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

Percentage Accurate: 82.9% → 97.1%

Time: 2.8s

Alternatives: 7

Speedup: 0.4×

• 73.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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: 82.9% accurate, 1.0× speedup

Math

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

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.1% accurate, 0.4× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \left(\frac{\frac{\mathsf{max}\left(\left|x\right|, \left|y\right|\right)}{z - -1}}{z} \cdot \frac{\mathsf{min}\left(\left|x\right|, \left|y\right|\right)}{z}\right)\right) \]

FPCore

(FPCore (x y z)
  :precision binary64
  (*
 (copysign 1.0 x)
 (*
  (copysign 1.0 y)
  (*
   (/ (/ (fmax (fabs x) (fabs y)) (- z -1.0)) z)
   (/ (fmin (fabs x) (fabs y)) z)))))

C

double code(double x, double y, double z) {
	return copysign(1.0, x) * (copysign(1.0, y) * (((fmax(fabs(x), fabs(y)) / (z - -1.0)) / z) * (fmin(fabs(x), fabs(y)) / z)));
}

Java

public static double code(double x, double y, double z) {
	return Math.copySign(1.0, x) * (Math.copySign(1.0, y) * (((fmax(Math.abs(x), Math.abs(y)) / (z - -1.0)) / z) * (fmin(Math.abs(x), Math.abs(y)) / z)));
}

Python

def code(x, y, z):
	return math.copysign(1.0, x) * (math.copysign(1.0, y) * (((fmax(math.fabs(x), math.fabs(y)) / (z - -1.0)) / z) * (fmin(math.fabs(x), math.fabs(y)) / z)))

Julia

function code(x, y, z)
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * Float64(Float64(Float64(fmax(abs(x), abs(y)) / Float64(z - -1.0)) / z) * Float64(fmin(abs(x), abs(y)) / z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (sign(x) * abs(1.0)) * ((sign(y) * abs(1.0)) * (((max(abs(x), abs(y)) / (z - -1.0)) / z) * (min(abs(x), abs(y)) / z)));
end

Wolfram

code[x_, y_, z_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[(N[(N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision] / N[(z - -1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * N[(N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.9%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
2. 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 94.2%

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

3. Applied rewrites 94.2%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. associate-*r/ N/A

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

   4. lift-fma.f64 N/A

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

   5. distribute-lft1-in N/A

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

   6. add-flip N/A

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

   7. metadata-eval N/A

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

   8. sub-negate-rev N/A

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

   9. lift--.f64 N/A

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

   10. times-frac N/A

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

   11. mult-flip N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lift--.f64 N/A

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

   15. sub-negate-rev N/A

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

   16. lower--.f64 96.2%

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

5. Applied rewrites 96.2%

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

Alternative 2: 95.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_1 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ t_2 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -50000000000000:\\ \;\;\;\;\frac{\frac{\frac{t\_0}{z} \cdot t\_1}{z}}{z}\\ \mathbf{elif}\;t\_2 \leq 10^{-62}:\\ \;\;\;\;\frac{\frac{t\_1}{1 \cdot z}}{z} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot t\_1\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fmax (fabs x) (fabs y)))
       (t_1 (fmin (fabs x) (fabs y)))
       (t_2 (* (* z z) (+ z 1.0))))
  (*
   (copysign 1.0 x)
   (*
    (copysign 1.0 y)
    (if (<= t_2 -50000000000000.0)
      (/ (/ (* (/ t_0 z) t_1) z) z)
      (if (<= t_2 1e-62)
        (* (/ (/ t_1 (* 1.0 z)) z) t_0)
        (* (/ t_0 (* (fma z z z) z)) t_1)))))))

C

double code(double x, double y, double z) {
	double t_0 = fmax(fabs(x), fabs(y));
	double t_1 = fmin(fabs(x), fabs(y));
	double t_2 = (z * z) * (z + 1.0);
	double tmp;
	if (t_2 <= -50000000000000.0) {
		tmp = (((t_0 / z) * t_1) / z) / z;
	} else if (t_2 <= 1e-62) {
		tmp = ((t_1 / (1.0 * z)) / z) * t_0;
	} else {
		tmp = (t_0 / (fma(z, z, z) * z)) * t_1;
	}
	return copysign(1.0, x) * (copysign(1.0, y) * tmp);
}

Julia

function code(x, y, z)
	t_0 = fmax(abs(x), abs(y))
	t_1 = fmin(abs(x), abs(y))
	t_2 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if (t_2 <= -50000000000000.0)
		tmp = Float64(Float64(Float64(Float64(t_0 / z) * t_1) / z) / z);
	elseif (t_2 <= 1e-62)
		tmp = Float64(Float64(Float64(t_1 / Float64(1.0 * z)) / z) * t_0);
	else
		tmp = Float64(Float64(t_0 / Float64(fma(z, z, z) * z)) * t_1);
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * tmp))
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, -50000000000000.0], N[(N[(N[(N[(t$95$0 / z), $MachinePrecision] * t$95$1), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$2, 1e-62], N[(N[(N[(t$95$1 / N[(1.0 * z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(t$95$0 / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -5e13

   1. Initial program 82.9%

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

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 93.8%

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 93.8%

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

   3. Applied rewrites 93.8%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 61.5%

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

   6. Applied rewrites 61.5%

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

   if -5e13 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 1e-62

   1. Initial program 82.9%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 70.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lower-/.f64 72.4%

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 72.4%

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

   6. Applied rewrites 72.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 73.9%

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

   8. Applied rewrites 73.9%

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

   if 1e-62 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 82.9%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. 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. lower-/.f64 83.7%

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 83.7%

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

   3. Applied rewrites 83.7%

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

Alternative 3: 94.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_1 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ t_2 := \frac{\frac{\frac{t\_0}{z} \cdot t\_1}{z}}{z}\\ t_3 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -50000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0.002:\\ \;\;\;\;\frac{\frac{t\_1}{1 \cdot z}}{z} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fmax (fabs x) (fabs y)))
       (t_1 (fmin (fabs x) (fabs y)))
       (t_2 (/ (/ (* (/ t_0 z) t_1) z) z))
       (t_3 (* (* z z) (+ z 1.0))))
  (*
   (copysign 1.0 x)
   (*
    (copysign 1.0 y)
    (if (<= t_3 -50000000000000.0)
      t_2
      (if (<= t_3 0.002) (* (/ (/ t_1 (* 1.0 z)) z) t_0) t_2))))))

C

double code(double x, double y, double z) {
	double t_0 = fmax(fabs(x), fabs(y));
	double t_1 = fmin(fabs(x), fabs(y));
	double t_2 = (((t_0 / z) * t_1) / z) / z;
	double t_3 = (z * z) * (z + 1.0);
	double tmp;
	if (t_3 <= -50000000000000.0) {
		tmp = t_2;
	} else if (t_3 <= 0.002) {
		tmp = ((t_1 / (1.0 * z)) / z) * t_0;
	} else {
		tmp = t_2;
	}
	return copysign(1.0, x) * (copysign(1.0, y) * tmp);
}

Java

public static double code(double x, double y, double z) {
	double t_0 = fmax(Math.abs(x), Math.abs(y));
	double t_1 = fmin(Math.abs(x), Math.abs(y));
	double t_2 = (((t_0 / z) * t_1) / z) / z;
	double t_3 = (z * z) * (z + 1.0);
	double tmp;
	if (t_3 <= -50000000000000.0) {
		tmp = t_2;
	} else if (t_3 <= 0.002) {
		tmp = ((t_1 / (1.0 * z)) / z) * t_0;
	} else {
		tmp = t_2;
	}
	return Math.copySign(1.0, x) * (Math.copySign(1.0, y) * tmp);
}

Python

def code(x, y, z):
	t_0 = fmax(math.fabs(x), math.fabs(y))
	t_1 = fmin(math.fabs(x), math.fabs(y))
	t_2 = (((t_0 / z) * t_1) / z) / z
	t_3 = (z * z) * (z + 1.0)
	tmp = 0
	if t_3 <= -50000000000000.0:
		tmp = t_2
	elif t_3 <= 0.002:
		tmp = ((t_1 / (1.0 * z)) / z) * t_0
	else:
		tmp = t_2
	return math.copysign(1.0, x) * (math.copysign(1.0, y) * tmp)

Julia

function code(x, y, z)
	t_0 = fmax(abs(x), abs(y))
	t_1 = fmin(abs(x), abs(y))
	t_2 = Float64(Float64(Float64(Float64(t_0 / z) * t_1) / z) / z)
	t_3 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if (t_3 <= -50000000000000.0)
		tmp = t_2;
	elseif (t_3 <= 0.002)
		tmp = Float64(Float64(Float64(t_1 / Float64(1.0 * z)) / z) * t_0);
	else
		tmp = t_2;
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * tmp))
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = max(abs(x), abs(y));
	t_1 = min(abs(x), abs(y));
	t_2 = (((t_0 / z) * t_1) / z) / z;
	t_3 = (z * z) * (z + 1.0);
	tmp = 0.0;
	if (t_3 <= -50000000000000.0)
		tmp = t_2;
	elseif (t_3 <= 0.002)
		tmp = ((t_1 / (1.0 * z)) / z) * t_0;
	else
		tmp = t_2;
	end
	tmp_2 = (sign(x) * abs(1.0)) * ((sign(y) * abs(1.0)) * tmp);
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(t$95$0 / z), $MachinePrecision] * t$95$1), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, -50000000000000.0], t$95$2, If[LessEqual[t$95$3, 0.002], N[(N[(N[(t$95$1 / N[(1.0 * z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * t$95$0), $MachinePrecision], t$95$2]]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -5e13 or 2e-3 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 82.9%

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

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 93.8%

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 93.8%

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

   3. Applied rewrites 93.8%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 61.5%

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

   6. Applied rewrites 61.5%

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

   if -5e13 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 2e-3

   1. Initial program 82.9%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 70.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lower-/.f64 72.4%

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 72.4%

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

   6. Applied rewrites 72.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 73.9%

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

   8. Applied rewrites 73.9%

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

Alternative 4: 94.5% accurate, 0.4× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \left(\frac{\mathsf{max}\left(\left|x\right|, \left|y\right|\right)}{\mathsf{fma}\left(z, z, z\right)} \cdot \frac{\mathsf{min}\left(\left|x\right|, \left|y\right|\right)}{z}\right)\right) \]

FPCore

(FPCore (x y z)
  :precision binary64
  (*
 (copysign 1.0 x)
 (*
  (copysign 1.0 y)
  (*
   (/ (fmax (fabs x) (fabs y)) (fma z z z))
   (/ (fmin (fabs x) (fabs y)) z)))))

C

double code(double x, double y, double z) {
	return copysign(1.0, x) * (copysign(1.0, y) * ((fmax(fabs(x), fabs(y)) / fma(z, z, z)) * (fmin(fabs(x), fabs(y)) / z)));
}

Julia

function code(x, y, z)
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * Float64(Float64(fmax(abs(x), abs(y)) / fma(z, z, z)) * Float64(fmin(abs(x), abs(y)) / z))))
end

Wolfram

code[x_, y_, z_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[(N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision] * N[(N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.9%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
2. 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 94.2%

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

3. Applied rewrites 94.2%

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

Alternative 5: 80.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ t_1 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 9.5 \cdot 10^{+30}:\\ \;\;\;\;\frac{t\_1}{1 \cdot z} \cdot \frac{t\_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(1 \cdot z\right) \cdot z} \cdot t\_1\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fmin (fabs x) (fabs y))) (t_1 (fmax (fabs x) (fabs y))))
  (*
   (copysign 1.0 x)
   (*
    (copysign 1.0 y)
    (if (<= t_1 9.5e+30)
      (* (/ t_1 (* 1.0 z)) (/ t_0 z))
      (* (/ t_0 (* (* 1.0 z) z)) t_1))))))

C

double code(double x, double y, double z) {
	double t_0 = fmin(fabs(x), fabs(y));
	double t_1 = fmax(fabs(x), fabs(y));
	double tmp;
	if (t_1 <= 9.5e+30) {
		tmp = (t_1 / (1.0 * z)) * (t_0 / z);
	} else {
		tmp = (t_0 / ((1.0 * z) * z)) * t_1;
	}
	return copysign(1.0, x) * (copysign(1.0, y) * tmp);
}

Java

public static double code(double x, double y, double z) {
	double t_0 = fmin(Math.abs(x), Math.abs(y));
	double t_1 = fmax(Math.abs(x), Math.abs(y));
	double tmp;
	if (t_1 <= 9.5e+30) {
		tmp = (t_1 / (1.0 * z)) * (t_0 / z);
	} else {
		tmp = (t_0 / ((1.0 * z) * z)) * t_1;
	}
	return Math.copySign(1.0, x) * (Math.copySign(1.0, y) * tmp);
}

Python

def code(x, y, z):
	t_0 = fmin(math.fabs(x), math.fabs(y))
	t_1 = fmax(math.fabs(x), math.fabs(y))
	tmp = 0
	if t_1 <= 9.5e+30:
		tmp = (t_1 / (1.0 * z)) * (t_0 / z)
	else:
		tmp = (t_0 / ((1.0 * z) * z)) * t_1
	return math.copysign(1.0, x) * (math.copysign(1.0, y) * tmp)

Julia

function code(x, y, z)
	t_0 = fmin(abs(x), abs(y))
	t_1 = fmax(abs(x), abs(y))
	tmp = 0.0
	if (t_1 <= 9.5e+30)
		tmp = Float64(Float64(t_1 / Float64(1.0 * z)) * Float64(t_0 / z));
	else
		tmp = Float64(Float64(t_0 / Float64(Float64(1.0 * z) * z)) * t_1);
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * tmp))
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = min(abs(x), abs(y));
	t_1 = max(abs(x), abs(y));
	tmp = 0.0;
	if (t_1 <= 9.5e+30)
		tmp = (t_1 / (1.0 * z)) * (t_0 / z);
	else
		tmp = (t_0 / ((1.0 * z) * z)) * t_1;
	end
	tmp_2 = (sign(x) * abs(1.0)) * ((sign(y) * abs(1.0)) * tmp);
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, 9.5e+30], N[(N[(t$95$1 / N[(1.0 * z), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / z), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(N[(1.0 * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 9.5000000000000003e30

   1. Initial program 82.9%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 70.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.5%

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

   6. Applied rewrites 74.5%

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

   if 9.5000000000000003e30 < y

   1. Initial program 82.9%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 70.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lower-/.f64 72.4%

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 72.4%

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

   6. Applied rewrites 72.4%

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

Alternative 6: 80.8% accurate, 0.5× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \left(\frac{\frac{\mathsf{min}\left(\left|x\right|, \left|y\right|\right)}{1 \cdot z}}{z} \cdot \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\right)\right) \]

FPCore

(FPCore (x y z)
  :precision binary64
  (*
 (copysign 1.0 x)
 (*
  (copysign 1.0 y)
  (*
   (/ (/ (fmin (fabs x) (fabs y)) (* 1.0 z)) z)
   (fmax (fabs x) (fabs y))))))

C

double code(double x, double y, double z) {
	return copysign(1.0, x) * (copysign(1.0, y) * (((fmin(fabs(x), fabs(y)) / (1.0 * z)) / z) * fmax(fabs(x), fabs(y))));
}

Java

public static double code(double x, double y, double z) {
	return Math.copySign(1.0, x) * (Math.copySign(1.0, y) * (((fmin(Math.abs(x), Math.abs(y)) / (1.0 * z)) / z) * fmax(Math.abs(x), Math.abs(y))));
}

Python

def code(x, y, z):
	return math.copysign(1.0, x) * (math.copysign(1.0, y) * (((fmin(math.fabs(x), math.fabs(y)) / (1.0 * z)) / z) * fmax(math.fabs(x), math.fabs(y))))

Julia

function code(x, y, z)
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * Float64(Float64(Float64(fmin(abs(x), abs(y)) / Float64(1.0 * z)) / z) * fmax(abs(x), abs(y)))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (sign(x) * abs(1.0)) * ((sign(y) * abs(1.0)) * (((min(abs(x), abs(y)) / (1.0 * z)) / z) * max(abs(x), abs(y))));
end

Wolfram

code[x_, y_, z_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[(N[(N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision] / N[(1.0 * z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.9%

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

2. Taylor expanded in z around 0

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

4. Applied rewrites 70.4%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. mult-flip-rev N/A

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

   9. lower-/.f64 72.4%

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 72.4%

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

6. Applied rewrites 72.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 73.9%

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

8. Applied rewrites 73.9%

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

Alternative 7: 75.5% accurate, 0.5× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \left(\frac{\mathsf{min}\left(\left|x\right|, \left|y\right|\right)}{\left(1 \cdot z\right) \cdot z} \cdot \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\right)\right) \]

FPCore

(FPCore (x y z)
  :precision binary64
  (*
 (copysign 1.0 x)
 (*
  (copysign 1.0 y)
  (*
   (/ (fmin (fabs x) (fabs y)) (* (* 1.0 z) z))
   (fmax (fabs x) (fabs y))))))

C

double code(double x, double y, double z) {
	return copysign(1.0, x) * (copysign(1.0, y) * ((fmin(fabs(x), fabs(y)) / ((1.0 * z) * z)) * fmax(fabs(x), fabs(y))));
}

Java

public static double code(double x, double y, double z) {
	return Math.copySign(1.0, x) * (Math.copySign(1.0, y) * ((fmin(Math.abs(x), Math.abs(y)) / ((1.0 * z) * z)) * fmax(Math.abs(x), Math.abs(y))));
}

Python

def code(x, y, z):
	return math.copysign(1.0, x) * (math.copysign(1.0, y) * ((fmin(math.fabs(x), math.fabs(y)) / ((1.0 * z) * z)) * fmax(math.fabs(x), math.fabs(y))))

Julia

function code(x, y, z)
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * Float64(Float64(fmin(abs(x), abs(y)) / Float64(Float64(1.0 * z) * z)) * fmax(abs(x), abs(y)))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (sign(x) * abs(1.0)) * ((sign(y) * abs(1.0)) * ((min(abs(x), abs(y)) / ((1.0 * z) * z)) * max(abs(x), abs(y))));
end

Wolfram

code[x_, y_, z_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[(N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision] / N[(N[(1.0 * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.9%

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

2. Taylor expanded in z around 0

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

4. Applied rewrites 70.4%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. mult-flip-rev N/A

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

   9. lower-/.f64 72.4%

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 72.4%

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

6. Applied rewrites 72.4%

   \[\leadsto \color{blue}{\frac{x}{\left(1 \cdot z\right) \cdot z} \cdot y} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (/ (* x y) (* (* z z) (+ z 1.0))))