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

Percentage Accurate: 83.2% → 98.5%

Time: 2.9s

Alternatives: 11

Speedup: 0.4×

• 25.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_1 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1 \cdot t\_0}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{t\_0}{z}}{\mathsf{fma}\left(z, z, z\right)} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \frac{\frac{t\_1}{z - -1}}{z}}{z}\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fmax (fabs x) (fabs y))) (t_1 (fmin (fabs x) (fabs y))))
   (*
    (copysign 1.0 x)
    (*
     (copysign 1.0 y)
     (if (<= (/ (* t_1 t_0) (* (* z z) (+ z 1.0))) 2e-133)
       (* (/ (/ t_0 z) (fma z z z)) t_1)
       (/ (* t_0 (/ (/ t_1 (- z -1.0)) z)) z))))))

C

double code(double x, double y, double z) {
	double t_0 = fmax(fabs(x), fabs(y));
	double t_1 = fmin(fabs(x), fabs(y));
	double tmp;
	if (((t_1 * t_0) / ((z * z) * (z + 1.0))) <= 2e-133) {
		tmp = ((t_0 / z) / fma(z, z, z)) * t_1;
	} else {
		tmp = (t_0 * ((t_1 / (z - -1.0)) / z)) / z;
	}
	return copysign(1.0, x) * (copysign(1.0, y) * tmp);
}

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-133)
		tmp = Float64(Float64(Float64(t_0 / z) / fma(z, z, z)) * t_1);
	else
		tmp = Float64(Float64(t_0 * Float64(Float64(t_1 / Float64(z - -1.0)) / z)) / z);
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * tmp))
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(t$95$1 * t$95$0), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-133], N[(N[(N[(t$95$0 / z), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(t$95$0 * N[(N[(t$95$1 / N[(z - -1.0), $MachinePrecision]), $MachinePrecision] / z), $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)))) < 2.0000000000000001e-133

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 84.8

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 84.8

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

   3. Applied rewrites 84.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. lower-/.f64 90.5

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

   5. Applied rewrites 90.5%

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

   if 2.0000000000000001e-133 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 94.7

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

   3. Applied rewrites 94.7%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. add-flip N/A

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

      5. metadata-eval N/A

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

      6. lift--.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 96.7

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

   5. Applied rewrites 96.7%

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

Alternative 2: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_1 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ t_2 := \frac{\frac{t\_0}{z}}{\mathsf{fma}\left(z, z, z\right)} \cdot t\_1\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-19}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 1.86 \cdot 10^{-75}:\\ \;\;\;\;\frac{t\_0 \cdot \frac{t\_1}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fmax (fabs x) (fabs y)))
        (t_1 (fmin (fabs x) (fabs y)))
        (t_2 (* (/ (/ t_0 z) (fma z z z)) t_1)))
   (*
    (copysign 1.0 x)
    (*
     (copysign 1.0 y)
     (if (<= z -1e-19)
       t_2
       (if (<= z 1.86e-75) (/ (* t_0 (/ t_1 z)) z) t_2))))))

C

double code(double x, double y, double z) {
	double t_0 = fmax(fabs(x), fabs(y));
	double t_1 = fmin(fabs(x), fabs(y));
	double t_2 = ((t_0 / z) / fma(z, z, z)) * t_1;
	double tmp;
	if (z <= -1e-19) {
		tmp = t_2;
	} else if (z <= 1.86e-75) {
		tmp = (t_0 * (t_1 / z)) / z;
	} else {
		tmp = t_2;
	}
	return copysign(1.0, x) * (copysign(1.0, y) * tmp);
}

Julia

function code(x, y, z)
	t_0 = fmax(abs(x), abs(y))
	t_1 = fmin(abs(x), abs(y))
	t_2 = Float64(Float64(Float64(t_0 / z) / fma(z, z, z)) * t_1)
	tmp = 0.0
	if (z <= -1e-19)
		tmp = t_2;
	elseif (z <= 1.86e-75)
		tmp = Float64(Float64(t_0 * Float64(t_1 / z)) / z);
	else
		tmp = t_2;
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * tmp))
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 / z), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[z, -1e-19], t$95$2, If[LessEqual[z, 1.86e-75], N[(N[(t$95$0 * N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$2]]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.9999999999999998e-20 or 1.86e-75 < z

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 84.8

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 84.8

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

   3. Applied rewrites 84.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. lower-/.f64 90.5

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

   5. Applied rewrites 90.5%

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

   if -9.9999999999999998e-20 < z < 1.86e-75

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 94.7

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

   3. Applied rewrites 94.7%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. add-flip N/A

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

      5. metadata-eval N/A

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

      6. lift--.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 74.1

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

   8. Applied rewrites 74.1%

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

Alternative 3: 97.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_1 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1 \cdot t\_0}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{t\_0}{z}}{\mathsf{fma}\left(z, z, z\right)} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \frac{t\_1}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fmax (fabs x) (fabs y))) (t_1 (fmin (fabs x) (fabs y))))
   (*
    (copysign 1.0 x)
    (*
     (copysign 1.0 y)
     (if (<= (/ (* t_1 t_0) (* (* z z) (+ z 1.0))) 2e-133)
       (* (/ (/ t_0 z) (fma z z z)) t_1)
       (/ (* t_0 (/ t_1 (fma z z z))) z))))))

C

double code(double x, double y, double z) {
	double t_0 = fmax(fabs(x), fabs(y));
	double t_1 = fmin(fabs(x), fabs(y));
	double tmp;
	if (((t_1 * t_0) / ((z * z) * (z + 1.0))) <= 2e-133) {
		tmp = ((t_0 / z) / fma(z, z, z)) * t_1;
	} else {
		tmp = (t_0 * (t_1 / fma(z, z, z))) / z;
	}
	return copysign(1.0, x) * (copysign(1.0, y) * tmp);
}

Julia

function code(x, y, z)
	t_0 = fmax(abs(x), abs(y))
	t_1 = fmin(abs(x), abs(y))
	tmp = 0.0
	if (Float64(Float64(t_1 * t_0) / Float64(Float64(z * z) * Float64(z + 1.0))) <= 2e-133)
		tmp = Float64(Float64(Float64(t_0 / z) / fma(z, z, z)) * t_1);
	else
		tmp = Float64(Float64(t_0 * Float64(t_1 / fma(z, z, z))) / z);
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * tmp))
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(t$95$1 * t$95$0), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-133], N[(N[(N[(t$95$0 / z), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(t$95$0 * N[(t$95$1 / N[(z * z + z), $MachinePrecision]), $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)))) < 2.0000000000000001e-133

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 84.8

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 84.8

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

   3. Applied rewrites 84.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. lower-/.f64 90.5

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

   5. Applied rewrites 90.5%

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

   if 2.0000000000000001e-133 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 94.7

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

   3. Applied rewrites 94.7%

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

Alternative 4: 96.5% accurate, 0.4× 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 1.0 x)
  (*
   (copysign 1.0 y)
   (/
    (* (/ (fmax (fabs x) (fabs y)) (- z -1.0)) (/ (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[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[(N[(N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision] / N[(z - -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   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 97.3

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

3. Applied rewrites 97.3%

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

Alternative 5: 93.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|x\right|, y\right)\\ t_1 := \mathsf{max}\left(\left|x\right|, y\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-19}:\\ \;\;\;\;\frac{t\_1}{z} \cdot \frac{t\_0}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{elif}\;z \leq 1.86 \cdot 10^{-75}:\\ \;\;\;\;\frac{t\_1 \cdot \frac{t\_0}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fmin (fabs x) y)) (t_1 (fmax (fabs x) y)))
   (*
    (copysign 1.0 x)
    (if (<= z -2e-19)
      (* (/ t_1 z) (/ t_0 (fma z z z)))
      (if (<= z 1.86e-75)
        (/ (* t_1 (/ t_0 z)) z)
        (* (/ t_1 (* (fma z z z) z)) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = fmin(fabs(x), y);
	double t_1 = fmax(fabs(x), y);
	double tmp;
	if (z <= -2e-19) {
		tmp = (t_1 / z) * (t_0 / fma(z, z, z));
	} else if (z <= 1.86e-75) {
		tmp = (t_1 * (t_0 / z)) / z;
	} else {
		tmp = (t_1 / (fma(z, z, z) * z)) * t_0;
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x, y, z)
	t_0 = fmin(abs(x), y)
	t_1 = fmax(abs(x), y)
	tmp = 0.0
	if (z <= -2e-19)
		tmp = Float64(Float64(t_1 / z) * Float64(t_0 / fma(z, z, z)));
	elseif (z <= 1.86e-75)
		tmp = Float64(Float64(t_1 * Float64(t_0 / z)) / z);
	else
		tmp = Float64(Float64(t_1 / Float64(fma(z, z, z) * z)) * t_0);
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Min[N[Abs[x], $MachinePrecision], y], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[x], $MachinePrecision], y], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[z, -2e-19], N[(N[(t$95$1 / z), $MachinePrecision] * N[(t$95$0 / N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.86e-75], N[(N[(t$95$1 * N[(t$95$0 / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(t$95$1 / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2e-19

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 94.7

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

   3. Applied rewrites 94.7%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. add-flip N/A

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

      5. metadata-eval N/A

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

      6. lift--.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 96.7

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

   5. Applied rewrites 96.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 96.6

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lift--.f64 N/A

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

      10. metadata-eval N/A

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

      11. add-flip N/A

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

      12. distribute-lft1-in N/A

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

      13. lift-fma.f64 N/A

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

      14. lower-/.f64 94.5

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

   7. Applied rewrites 94.5%

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

   if -2e-19 < z < 1.86e-75

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 94.7

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

   3. Applied rewrites 94.7%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. add-flip N/A

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

      5. metadata-eval N/A

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

      6. lift--.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 74.1

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

   8. Applied rewrites 74.1%

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

   if 1.86e-75 < z

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 84.8

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 84.8

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

   3. Applied rewrites 84.8%

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

Alternative 6: 93.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_1 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ t_2 := \frac{t\_0}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot t\_1\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-19}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 1.86 \cdot 10^{-75}:\\ \;\;\;\;\frac{t\_0 \cdot \frac{t\_1}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fmax (fabs x) (fabs y)))
        (t_1 (fmin (fabs x) (fabs y)))
        (t_2 (* (/ t_0 (* (fma z z z) z)) t_1)))
   (*
    (copysign 1.0 x)
    (*
     (copysign 1.0 y)
     (if (<= z -1e-19)
       t_2
       (if (<= z 1.86e-75) (/ (* t_0 (/ t_1 z)) z) t_2))))))

C

double code(double x, double y, double z) {
	double t_0 = fmax(fabs(x), fabs(y));
	double t_1 = fmin(fabs(x), fabs(y));
	double t_2 = (t_0 / (fma(z, z, z) * z)) * t_1;
	double tmp;
	if (z <= -1e-19) {
		tmp = t_2;
	} else if (z <= 1.86e-75) {
		tmp = (t_0 * (t_1 / z)) / z;
	} else {
		tmp = t_2;
	}
	return copysign(1.0, x) * (copysign(1.0, y) * tmp);
}

Julia

function code(x, y, z)
	t_0 = fmax(abs(x), abs(y))
	t_1 = fmin(abs(x), abs(y))
	t_2 = Float64(Float64(t_0 / Float64(fma(z, z, z) * z)) * t_1)
	tmp = 0.0
	if (z <= -1e-19)
		tmp = t_2;
	elseif (z <= 1.86e-75)
		tmp = Float64(Float64(t_0 * Float64(t_1 / z)) / z);
	else
		tmp = t_2;
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * tmp))
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[z, -1e-19], t$95$2, If[LessEqual[z, 1.86e-75], N[(N[(t$95$0 * N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$2]]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.9999999999999998e-20 or 1.86e-75 < z

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 84.8

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 84.8

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

   3. Applied rewrites 84.8%

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

   if -9.9999999999999998e-20 < z < 1.86e-75

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 94.7

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

   3. Applied rewrites 94.7%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. add-flip N/A

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

      5. metadata-eval N/A

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

      6. lift--.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 74.1

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

   8. Applied rewrites 74.1%

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

Alternative 7: 89.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_1 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1 \cdot t\_0}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 4.68 \cdot 10^{+277}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \frac{t\_1}{z}}{z}\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fmax (fabs x) (fabs y))) (t_1 (fmin (fabs x) (fabs y))))
   (*
    (copysign 1.0 x)
    (*
     (copysign 1.0 y)
     (if (<= (/ (* t_1 t_0) (* (* z z) (+ z 1.0))) 4.68e+277)
       (* (/ t_1 (* (fma z z z) z)) t_0)
       (/ (* t_0 (/ t_1 z)) z))))))

C

double code(double x, double y, double z) {
	double t_0 = fmax(fabs(x), fabs(y));
	double t_1 = fmin(fabs(x), fabs(y));
	double tmp;
	if (((t_1 * t_0) / ((z * z) * (z + 1.0))) <= 4.68e+277) {
		tmp = (t_1 / (fma(z, z, z) * z)) * t_0;
	} else {
		tmp = (t_0 * (t_1 / z)) / z;
	}
	return copysign(1.0, x) * (copysign(1.0, y) * tmp);
}

Julia

function code(x, y, z)
	t_0 = fmax(abs(x), abs(y))
	t_1 = fmin(abs(x), abs(y))
	tmp = 0.0
	if (Float64(Float64(t_1 * t_0) / Float64(Float64(z * z) * Float64(z + 1.0))) <= 4.68e+277)
		tmp = Float64(Float64(t_1 / Float64(fma(z, z, z) * z)) * t_0);
	else
		tmp = Float64(Float64(t_0 * Float64(t_1 / z)) / z);
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * tmp))
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(t$95$1 * t$95$0), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.68e+277], N[(N[(t$95$1 / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(t$95$0 * N[(t$95$1 / z), $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)))) < 4.67999999999999988e277

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lower-/.f64 84.5

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. distribute-lft-in N/A

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

      17. *-rgt-identity N/A

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

      18. lower-fma.f64 84.5

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

   3. Applied rewrites 84.5%

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

   if 4.67999999999999988e277 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 94.7

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

   3. Applied rewrites 94.7%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. add-flip N/A

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

      5. metadata-eval N/A

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

      6. lift--.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 74.1

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

   8. Applied rewrites 74.1%

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

Alternative 8: 79.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_1 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1 \cdot t\_0}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 10^{-147}:\\ \;\;\;\;\frac{t\_1}{\left(1 \cdot z\right) \cdot z} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \frac{t\_1}{z}}{z}\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fmax (fabs x) (fabs y))) (t_1 (fmin (fabs x) (fabs y))))
   (*
    (copysign 1.0 x)
    (*
     (copysign 1.0 y)
     (if (<= (/ (* t_1 t_0) (* (* z z) (+ z 1.0))) 1e-147)
       (* (/ t_1 (* (* 1.0 z) z)) t_0)
       (/ (* t_0 (/ t_1 z)) z))))))

C

double code(double x, double y, double z) {
	double t_0 = fmax(fabs(x), fabs(y));
	double t_1 = fmin(fabs(x), fabs(y));
	double tmp;
	if (((t_1 * t_0) / ((z * z) * (z + 1.0))) <= 1e-147) {
		tmp = (t_1 / ((1.0 * z) * z)) * t_0;
	} else {
		tmp = (t_0 * (t_1 / z)) / z;
	}
	return copysign(1.0, x) * (copysign(1.0, y) * tmp);
}

Java

public static double code(double x, double y, double z) {
	double t_0 = fmax(Math.abs(x), Math.abs(y));
	double t_1 = fmin(Math.abs(x), Math.abs(y));
	double tmp;
	if (((t_1 * t_0) / ((z * z) * (z + 1.0))) <= 1e-147) {
		tmp = (t_1 / ((1.0 * z) * z)) * t_0;
	} else {
		tmp = (t_0 * (t_1 / z)) / z;
	}
	return Math.copySign(1.0, x) * (Math.copySign(1.0, y) * tmp);
}

Python

def code(x, y, z):
	t_0 = fmax(math.fabs(x), math.fabs(y))
	t_1 = fmin(math.fabs(x), math.fabs(y))
	tmp = 0
	if ((t_1 * t_0) / ((z * z) * (z + 1.0))) <= 1e-147:
		tmp = (t_1 / ((1.0 * z) * z)) * t_0
	else:
		tmp = (t_0 * (t_1 / z)) / z
	return math.copysign(1.0, x) * (math.copysign(1.0, y) * tmp)

Julia

function code(x, y, z)
	t_0 = fmax(abs(x), abs(y))
	t_1 = fmin(abs(x), abs(y))
	tmp = 0.0
	if (Float64(Float64(t_1 * t_0) / Float64(Float64(z * z) * Float64(z + 1.0))) <= 1e-147)
		tmp = Float64(Float64(t_1 / Float64(Float64(1.0 * z) * z)) * t_0);
	else
		tmp = Float64(Float64(t_0 * Float64(t_1 / z)) / z);
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * tmp))
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = max(abs(x), abs(y));
	t_1 = min(abs(x), abs(y));
	tmp = 0.0;
	if (((t_1 * t_0) / ((z * z) * (z + 1.0))) <= 1e-147)
		tmp = (t_1 / ((1.0 * z) * z)) * t_0;
	else
		tmp = (t_0 * (t_1 / z)) / z;
	end
	tmp_2 = (sign(x) * abs(1.0)) * ((sign(y) * abs(1.0)) * tmp);
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(t$95$1 * t$95$0), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-147], N[(N[(t$95$1 / N[(N[(1.0 * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(t$95$0 * N[(t$95$1 / z), $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)))) < 9.9999999999999997e-148

   1. Initial program 83.2%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 69.9%

      \[\leadsto \frac{x \cdot y}{\left(z \cdot z\right) \cdot \color{blue}{1}} \]
   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 72.1

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

      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 72.1

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

   6. Applied rewrites 72.1%

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

   if 9.9999999999999997e-148 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 94.7

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

   3. Applied rewrites 94.7%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. add-flip N/A

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

      5. metadata-eval N/A

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

      6. lift--.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 74.1

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

   8. Applied rewrites 74.1%

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

Alternative 9: 78.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_1 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \cdot t\_0 \leq 10^{-36}:\\ \;\;\;\;\frac{t\_0 \cdot \frac{t\_1}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot t\_1}{z \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fmax (fabs x) (fabs y))) (t_1 (fmin (fabs x) (fabs y))))
   (*
    (copysign 1.0 x)
    (*
     (copysign 1.0 y)
     (if (<= (* t_1 t_0) 1e-36)
       (/ (* t_0 (/ t_1 z)) z)
       (/ (* t_0 t_1) (* z z)))))))

C

double code(double x, double y, double z) {
	double t_0 = fmax(fabs(x), fabs(y));
	double t_1 = fmin(fabs(x), fabs(y));
	double tmp;
	if ((t_1 * t_0) <= 1e-36) {
		tmp = (t_0 * (t_1 / z)) / z;
	} else {
		tmp = (t_0 * t_1) / (z * z);
	}
	return copysign(1.0, x) * (copysign(1.0, y) * tmp);
}

Java

public static double code(double x, double y, double z) {
	double t_0 = fmax(Math.abs(x), Math.abs(y));
	double t_1 = fmin(Math.abs(x), Math.abs(y));
	double tmp;
	if ((t_1 * t_0) <= 1e-36) {
		tmp = (t_0 * (t_1 / z)) / z;
	} else {
		tmp = (t_0 * t_1) / (z * z);
	}
	return Math.copySign(1.0, x) * (Math.copySign(1.0, y) * tmp);
}

Python

def code(x, y, z):
	t_0 = fmax(math.fabs(x), math.fabs(y))
	t_1 = fmin(math.fabs(x), math.fabs(y))
	tmp = 0
	if (t_1 * t_0) <= 1e-36:
		tmp = (t_0 * (t_1 / z)) / z
	else:
		tmp = (t_0 * t_1) / (z * z)
	return math.copysign(1.0, x) * (math.copysign(1.0, y) * tmp)

Julia

function code(x, y, z)
	t_0 = fmax(abs(x), abs(y))
	t_1 = fmin(abs(x), abs(y))
	tmp = 0.0
	if (Float64(t_1 * t_0) <= 1e-36)
		tmp = Float64(Float64(t_0 * Float64(t_1 / z)) / z);
	else
		tmp = Float64(Float64(t_0 * t_1) / Float64(z * z));
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * tmp))
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = max(abs(x), abs(y));
	t_1 = min(abs(x), abs(y));
	tmp = 0.0;
	if ((t_1 * t_0) <= 1e-36)
		tmp = (t_0 * (t_1 / z)) / z;
	else
		tmp = (t_0 * t_1) / (z * z);
	end
	tmp_2 = (sign(x) * abs(1.0)) * ((sign(y) * abs(1.0)) * tmp);
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(t$95$1 * t$95$0), $MachinePrecision], 1e-36], N[(N[(t$95$0 * N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(t$95$0 * t$95$1), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 9.9999999999999994e-37

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 94.7

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

   3. Applied rewrites 94.7%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. add-flip N/A

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

      5. metadata-eval N/A

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

      6. lift--.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 74.1

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

   8. Applied rewrites 74.1%

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

   if 9.9999999999999994e-37 < (*.f64 x y)

   1. Initial program 83.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 83.2

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 83.2

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

   3. Applied rewrites 83.2%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 69.9%

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

Alternative 10: 76.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(x, \left|y\right|\right)\\ t_1 := \mathsf{max}\left(x, \left|y\right|\right)\\ \mathsf{copysign}\left(1, y\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 10^{+75}:\\ \;\;\;\;\frac{\frac{t\_1}{z} \cdot t\_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 \cdot t\_0}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fmin x (fabs y))) (t_1 (fmax x (fabs y))))
   (*
    (copysign 1.0 y)
    (if (<= t_1 1e+75) (/ (* (/ t_1 z) t_0) z) (/ (* t_1 t_0) (* z z))))))

C

double code(double x, double y, double z) {
	double t_0 = fmin(x, fabs(y));
	double t_1 = fmax(x, fabs(y));
	double tmp;
	if (t_1 <= 1e+75) {
		tmp = ((t_1 / z) * t_0) / z;
	} else {
		tmp = (t_1 * t_0) / (z * z);
	}
	return copysign(1.0, y) * tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = fmin(x, Math.abs(y));
	double t_1 = fmax(x, Math.abs(y));
	double tmp;
	if (t_1 <= 1e+75) {
		tmp = ((t_1 / z) * t_0) / z;
	} else {
		tmp = (t_1 * t_0) / (z * z);
	}
	return Math.copySign(1.0, y) * tmp;
}

Python

def code(x, y, z):
	t_0 = fmin(x, math.fabs(y))
	t_1 = fmax(x, math.fabs(y))
	tmp = 0
	if t_1 <= 1e+75:
		tmp = ((t_1 / z) * t_0) / z
	else:
		tmp = (t_1 * t_0) / (z * z)
	return math.copysign(1.0, y) * tmp

Julia

function code(x, y, z)
	t_0 = fmin(x, abs(y))
	t_1 = fmax(x, abs(y))
	tmp = 0.0
	if (t_1 <= 1e+75)
		tmp = Float64(Float64(Float64(t_1 / z) * t_0) / z);
	else
		tmp = Float64(Float64(t_1 * t_0) / Float64(z * z));
	end
	return Float64(copysign(1.0, y) * tmp)
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = min(x, abs(y));
	t_1 = max(x, abs(y));
	tmp = 0.0;
	if (t_1 <= 1e+75)
		tmp = ((t_1 / z) * t_0) / z;
	else
		tmp = (t_1 * t_0) / (z * z);
	end
	tmp_2 = (sign(y) * abs(1.0)) * tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Min[x, N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[x, N[Abs[y], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, 1e+75], N[(N[(N[(t$95$1 / z), $MachinePrecision] * t$95$0), $MachinePrecision] / z), $MachinePrecision], N[(N[(t$95$1 * t$95$0), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 9.99999999999999927e74

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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 97.3

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

   3. Applied rewrites 97.3%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 70.7

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

   6. Applied rewrites 70.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 74.1

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

   8. Applied rewrites 74.1%

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

   if 9.99999999999999927e74 < y

   1. Initial program 83.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 83.2

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 83.2

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

   3. Applied rewrites 83.2%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 69.9%

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

Alternative 11: 69.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (y * x) / (z * 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 * x) / (z * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 83.2

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. distribute-lft-in N/A

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

   11. *-rgt-identity N/A

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

   12. lower-fma.f64 83.2

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

3. Applied rewrites 83.2%

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

4. Taylor expanded in z around 0

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

6. Applied rewrites 69.9%

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

Reproduce

herbie shell --seed 2025172 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (/ (* x y) (* (* z z) (+ z 1.0))))