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

Percentage Accurate: 83.2% → 96.3%

Time: 11.8s

Alternatives: 9

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: 83.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \frac{y}{z} \cdot \frac{\frac{x}{z}}{z + 1} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* (/ y z) (/ (/ x z) (+ z 1.0))))

C

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

Fortran

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

Java

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

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return (y / z) * ((x / z) / (z + 1.0))

Julia

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

MATLAB

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

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(y / z), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.0%

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

   1. *-commutative 85.0%

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

   2. frac-times 87.9%

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

   3. associate-*l/ 87.3%

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

   4. times-frac 97.2%

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

4. Applied egg-rr 97.2%

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

5. Taylor expanded in x around 0 95.6%

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

   1. associate-/r* 97.2%

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

   2. +-commutative 97.2%

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

7. Simplified 97.2%

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

8. Final simplification 97.2%

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

Alternative 2: 94.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 4.2 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 4.2e-15)))
   (* (/ y z) (/ (/ x z) z))
   (/ (* y (/ x z)) z)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 4.2e-15)) {
		tmp = (y / z) * ((x / z) / z);
	} else {
		tmp = (y * (x / z)) / z;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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 <= (-1.0d0)) .or. (.not. (z <= 4.2d-15))) then
        tmp = (y / z) * ((x / z) / z)
    else
        tmp = (y * (x / z)) / z
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 4.2e-15)) {
		tmp = (y / z) * ((x / z) / z);
	} else {
		tmp = (y * (x / z)) / z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 4.2e-15):
		tmp = (y / z) * ((x / z) / z)
	else:
		tmp = (y * (x / z)) / z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 4.2e-15))
		tmp = Float64(Float64(y / z) * Float64(Float64(x / z) / z));
	else
		tmp = Float64(Float64(y * Float64(x / z)) / z);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 4.2e-15)))
		tmp = (y / z) * ((x / z) / z);
	else
		tmp = (y * (x / z)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 4.2e-15]], $MachinePrecision]], N[(N[(y / z), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 4.19999999999999962e-15 < z

   1. Initial program 89.7%

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

      1. *-commutative 89.7%

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

      2. frac-times 95.8%

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

      3. associate-*l/ 94.3%

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

      4. times-frac 98.3%

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

   4. Applied egg-rr 98.3%

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

   5. Taylor expanded in z around inf 94.9%

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

   if -1 < z < 4.19999999999999962e-15

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-*l/ 80.5%

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

      3. *-commutative 80.5%

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

      4. sqr-neg 80.5%

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

      5. *-commutative 80.5%

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

      6. distribute-rgt1-in 80.5%

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

      7. sqr-neg 80.5%

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

      8. fma-def 80.5%

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

      9. sqr-neg 80.5%

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

      10. cube-unmult 80.5%

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

   3. Simplified 80.5%

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

      1. fma-def 80.5%

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

      2. associate-*r/ 80.7%

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

      3. *-commutative 80.7%

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

      4. cube-mult 80.7%

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

      5. distribute-rgt1-in 80.7%

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

      6. *-commutative 80.7%

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

      7. frac-times 80.5%

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

      8. associate-/r* 89.8%

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

      9. associate-*l/ 96.4%

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

   6. Applied egg-rr 96.4%

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

   7. Taylor expanded in z around 0 88.7%

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

      1. associate-*l/ 94.1%

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

      2. *-commutative 94.1%

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

   9. Simplified 94.1%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 4.2 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 4.2 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{\frac{y}{z} \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\frac{x}{z} - x\right)}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 4.2e-15)))
   (/ (* (/ y z) (/ x z)) z)
   (/ (* y (- (/ x z) x)) z)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 4.2e-15)) {
		tmp = ((y / z) * (x / z)) / z;
	} else {
		tmp = (y * ((x / z) - x)) / z;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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 <= (-1.0d0)) .or. (.not. (z <= 4.2d-15))) then
        tmp = ((y / z) * (x / z)) / z
    else
        tmp = (y * ((x / z) - x)) / z
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 4.2e-15)) {
		tmp = ((y / z) * (x / z)) / z;
	} else {
		tmp = (y * ((x / z) - x)) / z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 4.2e-15):
		tmp = ((y / z) * (x / z)) / z
	else:
		tmp = (y * ((x / z) - x)) / z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 4.2e-15))
		tmp = Float64(Float64(Float64(y / z) * Float64(x / z)) / z);
	else
		tmp = Float64(Float64(y * Float64(Float64(x / z) - x)) / z);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 4.2e-15)))
		tmp = ((y / z) * (x / z)) / z;
	else
		tmp = (y * ((x / z) - x)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 4.2e-15]], $MachinePrecision]], N[(N[(N[(y / z), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y * N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 4.19999999999999962e-15 < z

   1. Initial program 89.7%

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

      1. *-commutative 89.7%

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

      2. associate-*l/ 95.5%

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

      3. *-commutative 95.5%

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

      4. sqr-neg 95.5%

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

      5. *-commutative 95.5%

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

      6. distribute-rgt1-in 71.2%

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

      7. sqr-neg 71.2%

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

      8. fma-def 95.5%

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

      9. sqr-neg 95.5%

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

      10. cube-unmult 95.5%

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

   3. Simplified 95.5%

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

      1. fma-def 71.2%

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

      2. associate-*r/ 67.0%

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

      3. *-commutative 67.0%

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

      4. cube-mult 67.0%

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

      5. distribute-rgt1-in 89.7%

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

      6. *-commutative 89.7%

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

      7. frac-times 95.8%

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

      8. associate-/r* 99.1%

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

      9. associate-*l/ 98.4%

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

   6. Applied egg-rr 98.4%

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

   7. Taylor expanded in z around inf 95.0%

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

   if -1 < z < 4.19999999999999962e-15

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-*l/ 80.5%

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

      3. *-commutative 80.5%

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

      4. sqr-neg 80.5%

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

      5. *-commutative 80.5%

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

      6. distribute-rgt1-in 80.5%

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

      7. sqr-neg 80.5%

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

      8. fma-def 80.5%

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

      9. sqr-neg 80.5%

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

      10. cube-unmult 80.5%

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

   3. Simplified 80.5%

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

      1. fma-def 80.5%

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

      2. associate-*r/ 80.7%

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

      3. *-commutative 80.7%

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

      4. cube-mult 80.7%

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

      5. distribute-rgt1-in 80.7%

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

      6. *-commutative 80.7%

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

      7. frac-times 80.5%

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

      8. associate-/r* 89.8%

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

      9. associate-*l/ 96.4%

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

   6. Applied egg-rr 96.4%

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

   7. Taylor expanded in z around 0 85.5%

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

      1. mul-1-neg 85.5%

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

      2. distribute-lft-neg-out 85.5%

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

      3. associate-*l/ 91.0%

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

      4. distribute-rgt-out 94.8%

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

      5. +-commutative 94.8%

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

      6. unsub-neg 94.8%

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

   9. 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 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 4.2 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{\frac{y}{z} \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\frac{x}{z} - x\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (* (/ x z) (/ y (* z z)))
   (if (<= z 4.2e-15) (/ (* y (/ x z)) z) (* (/ y z) (/ (/ x z) z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = (x / z) * (y / (z * z));
	} else if (z <= 4.2e-15) {
		tmp = (y * (x / z)) / z;
	} else {
		tmp = (y / z) * ((x / z) / z);
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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 <= (-1.0d0)) then
        tmp = (x / z) * (y / (z * z))
    else if (z <= 4.2d-15) then
        tmp = (y * (x / z)) / z
    else
        tmp = (y / z) * ((x / z) / z)
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = (x / z) * (y / (z * z));
	} else if (z <= 4.2e-15) {
		tmp = (y * (x / z)) / z;
	} else {
		tmp = (y / z) * ((x / z) / z);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = (x / z) * (y / (z * z))
	elif z <= 4.2e-15:
		tmp = (y * (x / z)) / z
	else:
		tmp = (y / z) * ((x / z) / z)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(Float64(x / z) * Float64(y / Float64(z * z)));
	elseif (z <= 4.2e-15)
		tmp = Float64(Float64(y * Float64(x / z)) / z);
	else
		tmp = Float64(Float64(y / z) * Float64(Float64(x / z) / z));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = (x / z) * (y / (z * z));
	elseif (z <= 4.2e-15)
		tmp = (y * (x / z)) / z;
	else
		tmp = (y / z) * ((x / z) / z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, -1.0], N[(N[(x / z), $MachinePrecision] * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.2e-15], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 92.8%

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

      1. sqr-neg 92.8%

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

      2. *-commutative 92.8%

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

      3. times-frac 98.1%

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

      4. sqr-neg 98.1%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in z around inf 96.7%

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

   if -1 < z < 4.19999999999999962e-15

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-*l/ 80.5%

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

      3. *-commutative 80.5%

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

      4. sqr-neg 80.5%

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

      5. *-commutative 80.5%

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

      6. distribute-rgt1-in 80.5%

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

      7. sqr-neg 80.5%

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

      8. fma-def 80.5%

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

      9. sqr-neg 80.5%

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

      10. cube-unmult 80.5%

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

   3. Simplified 80.5%

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

      1. fma-def 80.5%

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

      2. associate-*r/ 80.7%

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

      3. *-commutative 80.7%

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

      4. cube-mult 80.7%

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

      5. distribute-rgt1-in 80.7%

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

      6. *-commutative 80.7%

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

      7. frac-times 80.5%

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

      8. associate-/r* 89.8%

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

      9. associate-*l/ 96.4%

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

   6. Applied egg-rr 96.4%

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

   7. Taylor expanded in z around 0 88.7%

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

      1. associate-*l/ 94.1%

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

      2. *-commutative 94.1%

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

   9. Simplified 94.1%

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

   if 4.19999999999999962e-15 < z

   1. Initial program 87.2%

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

      1. *-commutative 87.2%

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

      2. frac-times 93.9%

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

      3. associate-*l/ 92.6%

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

      4. times-frac 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in z around inf 93.5%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{y \cdot \left(\frac{x}{z} - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (* (/ x z) (/ y (* z z)))
   (if (<= z 4.2e-15) (/ (* y (- (/ x z) x)) z) (* (/ y z) (/ (/ x z) z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = (x / z) * (y / (z * z));
	} else if (z <= 4.2e-15) {
		tmp = (y * ((x / z) - x)) / z;
	} else {
		tmp = (y / z) * ((x / z) / z);
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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 <= (-1.0d0)) then
        tmp = (x / z) * (y / (z * z))
    else if (z <= 4.2d-15) then
        tmp = (y * ((x / z) - x)) / z
    else
        tmp = (y / z) * ((x / z) / z)
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = (x / z) * (y / (z * z));
	} else if (z <= 4.2e-15) {
		tmp = (y * ((x / z) - x)) / z;
	} else {
		tmp = (y / z) * ((x / z) / z);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = (x / z) * (y / (z * z))
	elif z <= 4.2e-15:
		tmp = (y * ((x / z) - x)) / z
	else:
		tmp = (y / z) * ((x / z) / z)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(Float64(x / z) * Float64(y / Float64(z * z)));
	elseif (z <= 4.2e-15)
		tmp = Float64(Float64(y * Float64(Float64(x / z) - x)) / z);
	else
		tmp = Float64(Float64(y / z) * Float64(Float64(x / z) / z));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = (x / z) * (y / (z * z));
	elseif (z <= 4.2e-15)
		tmp = (y * ((x / z) - x)) / z;
	else
		tmp = (y / z) * ((x / z) / z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, -1.0], N[(N[(x / z), $MachinePrecision] * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.2e-15], N[(N[(y * N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 92.8%

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

      1. sqr-neg 92.8%

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

      2. *-commutative 92.8%

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

      3. times-frac 98.1%

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

      4. sqr-neg 98.1%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in z around inf 96.7%

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

   if -1 < z < 4.19999999999999962e-15

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-*l/ 80.5%

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

      3. *-commutative 80.5%

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

      4. sqr-neg 80.5%

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

      5. *-commutative 80.5%

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

      6. distribute-rgt1-in 80.5%

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

      7. sqr-neg 80.5%

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

      8. fma-def 80.5%

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

      9. sqr-neg 80.5%

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

      10. cube-unmult 80.5%

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

   3. Simplified 80.5%

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

      1. fma-def 80.5%

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

      2. associate-*r/ 80.7%

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

      3. *-commutative 80.7%

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

      4. cube-mult 80.7%

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

      5. distribute-rgt1-in 80.7%

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

      6. *-commutative 80.7%

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

      7. frac-times 80.5%

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

      8. associate-/r* 89.8%

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

      9. associate-*l/ 96.4%

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

   6. Applied egg-rr 96.4%

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

   7. Taylor expanded in z around 0 85.5%

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

      1. mul-1-neg 85.5%

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

      2. distribute-lft-neg-out 85.5%

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

      3. associate-*l/ 91.0%

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

      4. distribute-rgt-out 94.8%

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

      5. +-commutative 94.8%

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

      6. unsub-neg 94.8%

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

   9. Simplified 94.8%

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

   if 4.19999999999999962e-15 < z

   1. Initial program 87.2%

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

      1. *-commutative 87.2%

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

      2. frac-times 93.9%

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

      3. associate-*l/ 92.6%

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

      4. times-frac 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in z around inf 93.5%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{y \cdot \left(\frac{x}{z} - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{y \cdot \frac{x}{z}}{z}}{z}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{y \cdot \left(\frac{x}{z} - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \frac{x}{z}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (/ (/ (* y (/ x z)) z) z)
   (if (<= z 4.2e-15) (/ (* y (- (/ x z) x)) z) (/ (* (/ y z) (/ x z)) z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = ((y * (x / z)) / z) / z;
	} else if (z <= 4.2e-15) {
		tmp = (y * ((x / z) - x)) / z;
	} else {
		tmp = ((y / z) * (x / z)) / z;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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 <= (-1.0d0)) then
        tmp = ((y * (x / z)) / z) / z
    else if (z <= 4.2d-15) then
        tmp = (y * ((x / z) - x)) / z
    else
        tmp = ((y / z) * (x / z)) / z
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = ((y * (x / z)) / z) / z;
	} else if (z <= 4.2e-15) {
		tmp = (y * ((x / z) - x)) / z;
	} else {
		tmp = ((y / z) * (x / z)) / z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = ((y * (x / z)) / z) / z
	elif z <= 4.2e-15:
		tmp = (y * ((x / z) - x)) / z
	else:
		tmp = ((y / z) * (x / z)) / z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(Float64(Float64(y * Float64(x / z)) / z) / z);
	elseif (z <= 4.2e-15)
		tmp = Float64(Float64(y * Float64(Float64(x / z) - x)) / z);
	else
		tmp = Float64(Float64(Float64(y / z) * Float64(x / z)) / z);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = ((y * (x / z)) / z) / z;
	elseif (z <= 4.2e-15)
		tmp = (y * ((x / z) - x)) / z;
	else
		tmp = ((y / z) * (x / z)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, -1.0], N[(N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 4.2e-15], N[(N[(y * N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(y / z), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 92.8%

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

      1. *-commutative 92.8%

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

      2. associate-*l/ 98.1%

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

      3. *-commutative 98.1%

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

      4. sqr-neg 98.1%

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

      5. *-commutative 98.1%

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

      6. distribute-rgt1-in 42.5%

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

      7. sqr-neg 42.5%

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

      8. fma-def 98.2%

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

      9. sqr-neg 98.2%

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

      10. cube-unmult 98.2%

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

   3. Simplified 98.2%

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

      1. fma-def 42.5%

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

      2. associate-*r/ 40.8%

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

      3. *-commutative 40.8%

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

      4. cube-mult 40.8%

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

      5. distribute-rgt1-in 92.8%

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

      6. *-commutative 92.8%

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

      7. frac-times 98.1%

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

      8. associate-*l/ 96.4%

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

      9. associate-/r* 98.2%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in z around inf 96.8%

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

   if -1 < z < 4.19999999999999962e-15

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-*l/ 80.5%

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

      3. *-commutative 80.5%

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

      4. sqr-neg 80.5%

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

      5. *-commutative 80.5%

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

      6. distribute-rgt1-in 80.5%

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

      7. sqr-neg 80.5%

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

      8. fma-def 80.5%

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

      9. sqr-neg 80.5%

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

      10. cube-unmult 80.5%

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

   3. Simplified 80.5%

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

      1. fma-def 80.5%

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

      2. associate-*r/ 80.7%

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

      3. *-commutative 80.7%

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

      4. cube-mult 80.7%

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

      5. distribute-rgt1-in 80.7%

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

      6. *-commutative 80.7%

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

      7. frac-times 80.5%

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

      8. associate-/r* 89.8%

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

      9. associate-*l/ 96.4%

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

   6. Applied egg-rr 96.4%

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

   7. Taylor expanded in z around 0 85.5%

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

      1. mul-1-neg 85.5%

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

      2. distribute-lft-neg-out 85.5%

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

      3. associate-*l/ 91.0%

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

      4. distribute-rgt-out 94.8%

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

      5. +-commutative 94.8%

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

      6. unsub-neg 94.8%

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

   9. Simplified 94.8%

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

   if 4.19999999999999962e-15 < z

   1. Initial program 87.2%

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

      1. *-commutative 87.2%

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

      2. associate-*l/ 93.4%

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

      3. *-commutative 93.4%

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

      4. sqr-neg 93.4%

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

      5. *-commutative 93.4%

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

      6. distribute-rgt1-in 93.4%

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

      7. sqr-neg 93.4%

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

      8. fma-def 93.4%

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

      9. sqr-neg 93.4%

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

      10. cube-unmult 93.4%

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

   3. Simplified 93.4%

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

      1. fma-def 93.4%

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

      2. associate-*r/ 87.2%

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

      3. *-commutative 87.2%

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

      4. cube-mult 87.2%

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

      5. distribute-rgt1-in 87.2%

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

      6. *-commutative 87.2%

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

      7. frac-times 93.9%

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

      8. associate-/r* 98.5%

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

      9. associate-*l/ 98.5%

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

   6. Applied egg-rr 98.5%

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

   7. Taylor expanded in z around inf 93.6%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{y \cdot \frac{x}{z}}{z}}{z}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{y \cdot \left(\frac{x}{z} - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \frac{x}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x 1.4e-138) (* (/ y z) (/ x z)) (* x (/ y (* z z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.4e-138) {
		tmp = (y / z) * (x / z);
	} else {
		tmp = x * (y / (z * z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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 (x <= 1.4d-138) then
        tmp = (y / z) * (x / z)
    else
        tmp = x * (y / (z * z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.4e-138) {
		tmp = (y / z) * (x / z);
	} else {
		tmp = x * (y / (z * z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if x <= 1.4e-138:
		tmp = (y / z) * (x / z)
	else:
		tmp = x * (y / (z * z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (x <= 1.4e-138)
		tmp = Float64(Float64(y / z) * Float64(x / z));
	else
		tmp = Float64(x * Float64(y / Float64(z * z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1.4e-138)
		tmp = (y / z) * (x / z);
	else
		tmp = x * (y / (z * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[x, 1.4e-138], N[(N[(y / z), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.4e-138

   1. Initial program 82.8%

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

      1. *-commutative 82.8%

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

      2. frac-times 85.4%

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

      3. associate-*l/ 84.5%

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

      4. times-frac 97.5%

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

   4. Applied egg-rr 97.5%

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

   5. Taylor expanded in z around 0 81.3%

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

   if 1.4e-138 < x

   1. Initial program 89.1%

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

      1. sqr-neg 89.1%

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

      2. *-commutative 89.1%

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

      3. times-frac 92.6%

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

      4. sqr-neg 92.6%

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

   3. Simplified 92.6%

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

   5. Taylor expanded in z around 0 79.3%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{-40}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1e-40) (* (/ y z) (/ x z)) (/ y (* z (/ z x)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e-40) {
		tmp = (y / z) * (x / z);
	} else {
		tmp = y / (z * (z / x));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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 (y <= 1d-40) then
        tmp = (y / z) * (x / z)
    else
        tmp = y / (z * (z / x))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e-40) {
		tmp = (y / z) * (x / z);
	} else {
		tmp = y / (z * (z / x));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 1e-40:
		tmp = (y / z) * (x / z)
	else:
		tmp = y / (z * (z / x))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1e-40)
		tmp = Float64(Float64(y / z) * Float64(x / z));
	else
		tmp = Float64(y / Float64(z * Float64(z / x)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1e-40)
		tmp = (y / z) * (x / z);
	else
		tmp = y / (z * (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 1e-40], N[(N[(y / z), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(y / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.9999999999999993e-41

   1. Initial program 85.5%

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

      1. *-commutative 85.5%

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

      2. frac-times 87.3%

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

      3. associate-*l/ 86.5%

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

      4. times-frac 97.4%

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

   4. Applied egg-rr 97.4%

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

   5. Taylor expanded in z around 0 84.2%

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

   if 9.9999999999999993e-41 < y

   1. Initial program 83.5%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. 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. frac-times 89.8%

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

      3. associate-*l/ 89.8%

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

      4. times-frac 96.7%

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

   4. Applied egg-rr 96.7%

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

   5. Taylor expanded in z around 0 59.9%

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

      1. associate-/r/ 66.5%

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

      2. div-inv 66.5%

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

      3. clear-num 66.5%

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

   7. Applied egg-rr 66.5%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-40}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \frac{y}{z} \cdot \frac{x}{z} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* (/ y z) (/ x z)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return (y / z) * (x / z);
}

Fortran

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

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	return (y / z) * (x / z);
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return (y / z) * (x / z)

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(Float64(y / z) * Float64(x / z))
end

MATLAB

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

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(y / z), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.0%

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

   1. *-commutative 85.0%

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

   2. frac-times 87.9%

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

   3. associate-*l/ 87.3%

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

   4. times-frac 97.2%

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

4. Applied egg-rr 97.2%

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

5. Taylor expanded in z around 0 78.5%

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

6. Final simplification 78.5%

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

Developer target: 95.5% 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 2024021 
(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))))