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

Percentage Accurate: 83.1% → 96.5%

Time: 8.6s

Alternatives: 14

Speedup: 0.9×

• 25.2% 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: 83.1% 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: 96.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return (((x / z) * y) / (1.0 + z)) / 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 = (((x / z) * y) / (1.0d0 + z)) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 98.6

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 98.6

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

4. Applied rewrites 98.6%

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

5. Final simplification 98.6%

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

Alternative 2: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(1 + z\right)\\ t_1 := \frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* z z) (+ 1.0 z))) (t_1 (* (/ y (* (fma z z z) z)) x)))
   (if (<= t_0 -1e+21) t_1 (if (<= t_0 2e-37) (/ (* (/ x z) y) z) t_1))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = (z * z) * (1.0 + z);
	double t_1 = (y / (fma(z, z, z) * z)) * x;
	double tmp;
	if (t_0 <= -1e+21) {
		tmp = t_1;
	} else if (t_0 <= 2e-37) {
		tmp = ((x / z) * y) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(Float64(z * z) * Float64(1.0 + z))
	t_1 = Float64(Float64(y / Float64(fma(z, z, z) * z)) * x)
	tmp = 0.0
	if (t_0 <= -1e+21)
		tmp = t_1;
	elseif (t_0 <= 2e-37)
		tmp = Float64(Float64(Float64(x / z) * y) / z);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * z), $MachinePrecision] * N[(1.0 + z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+21], t$95$1, If[LessEqual[t$95$0, 2e-37], N[(N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -1e21 or 2.00000000000000013e-37 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 89.3%

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

   3. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 55.6

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

   5. Applied rewrites 55.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 61.6

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

   7. Applied rewrites 61.6%

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

   8. Taylor expanded in z around 0

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

      1. distribute-lft-in N/A

         \[\leadsto \frac{y}{\color{blue}{{z}^{2} \cdot 1 + {z}^{2} \cdot z}} \cdot x \]

      2. *-rgt-identity N/A

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

      3. unpow2 N/A

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

      4. distribute-rgt-in N/A

         \[\leadsto \frac{y}{\color{blue}{z \cdot \left(z + {z}^{2}\right)}} \cdot x \]

      5. *-commutative N/A

         \[\leadsto \frac{y}{\color{blue}{\left(z + {z}^{2}\right) \cdot z}} \cdot x \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(z + {z}^{2}\right) \cdot z}} \cdot x \]

      7. +-commutative N/A

         \[\leadsto \frac{y}{\color{blue}{\left({z}^{2} + z\right)} \cdot z} \cdot x \]

      8. unpow2 N/A

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

      9. lower-fma.f64 89.1

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

   10. Applied rewrites 89.1%

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

   if -1e21 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 2.00000000000000013e-37

   1. Initial program 90.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 98.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 98.8

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 94.6

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

   7. Applied rewrites 94.6%

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

   9. Applied rewrites 98.8%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(1 + z\right) \leq -1 \cdot 10^{+21}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(1 + z\right) \leq 2 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq 10^{-187}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(1 + z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) 1e-187)
   (/ (/ y z) (/ z x))
   (/ (* x y) (* (* z z) (+ 1.0 z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((x * y) <= 1e-187) {
		tmp = (y / z) / (z / x);
	} else {
		tmp = (x * y) / ((z * z) * (1.0 + 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 * y) <= 1d-187) then
        tmp = (y / z) / (z / x)
    else
        tmp = (x * y) / ((z * z) * (1.0d0 + 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 * y) <= 1e-187) {
		tmp = (y / z) / (z / x);
	} else {
		tmp = (x * y) / ((z * z) * (1.0 + z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (x * y) <= 1e-187:
		tmp = (y / z) / (z / x)
	else:
		tmp = (x * y) / ((z * z) * (1.0 + z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(x * y) <= 1e-187)
		tmp = Float64(Float64(y / z) / Float64(z / x));
	else
		tmp = Float64(Float64(x * y) / Float64(Float64(z * z) * Float64(1.0 + 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 * y) <= 1e-187)
		tmp = (y / z) / (z / x);
	else
		tmp = (x * y) / ((z * z) * (1.0 + 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[N[(x * y), $MachinePrecision], 1e-187], N[(N[(y / z), $MachinePrecision] / N[(z / x), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 1e-187

   1. Initial program 88.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 98.5

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 74.0

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

   7. Applied rewrites 74.0%

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

   9. Applied rewrites 80.3%

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

   if 1e-187 < (*.f64 x y)

   1. Initial program 92.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq 10^{-187}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(1 + z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return (y / (1.0 + z)) / ((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 / (1.0d0 + z)) / ((z / x) * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

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. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-/l* N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 96.8

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

4. Applied rewrites 96.8%

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

5. Final simplification 96.8%

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

Alternative 5: 93.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return ((x / z) / z) * (y / (1.0 + 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 = ((x / z) / z) * (y / (1.0d0 + z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 98.6

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 98.6

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

4. Applied rewrites 98.6%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 95.9

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

6. Applied rewrites 95.9%

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

Alternative 6: 81.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq 10^{-187}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(1 + z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) 1e-187)
   (* (/ y z) (/ x z))
   (/ (* x y) (* (* z z) (+ 1.0 z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((x * y) <= 1e-187) {
		tmp = (y / z) * (x / z);
	} else {
		tmp = (x * y) / ((z * z) * (1.0 + 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 * y) <= 1d-187) then
        tmp = (y / z) * (x / z)
    else
        tmp = (x * y) / ((z * z) * (1.0d0 + 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 * y) <= 1e-187) {
		tmp = (y / z) * (x / z);
	} else {
		tmp = (x * y) / ((z * z) * (1.0 + z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (x * y) <= 1e-187:
		tmp = (y / z) * (x / z)
	else:
		tmp = (x * y) / ((z * z) * (1.0 + z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(x * y) <= 1e-187)
		tmp = Float64(Float64(y / z) * Float64(x / z));
	else
		tmp = Float64(Float64(x * y) / Float64(Float64(z * z) * Float64(1.0 + 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 * y) <= 1e-187)
		tmp = (y / z) * (x / z);
	else
		tmp = (x * y) / ((z * z) * (1.0 + 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[N[(x * y), $MachinePrecision], 1e-187], N[(N[(y / z), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 1e-187

   1. Initial program 88.3%

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

   3. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 80.3

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

   5. Applied rewrites 80.3%

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

   if 1e-187 < (*.f64 x y)

   1. Initial program 92.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq 10^{-187}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(1 + z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. times-frac N/A

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

   8. associate-/r* N/A

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

   9. clear-num N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-+.f64 N/A

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

   15. distribute-lft1-in N/A

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

   16. lower-fma.f64 95.7

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

4. Applied rewrites 95.7%

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

Alternative 8: 81.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq 10^{-187}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(z, z, z\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) 1e-187) (* (/ y z) (/ x z)) (/ (* x y) (* (fma z z z) z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((x * y) <= 1e-187) {
		tmp = (y / z) * (x / z);
	} else {
		tmp = (x * y) / (fma(z, z, z) * z);
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(x * y) <= 1e-187)
		tmp = Float64(Float64(y / z) * Float64(x / z));
	else
		tmp = Float64(Float64(x * y) / Float64(fma(z, z, z) * z));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(x * y), $MachinePrecision], 1e-187], N[(N[(y / z), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 1e-187

   1. Initial program 88.3%

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

   3. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 80.3

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

   5. Applied rewrites 80.3%

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

   if 1e-187 < (*.f64 x y)

   1. Initial program 92.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 92.0

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. distribute-lft1-in N/A

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

      12. lower-fma.f64 92.0

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

   4. Applied rewrites 92.0%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq 10^{-187}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(z, z, z\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq 10^{-187}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) 1e-187) (* (/ y z) (/ x z)) (* (/ x (* (fma z z z) z)) y)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((x * y) <= 1e-187) {
		tmp = (y / z) * (x / z);
	} else {
		tmp = (x / (fma(z, z, z) * z)) * y;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(x * y) <= 1e-187)
		tmp = Float64(Float64(y / z) * Float64(x / z));
	else
		tmp = Float64(Float64(x / Float64(fma(z, z, z) * z)) * y);
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(x * y), $MachinePrecision], 1e-187], N[(N[(y / z), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 1e-187

   1. Initial program 88.3%

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

   3. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 80.3

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

   5. Applied rewrites 80.3%

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

   if 1e-187 < (*.f64 x y)

   1. Initial program 92.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 98.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 98.8

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

   4. Applied rewrites 98.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 93.8

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

   6. Applied rewrites 93.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l/ N/A

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

      15. lift-*.f64 N/A

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

      16. remove-double-neg N/A

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

      17. lift-neg.f64 N/A

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

      18. neg-mul-1 N/A

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

      19. associate-*r* N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. associate-*l/ N/A

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

      23. lift-/.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. associate-*r* N/A

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

   8. Applied rewrites 89.6%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq 10^{-187}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+26}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot z} \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2e+26) (* (/ y z) (/ x z)) (* (/ x (* z z)) y)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e+26) {
		tmp = (y / z) * (x / z);
	} else {
		tmp = (x / (z * z)) * y;
	}
	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 <= 2d+26) then
        tmp = (y / z) * (x / z)
    else
        tmp = (x / (z * z)) * y
    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 <= 2e+26) {
		tmp = (y / z) * (x / z);
	} else {
		tmp = (x / (z * z)) * y;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 2e+26:
		tmp = (y / z) * (x / z)
	else:
		tmp = (x / (z * z)) * y
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 2e+26)
		tmp = Float64(Float64(y / z) * Float64(x / z));
	else
		tmp = Float64(Float64(x / Float64(z * z)) * y);
	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 <= 2e+26)
		tmp = (y / z) * (x / z);
	else
		tmp = (x / (z * z)) * y;
	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, 2e+26], N[(N[(y / z), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.0000000000000001e26

   1. Initial program 88.9%

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

   3. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 77.9

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

   5. Applied rewrites 77.9%

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

   if 2.0000000000000001e26 < y

   1. Initial program 93.7%

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

   3. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 78.3

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

   5. Applied rewrites 78.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 80.4

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

   7. Applied rewrites 80.4%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+26}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 94.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. frac-times N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-+.f64 N/A

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

   13. distribute-lft1-in N/A

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

   14. lower-fma.f64 96.8

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

4. Applied rewrites 96.8%

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

Alternative 12: 94.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-+.f64 N/A

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

   13. distribute-lft1-in N/A

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

   14. lower-fma.f64 96.3

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

4. Applied rewrites 96.3%

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

5. Final simplification 96.3%

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

Alternative 13: 74.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

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 = ((x / z) / z) * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 98.6

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 98.6

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

4. Applied rewrites 98.6%

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

5. Taylor expanded in z around 0

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

   1. unpow2 N/A

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

   2. associate-/r* N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 73.0

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

7. Applied rewrites 73.0%

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

9. Applied rewrites 74.8%

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

Alternative 14: 72.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

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 = (x / (z * z)) * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

3. Taylor expanded in z around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 73.6

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

5. Applied rewrites 73.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 75.4

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

7. Applied rewrites 75.4%

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

8. Final simplification 75.4%

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

Developer Target 1: 95.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024332 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

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

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