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

Percentage Accurate: 82.8% → 94.7%

Time: 7.1s

Alternatives: 8

Speedup: 0.9×

• 25.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.8% 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: 94.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.5%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
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. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. distribute-lft-in N/A

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

   12. *-rgt-identity N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-/.f64 95.6

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

4. Applied rewrites 95.6%

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

Alternative 2: 92.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+35} \lor \neg \left(t\_0 \leq 10^{-21}\right):\\ \;\;\;\;\frac{y\_m}{z \cdot z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \left(\frac{x}{z} - x\right)}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* (* z z) (+ z 1.0))))
   (*
    y_s
    (if (or (<= t_0 -2e+35) (not (<= t_0 1e-21)))
      (* (/ y_m (* z z)) (/ x z))
      (/ (* y_m (- (/ x z) x)) z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (z * z) * (z + 1.0);
	double tmp;
	if ((t_0 <= -2e+35) || !(t_0 <= 1e-21)) {
		tmp = (y_m / (z * z)) * (x / z);
	} else {
		tmp = (y_m * ((x / z) - x)) / z;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (z * z) * (z + 1.0d0)
    if ((t_0 <= (-2d+35)) .or. (.not. (t_0 <= 1d-21))) then
        tmp = (y_m / (z * z)) * (x / z)
    else
        tmp = (y_m * ((x / z) - x)) / z
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = (z * z) * (z + 1.0);
	double tmp;
	if ((t_0 <= -2e+35) || !(t_0 <= 1e-21)) {
		tmp = (y_m / (z * z)) * (x / z);
	} else {
		tmp = (y_m * ((x / z) - x)) / z;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	t_0 = (z * z) * (z + 1.0)
	tmp = 0
	if (t_0 <= -2e+35) or not (t_0 <= 1e-21):
		tmp = (y_m / (z * z)) * (x / z)
	else:
		tmp = (y_m * ((x / z) - x)) / z
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if ((t_0 <= -2e+35) || !(t_0 <= 1e-21))
		tmp = Float64(Float64(y_m / Float64(z * z)) * Float64(x / z));
	else
		tmp = Float64(Float64(y_m * Float64(Float64(x / z) - x)) / z);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = (z * z) * (z + 1.0);
	tmp = 0.0;
	if ((t_0 <= -2e+35) || ~((t_0 <= 1e-21)))
		tmp = (y_m / (z * z)) * (x / z);
	else
		tmp = (y_m * ((x / z) - x)) / z;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[Or[LessEqual[t$95$0, -2e+35], N[Not[LessEqual[t$95$0, 1e-21]], $MachinePrecision]], N[(N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -1.9999999999999999e35 or 9.99999999999999908e-22 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 83.0%

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. distribute-lft-in N/A

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

      12. *-rgt-identity N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 93.4

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

   4. Applied rewrites 93.4%

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

   5. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 95.1

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

   7. Applied rewrites 95.1%

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

   9. Applied rewrites 93.1%

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

   if -1.9999999999999999e35 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 9.99999999999999908e-22

   1. Initial program 79.8%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 96.5

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in z around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. div-sub N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 96.5

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

   7. Applied rewrites 96.5%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq -2 \cdot 10^{+35} \lor \neg \left(\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 10^{-21}\right):\\ \;\;\;\;\frac{y}{z \cdot z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\frac{x}{z} - x\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+35} \lor \neg \left(t\_0 \leq 2 \cdot 10^{-207}\right):\\ \;\;\;\;\frac{y\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot y\_m}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* (* z z) (+ z 1.0))))
   (*
    y_s
    (if (or (<= t_0 -2e+35) (not (<= t_0 2e-207)))
      (* (/ y_m (* (fma z z z) z)) x)
      (/ (* (/ x z) y_m) z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (z * z) * (z + 1.0);
	double tmp;
	if ((t_0 <= -2e+35) || !(t_0 <= 2e-207)) {
		tmp = (y_m / (fma(z, z, z) * z)) * x;
	} else {
		tmp = ((x / z) * y_m) / z;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if ((t_0 <= -2e+35) || !(t_0 <= 2e-207))
		tmp = Float64(Float64(y_m / Float64(fma(z, z, z) * z)) * x);
	else
		tmp = Float64(Float64(Float64(x / z) * y_m) / z);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[Or[LessEqual[t$95$0, -2e+35], N[Not[LessEqual[t$95$0, 2e-207]], $MachinePrecision]], N[(N[(y$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(x / z), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -1.9999999999999999e35 or 1.99999999999999985e-207 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 84.7%

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

      1. 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. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft-in N/A

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

      15. *-rgt-identity N/A

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

      16. lower-fma.f64 94.2

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

   4. Applied rewrites 94.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 91.2

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

   6. Applied rewrites 91.2%

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

   if -1.9999999999999999e35 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 1.99999999999999985e-207

   1. Initial program 74.9%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 96.7

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

   4. Applied rewrites 96.7%

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

   5. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 98.7

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

   7. Applied rewrites 98.7%

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

   9. Applied rewrites 96.7%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq -2 \cdot 10^{+35} \lor \neg \left(\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 2 \cdot 10^{-207}\right):\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+35} \lor \neg \left(t\_0 \leq 2 \cdot 10^{-207}\right):\\ \;\;\;\;\frac{y\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y\_m}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* (* z z) (+ z 1.0))))
   (*
    y_s
    (if (or (<= t_0 -2e+35) (not (<= t_0 2e-207)))
      (* (/ y_m (* (fma z z z) z)) x)
      (* (/ x z) (/ y_m z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (z * z) * (z + 1.0);
	double tmp;
	if ((t_0 <= -2e+35) || !(t_0 <= 2e-207)) {
		tmp = (y_m / (fma(z, z, z) * z)) * x;
	} else {
		tmp = (x / z) * (y_m / z);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if ((t_0 <= -2e+35) || !(t_0 <= 2e-207))
		tmp = Float64(Float64(y_m / Float64(fma(z, z, z) * z)) * x);
	else
		tmp = Float64(Float64(x / z) * Float64(y_m / z));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[Or[LessEqual[t$95$0, -2e+35], N[Not[LessEqual[t$95$0, 2e-207]], $MachinePrecision]], N[(N[(y$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -1.9999999999999999e35 or 1.99999999999999985e-207 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 84.7%

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

      1. 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. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft-in N/A

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

      15. *-rgt-identity N/A

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

      16. lower-fma.f64 94.2

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

   4. Applied rewrites 94.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 91.2

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

   6. Applied rewrites 91.2%

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

   if -1.9999999999999999e35 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 1.99999999999999985e-207

   1. Initial program 74.9%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 97.8

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

   5. Applied rewrites 97.8%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq -2 \cdot 10^{+35} \lor \neg \left(\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 2 \cdot 10^{-207}\right):\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+35} \lor \neg \left(t\_0 \leq 4 \cdot 10^{-295}\right):\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y\_m}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* (* z z) (+ z 1.0))))
   (*
    y_s
    (if (or (<= t_0 -2e+35) (not (<= t_0 4e-295)))
      (* (/ x (* (fma z z z) z)) y_m)
      (* (/ x z) (/ y_m z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (z * z) * (z + 1.0);
	double tmp;
	if ((t_0 <= -2e+35) || !(t_0 <= 4e-295)) {
		tmp = (x / (fma(z, z, z) * z)) * y_m;
	} else {
		tmp = (x / z) * (y_m / z);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if ((t_0 <= -2e+35) || !(t_0 <= 4e-295))
		tmp = Float64(Float64(x / Float64(fma(z, z, z) * z)) * y_m);
	else
		tmp = Float64(Float64(x / z) * Float64(y_m / z));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[Or[LessEqual[t$95$0, -2e+35], N[Not[LessEqual[t$95$0, 4e-295]], $MachinePrecision]], N[(N[(x / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -1.9999999999999999e35 or 4.00000000000000024e-295 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 84.8%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft-in N/A

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

      16. *-rgt-identity N/A

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

      17. lower-fma.f64 94.8

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

   4. Applied rewrites 94.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 90.7

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

   6. Applied rewrites 90.7%

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

   if -1.9999999999999999e35 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 4.00000000000000024e-295

   1. Initial program 72.6%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq -2 \cdot 10^{+35} \lor \neg \left(\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 4 \cdot 10^{-295}\right):\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y\_m \leq 2 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot z} \cdot y\_m\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= (* x y_m) 2e-84) (* (/ x z) (/ y_m z)) (* (/ x (* z z)) y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((x * y_m) <= 2e-84) {
		tmp = (x / z) * (y_m / z);
	} else {
		tmp = (x / (z * z)) * y_m;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * y_m) <= 2d-84) then
        tmp = (x / z) * (y_m / z)
    else
        tmp = (x / (z * z)) * y_m
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((x * y_m) <= 2e-84) {
		tmp = (x / z) * (y_m / z);
	} else {
		tmp = (x / (z * z)) * y_m;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if (x * y_m) <= 2e-84:
		tmp = (x / z) * (y_m / z)
	else:
		tmp = (x / (z * z)) * y_m
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(x * y_m) <= 2e-84)
		tmp = Float64(Float64(x / z) * Float64(y_m / z));
	else
		tmp = Float64(Float64(x / Float64(z * z)) * y_m);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if ((x * y_m) <= 2e-84)
		tmp = (x / z) * (y_m / z);
	else
		tmp = (x / (z * z)) * y_m;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(x * y$95$m), $MachinePrecision], 2e-84], N[(N[(x / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 2.0000000000000001e-84

   1. Initial program 80.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 79.8

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

   5. Applied rewrites 79.8%

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

   if 2.0000000000000001e-84 < (*.f64 x y)

   1. Initial program 84.4%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 71.6

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

   5. Applied rewrites 71.6%

      \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot z}} \]
   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. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 76.4

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

   7. Applied rewrites 76.4%

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

Alternative 7: 77.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.5%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
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. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. distribute-lft-in N/A

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

   16. *-rgt-identity N/A

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

   17. lower-fma.f64 92.2

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

4. Applied rewrites 92.2%

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

5. Taylor expanded in z around 0

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

   1. unpow2 N/A

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

   2. associate-/r* N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 75.8

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

7. Applied rewrites 75.8%

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

Alternative 8: 74.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.5%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
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 69.8

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

5. Applied rewrites 69.8%

   \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot z}} \]
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. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 74.8

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

7. Applied rewrites 74.8%

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

Developer Target 1: 95.9% 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 2024320 
(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))))