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

Percentage Accurate: 83.2% → 97.5%

Time: 2.6s

Alternatives: 10

Speedup: 0.7×

• 74.2% 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: 83.2% 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: 97.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_1 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1 \cdot t\_0}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 5 \cdot 10^{-300}:\\ \;\;\;\;\frac{\frac{t\_0}{\left(z - -1\right) \cdot z}}{z} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0 \cdot \frac{t\_1}{z}}{z - -1}}{z}\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fmax (fabs x) (fabs y))) (t_1 (fmin (fabs x) (fabs y))))
  (*
   (copysign 1.0 x)
   (*
    (copysign 1.0 y)
    (if (<= (/ (* t_1 t_0) (* (* z z) (+ z 1.0))) 5e-300)
      (* (/ (/ t_0 (* (- z -1.0) z)) z) t_1)
      (/ (/ (* t_0 (/ t_1 z)) (- z -1.0)) z))))))

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 tmp;
	if (((t_1 * t_0) / ((z * z) * (z + 1.0))) <= 5e-300) {
		tmp = ((t_0 / ((z - -1.0) * z)) / z) * t_1;
	} else {
		tmp = ((t_0 * (t_1 / z)) / (z - -1.0)) / z;
	}
	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 tmp;
	if (((t_1 * t_0) / ((z * z) * (z + 1.0))) <= 5e-300) {
		tmp = ((t_0 / ((z - -1.0) * z)) / z) * t_1;
	} else {
		tmp = ((t_0 * (t_1 / z)) / (z - -1.0)) / z;
	}
	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))
	tmp = 0
	if ((t_1 * t_0) / ((z * z) * (z + 1.0))) <= 5e-300:
		tmp = ((t_0 / ((z - -1.0) * z)) / z) * t_1
	else:
		tmp = ((t_0 * (t_1 / z)) / (z - -1.0)) / z
	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))
	tmp = 0.0
	if (Float64(Float64(t_1 * t_0) / Float64(Float64(z * z) * Float64(z + 1.0))) <= 5e-300)
		tmp = Float64(Float64(Float64(t_0 / Float64(Float64(z - -1.0) * z)) / z) * t_1);
	else
		tmp = Float64(Float64(Float64(t_0 * Float64(t_1 / z)) / Float64(z - -1.0)) / z);
	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));
	tmp = 0.0;
	if (((t_1 * t_0) / ((z * z) * (z + 1.0))) <= 5e-300)
		tmp = ((t_0 / ((z - -1.0) * z)) / z) * t_1;
	else
		tmp = ((t_0 * (t_1 / z)) / (z - -1.0)) / z;
	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]}, 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[N[(N[(t$95$1 * t$95$0), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-300], N[(N[(N[(t$95$0 / N[(N[(z - -1.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(t$95$0 * N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision] / N[(z - -1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < 5e-300

   1. Initial program 83.2%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 85.0%

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

      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. *-commutative N/A

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

      13. lower-*.f64 85.0%

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 85.0%

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

   3. Applied rewrites 85.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-/.f64 N/A

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

      5. lower-/.f64 90.7%

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

   5. Applied rewrites 90.7%

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

   if 5e-300 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 83.2%

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

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 96.2%

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 96.2%

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

   3. Applied rewrites 96.2%

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

Alternative 2: 96.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_1 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{\frac{t\_0}{z} \cdot t\_1}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(z - -1\right) \cdot z} \cdot \frac{t\_1}{z}\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fmax (fabs x) (fabs y))) (t_1 (fmin (fabs x) (fabs y))))
  (*
   (copysign 1.0 x)
   (*
    (copysign 1.0 y)
    (if (<= z -1.1e+29)
      (/ (/ (* (/ t_0 z) t_1) z) z)
      (* (/ t_0 (* (- z -1.0) z)) (/ t_1 z)))))))

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 tmp;
	if (z <= -1.1e+29) {
		tmp = (((t_0 / z) * t_1) / z) / z;
	} else {
		tmp = (t_0 / ((z - -1.0) * z)) * (t_1 / z);
	}
	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 tmp;
	if (z <= -1.1e+29) {
		tmp = (((t_0 / z) * t_1) / z) / z;
	} else {
		tmp = (t_0 / ((z - -1.0) * z)) * (t_1 / z);
	}
	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))
	tmp = 0
	if z <= -1.1e+29:
		tmp = (((t_0 / z) * t_1) / z) / z
	else:
		tmp = (t_0 / ((z - -1.0) * z)) * (t_1 / z)
	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))
	tmp = 0.0
	if (z <= -1.1e+29)
		tmp = Float64(Float64(Float64(Float64(t_0 / z) * t_1) / z) / z);
	else
		tmp = Float64(Float64(t_0 / Float64(Float64(z - -1.0) * z)) * Float64(t_1 / z));
	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));
	tmp = 0.0;
	if (z <= -1.1e+29)
		tmp = (((t_0 / z) * t_1) / z) / z;
	else
		tmp = (t_0 / ((z - -1.0) * z)) * (t_1 / z);
	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]}, 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[z, -1.1e+29], N[(N[(N[(N[(t$95$0 / z), $MachinePrecision] * t$95$1), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision], N[(N[(t$95$0 / N[(N[(z - -1.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.1000000000000001e29

   1. Initial program 83.2%

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

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 96.2%

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 96.2%

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

   3. Applied rewrites 96.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*l/ N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 87.9%

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

   5. Applied rewrites 87.9%

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

   6. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 57.8%

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

   8. Applied rewrites 57.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 58.6%

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 62.4%

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

   10. Applied rewrites 62.4%

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

   if -1.1000000000000001e29 < z

   1. Initial program 83.2%

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

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 96.2%

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 96.2%

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

   3. Applied rewrites 96.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 94.1%

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

   5. Applied rewrites 94.1%

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

Alternative 3: 95.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_1 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1 \cdot t\_0}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 2 \cdot 10^{+52}:\\ \;\;\;\;\frac{\frac{t\_0}{\left(z - -1\right) \cdot z}}{z} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(-t\_0\right) \cdot \frac{\frac{t\_1}{z}}{\left(-1 - z\right) \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fmax (fabs x) (fabs y))) (t_1 (fmin (fabs x) (fabs y))))
  (*
   (copysign 1.0 x)
   (*
    (copysign 1.0 y)
    (if (<= (/ (* t_1 t_0) (* (* z z) (+ z 1.0))) 2e+52)
      (* (/ (/ t_0 (* (- z -1.0) z)) z) t_1)
      (* (- t_0) (/ (/ t_1 z) (* (- -1.0 z) z))))))))

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 tmp;
	if (((t_1 * t_0) / ((z * z) * (z + 1.0))) <= 2e+52) {
		tmp = ((t_0 / ((z - -1.0) * z)) / z) * t_1;
	} else {
		tmp = -t_0 * ((t_1 / z) / ((-1.0 - z) * z));
	}
	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 tmp;
	if (((t_1 * t_0) / ((z * z) * (z + 1.0))) <= 2e+52) {
		tmp = ((t_0 / ((z - -1.0) * z)) / z) * t_1;
	} else {
		tmp = -t_0 * ((t_1 / z) / ((-1.0 - z) * z));
	}
	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))
	tmp = 0
	if ((t_1 * t_0) / ((z * z) * (z + 1.0))) <= 2e+52:
		tmp = ((t_0 / ((z - -1.0) * z)) / z) * t_1
	else:
		tmp = -t_0 * ((t_1 / z) / ((-1.0 - z) * z))
	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))
	tmp = 0.0
	if (Float64(Float64(t_1 * t_0) / Float64(Float64(z * z) * Float64(z + 1.0))) <= 2e+52)
		tmp = Float64(Float64(Float64(t_0 / Float64(Float64(z - -1.0) * z)) / z) * t_1);
	else
		tmp = Float64(Float64(-t_0) * Float64(Float64(t_1 / z) / Float64(Float64(-1.0 - z) * z)));
	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));
	tmp = 0.0;
	if (((t_1 * t_0) / ((z * z) * (z + 1.0))) <= 2e+52)
		tmp = ((t_0 / ((z - -1.0) * z)) / z) * t_1;
	else
		tmp = -t_0 * ((t_1 / z) / ((-1.0 - z) * z));
	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]}, 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[N[(N[(t$95$1 * t$95$0), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+52], N[(N[(N[(t$95$0 / N[(N[(z - -1.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * t$95$1), $MachinePrecision], N[((-t$95$0) * N[(N[(t$95$1 / z), $MachinePrecision] / N[(N[(-1.0 - z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < 2e52

   1. Initial program 83.2%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 85.0%

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

      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. *-commutative N/A

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

      13. lower-*.f64 85.0%

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 85.0%

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

   3. Applied rewrites 85.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-/.f64 N/A

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

      5. lower-/.f64 90.7%

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

   5. Applied rewrites 90.7%

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

   if 2e52 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 83.2%

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

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 96.2%

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 96.2%

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

   3. Applied rewrites 96.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-lft-neg-in N/A

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

      14. lower-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. lower--.f64 90.1%

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

   5. Applied rewrites 90.1%

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

Alternative 4: 94.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_1 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ t_2 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -20000000000:\\ \;\;\;\;\frac{\frac{t\_0}{z} \cdot t\_1}{z \cdot z}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-49}:\\ \;\;\;\;\frac{\frac{t\_1}{1 \cdot z}}{z} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(\left(z - -1\right) \cdot z\right) \cdot z} \cdot t\_1\\ \end{array}\right) \end{array} \]

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 -20000000000.0)
      (/ (* (/ t_0 z) t_1) (* z z))
      (if (<= t_2 5e-49)
        (* (/ (/ t_1 (* 1.0 z)) z) t_0)
        (* (/ t_0 (* (* (- z -1.0) 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 <= -20000000000.0) {
		tmp = ((t_0 / z) * t_1) / (z * z);
	} else if (t_2 <= 5e-49) {
		tmp = ((t_1 / (1.0 * z)) / z) * t_0;
	} else {
		tmp = (t_0 / (((z - -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 = fmax(Math.abs(x), Math.abs(y));
	double t_1 = fmin(Math.abs(x), Math.abs(y));
	double t_2 = (z * z) * (z + 1.0);
	double tmp;
	if (t_2 <= -20000000000.0) {
		tmp = ((t_0 / z) * t_1) / (z * z);
	} else if (t_2 <= 5e-49) {
		tmp = ((t_1 / (1.0 * z)) / z) * t_0;
	} else {
		tmp = (t_0 / (((z - -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 = fmax(math.fabs(x), math.fabs(y))
	t_1 = fmin(math.fabs(x), math.fabs(y))
	t_2 = (z * z) * (z + 1.0)
	tmp = 0
	if t_2 <= -20000000000.0:
		tmp = ((t_0 / z) * t_1) / (z * z)
	elif t_2 <= 5e-49:
		tmp = ((t_1 / (1.0 * z)) / z) * t_0
	else:
		tmp = (t_0 / (((z - -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 = 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 <= -20000000000.0)
		tmp = Float64(Float64(Float64(t_0 / z) * t_1) / Float64(z * z));
	elseif (t_2 <= 5e-49)
		tmp = Float64(Float64(Float64(t_1 / Float64(1.0 * z)) / z) * t_0);
	else
		tmp = Float64(Float64(t_0 / Float64(Float64(Float64(z - -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 = max(abs(x), abs(y));
	t_1 = min(abs(x), abs(y));
	t_2 = (z * z) * (z + 1.0);
	tmp = 0.0;
	if (t_2 <= -20000000000.0)
		tmp = ((t_0 / z) * t_1) / (z * z);
	elseif (t_2 <= 5e-49)
		tmp = ((t_1 / (1.0 * z)) / z) * t_0;
	else
		tmp = (t_0 / (((z - -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[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, -20000000000.0], N[(N[(N[(t$95$0 / z), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-49], 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[(N[(z - -1.0), $MachinePrecision] * 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))) < -2e10

   1. Initial program 83.2%

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

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 96.2%

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 96.2%

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

   3. Applied rewrites 96.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*l/ N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 87.9%

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

   5. Applied rewrites 87.9%

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

   6. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 57.8%

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

   8. Applied rewrites 57.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 60.3%

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

   10. Applied rewrites 60.3%

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

   if -2e10 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 4.9999999999999999e-49

   1. Initial program 83.2%

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

   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 70.0%

      \[\leadsto \frac{x \cdot y}{\left(z \cdot z\right) \cdot \color{blue}{1}} \]
   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 72.2%

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

      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 72.2%

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

   6. Applied rewrites 72.2%

      \[\leadsto \color{blue}{\frac{x}{\left(1 \cdot z\right) \cdot z} \cdot y} \]
   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 73.6%

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

   8. Applied rewrites 73.6%

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

   if 4.9999999999999999e-49 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 83.2%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 85.0%

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

      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. *-commutative N/A

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

      13. lower-*.f64 85.0%

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 85.0%

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

   3. Applied rewrites 85.0%

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

Alternative 5: 94.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_1 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ t_2 := \frac{t\_0}{\left(\left(z - -1\right) \cdot z\right) \cdot z} \cdot t\_1\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-52}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 10^{-24}:\\ \;\;\;\;\frac{\frac{t\_1}{1 \cdot z}}{z} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \]

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 -1.0) z) z)) t_1)))
  (*
   (copysign 1.0 x)
   (*
    (copysign 1.0 y)
    (if (<= z -5e-52)
      t_2
      (if (<= z 1e-24) (* (/ (/ 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 - -1.0) * z) * z)) * t_1;
	double tmp;
	if (z <= -5e-52) {
		tmp = t_2;
	} else if (z <= 1e-24) {
		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 - -1.0) * z) * z)) * t_1;
	double tmp;
	if (z <= -5e-52) {
		tmp = t_2;
	} else if (z <= 1e-24) {
		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 - -1.0) * z) * z)) * t_1
	tmp = 0
	if z <= -5e-52:
		tmp = t_2
	elif z <= 1e-24:
		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(t_0 / Float64(Float64(Float64(z - -1.0) * z) * z)) * t_1)
	tmp = 0.0
	if (z <= -5e-52)
		tmp = t_2;
	elseif (z <= 1e-24)
		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 - -1.0) * z) * z)) * t_1;
	tmp = 0.0;
	if (z <= -5e-52)
		tmp = t_2;
	elseif (z <= 1e-24)
		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[(t$95$0 / N[(N[(N[(z - -1.0), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * t$95$1), $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[z, -5e-52], t$95$2, If[LessEqual[z, 1e-24], 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 z < -5e-52 or 9.9999999999999992e-25 < z

   1. Initial program 83.2%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 85.0%

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

      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. *-commutative N/A

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

      13. lower-*.f64 85.0%

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 85.0%

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

   3. Applied rewrites 85.0%

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

   if -5e-52 < z < 9.9999999999999992e-25

   1. Initial program 83.2%

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

   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 70.0%

      \[\leadsto \frac{x \cdot y}{\left(z \cdot z\right) \cdot \color{blue}{1}} \]
   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 72.2%

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

      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 72.2%

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

   6. Applied rewrites 72.2%

      \[\leadsto \color{blue}{\frac{x}{\left(1 \cdot z\right) \cdot z} \cdot y} \]
   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 73.6%

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

   8. Applied rewrites 73.6%

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

Alternative 6: 93.2% accurate, 0.1× speedup

Math

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

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(fmax(abs(x), abs(y)) / Float64(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[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision] / N[(N[(z - -1.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * N[(N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.2%

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

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 96.2%

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

   15. lift-+.f64 N/A

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

   16. add-flip N/A

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

   17. lower--.f64 N/A

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

   18. metadata-eval 96.2%

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

3. Applied rewrites 96.2%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l/ N/A

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

   7. lift-/.f64 N/A

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

   8. lower-*.f64 94.1%

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

5. Applied rewrites 94.1%

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

Alternative 7: 92.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_1 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ t_2 := \frac{t\_0}{\left(\left(z - -1\right) \cdot z\right) \cdot z} \cdot t\_1\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-32}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 10^{-24}:\\ \;\;\;\;\frac{t\_0}{z} \cdot \frac{t\_1}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \]

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 -1.0) z) z)) t_1)))
  (*
   (copysign 1.0 x)
   (*
    (copysign 1.0 y)
    (if (<= z -2e-32)
      t_2
      (if (<= z 1e-24) (* (/ t_0 z) (/ t_1 z)) 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 - -1.0) * z) * z)) * t_1;
	double tmp;
	if (z <= -2e-32) {
		tmp = t_2;
	} else if (z <= 1e-24) {
		tmp = (t_0 / z) * (t_1 / z);
	} 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 - -1.0) * z) * z)) * t_1;
	double tmp;
	if (z <= -2e-32) {
		tmp = t_2;
	} else if (z <= 1e-24) {
		tmp = (t_0 / z) * (t_1 / z);
	} 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 - -1.0) * z) * z)) * t_1
	tmp = 0
	if z <= -2e-32:
		tmp = t_2
	elif z <= 1e-24:
		tmp = (t_0 / z) * (t_1 / z)
	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(t_0 / Float64(Float64(Float64(z - -1.0) * z) * z)) * t_1)
	tmp = 0.0
	if (z <= -2e-32)
		tmp = t_2;
	elseif (z <= 1e-24)
		tmp = Float64(Float64(t_0 / z) * Float64(t_1 / z));
	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 - -1.0) * z) * z)) * t_1;
	tmp = 0.0;
	if (z <= -2e-32)
		tmp = t_2;
	elseif (z <= 1e-24)
		tmp = (t_0 / z) * (t_1 / z);
	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[(t$95$0 / N[(N[(N[(z - -1.0), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * t$95$1), $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[z, -2e-32], t$95$2, If[LessEqual[z, 1e-24], N[(N[(t$95$0 / z), $MachinePrecision] * N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision], t$95$2]]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.0000000000000001e-32 or 9.9999999999999992e-25 < z

   1. Initial program 83.2%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 85.0%

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

      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. *-commutative N/A

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

      13. lower-*.f64 85.0%

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 85.0%

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

   3. Applied rewrites 85.0%

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

   if -2.0000000000000001e-32 < z < 9.9999999999999992e-25

   1. Initial program 83.2%

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

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 96.2%

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 96.2%

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

   3. Applied rewrites 96.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 94.1%

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

   5. Applied rewrites 94.1%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 73.9%

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

   8. Applied rewrites 73.9%

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

Alternative 8: 90.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \frac{x}{\left(\left(z - -1\right) \cdot z\right) \cdot z} \cdot y\\ \mathbf{if}\;z \leq -2 \cdot 10^{-32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (* (/ x (* (* (- z -1.0) z) z)) y)))
  (if (<= z -2e-32) t_0 (if (<= z 2e-133) (* (/ y z) (/ x z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x / (((z - -1.0) * z) * z)) * y;
	double tmp;
	if (z <= -2e-32) {
		tmp = t_0;
	} else if (z <= 2e-133) {
		tmp = (y / z) * (x / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / (((z - (-1.0d0)) * z) * z)) * y
    if (z <= (-2d-32)) then
        tmp = t_0
    else if (z <= 2d-133) then
        tmp = (y / z) * (x / z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x / (((z - -1.0) * z) * z)) * y;
	double tmp;
	if (z <= -2e-32) {
		tmp = t_0;
	} else if (z <= 2e-133) {
		tmp = (y / z) * (x / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x / (((z - -1.0) * z) * z)) * y
	tmp = 0
	if z <= -2e-32:
		tmp = t_0
	elif z <= 2e-133:
		tmp = (y / z) * (x / z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x / Float64(Float64(Float64(z - -1.0) * z) * z)) * y)
	tmp = 0.0
	if (z <= -2e-32)
		tmp = t_0;
	elseif (z <= 2e-133)
		tmp = Float64(Float64(y / z) * Float64(x / z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x / (((z - -1.0) * z) * z)) * y;
	tmp = 0.0;
	if (z <= -2e-32)
		tmp = t_0;
	elseif (z <= 2e-133)
		tmp = (y / z) * (x / z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / N[(N[(N[(z - -1.0), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[z, -2e-32], t$95$0, If[LessEqual[z, 2e-133], N[(N[(y / z), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.0000000000000001e-32 or 2.0000000000000001e-133 < z

   1. Initial program 83.2%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lower-/.f64 84.4%

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 84.4%

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

      17. lift-+.f64 N/A

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

      18. add-flip N/A

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

      19. lower--.f64 N/A

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

      20. metadata-eval 84.4%

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

   3. Applied rewrites 84.4%

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

   if -2.0000000000000001e-32 < z < 2.0000000000000001e-133

   1. Initial program 83.2%

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

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 96.2%

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 96.2%

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

   3. Applied rewrites 96.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 94.1%

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

   5. Applied rewrites 94.1%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 73.9%

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

   8. Applied rewrites 73.9%

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

Alternative 9: 80.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_1 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \cdot t\_0 \leq 5 \cdot 10^{-94}:\\ \;\;\;\;\frac{t\_0}{z} \cdot \frac{t\_1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\left(1 \cdot z\right) \cdot z} \cdot t\_0\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fmax (fabs x) (fabs y))) (t_1 (fmin (fabs x) (fabs y))))
  (*
   (copysign 1.0 x)
   (*
    (copysign 1.0 y)
    (if (<= (* t_1 t_0) 5e-94)
      (* (/ t_0 z) (/ t_1 z))
      (* (/ t_1 (* (* 1.0 z) z)) t_0))))))

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 tmp;
	if ((t_1 * t_0) <= 5e-94) {
		tmp = (t_0 / z) * (t_1 / z);
	} else {
		tmp = (t_1 / ((1.0 * z) * z)) * t_0;
	}
	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 tmp;
	if ((t_1 * t_0) <= 5e-94) {
		tmp = (t_0 / z) * (t_1 / z);
	} else {
		tmp = (t_1 / ((1.0 * z) * z)) * t_0;
	}
	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))
	tmp = 0
	if (t_1 * t_0) <= 5e-94:
		tmp = (t_0 / z) * (t_1 / z)
	else:
		tmp = (t_1 / ((1.0 * z) * z)) * t_0
	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))
	tmp = 0.0
	if (Float64(t_1 * t_0) <= 5e-94)
		tmp = Float64(Float64(t_0 / z) * Float64(t_1 / z));
	else
		tmp = Float64(Float64(t_1 / Float64(Float64(1.0 * z) * z)) * t_0);
	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));
	tmp = 0.0;
	if ((t_1 * t_0) <= 5e-94)
		tmp = (t_0 / z) * (t_1 / z);
	else
		tmp = (t_1 / ((1.0 * z) * z)) * t_0;
	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]}, 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[N[(t$95$1 * t$95$0), $MachinePrecision], 5e-94], N[(N[(t$95$0 / z), $MachinePrecision] * N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[(N[(1.0 * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 4.9999999999999995e-94

   1. Initial program 83.2%

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

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 96.2%

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 96.2%

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

   3. Applied rewrites 96.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 94.1%

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

   5. Applied rewrites 94.1%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 73.9%

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

   8. Applied rewrites 73.9%

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

   if 4.9999999999999995e-94 < (*.f64 x y)

   1. Initial program 83.2%

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

   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 70.0%

      \[\leadsto \frac{x \cdot y}{\left(z \cdot z\right) \cdot \color{blue}{1}} \]
   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 72.2%

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

      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 72.2%

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

   6. Applied rewrites 72.2%

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

Alternative 10: 75.2% accurate, 0.1× 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 83.2%

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

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 70.0%

   \[\leadsto \frac{x \cdot y}{\left(z \cdot z\right) \cdot \color{blue}{1}} \]
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 72.2%

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

   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 72.2%

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

6. Applied rewrites 72.2%

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

Reproduce

herbie shell --seed 2025258 
(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))))