)} \cdot \left(z + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{\left(z \cdot \left(z + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(z + 1\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(z + 1\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 84.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(z, \color{blue}{1}\right)\right)\right)\right) \]

   3. Simplified 84.1%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \color{blue}{\left({z}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{z}\right)\right) \]

      2. *-lowering-*.f64 83.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right) \]

   7. Simplified 83.2%

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

      1. times-frac N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{z \cdot z}{x}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z\right), \color{blue}{x}\right)\right) \]

      8. *-lowering-*.f64 82.4%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), x\right)\right) \]

   9. Applied egg-rr 82.4%

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

       1. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(y, \left(\frac{z}{x} \cdot \color{blue}{z}\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\frac{z}{x}\right), \color{blue}{z}\right)\right) \]

       3. /-lowering-/.f64 88.2%

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, x\right), z\right)\right) \]

   11. Applied egg-rr 88.2%

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

       1. associate-/l/ N/A

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

       2. clear-num N/A

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

       3. associate-/l/ N/A

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

       4. associate-/r/ N/A

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

       5. frac-2neg N/A

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

       6. associate-/r/ N/A

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

       7. inv-pow N/A

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

       8. metadata-eval N/A

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

       9. metadata-eval N/A

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

       10. pow-pow N/A

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

       11. pow2 N/A

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

       12. sqr-neg N/A

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

       13. pow-prod-down N/A

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

       14. sqr-pow N/A

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

       15. inv-pow N/A

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

       16. associate-/r/ N/A

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

       17. clear-num N/A

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

       18. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{x}{\frac{z}{y}}\right)\right), \color{blue}{z}\right) \]

   13. Applied egg-rr 95.6%

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

Alternative 4: 80.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2.6 \cdot 10^{+56}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= y_m 2.6e+56) (* (/ x_m z) (/ y_m z)) (* y_m (/ x_m (* z z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 2.6e+56) {
		tmp = (x_m / z) * (y_m / z);
	} else {
		tmp = y_m * (x_m / (z * z));
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 2.6d+56) then
        tmp = (x_m / z) * (y_m / z)
    else
        tmp = y_m * (x_m / (z * z))
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 2.6e+56) {
		tmp = (x_m / z) * (y_m / z);
	} else {
		tmp = y_m * (x_m / (z * z));
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if y_m <= 2.6e+56:
		tmp = (x_m / z) * (y_m / z)
	else:
		tmp = y_m * (x_m / (z * z))
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 2.6e+56)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	else
		tmp = Float64(y_m * Float64(x_m / Float64(z * z)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (y_m <= 2.6e+56)
		tmp = (x_m / z) * (y_m / z);
	else
		tmp = y_m * (x_m / (z * z));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, 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[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 2.6e+56], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.60000000000000011e56

   1. Initial program 82.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(z \cdot z\right) \cdot \left(z + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(z \cdot z\right)} \cdot \left(z + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{\left(z \cdot \left(z + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(z + 1\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(z + 1\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 82.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(z, \color{blue}{1}\right)\right)\right)\right) \]

   3. Simplified 82.4%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \color{blue}{\left({z}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{z}\right)\right) \]

      2. *-lowering-*.f64 70.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right) \]

   7. Simplified 70.2%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{y}{z}\right), \color{blue}{\left(\frac{x}{z}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \left(\frac{\color{blue}{x}}{z}\right)\right) \]

      5. /-lowering-/.f64 77.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \color{blue}{z}\right)\right) \]

   9. Applied egg-rr 77.4%

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

   if 2.60000000000000011e56 < y

   1. Initial program 89.2%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(z \cdot z\right) \cdot \left(z + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(z \cdot z\right)} \cdot \left(z + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{\left(z \cdot \left(z + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(z + 1\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(z + 1\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 89.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(z, \color{blue}{1}\right)\right)\right)\right) \]

   3. Simplified 89.3%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \color{blue}{\left({z}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{z}\right)\right) \]

      2. *-lowering-*.f64 73.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right) \]

   7. Simplified 73.8%

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

      1. times-frac N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{z}\right), \color{blue}{\left(\frac{z}{y}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \left(\frac{\color{blue}{z}}{y}\right)\right) \]

      6. /-lowering-/.f64 70.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{/.f64}\left(z, \color{blue}{y}\right)\right) \]

   9. Applied egg-rr 70.6%

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

       1. associate-/r/ N/A

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

       2. associate-/r* N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{z \cdot z}\right), \color{blue}{y}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(z \cdot z\right)\right), y\right) \]

       5. *-lowering-*.f64 79.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, z\right)\right), y\right) \]

   11. Applied egg-rr 79.3%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{+56}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(z \cdot z\right) \cdot \left(z + 1\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(z \cdot z\right)} \cdot \left(z + 1\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{\left(z \cdot \left(z + 1\right)\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(z + 1\right)\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(z + 1\right)}\right)\right)\right) \]

   6. +-lowering-+.f64 83.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(z, \color{blue}{1}\right)\right)\right)\right) \]

3. Simplified 83.9%

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

   1. associate-*r* N/A

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

   2. times-frac N/A

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

   3. *-commutative N/A

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

   4. clear-num N/A

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

   5. associate-/r* N/A

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

   6. frac-times N/A

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

   7. clear-num N/A

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

   8. div-inv N/A

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

   9. clear-num N/A

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

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{z}\right), \color{blue}{\left(\frac{z + 1}{y} \cdot z\right)}\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \left(\color{blue}{\frac{z + 1}{y}} \cdot z\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{*.f64}\left(\left(\frac{z + 1}{y}\right), \color{blue}{z}\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(z + 1\right), y\right), z\right)\right) \]

   14. +-lowering-+.f64 95.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(z, 1\right), y\right), z\right)\right) \]

6. Applied egg-rr 95.8%

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

7. Final simplification 95.8%

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

Alternative 6: 80.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(z \cdot z\right) \cdot \left(z + 1\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(z \cdot z\right)} \cdot \left(z + 1\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{\left(z \cdot \left(z + 1\right)\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(z + 1\right)\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(z + 1\right)}\right)\right)\right) \]

   6. +-lowering-+.f64 83.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(z, \color{blue}{1}\right)\right)\right)\right) \]

3. Simplified 83.9%

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

5. Taylor expanded in z around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \color{blue}{\left({z}^{2}\right)}\right) \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{z}\right)\right) \]

   2. *-lowering-*.f64 71.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right) \]

7. Simplified 71.0%

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

   1. times-frac N/A

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

   2. clear-num N/A

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

   3. frac-times N/A

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

   4. *-lft-identity N/A

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

   5. associate-*l/ N/A

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

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{z \cdot z}{x}\right)}\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z\right), \color{blue}{x}\right)\right) \]

   8. *-lowering-*.f64 72.6%

      \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), x\right)\right) \]

9. Applied egg-rr 72.6%

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

    1. associate-*l/ N/A

       \[\leadsto \mathsf{/.f64}\left(y, \left(\frac{z}{x} \cdot \color{blue}{z}\right)\right) \]

    2. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(\frac{z}{x}\right), \color{blue}{z}\right)\right) \]

    3. /-lowering-/.f64 74.1%

       \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, x\right), z\right)\right) \]

11. Applied egg-rr 74.1%

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

12. Final simplification 74.1%

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

Alternative 7: 75.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(z \cdot z\right) \cdot \left(z + 1\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(z \cdot z\right)} \cdot \left(z + 1\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{\left(z \cdot \left(z + 1\right)\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(z + 1\right)\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(z + 1\right)}\right)\right)\right) \]

   6. +-lowering-+.f64 83.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(z, \color{blue}{1}\right)\right)\right)\right) \]

3. Simplified 83.9%

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

5. Taylor expanded in z around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \color{blue}{\left({z}^{2}\right)}\right) \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{z}\right)\right) \]

   2. *-lowering-*.f64 71.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right) \]

7. Simplified 71.0%

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

   1. times-frac N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{z}\right), \color{blue}{\left(\frac{z}{y}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \left(\frac{\color{blue}{z}}{y}\right)\right) \]

   6. /-lowering-/.f64 75.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{/.f64}\left(z, \color{blue}{y}\right)\right) \]

9. Applied egg-rr 75.9%

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

    1. associate-/r/ N/A

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

    2. associate-/r* N/A

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

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{z \cdot z}\right), \color{blue}{y}\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(z \cdot z\right)\right), y\right) \]

    5. *-lowering-*.f64 72.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, z\right)\right), y\right) \]

11. Applied egg-rr 72.6%

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

12. Final simplification 72.6%

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

Developer Target 1: 96.9% accurate, 0.7× 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 2024138 
(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))))