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

Percentage Accurate: 82.7% → 97.7%

Time: 13.5s

Alternatives: 13

Speedup: 1.0×

• 25.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.7% 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: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.9%

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

   1. sqr-neg 83.9%

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

   2. times-frac 87.7%

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

   3. sqr-neg 87.7%

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

3. Simplified 87.7%

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

   1. *-commutative 87.7%

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

   2. associate-/r* 93.1%

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

   3. associate-*r/ 96.9%

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

6. Applied egg-rr 96.9%

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

7. Final simplification 96.9%

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

Alternative 2: 91.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -18000000000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{x}{z \cdot z} \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (or (<= z -18000000000000.0) (not (<= z 1.0)))
    (* (/ x (* z z)) (/ y_m z))
    (* y_m (/ (/ x z) z)))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z <= -18000000000000.0) || !(z <= 1.0)) {
		tmp = (x / (z * z)) * (y_m / z);
	} else {
		tmp = y_m * ((x / z) / z);
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-18000000000000.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = (x / (z * z)) * (y_m / z)
    else
        tmp = y_m * ((x / z) / z)
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z <= -18000000000000.0) || !(z <= 1.0)) {
		tmp = (x / (z * z)) * (y_m / z);
	} else {
		tmp = y_m * ((x / z) / z);
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if (z <= -18000000000000.0) or not (z <= 1.0):
		tmp = (x / (z * z)) * (y_m / z)
	else:
		tmp = y_m * ((x / z) / z)
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if ((z <= -18000000000000.0) || !(z <= 1.0))
		tmp = Float64(Float64(x / Float64(z * z)) * Float64(y_m / z));
	else
		tmp = Float64(y_m * Float64(Float64(x / z) / z));
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if ((z <= -18000000000000.0) || ~((z <= 1.0)))
		tmp = (x / (z * z)) * (y_m / z);
	else
		tmp = y_m * ((x / z) / z);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[Or[LessEqual[z, -18000000000000.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.8e13 or 1 < z

   1. Initial program 81.7%

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

      1. sqr-neg 81.7%

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

      2. times-frac 91.6%

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

      3. sqr-neg 91.6%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in z around inf 90.1%

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

   if -1.8e13 < z < 1

   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. sqr-neg 85.7%

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

      2. times-frac 84.4%

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

      3. sqr-neg 84.4%

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

   3. Simplified 84.4%

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

      1. associate-/r* 89.9%

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

      2. div-inv 89.8%

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

   6. Applied egg-rr 89.8%

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

      1. associate-*r/ 89.9%

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

      2. *-rgt-identity 89.9%

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

   8. Simplified 89.9%

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

   9. Taylor expanded in z around 0 87.5%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -18000000000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{x}{z \cdot z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -18000000000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{y\_m}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (or (<= z -18000000000000.0) (not (<= z 1.0)))
    (* (/ x z) (/ y_m (* z z)))
    (* y_m (/ (/ x z) z)))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z <= -18000000000000.0) || !(z <= 1.0)) {
		tmp = (x / z) * (y_m / (z * z));
	} else {
		tmp = y_m * ((x / z) / z);
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-18000000000000.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = (x / z) * (y_m / (z * z))
    else
        tmp = y_m * ((x / z) / z)
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z <= -18000000000000.0) || !(z <= 1.0)) {
		tmp = (x / z) * (y_m / (z * z));
	} else {
		tmp = y_m * ((x / z) / z);
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if (z <= -18000000000000.0) or not (z <= 1.0):
		tmp = (x / z) * (y_m / (z * z))
	else:
		tmp = y_m * ((x / z) / z)
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if ((z <= -18000000000000.0) || !(z <= 1.0))
		tmp = Float64(Float64(x / z) * Float64(y_m / Float64(z * z)));
	else
		tmp = Float64(y_m * Float64(Float64(x / z) / z));
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if ((z <= -18000000000000.0) || ~((z <= 1.0)))
		tmp = (x / z) * (y_m / (z * z));
	else
		tmp = y_m * ((x / z) / z);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[Or[LessEqual[z, -18000000000000.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.8e13 or 1 < z

   1. Initial program 81.7%

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

      1. *-commutative 81.7%

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

      2. sqr-neg 81.7%

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

      3. times-frac 92.5%

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

      4. sqr-neg 92.5%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in z around inf 90.9%

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

   if -1.8e13 < z < 1

   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. sqr-neg 85.7%

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

      2. times-frac 84.4%

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

      3. sqr-neg 84.4%

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

   3. Simplified 84.4%

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

      1. associate-/r* 89.9%

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

      2. div-inv 89.8%

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

   6. Applied egg-rr 89.8%

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

      1. associate-*r/ 89.9%

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

      2. *-rgt-identity 89.9%

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

   8. Simplified 89.9%

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

   9. Taylor expanded in z around 0 87.5%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -18000000000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.75\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{y\_m}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(\frac{x}{z} - x\right)}{z}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (or (<= z -1.0) (not (<= z 0.75)))
    (* (/ x z) (/ y_m (* z z)))
    (/ (* y_m (- (/ x z) x)) z))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 0.75)) {
		tmp = (x / z) * (y_m / (z * z));
	} else {
		tmp = (y_m * ((x / z) - x)) / z;
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 0.75d0))) then
        tmp = (x / z) * (y_m / (z * z))
    else
        tmp = (y_m * ((x / z) - x)) / z
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 0.75)) {
		tmp = (x / z) * (y_m / (z * z));
	} else {
		tmp = (y_m * ((x / z) - x)) / z;
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 0.75):
		tmp = (x / z) * (y_m / (z * z))
	else:
		tmp = (y_m * ((x / z) - x)) / z
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 0.75))
		tmp = Float64(Float64(x / z) * Float64(y_m / Float64(z * z)));
	else
		tmp = Float64(Float64(y_m * Float64(Float64(x / z) - x)) / z);
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 0.75)))
		tmp = (x / z) * (y_m / (z * z));
	else
		tmp = (y_m * ((x / z) - x)) / z;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 0.75]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 0.75 < z

   1. Initial program 81.9%

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

      1. *-commutative 81.9%

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

      2. sqr-neg 81.9%

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

      3. times-frac 92.5%

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

      4. sqr-neg 92.5%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in z around inf 91.0%

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

   if -1 < z < 0.75

   1. Initial program 85.6%

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

      1. sqr-neg 85.6%

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

      2. times-frac 84.3%

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

      3. sqr-neg 84.3%

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

   3. Simplified 84.3%

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

      1. *-commutative 84.3%

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

      2. associate-/r* 89.8%

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

      3. associate-*r/ 96.3%

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

   6. Applied egg-rr 96.3%

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

      1. frac-times 92.3%

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

      2. associate-/l* 96.7%

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

      3. associate-*l/ 96.7%

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

      4. *-commutative 96.7%

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

   8. Applied egg-rr 96.7%

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

   9. Taylor expanded in z around 0 83.6%

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

       1. +-commutative 83.6%

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

       2. associate-*l/ 87.6%

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

       3. associate-*r* 87.6%

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

       4. neg-mul-1 87.6%

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

       5. distribute-rgt-out 94.8%

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

       6. unsub-neg 94.8%

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

   11. Simplified 94.8%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.75\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\frac{x}{z} - x\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.75\right):\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y\_m}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(\frac{x}{z} - x\right)}{z}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (or (<= z -1.0) (not (<= z 0.75)))
    (/ (* (/ x z) (/ y_m z)) z)
    (/ (* y_m (- (/ x z) x)) z))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 0.75)) {
		tmp = ((x / z) * (y_m / z)) / z;
	} else {
		tmp = (y_m * ((x / z) - x)) / z;
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 0.75d0))) then
        tmp = ((x / z) * (y_m / z)) / z
    else
        tmp = (y_m * ((x / z) - x)) / z
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 0.75)) {
		tmp = ((x / z) * (y_m / z)) / z;
	} else {
		tmp = (y_m * ((x / z) - x)) / z;
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 0.75):
		tmp = ((x / z) * (y_m / z)) / z
	else:
		tmp = (y_m * ((x / z) - x)) / z
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 0.75))
		tmp = Float64(Float64(Float64(x / z) * Float64(y_m / z)) / z);
	else
		tmp = Float64(Float64(y_m * Float64(Float64(x / z) - x)) / z);
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 0.75)))
		tmp = ((x / z) * (y_m / z)) / z;
	else
		tmp = (y_m * ((x / z) - x)) / z;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 0.75]], $MachinePrecision]], N[(N[(N[(x / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m * N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 0.75 < z

   1. Initial program 81.9%

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

      1. sqr-neg 81.9%

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

      2. times-frac 91.7%

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

      3. sqr-neg 91.7%

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

   3. Simplified 91.7%

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

   5. Taylor expanded in z around inf 90.2%

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

      1. *-commutative 90.2%

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

      2. associate-/r* 95.5%

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

      3. associate-*r/ 96.2%

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

   7. Applied egg-rr 96.2%

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

   if -1 < z < 0.75

   1. Initial program 85.6%

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

      1. sqr-neg 85.6%

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

      2. times-frac 84.3%

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

      3. sqr-neg 84.3%

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

   3. Simplified 84.3%

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

      1. *-commutative 84.3%

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

      2. associate-/r* 89.8%

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

      3. associate-*r/ 96.3%

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

   6. Applied egg-rr 96.3%

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

      1. frac-times 92.3%

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

      2. associate-/l* 96.7%

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

      3. associate-*l/ 96.7%

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

      4. *-commutative 96.7%

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

   8. Applied egg-rr 96.7%

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

   9. Taylor expanded in z around 0 83.6%

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

       1. +-commutative 83.6%

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

       2. associate-*l/ 87.6%

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

       3. associate-*r* 87.6%

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

       4. neg-mul-1 87.6%

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

       5. distribute-rgt-out 94.8%

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

       6. unsub-neg 94.8%

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

   11. Simplified 94.8%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.75\right):\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\frac{x}{z} - x\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y\_m}{z}}{z}\\ \mathbf{elif}\;z \leq 0.76:\\ \;\;\;\;\frac{y\_m \cdot \left(\frac{x}{z} - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{\frac{z}{y\_m}}}{z}}{z}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= z -1.0)
    (/ (* (/ x z) (/ y_m z)) z)
    (if (<= z 0.76) (/ (* y_m (- (/ x z) x)) z) (/ (/ (/ x (/ z y_m)) z) z)))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = ((x / z) * (y_m / z)) / z;
	} else if (z <= 0.76) {
		tmp = (y_m * ((x / z) - x)) / z;
	} else {
		tmp = ((x / (z / y_m)) / z) / z;
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.0d0)) then
        tmp = ((x / z) * (y_m / z)) / z
    else if (z <= 0.76d0) then
        tmp = (y_m * ((x / z) - x)) / z
    else
        tmp = ((x / (z / y_m)) / z) / z
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = ((x / z) * (y_m / z)) / z;
	} else if (z <= 0.76) {
		tmp = (y_m * ((x / z) - x)) / z;
	} else {
		tmp = ((x / (z / y_m)) / z) / z;
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if z <= -1.0:
		tmp = ((x / z) * (y_m / z)) / z
	elif z <= 0.76:
		tmp = (y_m * ((x / z) - x)) / z
	else:
		tmp = ((x / (z / y_m)) / z) / z
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(Float64(Float64(x / z) * Float64(y_m / z)) / z);
	elseif (z <= 0.76)
		tmp = Float64(Float64(y_m * Float64(Float64(x / z) - x)) / z);
	else
		tmp = Float64(Float64(Float64(x / Float64(z / y_m)) / z) / z);
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = ((x / z) * (y_m / z)) / z;
	elseif (z <= 0.76)
		tmp = (y_m * ((x / z) - x)) / z;
	else
		tmp = ((x / (z / y_m)) / z) / z;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[z, -1.0], N[(N[(N[(x / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 0.76], N[(N[(y$95$m * N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(x / N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 80.8%

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

      1. sqr-neg 80.8%

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

      2. times-frac 96.7%

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

      3. sqr-neg 96.7%

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

   3. Simplified 96.7%

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

   5. Taylor expanded in z around inf 96.6%

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

      1. *-commutative 96.6%

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

      2. associate-/r* 98.1%

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

      3. associate-*r/ 98.2%

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

   7. Applied egg-rr 98.2%

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

   if -1 < z < 0.76000000000000001

   1. Initial program 85.6%

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

      1. sqr-neg 85.6%

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

      2. times-frac 84.3%

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

      3. sqr-neg 84.3%

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

   3. Simplified 84.3%

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

      1. *-commutative 84.3%

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

      2. associate-/r* 89.8%

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

      3. associate-*r/ 96.3%

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

   6. Applied egg-rr 96.3%

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

      1. frac-times 92.3%

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

      2. associate-/l* 96.7%

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

      3. associate-*l/ 96.7%

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

      4. *-commutative 96.7%

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

   8. Applied egg-rr 96.7%

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

   9. Taylor expanded in z around 0 83.6%

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

       1. +-commutative 83.6%

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

       2. associate-*l/ 87.6%

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

       3. associate-*r* 87.6%

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

       4. neg-mul-1 87.6%

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

       5. distribute-rgt-out 94.8%

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

       6. unsub-neg 94.8%

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

   11. Simplified 94.8%

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

   if 0.76000000000000001 < z

   1. Initial program 83.0%

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

      1. sqr-neg 83.0%

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

      2. times-frac 86.5%

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

      3. sqr-neg 86.5%

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

   3. Simplified 86.5%

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

   5. Taylor expanded in z around inf 83.5%

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

      1. *-commutative 83.5%

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

      2. associate-/r* 92.8%

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

      3. associate-*r/ 94.0%

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

   7. Applied egg-rr 94.0%

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

      1. associate-*l/ 96.3%

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

      2. associate-*r/ 81.0%

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

      3. *-commutative 81.0%

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

      4. associate-/l* 93.1%

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

   9. Simplified 93.1%

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

4. Final simplification 95.2%

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

Alternative 7: 95.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -6.1 \cdot 10^{-121}:\\ \;\;\;\;\frac{y\_m}{z + 1} \cdot \frac{x}{z \cdot z}\\ \mathbf{elif}\;z \leq 0.76:\\ \;\;\;\;\frac{y\_m \cdot \left(\frac{x}{z} - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{\frac{z}{y\_m}}}{z}}{z}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= z -6.1e-121)
    (* (/ y_m (+ z 1.0)) (/ x (* z z)))
    (if (<= z 0.76) (/ (* y_m (- (/ x z) x)) z) (/ (/ (/ x (/ z y_m)) z) z)))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= -6.1e-121) {
		tmp = (y_m / (z + 1.0)) * (x / (z * z));
	} else if (z <= 0.76) {
		tmp = (y_m * ((x / z) - x)) / z;
	} else {
		tmp = ((x / (z / y_m)) / z) / z;
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-6.1d-121)) then
        tmp = (y_m / (z + 1.0d0)) * (x / (z * z))
    else if (z <= 0.76d0) then
        tmp = (y_m * ((x / z) - x)) / z
    else
        tmp = ((x / (z / y_m)) / z) / z
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= -6.1e-121) {
		tmp = (y_m / (z + 1.0)) * (x / (z * z));
	} else if (z <= 0.76) {
		tmp = (y_m * ((x / z) - x)) / z;
	} else {
		tmp = ((x / (z / y_m)) / z) / z;
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if z <= -6.1e-121:
		tmp = (y_m / (z + 1.0)) * (x / (z * z))
	elif z <= 0.76:
		tmp = (y_m * ((x / z) - x)) / z
	else:
		tmp = ((x / (z / y_m)) / z) / z
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (z <= -6.1e-121)
		tmp = Float64(Float64(y_m / Float64(z + 1.0)) * Float64(x / Float64(z * z)));
	elseif (z <= 0.76)
		tmp = Float64(Float64(y_m * Float64(Float64(x / z) - x)) / z);
	else
		tmp = Float64(Float64(Float64(x / Float64(z / y_m)) / z) / z);
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (z <= -6.1e-121)
		tmp = (y_m / (z + 1.0)) * (x / (z * z));
	elseif (z <= 0.76)
		tmp = (y_m * ((x / z) - x)) / z;
	else
		tmp = ((x / (z / y_m)) / z) / z;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[z, -6.1e-121], N[(N[(y$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.76], N[(N[(y$95$m * N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(x / N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -6.09999999999999978e-121

   1. Initial program 85.1%

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

      1. sqr-neg 85.1%

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

      2. times-frac 91.6%

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

      3. sqr-neg 91.6%

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

   3. Simplified 91.6%

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

   if -6.09999999999999978e-121 < z < 0.76000000000000001

   1. Initial program 83.3%

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

      1. sqr-neg 83.3%

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

      2. times-frac 85.0%

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

      3. sqr-neg 85.0%

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

   3. Simplified 85.0%

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

      1. *-commutative 85.0%

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

      2. associate-/r* 92.1%

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

      3. associate-*r/ 97.9%

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

   6. Applied egg-rr 97.9%

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

      1. frac-times 91.9%

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

      2. associate-/l* 98.2%

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

      3. associate-*l/ 98.2%

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

      4. *-commutative 98.2%

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

   8. Applied egg-rr 98.2%

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

   9. Taylor expanded in z around 0 81.4%

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

       1. +-commutative 81.4%

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

       2. associate-*l/ 87.4%

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

       3. associate-*r* 87.4%

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

       4. neg-mul-1 87.4%

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

       5. distribute-rgt-out 96.7%

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

       6. unsub-neg 96.7%

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

   11. Simplified 96.7%

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

   if 0.76000000000000001 < z

   1. Initial program 83.0%

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

      1. sqr-neg 83.0%

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

      2. times-frac 86.5%

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

      3. sqr-neg 86.5%

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

   3. Simplified 86.5%

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

   5. Taylor expanded in z around inf 83.5%

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

      1. *-commutative 83.5%

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

      2. associate-/r* 92.8%

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

      3. associate-*r/ 94.0%

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

   7. Applied egg-rr 94.0%

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

      1. associate-*l/ 96.3%

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

      2. associate-*r/ 81.0%

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

      3. *-commutative 81.0%

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

      4. associate-/l* 93.1%

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

   9. Simplified 93.1%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.1 \cdot 10^{-121}:\\ \;\;\;\;\frac{y}{z + 1} \cdot \frac{x}{z \cdot z}\\ \mathbf{elif}\;z \leq 0.76:\\ \;\;\;\;\frac{y \cdot \left(\frac{x}{z} - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{\frac{z}{y}}}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-34}:\\ \;\;\;\;\frac{y\_m}{z \cdot z} \cdot \frac{x}{z + 1}\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{y\_m \cdot \left(\frac{x}{z} - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{\frac{z}{y\_m}}}{z}}{z}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= z -7.5e-34)
    (* (/ y_m (* z z)) (/ x (+ z 1.0)))
    (if (<= z 0.75) (/ (* y_m (- (/ x z) x)) z) (/ (/ (/ x (/ z y_m)) z) z)))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= -7.5e-34) {
		tmp = (y_m / (z * z)) * (x / (z + 1.0));
	} else if (z <= 0.75) {
		tmp = (y_m * ((x / z) - x)) / z;
	} else {
		tmp = ((x / (z / y_m)) / z) / z;
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-7.5d-34)) then
        tmp = (y_m / (z * z)) * (x / (z + 1.0d0))
    else if (z <= 0.75d0) then
        tmp = (y_m * ((x / z) - x)) / z
    else
        tmp = ((x / (z / y_m)) / z) / z
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= -7.5e-34) {
		tmp = (y_m / (z * z)) * (x / (z + 1.0));
	} else if (z <= 0.75) {
		tmp = (y_m * ((x / z) - x)) / z;
	} else {
		tmp = ((x / (z / y_m)) / z) / z;
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if z <= -7.5e-34:
		tmp = (y_m / (z * z)) * (x / (z + 1.0))
	elif z <= 0.75:
		tmp = (y_m * ((x / z) - x)) / z
	else:
		tmp = ((x / (z / y_m)) / z) / z
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (z <= -7.5e-34)
		tmp = Float64(Float64(y_m / Float64(z * z)) * Float64(x / Float64(z + 1.0)));
	elseif (z <= 0.75)
		tmp = Float64(Float64(y_m * Float64(Float64(x / z) - x)) / z);
	else
		tmp = Float64(Float64(Float64(x / Float64(z / y_m)) / z) / z);
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (z <= -7.5e-34)
		tmp = (y_m / (z * z)) * (x / (z + 1.0));
	elseif (z <= 0.75)
		tmp = (y_m * ((x / z) - x)) / z;
	else
		tmp = ((x / (z / y_m)) / z) / z;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[z, -7.5e-34], N[(N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(x / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.75], N[(N[(y$95$m * N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(x / N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -7.5000000000000004e-34

   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. sqr-neg 82.4%

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

      3. times-frac 98.3%

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

      4. sqr-neg 98.3%

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

   3. Simplified 98.3%

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

   if -7.5000000000000004e-34 < z < 0.75

   1. Initial program 85.2%

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

      1. sqr-neg 85.2%

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

      2. times-frac 83.0%

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

      3. sqr-neg 83.0%

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

   3. Simplified 83.0%

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

      1. *-commutative 83.0%

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

      2. associate-/r* 89.0%

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

      3. associate-*r/ 96.0%

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

   6. Applied egg-rr 96.0%

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

      1. frac-times 92.4%

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

      2. associate-/l* 96.4%

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

      3. associate-*l/ 96.4%

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

      4. *-commutative 96.4%

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

   8. Applied egg-rr 96.4%

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

   9. Taylor expanded in z around 0 83.5%

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

       1. +-commutative 83.5%

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

       2. associate-*l/ 87.1%

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

       3. associate-*r* 87.1%

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

       4. neg-mul-1 87.1%

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

       5. distribute-rgt-out 95.0%

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

       6. unsub-neg 95.0%

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

   11. Simplified 95.0%

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

   if 0.75 < z

   1. Initial program 83.0%

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

      1. sqr-neg 83.0%

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

      2. times-frac 86.5%

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

      3. sqr-neg 86.5%

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

   3. Simplified 86.5%

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

   5. Taylor expanded in z around inf 83.5%

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

      1. *-commutative 83.5%

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

      2. associate-/r* 92.8%

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

      3. associate-*r/ 94.0%

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

   7. Applied egg-rr 94.0%

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

      1. associate-*l/ 96.3%

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

      2. associate-*r/ 81.0%

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

      3. *-commutative 81.0%

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

      4. associate-/l* 93.1%

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

   9. Simplified 93.1%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-34}:\\ \;\;\;\;\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{y \cdot \left(\frac{x}{z} - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{\frac{z}{y}}}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.9%

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

   1. sqr-neg 83.9%

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

   2. times-frac 87.7%

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

   3. sqr-neg 87.7%

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

3. Simplified 87.7%

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

   1. associate-/r* 93.1%

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

   2. div-inv 93.1%

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

6. Applied egg-rr 93.1%

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

   1. associate-*r/ 93.1%

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

   2. *-rgt-identity 93.1%

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

8. Simplified 93.1%

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

9. Final simplification 93.1%

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

Alternative 10: 73.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.9%

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

   1. sqr-neg 83.9%

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

   2. times-frac 87.7%

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

   3. sqr-neg 87.7%

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

3. Simplified 87.7%

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

5. Taylor expanded in z around 0 72.9%

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

6. Final simplification 72.9%

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

Alternative 11: 76.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.9%

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

   1. sqr-neg 83.9%

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

   2. times-frac 87.7%

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

   3. sqr-neg 87.7%

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

3. Simplified 87.7%

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

   1. associate-/r* 93.1%

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

   2. div-inv 93.1%

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

6. Applied egg-rr 93.1%

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

   1. associate-*r/ 93.1%

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

   2. *-rgt-identity 93.1%

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

8. Simplified 93.1%

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

9. Taylor expanded in z around 0 74.3%

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

10. Final simplification 74.3%

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

Alternative 12: 28.8% accurate, 1.8× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z) :precision binary64 (* y_s (* x (/ (- y_m) z))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	return y_s * (x * (-y_m / z));
}

Fortran

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

Java

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

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	return y_s * (x * (-y_m / z))

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(x * Float64(Float64(-y_m) / z)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.9%

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

   1. sqr-neg 83.9%

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

   2. times-frac 87.7%

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

   3. sqr-neg 87.7%

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

3. Simplified 87.7%

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

   1. associate-/r* 93.1%

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

   2. div-inv 93.1%

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

6. Applied egg-rr 93.1%

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

   1. associate-*r/ 93.1%

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

   2. *-rgt-identity 93.1%

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

8. Simplified 93.1%

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

   1. associate-*l/ 96.9%

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

   2. *-commutative 96.9%

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

   3. frac-times 89.4%

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

   4. associate-/l* 94.6%

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

   5. associate-*l/ 97.0%

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

   6. associate-/r* 92.3%

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

   7. associate-/l/ 96.5%

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

   8. div-inv 96.3%

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

   9. clear-num 96.1%

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

   10. associate-*l/ 96.2%

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

   11. *-un-lft-identity 96.2%

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

   12. associate-/r* 96.2%

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

   13. clear-num 96.2%

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

10. Applied egg-rr 96.2%

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

11. Taylor expanded in z around 0 66.2%

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

    1. neg-mul-1 66.2%

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

    2. +-commutative 66.2%

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

    3. unsub-neg 66.2%

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

13. Simplified 66.2%

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

14. Taylor expanded in z around inf 23.1%

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

    1. mul-1-neg 23.1%

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

    2. *-commutative 23.1%

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

    3. associate-*l/ 25.6%

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

    4. *-commutative 25.6%

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

    5. distribute-rgt-neg-in 25.6%

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

    6. distribute-frac-neg 25.6%

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

16. Simplified 25.6%

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

17. Final simplification 25.6%

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

Alternative 13: 30.3% accurate, 1.8× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z) :precision binary64 (* y_s (* y_m (/ (- x) z))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	return y_s * (y_m * (-x / z));
}

Fortran

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

Java

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

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	return y_s * (y_m * (-x / z))

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(y_m * Float64(Float64(-x) / z)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.9%

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

   1. sqr-neg 83.9%

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

   2. times-frac 87.7%

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

   3. sqr-neg 87.7%

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

3. Simplified 87.7%

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

   1. associate-/r* 93.1%

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

   2. div-inv 93.1%

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

6. Applied egg-rr 93.1%

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

   1. associate-*r/ 93.1%

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

   2. *-rgt-identity 93.1%

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

8. Simplified 93.1%

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

   1. associate-*l/ 96.9%

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

   2. *-commutative 96.9%

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

   3. frac-times 89.4%

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

   4. associate-/l* 94.6%

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

   5. associate-*l/ 97.0%

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

   6. associate-/r* 92.3%

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

   7. associate-/l/ 96.5%

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

   8. div-inv 96.3%

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

   9. clear-num 96.1%

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

   10. associate-*l/ 96.2%

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

   11. *-un-lft-identity 96.2%

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

   12. associate-/r* 96.2%

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

   13. clear-num 96.2%

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

10. Applied egg-rr 96.2%

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

11. Taylor expanded in z around 0 66.2%

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

    1. neg-mul-1 66.2%

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

    2. +-commutative 66.2%

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

    3. unsub-neg 66.2%

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

13. Simplified 66.2%

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

14. Taylor expanded in z around inf 23.1%

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

    1. associate-*r/ 23.1%

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

    2. mul-1-neg 23.1%

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

    3. distribute-rgt-neg-out 23.1%

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

    4. associate-*l/ 28.8%

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

16. Simplified 28.8%

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

17. Final simplification 28.8%

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

Developer target: 96.1% 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 2024034 
(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))))