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

Percentage Accurate: 83.3% → 95.0%

Time: 16.7s

Alternatives: 13

Speedup: 1.0×

• 26.3% 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: 83.3% 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: 95.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot x\_m \leq 10^{-312}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{\frac{z}{x\_m}}\\ \mathbf{elif}\;y\_m \cdot x\_m \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\frac{y\_m \cdot x\_m}{\left(z + 1\right) \cdot \left(z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z + 1} \cdot \frac{\frac{y\_m}{z}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 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 x_m) 1e-312)
     (/ (/ y_m z) (/ z x_m))
     (if (<= (* y_m x_m) 2e+152)
       (/ (* y_m x_m) (* (+ z 1.0) (* z z)))
       (* (/ x_m (+ z 1.0)) (/ (/ y_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 * x_m) <= 1e-312) {
		tmp = (y_m / z) / (z / x_m);
	} else if ((y_m * x_m) <= 2e+152) {
		tmp = (y_m * x_m) / ((z + 1.0) * (z * z));
	} else {
		tmp = (x_m / (z + 1.0)) * ((y_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 * x_m) <= 1d-312) then
        tmp = (y_m / z) / (z / x_m)
    else if ((y_m * x_m) <= 2d+152) then
        tmp = (y_m * x_m) / ((z + 1.0d0) * (z * z))
    else
        tmp = (x_m / (z + 1.0d0)) * ((y_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 * x_m) <= 1e-312) {
		tmp = (y_m / z) / (z / x_m);
	} else if ((y_m * x_m) <= 2e+152) {
		tmp = (y_m * x_m) / ((z + 1.0) * (z * z));
	} else {
		tmp = (x_m / (z + 1.0)) * ((y_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 * x_m) <= 1e-312:
		tmp = (y_m / z) / (z / x_m)
	elif (y_m * x_m) <= 2e+152:
		tmp = (y_m * x_m) / ((z + 1.0) * (z * z))
	else:
		tmp = (x_m / (z + 1.0)) * ((y_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 (Float64(y_m * x_m) <= 1e-312)
		tmp = Float64(Float64(y_m / z) / Float64(z / x_m));
	elseif (Float64(y_m * x_m) <= 2e+152)
		tmp = Float64(Float64(y_m * x_m) / Float64(Float64(z + 1.0) * Float64(z * z)));
	else
		tmp = Float64(Float64(x_m / Float64(z + 1.0)) * Float64(Float64(y_m / 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 * x_m) <= 1e-312)
		tmp = (y_m / z) / (z / x_m);
	elseif ((y_m * x_m) <= 2e+152)
		tmp = (y_m * x_m) / ((z + 1.0) * (z * z));
	else
		tmp = (x_m / (z + 1.0)) * ((y_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[N[(y$95$m * x$95$m), $MachinePrecision], 1e-312], N[(N[(y$95$m / z), $MachinePrecision] / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 2e+152], N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < 9.9999999999847e-313

   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. *-commutative 82.4%

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

      2. associate-*l/ 86.1%

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

      3. *-commutative 86.1%

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

      4. sqr-neg 86.1%

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

      5. *-commutative 86.1%

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

      6. distribute-rgt1-in 74.5%

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

      7. sqr-neg 74.5%

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

      8. fma-define 86.1%

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

      9. sqr-neg 86.1%

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

      10. cube-unmult 86.1%

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

   3. Simplified 86.1%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-define 74.5%

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

      2. associate-*r/ 71.6%

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

      3. *-commutative 71.6%

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

      4. cube-mult 71.6%

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

      5. distribute-rgt1-in 82.4%

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

      6. *-commutative 82.4%

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

      7. frac-times 89.0%

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

      8. associate-/r* 94.0%

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

      9. associate-*l/ 97.7%

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

   6. Applied egg-rr 97.7%

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

   7. Taylor expanded in z around 0 66.7%

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

      1. *-commutative 66.7%

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

      2. associate-/l* 75.6%

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

      3. associate-/r/ 75.6%

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

   9. Simplified 75.6%

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

       1. associate-/l* 77.7%

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

       2. associate-/r/ 75.9%

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

   11. Applied egg-rr 75.9%

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

       1. associate-/r/ 77.7%

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

   13. Applied egg-rr 77.7%

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

   if 9.9999999999847e-313 < (*.f64 x y) < 2.0000000000000001e152

   1. Initial program 97.4%

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

   if 2.0000000000000001e152 < (*.f64 x y)

   1. Initial program 68.2%

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

      1. sqr-neg 68.2%

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

      2. *-commutative 68.2%

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

      3. times-frac 83.2%

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

      4. sqr-neg 83.2%

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

   3. Simplified 83.2%

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

      1. associate-/r* 97.2%

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

      2. div-inv 97.1%

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

   6. Applied egg-rr 97.1%

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

      1. un-div-inv 97.2%

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

   8. Applied egg-rr 97.2%

      \[\leadsto \color{blue}{\frac{\frac{y}{z}}{z}} \cdot \frac{x}{z + 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq 10^{-312}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{x}}\\ \mathbf{elif}\;y \cdot x \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\frac{y \cdot x}{\left(z + 1\right) \cdot \left(z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z + 1} \cdot \frac{\frac{y}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{y\_m}}{z}\\ y\_s \cdot \left(x\_s \cdot \left(t\_0 \cdot \left(t\_0 \cdot \frac{x\_m}{z + 1}\right)\right)\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 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 (/ (sqrt y_m) z)))
   (* y_s (* x_s (* t_0 (* t_0 (/ x_m (+ z 1.0))))))))

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 = sqrt(y_m) / z;
	return y_s * (x_s * (t_0 * (t_0 * (x_m / (z + 1.0)))));
}

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
    t_0 = sqrt(y_m) / z
    code = y_s * (x_s * (t_0 * (t_0 * (x_m / (z + 1.0d0)))))
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 = Math.sqrt(y_m) / z;
	return y_s * (x_s * (t_0 * (t_0 * (x_m / (z + 1.0)))));
}

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 = math.sqrt(y_m) / z
	return y_s * (x_s * (t_0 * (t_0 * (x_m / (z + 1.0)))))

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(sqrt(y_m) / z)
	return Float64(y_s * Float64(x_s * Float64(t_0 * Float64(t_0 * Float64(x_m / Float64(z + 1.0))))))
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)
	t_0 = sqrt(y_m) / z;
	tmp = y_s * (x_s * (t_0 * (t_0 * (x_m / (z + 1.0)))));
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[Sqrt[y$95$m], $MachinePrecision] / z), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * N[(t$95$0 * N[(t$95$0 * N[(x$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 84.7%

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

   1. *-commutative 84.7%

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

   2. frac-times 88.8%

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

   3. add-sqr-sqrt 59.1%

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

   4. associate-*l* 59.1%

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

   5. sqrt-div 45.7%

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

   6. sqrt-prod 17.5%

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

   7. add-sqr-sqrt 26.9%

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

   8. sqrt-div 28.0%

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

   9. sqrt-prod 20.4%

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

   10. add-sqr-sqrt 50.4%

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

4. Applied egg-rr 50.4%

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

5. Final simplification 50.4%

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

Alternative 3: 94.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.46\right):\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{x\_m}{z}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 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 (or (<= z -1.0) (not (<= z 0.46)))
     (* (/ x_m z) (/ y_m (* z 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 ((z <= -1.0) || !(z <= 0.46)) {
		tmp = (x_m / z) * (y_m / (z * 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 ((z <= (-1.0d0)) .or. (.not. (z <= 0.46d0))) then
        tmp = (x_m / z) * (y_m / (z * 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 ((z <= -1.0) || !(z <= 0.46)) {
		tmp = (x_m / z) * (y_m / (z * 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 (z <= -1.0) or not (z <= 0.46):
		tmp = (x_m / z) * (y_m / (z * 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 ((z <= -1.0) || !(z <= 0.46))
		tmp = Float64(Float64(x_m / z) * Float64(y_m / Float64(z * z)));
	else
		tmp = Float64(y_m * Float64(Float64(x_m / 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 ((z <= -1.0) || ~((z <= 0.46)))
		tmp = (x_m / z) * (y_m / (z * 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[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 0.46]], $MachinePrecision]], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 0.46000000000000002 < z

   1. Initial program 82.2%

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

      1. sqr-neg 82.2%

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

      2. *-commutative 82.2%

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

      3. times-frac 88.3%

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

      4. sqr-neg 88.3%

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

   3. Simplified 88.3%

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

   5. Taylor expanded in z around inf 85.2%

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

   if -1 < z < 0.46000000000000002

   1. Initial program 87.3%

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

      1. *-commutative 87.3%

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

      2. associate-*l/ 89.3%

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

      3. *-commutative 89.3%

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

      4. sqr-neg 89.3%

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

      5. *-commutative 89.3%

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

      6. distribute-rgt1-in 89.3%

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

      7. sqr-neg 89.3%

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

      8. fma-define 89.3%

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

      9. sqr-neg 89.3%

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

      10. cube-unmult 89.3%

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

   3. Simplified 89.3%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-define 89.3%

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

      2. associate-*r/ 87.3%

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

      3. *-commutative 87.3%

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

      4. cube-mult 87.3%

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

      5. distribute-rgt1-in 87.3%

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

      6. *-commutative 87.3%

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

      7. frac-times 89.3%

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

      8. associate-/r* 93.1%

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

      9. associate-*l/ 97.4%

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

   6. Applied egg-rr 97.4%

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

   7. Taylor expanded in z around 0 85.7%

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

      1. *-commutative 85.7%

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

      2. associate-/l* 93.4%

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

      3. associate-/r/ 94.2%

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

   9. Simplified 94.2%

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

       1. associate-*r/ 94.2%

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

       2. associate-*l/ 93.4%

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

       3. *-un-lft-identity 93.4%

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

       4. times-frac 90.5%

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

   11. Applied egg-rr 90.5%

       \[\leadsto \color{blue}{\frac{y}{1} \cdot \frac{\frac{x}{z}}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.46\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.46\right):\\ \;\;\;\;\frac{\frac{y\_m}{z}}{z} \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{x\_m}{z}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 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 (or (<= z -1.0) (not (<= z 0.46)))
     (* (/ (/ y_m z) z) (/ x_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 ((z <= -1.0) || !(z <= 0.46)) {
		tmp = ((y_m / z) / z) * (x_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 ((z <= (-1.0d0)) .or. (.not. (z <= 0.46d0))) then
        tmp = ((y_m / z) / z) * (x_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 ((z <= -1.0) || !(z <= 0.46)) {
		tmp = ((y_m / z) / z) * (x_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 (z <= -1.0) or not (z <= 0.46):
		tmp = ((y_m / z) / z) * (x_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 ((z <= -1.0) || !(z <= 0.46))
		tmp = Float64(Float64(Float64(y_m / z) / z) * Float64(x_m / z));
	else
		tmp = Float64(y_m * Float64(Float64(x_m / 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 ((z <= -1.0) || ~((z <= 0.46)))
		tmp = ((y_m / z) / z) * (x_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[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 0.46]], $MachinePrecision]], N[(N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 0.46000000000000002 < z

   1. Initial program 82.2%

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

      1. *-commutative 82.2%

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

      2. associate-*l/ 83.0%

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

      3. *-commutative 83.0%

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

      4. sqr-neg 83.0%

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

      5. *-commutative 83.0%

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

      6. distribute-rgt1-in 60.5%

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

      7. sqr-neg 60.5%

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

      8. fma-define 83.0%

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

      9. sqr-neg 83.0%

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

      10. cube-unmult 83.0%

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

   3. Simplified 83.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-define 60.5%

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

      2. associate-*r/ 60.7%

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

      3. *-commutative 60.7%

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

      4. cube-mult 60.7%

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

      5. distribute-rgt1-in 82.2%

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

      6. *-commutative 82.2%

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

      7. frac-times 88.3%

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

      8. associate-/r* 94.3%

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

      9. associate-*l/ 96.2%

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

   6. Applied egg-rr 96.2%

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

   7. Taylor expanded in z around inf 93.0%

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

      1. associate-/l* 91.2%

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

      2. associate-/r/ 91.3%

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

   9. Applied egg-rr 91.3%

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

   if -1 < z < 0.46000000000000002

   1. Initial program 87.3%

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

      1. *-commutative 87.3%

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

      2. associate-*l/ 89.3%

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

      3. *-commutative 89.3%

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

      4. sqr-neg 89.3%

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

      5. *-commutative 89.3%

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

      6. distribute-rgt1-in 89.3%

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

      7. sqr-neg 89.3%

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

      8. fma-define 89.3%

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

      9. sqr-neg 89.3%

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

      10. cube-unmult 89.3%

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

   3. Simplified 89.3%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-define 89.3%

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

      2. associate-*r/ 87.3%

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

      3. *-commutative 87.3%

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

      4. cube-mult 87.3%

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

      5. distribute-rgt1-in 87.3%

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

      6. *-commutative 87.3%

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

      7. frac-times 89.3%

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

      8. associate-/r* 93.1%

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

      9. associate-*l/ 97.4%

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

   6. Applied egg-rr 97.4%

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

   7. Taylor expanded in z around 0 85.7%

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

      1. *-commutative 85.7%

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

      2. associate-/l* 93.4%

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

      3. associate-/r/ 94.2%

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

   9. Simplified 94.2%

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

       1. associate-*r/ 94.2%

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

       2. associate-*l/ 93.4%

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

       3. *-un-lft-identity 93.4%

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

       4. times-frac 90.5%

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

   11. Applied egg-rr 90.5%

       \[\leadsto \color{blue}{\frac{y}{1} \cdot \frac{\frac{x}{z}}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.46\right):\\ \;\;\;\;\frac{\frac{y}{z}}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.46\right):\\ \;\;\;\;\frac{\frac{y\_m}{z}}{z} \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(\frac{x\_m}{z} - x\_m\right)}{z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 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 (or (<= z -1.0) (not (<= z 0.46)))
     (* (/ (/ y_m z) z) (/ x_m z))
     (/ (* y_m (- (/ x_m z) x_m)) 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 ((z <= -1.0) || !(z <= 0.46)) {
		tmp = ((y_m / z) / z) * (x_m / z);
	} else {
		tmp = (y_m * ((x_m / z) - x_m)) / 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 ((z <= (-1.0d0)) .or. (.not. (z <= 0.46d0))) then
        tmp = ((y_m / z) / z) * (x_m / z)
    else
        tmp = (y_m * ((x_m / z) - x_m)) / 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 ((z <= -1.0) || !(z <= 0.46)) {
		tmp = ((y_m / z) / z) * (x_m / z);
	} else {
		tmp = (y_m * ((x_m / z) - x_m)) / 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 (z <= -1.0) or not (z <= 0.46):
		tmp = ((y_m / z) / z) * (x_m / z)
	else:
		tmp = (y_m * ((x_m / z) - x_m)) / 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 ((z <= -1.0) || !(z <= 0.46))
		tmp = Float64(Float64(Float64(y_m / z) / z) * Float64(x_m / z));
	else
		tmp = Float64(Float64(y_m * Float64(Float64(x_m / z) - x_m)) / 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 ((z <= -1.0) || ~((z <= 0.46)))
		tmp = ((y_m / z) / z) * (x_m / z);
	else
		tmp = (y_m * ((x_m / z) - x_m)) / 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[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 0.46]], $MachinePrecision]], N[(N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 0.46000000000000002 < z

   1. Initial program 82.2%

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

      1. *-commutative 82.2%

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

      2. associate-*l/ 83.0%

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

      3. *-commutative 83.0%

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

      4. sqr-neg 83.0%

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

      5. *-commutative 83.0%

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

      6. distribute-rgt1-in 60.5%

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

      7. sqr-neg 60.5%

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

      8. fma-define 83.0%

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

      9. sqr-neg 83.0%

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

      10. cube-unmult 83.0%

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

   3. Simplified 83.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-define 60.5%

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

      2. associate-*r/ 60.7%

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

      3. *-commutative 60.7%

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

      4. cube-mult 60.7%

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

      5. distribute-rgt1-in 82.2%

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

      6. *-commutative 82.2%

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

      7. frac-times 88.3%

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

      8. associate-/r* 94.3%

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

      9. associate-*l/ 96.2%

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

   6. Applied egg-rr 96.2%

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

   7. Taylor expanded in z around inf 93.0%

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

      1. associate-/l* 91.2%

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

      2. associate-/r/ 91.3%

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

   9. Applied egg-rr 91.3%

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

   if -1 < z < 0.46000000000000002

   1. Initial program 87.3%

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

      1. *-commutative 87.3%

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

      2. associate-*l/ 89.3%

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

      3. *-commutative 89.3%

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

      4. sqr-neg 89.3%

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

      5. *-commutative 89.3%

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

      6. distribute-rgt1-in 89.3%

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

      7. sqr-neg 89.3%

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

      8. fma-define 89.3%

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

      9. sqr-neg 89.3%

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

      10. cube-unmult 89.3%

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

   3. Simplified 89.3%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-define 89.3%

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

      2. associate-*r/ 87.3%

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

      3. *-commutative 87.3%

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

      4. cube-mult 87.3%

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

      5. distribute-rgt1-in 87.3%

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

      6. *-commutative 87.3%

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

      7. frac-times 89.3%

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

      8. associate-/r* 93.1%

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

      9. associate-*l/ 97.4%

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

   6. Applied egg-rr 97.4%

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

      1. clear-num 97.3%

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

      2. frac-times 97.6%

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

      3. *-un-lft-identity 97.6%

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

   8. Applied egg-rr 97.6%

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

   9. Taylor expanded in z around 0 79.8%

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

       1. associate-*r* 79.8%

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

       2. neg-mul-1 79.8%

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

       3. associate-*l/ 87.5%

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

       4. distribute-rgt-out 94.7%

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

       5. +-commutative 94.7%

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

       6. unsub-neg 94.7%

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

   11. Simplified 94.7%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.46\right):\\ \;\;\;\;\frac{\frac{y}{z}}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\frac{x}{z} - x\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{z} \cdot \frac{x\_m}{z}\\ \mathbf{elif}\;z \leq 0.46:\\ \;\;\;\;\frac{y\_m \cdot \left(\frac{x\_m}{z} - x\_m\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m \cdot \frac{y\_m}{z}}{z}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 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 (<= z -1.0)
     (* (/ (/ y_m z) z) (/ x_m z))
     (if (<= z 0.46)
       (/ (* y_m (- (/ x_m z) x_m)) z)
       (/ (/ (* x_m (/ y_m z)) 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 (z <= -1.0) {
		tmp = ((y_m / z) / z) * (x_m / z);
	} else if (z <= 0.46) {
		tmp = (y_m * ((x_m / z) - x_m)) / z;
	} else {
		tmp = ((x_m * (y_m / z)) / 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 (z <= (-1.0d0)) then
        tmp = ((y_m / z) / z) * (x_m / z)
    else if (z <= 0.46d0) then
        tmp = (y_m * ((x_m / z) - x_m)) / z
    else
        tmp = ((x_m * (y_m / z)) / 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 (z <= -1.0) {
		tmp = ((y_m / z) / z) * (x_m / z);
	} else if (z <= 0.46) {
		tmp = (y_m * ((x_m / z) - x_m)) / z;
	} else {
		tmp = ((x_m * (y_m / z)) / 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 z <= -1.0:
		tmp = ((y_m / z) / z) * (x_m / z)
	elif z <= 0.46:
		tmp = (y_m * ((x_m / z) - x_m)) / z
	else:
		tmp = ((x_m * (y_m / z)) / 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 (z <= -1.0)
		tmp = Float64(Float64(Float64(y_m / z) / z) * Float64(x_m / z));
	elseif (z <= 0.46)
		tmp = Float64(Float64(y_m * Float64(Float64(x_m / z) - x_m)) / z);
	else
		tmp = Float64(Float64(Float64(x_m * Float64(y_m / z)) / 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 (z <= -1.0)
		tmp = ((y_m / z) / z) * (x_m / z);
	elseif (z <= 0.46)
		tmp = (y_m * ((x_m / z) - x_m)) / z;
	else
		tmp = ((x_m * (y_m / z)) / 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[z, -1.0], N[(N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.46], N[(N[(y$95$m * N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(x$95$m * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 81.5%

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

      1. *-commutative 81.5%

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

      2. associate-*l/ 82.6%

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

      3. *-commutative 82.6%

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

      4. sqr-neg 82.6%

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

      5. *-commutative 82.6%

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

      6. distribute-rgt1-in 47.5%

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

      7. sqr-neg 47.5%

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

      8. fma-define 82.6%

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

      9. sqr-neg 82.6%

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

      10. cube-unmult 82.6%

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

   3. Simplified 82.6%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-define 47.5%

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

      2. associate-*r/ 47.9%

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

      3. *-commutative 47.9%

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

      4. cube-mult 47.9%

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

      5. distribute-rgt1-in 81.5%

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

      6. *-commutative 81.5%

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

      7. frac-times 89.7%

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

      8. associate-/r* 96.3%

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

      9. associate-*l/ 96.5%

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

   6. Applied egg-rr 96.5%

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

   7. Taylor expanded in z around inf 91.5%

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

      1. associate-/l* 90.8%

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

      2. associate-/r/ 91.6%

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

   9. Applied egg-rr 91.6%

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

   if -1 < z < 0.46000000000000002

   1. Initial program 87.3%

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

      1. *-commutative 87.3%

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

      2. associate-*l/ 89.3%

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

      3. *-commutative 89.3%

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

      4. sqr-neg 89.3%

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

      5. *-commutative 89.3%

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

      6. distribute-rgt1-in 89.3%

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

      7. sqr-neg 89.3%

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

      8. fma-define 89.3%

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

      9. sqr-neg 89.3%

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

      10. cube-unmult 89.3%

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

   3. Simplified 89.3%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-define 89.3%

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

      2. associate-*r/ 87.3%

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

      3. *-commutative 87.3%

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

      4. cube-mult 87.3%

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

      5. distribute-rgt1-in 87.3%

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

      6. *-commutative 87.3%

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

      7. frac-times 89.3%

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

      8. associate-/r* 93.1%

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

      9. associate-*l/ 97.4%

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

   6. Applied egg-rr 97.4%

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

      1. clear-num 97.3%

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

      2. frac-times 97.6%

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

      3. *-un-lft-identity 97.6%

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

   8. Applied egg-rr 97.6%

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

   9. Taylor expanded in z around 0 79.8%

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

       1. associate-*r* 79.8%

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

       2. neg-mul-1 79.8%

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

       3. associate-*l/ 87.5%

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

       4. distribute-rgt-out 94.7%

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

       5. +-commutative 94.7%

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

       6. unsub-neg 94.7%

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

   11. Simplified 94.7%

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

   if 0.46000000000000002 < z

   1. Initial program 83.5%

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

      1. *-commutative 83.5%

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

      2. associate-*l/ 83.8%

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

      3. *-commutative 83.8%

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

      4. sqr-neg 83.8%

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

      5. *-commutative 83.8%

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

      6. distribute-rgt1-in 83.8%

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

      7. sqr-neg 83.8%

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

      8. fma-define 83.8%

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

      9. sqr-neg 83.8%

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

      10. cube-unmult 83.8%

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

   3. Simplified 83.8%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-define 83.8%

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

      2. associate-*r/ 83.5%

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

      3. *-commutative 83.5%

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

      4. cube-mult 83.5%

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

      5. distribute-rgt1-in 83.5%

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

      6. *-commutative 83.5%

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

      7. frac-times 85.9%

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

      8. associate-/r* 90.8%

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

      9. associate-*l/ 95.7%

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

   6. Applied egg-rr 95.7%

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

   7. Taylor expanded in z around inf 95.5%

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

      1. *-commutative 95.5%

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

      2. associate-*l/ 97.5%

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

   9. Applied egg-rr 97.5%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{y}{z}}{z} \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 0.46:\\ \;\;\;\;\frac{y \cdot \left(\frac{x}{z} - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \frac{y}{z}}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{-49}:\\ \;\;\;\;\frac{y\_m \cdot \frac{x\_m}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{z \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 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 (<= x_m 1e-49) (/ (* y_m (/ x_m z)) z) (* x_m (/ y_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 (x_m <= 1e-49) {
		tmp = (y_m * (x_m / z)) / z;
	} else {
		tmp = x_m * (y_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 (x_m <= 1d-49) then
        tmp = (y_m * (x_m / z)) / z
    else
        tmp = x_m * (y_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 (x_m <= 1e-49) {
		tmp = (y_m * (x_m / z)) / z;
	} else {
		tmp = x_m * (y_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 x_m <= 1e-49:
		tmp = (y_m * (x_m / z)) / z
	else:
		tmp = x_m * (y_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 (x_m <= 1e-49)
		tmp = Float64(Float64(y_m * Float64(x_m / z)) / z);
	else
		tmp = Float64(x_m * Float64(y_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 (x_m <= 1e-49)
		tmp = (y_m * (x_m / z)) / z;
	else
		tmp = x_m * (y_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[x$95$m, 1e-49], N[(N[(y$95$m * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x$95$m * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999936e-50

   1. Initial program 85.3%

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

      1. sqr-neg 85.3%

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

      2. *-commutative 85.3%

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

      3. times-frac 88.8%

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

      4. sqr-neg 88.8%

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

   3. Simplified 88.8%

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

   5. Taylor expanded in z around 0 73.4%

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

      1. associate-*l/ 70.9%

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

      2. associate-/r* 68.5%

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

   7. Applied egg-rr 68.5%

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

      1. associate-*r/ 76.3%

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

      2. *-commutative 76.3%

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

   9. Applied egg-rr 76.3%

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

   if 9.99999999999999936e-50 < x

   1. Initial program 83.1%

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

      1. sqr-neg 83.1%

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

      2. *-commutative 83.1%

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

      3. times-frac 88.7%

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

      4. sqr-neg 88.7%

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

   3. Simplified 88.7%

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

   5. Taylor expanded in z around 0 64.8%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-49}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 7 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{y\_m}{\frac{z}{x\_m}}}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{z \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 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 (<= x_m 7e-16) (/ (/ y_m (/ z x_m)) z) (* x_m (/ y_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 (x_m <= 7e-16) {
		tmp = (y_m / (z / x_m)) / z;
	} else {
		tmp = x_m * (y_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 (x_m <= 7d-16) then
        tmp = (y_m / (z / x_m)) / z
    else
        tmp = x_m * (y_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 (x_m <= 7e-16) {
		tmp = (y_m / (z / x_m)) / z;
	} else {
		tmp = x_m * (y_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 x_m <= 7e-16:
		tmp = (y_m / (z / x_m)) / z
	else:
		tmp = x_m * (y_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 (x_m <= 7e-16)
		tmp = Float64(Float64(y_m / Float64(z / x_m)) / z);
	else
		tmp = Float64(x_m * Float64(y_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 (x_m <= 7e-16)
		tmp = (y_m / (z / x_m)) / z;
	else
		tmp = x_m * (y_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[x$95$m, 7e-16], N[(N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x$95$m * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 7.00000000000000035e-16

   1. Initial program 85.7%

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

      1. *-commutative 85.7%

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

      2. associate-*l/ 86.7%

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

      3. *-commutative 86.7%

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

      4. sqr-neg 86.7%

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

      5. *-commutative 86.7%

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

      6. distribute-rgt1-in 75.2%

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

      7. sqr-neg 75.2%

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

      8. fma-define 86.7%

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

      9. sqr-neg 86.7%

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

      10. cube-unmult 86.7%

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

   3. Simplified 86.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-define 75.2%

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

      2. associate-*r/ 74.3%

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

      3. *-commutative 74.3%

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

      4. cube-mult 74.3%

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

      5. distribute-rgt1-in 85.7%

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

      6. *-commutative 85.7%

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

      7. frac-times 89.1%

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

      8. associate-/r* 93.8%

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

      9. associate-*l/ 96.3%

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

   6. Applied egg-rr 96.3%

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

   7. Taylor expanded in z around 0 68.1%

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

      1. *-commutative 68.1%

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

      2. associate-/l* 75.7%

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

      3. associate-/r/ 74.6%

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

   9. Simplified 74.6%

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

       1. associate-/r/ 75.7%

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

   11. Applied egg-rr 75.7%

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

   if 7.00000000000000035e-16 < x

   1. Initial program 81.5%

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

      1. sqr-neg 81.5%

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

      2. *-commutative 81.5%

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

      3. times-frac 87.6%

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

      4. sqr-neg 87.6%

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

   3. Simplified 87.6%

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

   5. Taylor expanded in z around 0 65.6%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.9% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 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) (/ (/ x_m (+ z 1.0)) 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 / z) * ((x_m / (z + 1.0)) / 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 / z) * ((x_m / (z + 1.0d0)) / 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 / z) * ((x_m / (z + 1.0)) / 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 / z) * ((x_m / (z + 1.0)) / 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(Float64(y_m / z) * Float64(Float64(x_m / Float64(z + 1.0)) / 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 / z) * ((x_m / (z + 1.0)) / 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[(N[(y$95$m / z), $MachinePrecision] * N[(N[(x$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.7%

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

   1. *-commutative 84.7%

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

   2. frac-times 88.8%

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

   3. associate-*l/ 87.9%

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

   4. times-frac 96.2%

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

4. Applied egg-rr 96.2%

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

5. Final simplification 96.2%

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

Alternative 10: 96.5% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 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 1.0)) (/ y_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 * (((x_m / (z + 1.0)) * (y_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 * (((x_m / (z + 1.0d0)) * (y_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 * (((x_m / (z + 1.0)) * (y_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 * (((x_m / (z + 1.0)) * (y_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(Float64(Float64(x_m / Float64(z + 1.0)) * Float64(y_m / 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 * (((x_m / (z + 1.0)) * (y_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[(N[(N[(x$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.7%

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

   1. *-commutative 84.7%

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

   2. associate-*l/ 86.1%

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

   3. *-commutative 86.1%

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

   4. sqr-neg 86.1%

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

   5. *-commutative 86.1%

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

   6. distribute-rgt1-in 74.6%

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

   7. sqr-neg 74.6%

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

   8. fma-define 86.1%

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

   9. sqr-neg 86.1%

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

   10. cube-unmult 86.1%

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

3. Simplified 86.1%

   \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-define 74.6%

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

   2. associate-*r/ 73.7%

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

   3. *-commutative 73.7%

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

   4. cube-mult 73.7%

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

   5. distribute-rgt1-in 84.7%

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

   6. *-commutative 84.7%

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

   7. frac-times 88.8%

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

   8. associate-/r* 93.7%

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

   9. associate-*l/ 96.8%

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

6. Applied egg-rr 96.8%

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

7. Final simplification 96.8%

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

Alternative 11: 97.3% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 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 1.0) (/ z y_m))) 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 * ((x_m / ((z + 1.0) * (z / y_m))) / 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 * ((x_m / ((z + 1.0d0) * (z / y_m))) / 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 * ((x_m / ((z + 1.0) * (z / y_m))) / 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 * ((x_m / ((z + 1.0) * (z / y_m))) / 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(Float64(x_m / Float64(Float64(z + 1.0) * Float64(z / y_m))) / 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 * ((x_m / ((z + 1.0) * (z / y_m))) / 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[(N[(x$95$m / N[(N[(z + 1.0), $MachinePrecision] * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.7%

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

   1. *-commutative 84.7%

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

   2. associate-*l/ 86.1%

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

   3. *-commutative 86.1%

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

   4. sqr-neg 86.1%

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

   5. *-commutative 86.1%

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

   6. distribute-rgt1-in 74.6%

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

   7. sqr-neg 74.6%

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

   8. fma-define 86.1%

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

   9. sqr-neg 86.1%

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

   10. cube-unmult 86.1%

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

3. Simplified 86.1%

   \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-define 74.6%

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

   2. associate-*r/ 73.7%

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

   3. *-commutative 73.7%

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

   4. cube-mult 73.7%

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

   5. distribute-rgt1-in 84.7%

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

   6. *-commutative 84.7%

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

   7. frac-times 88.8%

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

   8. associate-/r* 93.7%

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

   9. associate-*l/ 96.8%

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

6. Applied egg-rr 96.8%

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

   1. clear-num 96.5%

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

   2. frac-times 96.3%

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

   3. *-un-lft-identity 96.3%

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

8. Applied egg-rr 96.3%

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

9. Final simplification 96.3%

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

Alternative 12: 70.3% accurate, 1.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 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 (/ y_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 * (x_m * (y_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 * (x_m * (y_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 * (x_m * (y_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 * (x_m * (y_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(x_m * Float64(y_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 * (x_m * (y_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[(x$95$m * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.7%

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

   1. sqr-neg 84.7%

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

   2. *-commutative 84.7%

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

   3. times-frac 88.8%

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

   4. sqr-neg 88.8%

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

3. Simplified 88.8%

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

5. Taylor expanded in z around 0 71.1%

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

6. Final simplification 71.1%

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

Alternative 13: 80.5% accurate, 1.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 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(Float64(x_m / 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[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.7%

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

   1. *-commutative 84.7%

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

   2. associate-*l/ 86.1%

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

   3. *-commutative 86.1%

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

   4. sqr-neg 86.1%

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

   5. *-commutative 86.1%

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

   6. distribute-rgt1-in 74.6%

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

   7. sqr-neg 74.6%

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

   8. fma-define 86.1%

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

   9. sqr-neg 86.1%

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

   10. cube-unmult 86.1%

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

3. Simplified 86.1%

   \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-define 74.6%

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

   2. associate-*r/ 73.7%

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

   3. *-commutative 73.7%

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

   4. cube-mult 73.7%

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

   5. distribute-rgt1-in 84.7%

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

   6. *-commutative 84.7%

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

   7. frac-times 88.8%

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

   8. associate-/r* 93.7%

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

   9. associate-*l/ 96.8%

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

6. Applied egg-rr 96.8%

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

7. Taylor expanded in z around 0 65.6%

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

   1. *-commutative 65.6%

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

   2. associate-/l* 70.6%

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

   3. associate-/r/ 70.8%

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

9. Simplified 70.8%

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

    1. associate-*r/ 72.0%

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

    2. associate-*l/ 70.5%

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

    3. *-un-lft-identity 70.5%

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

    4. times-frac 72.0%

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

11. Applied egg-rr 72.0%

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

12. Final simplification 72.0%

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

Developer target: 97.0% 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 2024039 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1.0 z)) x) z))

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