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

Percentage Accurate: 83.0% → 98.0%

Time: 12.1s

Alternatives: 9

Speedup: 0.8×

• 74.8% 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))))

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), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 83.0% 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))))

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), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.0% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x y z)
  :precision binary64
  (*
 (copysign 1 x)
 (*
  (copysign 1 y)
  (/
   (*
    (/ (fmax (fabs x) (fabs y)) (- z -1))
    (/ (fmin (fabs x) (fabs y)) z))
   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)) * (fmin(fabs(x), fabs(y)) / z)) / 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)) * (fmin(Math.abs(x), Math.abs(y)) / z)) / 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)) * (fmin(math.fabs(x), math.fabs(y)) / z)) / z))

Julia

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

Wolfram

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

Derivation

1. Initial program 83.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. add-flip N/A

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

   14. lower--.f64 N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 97.0%

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

3. Applied rewrites 97.0%

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

Alternative 2: 96.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_1 := \left(z - -1\right) \cdot z\\ t_2 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_2 \cdot t\_0}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 10000000000000000213204190094543968723012578712679649467743338496:\\ \;\;\;\;\frac{\frac{t\_0}{t\_1}}{z} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_2}{t\_1} \cdot t\_0}{z}\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fmax (fabs x) (fabs y)))
       (t_1 (* (- z -1) z))
       (t_2 (fmin (fabs x) (fabs y))))
  (*
   (copysign 1 x)
   (*
    (copysign 1 y)
    (if (<=
         (/ (* t_2 t_0) (* (* z z) (+ z 1)))
         10000000000000000213204190094543968723012578712679649467743338496)
      (* (/ (/ t_0 t_1) z) t_2)
      (/ (* (/ t_2 t_1) t_0) z))))))

C

double code(double x, double y, double z) {
	double t_0 = fmax(fabs(x), fabs(y));
	double t_1 = (z - -1.0) * z;
	double t_2 = fmin(fabs(x), fabs(y));
	double tmp;
	if (((t_2 * t_0) / ((z * z) * (z + 1.0))) <= 1e+64) {
		tmp = ((t_0 / t_1) / z) * t_2;
	} else {
		tmp = ((t_2 / t_1) * t_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 = (z - -1.0) * z;
	double t_2 = fmin(Math.abs(x), Math.abs(y));
	double tmp;
	if (((t_2 * t_0) / ((z * z) * (z + 1.0))) <= 1e+64) {
		tmp = ((t_0 / t_1) / z) * t_2;
	} else {
		tmp = ((t_2 / t_1) * t_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 = (z - -1.0) * z
	t_2 = fmin(math.fabs(x), math.fabs(y))
	tmp = 0
	if ((t_2 * t_0) / ((z * z) * (z + 1.0))) <= 1e+64:
		tmp = ((t_0 / t_1) / z) * t_2
	else:
		tmp = ((t_2 / t_1) * t_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 = Float64(Float64(z - -1.0) * z)
	t_2 = fmin(abs(x), abs(y))
	tmp = 0.0
	if (Float64(Float64(t_2 * t_0) / Float64(Float64(z * z) * Float64(z + 1.0))) <= 1e+64)
		tmp = Float64(Float64(Float64(t_0 / t_1) / z) * t_2);
	else
		tmp = Float64(Float64(Float64(t_2 / t_1) * t_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 = (z - -1.0) * z;
	t_2 = min(abs(x), abs(y));
	tmp = 0.0;
	if (((t_2 * t_0) / ((z * z) * (z + 1.0))) <= 1e+64)
		tmp = ((t_0 / t_1) / z) * t_2;
	else
		tmp = ((t_2 / t_1) * t_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[(N[(z - -1), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(t$95$2 * t$95$0), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 10000000000000000213204190094543968723012578712679649467743338496], N[(N[(N[(t$95$0 / t$95$1), $MachinePrecision] / z), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(N[(t$95$2 / t$95$1), $MachinePrecision] * t$95$0), $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)))) < 1e64

   1. Initial program 83.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 84.7%

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 84.7%

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

      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 84.7%

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

   3. Applied rewrites 84.7%

      \[\leadsto \color{blue}{\frac{y}{\left(\left(z - -1\right) \cdot z\right) \cdot z} \cdot x} \]
   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. lower-/.f64 N/A

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

      5. lower-/.f64 90.9%

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

   5. Applied rewrites 90.9%

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

   if 1e64 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 83.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. lower--.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 97.0%

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

   3. Applied rewrites 97.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 97.0%

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

   5. Applied rewrites 97.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

      12. frac-times N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. *-rgt-identity N/A

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

      16. *-commutative N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 94.2%

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

   7. Applied rewrites 94.2%

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

Alternative 3: 95.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \left(z - -1\right) \cdot z\\ \mathbf{if}\;\mathsf{max}\left(x, y\right) \leq 20000000000000001016445696805993759409582178897019679576898416057743923428824704540156776745107920382581920574891563668662589154296936754315264:\\ \;\;\;\;\frac{\mathsf{max}\left(x, y\right)}{z} \cdot \frac{\mathsf{min}\left(x, y\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{min}\left(x, y\right) \cdot \frac{\mathsf{max}\left(x, y\right)}{t\_0}}{z}\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (* (- z -1) z)))
  (if (<=
       (fmax x y)
       20000000000000001016445696805993759409582178897019679576898416057743923428824704540156776745107920382581920574891563668662589154296936754315264)
    (* (/ (fmax x y) z) (/ (fmin x y) t_0))
    (/ (* (fmin x y) (/ (fmax x y) t_0)) z))))

C

double code(double x, double y, double z) {
	double t_0 = (z - -1.0) * z;
	double tmp;
	if (fmax(x, y) <= 2e+142) {
		tmp = (fmax(x, y) / z) * (fmin(x, y) / t_0);
	} else {
		tmp = (fmin(x, y) * (fmax(x, y) / t_0)) / z;
	}
	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 = (z - (-1.0d0)) * z
    if (fmax(x, y) <= 2d+142) then
        tmp = (fmax(x, y) / z) * (fmin(x, y) / t_0)
    else
        tmp = (fmin(x, y) * (fmax(x, y) / t_0)) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (z - -1.0) * z;
	double tmp;
	if (fmax(x, y) <= 2e+142) {
		tmp = (fmax(x, y) / z) * (fmin(x, y) / t_0);
	} else {
		tmp = (fmin(x, y) * (fmax(x, y) / t_0)) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z - -1.0) * z
	tmp = 0
	if fmax(x, y) <= 2e+142:
		tmp = (fmax(x, y) / z) * (fmin(x, y) / t_0)
	else:
		tmp = (fmin(x, y) * (fmax(x, y) / t_0)) / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z - -1.0) * z)
	tmp = 0.0
	if (fmax(x, y) <= 2e+142)
		tmp = Float64(Float64(fmax(x, y) / z) * Float64(fmin(x, y) / t_0));
	else
		tmp = Float64(Float64(fmin(x, y) * Float64(fmax(x, y) / t_0)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z - -1.0) * z;
	tmp = 0.0;
	if (max(x, y) <= 2e+142)
		tmp = (max(x, y) / z) * (min(x, y) / t_0);
	else
		tmp = (min(x, y) * (max(x, y) / t_0)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z - -1), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[N[Max[x, y], $MachinePrecision], 20000000000000001016445696805993759409582178897019679576898416057743923428824704540156776745107920382581920574891563668662589154296936754315264], N[(N[(N[Max[x, y], $MachinePrecision] / z), $MachinePrecision] * N[(N[Min[x, y], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Min[x, y], $MachinePrecision] * N[(N[Max[x, y], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.0000000000000001e142

   1. Initial program 83.0%

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 94.5%

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

      13. lift-+.f64 N/A

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

      14. add-flip N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 94.5%

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

   3. Applied rewrites 94.5%

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

   if 2.0000000000000001e142 < y

   1. Initial program 83.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 94.4%

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

      13. lift-+.f64 N/A

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

      14. add-flip N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 94.4%

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

   3. Applied rewrites 94.4%

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

Alternative 4: 95.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_1 := \left(z - -1\right) \cdot z\\ t_2 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_2 \cdot t\_0}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq \frac{4784065733063811}{4784065733063810973581885157618788676291241975216665977767007373648750357731006099232824032039924032894289638403441329240212719241920971274455782595989040464660523567661989180298099889009174801022976}:\\ \;\;\;\;\frac{\frac{t\_0}{t\_1}}{z} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{z} \cdot \frac{t\_2}{t\_1}\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fmax (fabs x) (fabs y)))
       (t_1 (* (- z -1) z))
       (t_2 (fmin (fabs x) (fabs y))))
  (*
   (copysign 1 x)
   (*
    (copysign 1 y)
    (if (<=
         (/ (* t_2 t_0) (* (* z z) (+ z 1)))
         4784065733063811/4784065733063810973581885157618788676291241975216665977767007373648750357731006099232824032039924032894289638403441329240212719241920971274455782595989040464660523567661989180298099889009174801022976)
      (* (/ (/ t_0 t_1) z) t_2)
      (* (/ t_0 z) (/ t_2 t_1)))))))

C

double code(double x, double y, double z) {
	double t_0 = fmax(fabs(x), fabs(y));
	double t_1 = (z - -1.0) * z;
	double t_2 = fmin(fabs(x), fabs(y));
	double tmp;
	if (((t_2 * t_0) / ((z * z) * (z + 1.0))) <= 1e-183) {
		tmp = ((t_0 / t_1) / z) * t_2;
	} else {
		tmp = (t_0 / z) * (t_2 / 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 = (z - -1.0) * z;
	double t_2 = fmin(Math.abs(x), Math.abs(y));
	double tmp;
	if (((t_2 * t_0) / ((z * z) * (z + 1.0))) <= 1e-183) {
		tmp = ((t_0 / t_1) / z) * t_2;
	} else {
		tmp = (t_0 / z) * (t_2 / 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 = (z - -1.0) * z
	t_2 = fmin(math.fabs(x), math.fabs(y))
	tmp = 0
	if ((t_2 * t_0) / ((z * z) * (z + 1.0))) <= 1e-183:
		tmp = ((t_0 / t_1) / z) * t_2
	else:
		tmp = (t_0 / z) * (t_2 / 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 = Float64(Float64(z - -1.0) * z)
	t_2 = fmin(abs(x), abs(y))
	tmp = 0.0
	if (Float64(Float64(t_2 * t_0) / Float64(Float64(z * z) * Float64(z + 1.0))) <= 1e-183)
		tmp = Float64(Float64(Float64(t_0 / t_1) / z) * t_2);
	else
		tmp = Float64(Float64(t_0 / z) * Float64(t_2 / 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 = (z - -1.0) * z;
	t_2 = min(abs(x), abs(y));
	tmp = 0.0;
	if (((t_2 * t_0) / ((z * z) * (z + 1.0))) <= 1e-183)
		tmp = ((t_0 / t_1) / z) * t_2;
	else
		tmp = (t_0 / z) * (t_2 / 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[(N[(z - -1), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(t$95$2 * t$95$0), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4784065733063811/4784065733063810973581885157618788676291241975216665977767007373648750357731006099232824032039924032894289638403441329240212719241920971274455782595989040464660523567661989180298099889009174801022976], N[(N[(N[(t$95$0 / t$95$1), $MachinePrecision] / z), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(t$95$0 / z), $MachinePrecision] * N[(t$95$2 / t$95$1), $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)))) < 1e-183

   1. Initial program 83.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 84.7%

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 84.7%

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

      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 84.7%

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

   3. Applied rewrites 84.7%

      \[\leadsto \color{blue}{\frac{y}{\left(\left(z - -1\right) \cdot z\right) \cdot z} \cdot x} \]
   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. lower-/.f64 N/A

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

      5. lower-/.f64 90.9%

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

   5. Applied rewrites 90.9%

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

   if 1e-183 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 83.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 94.5%

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

      13. lift-+.f64 N/A

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

      14. add-flip N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 94.5%

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

   3. Applied rewrites 94.5%

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

Alternative 5: 94.5% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y z)
  :precision binary64
  (* (/ y z) (/ x (* (- z -1) z))))

C

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

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 = (y / z) * (x / ((z - (-1.0d0)) * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(y / z), $MachinePrecision] * N[(x / N[(N[(z - -1), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.0%

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

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 94.5%

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

   13. lift-+.f64 N/A

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

   14. add-flip N/A

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

   15. lower--.f64 N/A

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

   16. metadata-eval 94.5%

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

3. Applied rewrites 94.5%

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

Alternative 6: 90.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_1 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1 \cdot t\_0}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 499999999999999980914845420907469931724617668138392575722702061727550202027827845338095855082297280184351144790266035545655630691827712:\\ \;\;\;\;\frac{t\_0}{\left(\left(z - -1\right) \cdot z\right) \cdot z} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{z} \cdot t\_0}{z}\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fmax (fabs x) (fabs y))) (t_1 (fmin (fabs x) (fabs y))))
  (*
   (copysign 1 x)
   (*
    (copysign 1 y)
    (if (<=
         (/ (* t_1 t_0) (* (* z z) (+ z 1)))
         499999999999999980914845420907469931724617668138392575722702061727550202027827845338095855082297280184351144790266035545655630691827712)
      (* (/ t_0 (* (* (- z -1) z) z)) t_1)
      (/ (* (/ t_1 z) t_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+134) {
		tmp = (t_0 / (((z - -1.0) * z) * z)) * t_1;
	} else {
		tmp = ((t_1 / z) * t_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+134) {
		tmp = (t_0 / (((z - -1.0) * z) * z)) * t_1;
	} else {
		tmp = ((t_1 / z) * t_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+134:
		tmp = (t_0 / (((z - -1.0) * z) * z)) * t_1
	else:
		tmp = ((t_1 / z) * t_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+134)
		tmp = Float64(Float64(t_0 / Float64(Float64(Float64(z - -1.0) * z) * z)) * t_1);
	else
		tmp = Float64(Float64(Float64(t_1 / z) * t_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+134)
		tmp = (t_0 / (((z - -1.0) * z) * z)) * t_1;
	else
		tmp = ((t_1 / z) * t_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], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1], 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), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 499999999999999980914845420907469931724617668138392575722702061727550202027827845338095855082297280184351144790266035545655630691827712], N[(N[(t$95$0 / N[(N[(N[(z - -1), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(t$95$1 / z), $MachinePrecision] * t$95$0), $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)))) < 4.9999999999999998e134

   1. Initial program 83.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 84.7%

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 84.7%

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

      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 84.7%

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

   3. Applied rewrites 84.7%

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

   if 4.9999999999999998e134 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 83.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. lower--.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 97.0%

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

   3. Applied rewrites 97.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 97.0%

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

   5. Applied rewrites 97.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

      12. frac-times N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. *-rgt-identity N/A

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

      16. *-commutative N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 94.2%

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

   7. Applied rewrites 94.2%

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

   8. Taylor expanded in z around 0

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

      1. lower-/.f64 74.2%

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

   10. Applied rewrites 74.2%

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

Alternative 7: 88.7% accurate, 0.1× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{\left|x\right| \cdot \left|y\right|}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 399999999999999987819614071793278008371859228978996844859369901040438779531494853410754618301220342856148729636899364538023571524733514824321012998076331615720432379158562133553406184339792027801304062955170675602258086854656:\\ \;\;\;\;\frac{\left|x\right|}{\left(\left(z - -1\right) \cdot z\right) \cdot z} \cdot \left|y\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left|x\right|}{z} \cdot \left|y\right|}{z}\\ \end{array}\right) \]

FPCore

(FPCore (x y z)
  :precision binary64
  (*
 (copysign 1 x)
 (*
  (copysign 1 y)
  (if (<=
       (/ (* (fabs x) (fabs y)) (* (* z z) (+ z 1)))
       399999999999999987819614071793278008371859228978996844859369901040438779531494853410754618301220342856148729636899364538023571524733514824321012998076331615720432379158562133553406184339792027801304062955170675602258086854656)
    (* (/ (fabs x) (* (* (- z -1) z) z)) (fabs y))
    (/ (* (/ (fabs x) z) (fabs y)) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((fabs(x) * fabs(y)) / ((z * z) * (z + 1.0))) <= 4e+224) {
		tmp = (fabs(x) / (((z - -1.0) * z) * z)) * fabs(y);
	} else {
		tmp = ((fabs(x) / z) * fabs(y)) / z;
	}
	return copysign(1.0, x) * (copysign(1.0, y) * tmp);
}

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((Math.abs(x) * Math.abs(y)) / ((z * z) * (z + 1.0))) <= 4e+224) {
		tmp = (Math.abs(x) / (((z - -1.0) * z) * z)) * Math.abs(y);
	} else {
		tmp = ((Math.abs(x) / z) * Math.abs(y)) / z;
	}
	return Math.copySign(1.0, x) * (Math.copySign(1.0, y) * tmp);
}

Python

def code(x, y, z):
	tmp = 0
	if ((math.fabs(x) * math.fabs(y)) / ((z * z) * (z + 1.0))) <= 4e+224:
		tmp = (math.fabs(x) / (((z - -1.0) * z) * z)) * math.fabs(y)
	else:
		tmp = ((math.fabs(x) / z) * math.fabs(y)) / z
	return math.copysign(1.0, x) * (math.copysign(1.0, y) * tmp)

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(abs(x) * abs(y)) / Float64(Float64(z * z) * Float64(z + 1.0))) <= 4e+224)
		tmp = Float64(Float64(abs(x) / Float64(Float64(Float64(z - -1.0) * z) * z)) * abs(y));
	else
		tmp = Float64(Float64(Float64(abs(x) / z) * abs(y)) / z);
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * tmp))
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((abs(x) * abs(y)) / ((z * z) * (z + 1.0))) <= 4e+224)
		tmp = (abs(x) / (((z - -1.0) * z) * z)) * abs(y);
	else
		tmp = ((abs(x) / z) * abs(y)) / z;
	end
	tmp_2 = (sign(x) * abs(1.0)) * ((sign(y) * abs(1.0)) * tmp);
end

Wolfram

code[x_, y_, z_] := N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[y], $MachinePrecision]), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 399999999999999987819614071793278008371859228978996844859369901040438779531494853410754618301220342856148729636899364538023571524733514824321012998076331615720432379158562133553406184339792027801304062955170675602258086854656], N[(N[(N[Abs[x], $MachinePrecision] / N[(N[(N[(z - -1), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * N[Abs[y], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Abs[x], $MachinePrecision] / z), $MachinePrecision] * N[Abs[y], $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)))) < 3.9999999999999999e224

   1. Initial program 83.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. lower--.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 97.0%

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

   3. Applied rewrites 97.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lift-/.f64 N/A

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

      13. mult-flip N/A

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

      14. associate-*l* N/A

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

      15. mult-flip N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. associate-/l/ N/A

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

      20. lift-*.f64 N/A

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

      21. lower-/.f64 84.5%

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

   5. Applied rewrites 84.5%

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

   if 3.9999999999999999e224 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 83.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. lower--.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 97.0%

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

   3. Applied rewrites 97.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 97.0%

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

   5. Applied rewrites 97.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

      12. frac-times N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. *-rgt-identity N/A

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

      16. *-commutative N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 94.2%

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

   7. Applied rewrites 94.2%

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

   8. Taylor expanded in z around 0

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

      1. lower-/.f64 74.2%

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

   10. Applied rewrites 74.2%

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

Alternative 8: 80.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_1 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1 \cdot t\_0}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq \frac{1668739871813211}{8343699359066055009355553539724812947666814540455674882605631280555545803830627148527195652096}:\\ \;\;\;\;\frac{t\_1}{\left(1 \cdot z\right) \cdot z} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{z} \cdot t\_0}{z}\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fmax (fabs x) (fabs y))) (t_1 (fmin (fabs x) (fabs y))))
  (*
   (copysign 1 x)
   (*
    (copysign 1 y)
    (if (<=
         (/ (* t_1 t_0) (* (* z z) (+ z 1)))
         1668739871813211/8343699359066055009355553539724812947666814540455674882605631280555545803830627148527195652096)
      (* (/ t_1 (* (* 1 z) z)) t_0)
      (/ (* (/ t_1 z) t_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))) <= 2e-79) {
		tmp = (t_1 / ((1.0 * z) * z)) * t_0;
	} else {
		tmp = ((t_1 / z) * t_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))) <= 2e-79) {
		tmp = (t_1 / ((1.0 * z) * z)) * t_0;
	} else {
		tmp = ((t_1 / z) * t_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))) <= 2e-79:
		tmp = (t_1 / ((1.0 * z) * z)) * t_0
	else:
		tmp = ((t_1 / z) * t_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))) <= 2e-79)
		tmp = Float64(Float64(t_1 / Float64(Float64(1.0 * z) * z)) * t_0);
	else
		tmp = Float64(Float64(Float64(t_1 / z) * t_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))) <= 2e-79)
		tmp = (t_1 / ((1.0 * z) * z)) * t_0;
	else
		tmp = ((t_1 / z) * t_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], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1], 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), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1668739871813211/8343699359066055009355553539724812947666814540455674882605631280555545803830627148527195652096], N[(N[(t$95$1 / N[(N[(1 * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(t$95$1 / z), $MachinePrecision] * t$95$0), $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)))) < 2e-79

   1. Initial program 83.0%

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

   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 70.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lower-/.f64 72.4%

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 72.4%

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

   6. Applied rewrites 72.4%

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

   if 2e-79 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 83.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. lower--.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 97.0%

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

   3. Applied rewrites 97.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 97.0%

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

   5. Applied rewrites 97.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

      12. frac-times N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. *-rgt-identity N/A

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

      16. *-commutative N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 94.2%

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

   7. Applied rewrites 94.2%

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

   8. Taylor expanded in z around 0

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

      1. lower-/.f64 74.2%

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

   10. Applied rewrites 74.2%

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

Alternative 9: 75.5% 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 x)
 (*
  (copysign 1 y)
  (*
   (/ (fmin (fabs x) (fabs y)) (* (* 1 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], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1], 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 * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.0%

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

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 70.5%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. mult-flip-rev N/A

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

   9. lower-/.f64 72.4%

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 72.4%

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

6. Applied rewrites 72.4%

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

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (/ (* x y) (* (* z z) (+ z 1))))