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

Percentage Accurate: 82.4% → 97.4%

Time: 6.2s

Alternatives: 8

Speedup: 0.9×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot y\_m \leq 2 \cdot 10^{-311}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \left(\frac{y\_m}{z} - y\_m\right)\\ \mathbf{elif}\;x\_m \cdot y\_m \leq 2 \cdot 10^{+247}:\\ \;\;\;\;\frac{\frac{y\_m \cdot x\_m}{z}}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot x\_m\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* x_m y_m) 2e-311)
     (* (/ x_m z) (- (/ y_m z) y_m))
     (if (<= (* x_m y_m) 2e+247)
       (/ (/ (* y_m x_m) z) (fma z z z))
       (* (/ (/ y_m (fma z z z)) z) x_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((x_m * y_m) <= 2e-311) {
		tmp = (x_m / z) * ((y_m / z) - y_m);
	} else if ((x_m * y_m) <= 2e+247) {
		tmp = ((y_m * x_m) / z) / fma(z, z, z);
	} else {
		tmp = ((y_m / fma(z, z, z)) / z) * x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(x_m * y_m) <= 2e-311)
		tmp = Float64(Float64(x_m / z) * Float64(Float64(y_m / z) - y_m));
	elseif (Float64(x_m * y_m) <= 2e+247)
		tmp = Float64(Float64(Float64(y_m * x_m) / z) / fma(z, z, z));
	else
		tmp = Float64(Float64(Float64(y_m / fma(z, z, z)) / z) * x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(x$95$m * y$95$m), $MachinePrecision], 2e-311], N[(N[(x$95$m / z), $MachinePrecision] * N[(N[(y$95$m / z), $MachinePrecision] - y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x$95$m * y$95$m), $MachinePrecision], 2e+247], N[(N[(N[(y$95$m * x$95$m), $MachinePrecision] / z), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < 1.9999999999999e-311

   1. Initial program 71.6%

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

   3. Taylor expanded in z around 0

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

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

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right) - \left(\mathsf{neg}\left(x\right)\right) \cdot y}}{{z}^{2}} \]

      2. div-sub N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right)}{{z}^{2}} - \frac{\left(\mathsf{neg}\left(x\right)\right) \cdot y}{{z}^{2}}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right)}{{z}^{2}} - \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{y}{{z}^{2}}} \]

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

         \[\leadsto \frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right)}{{z}^{2}} - \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{y}{{z}^{2}}\right)\right)} \]

      5. associate-/l* N/A

         \[\leadsto \frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right)}{{z}^{2}} - \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot y}{{z}^{2}}}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right)}{{z}^{2}} - \color{blue}{-1 \cdot \frac{x \cdot y}{{z}^{2}}} \]

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

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right)}{{z}^{2}} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x \cdot y}{{z}^{2}}} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \left(y \cdot z\right)}{{z}^{2}}} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x \cdot y}{{z}^{2}} \]

      9. associate-*r* N/A

         \[\leadsto -1 \cdot \frac{\color{blue}{\left(x \cdot y\right) \cdot z}}{{z}^{2}} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x \cdot y}{{z}^{2}} \]

      10. associate-/l* N/A

          \[\leadsto -1 \cdot \color{blue}{\left(\left(x \cdot y\right) \cdot \frac{z}{{z}^{2}}\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x \cdot y}{{z}^{2}} \]

      11. unpow2 N/A

          \[\leadsto -1 \cdot \left(\left(x \cdot y\right) \cdot \frac{z}{\color{blue}{z \cdot z}}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x \cdot y}{{z}^{2}} \]

      12. associate-/r* N/A

          \[\leadsto -1 \cdot \left(\left(x \cdot y\right) \cdot \color{blue}{\frac{\frac{z}{z}}{z}}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x \cdot y}{{z}^{2}} \]

      13. *-inverses N/A

          \[\leadsto -1 \cdot \left(\left(x \cdot y\right) \cdot \frac{\color{blue}{1}}{z}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x \cdot y}{{z}^{2}} \]

      14. associate-/l* N/A

          \[\leadsto -1 \cdot \color{blue}{\frac{\left(x \cdot y\right) \cdot 1}{z}} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x \cdot y}{{z}^{2}} \]

      15. *-rgt-identity N/A

          \[\leadsto -1 \cdot \frac{\color{blue}{x \cdot y}}{z} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x \cdot y}{{z}^{2}} \]

      16. metadata-eval N/A

          \[\leadsto -1 \cdot \frac{x \cdot y}{z} + \color{blue}{1} \cdot \frac{x \cdot y}{{z}^{2}} \]

      17. *-lft-identity N/A

          \[\leadsto -1 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]

   5. Applied rewrites 72.3%

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

   if 1.9999999999999e-311 < (*.f64 x y) < 1.9999999999999999e247

   1. Initial program 89.8%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. associate-/r* N/A

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

      4. *-rgt-identity N/A

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

      5. rgt-mult-inverse N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

         \[\leadsto \frac{\frac{x \cdot y}{z}}{\color{blue}{{z}^{2}} \cdot \left(1 + \frac{1}{z}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{x \cdot y}{z}}{\color{blue}{\left(1 + \frac{1}{z}\right) \cdot {z}^{2}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{z}}{\left(1 + \frac{1}{z}\right) \cdot {z}^{2}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{z}}}{\left(1 + \frac{1}{z}\right) \cdot {z}^{2}} \]

      13. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{z}}{\left(1 + \frac{1}{z}\right) \cdot {z}^{2}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{z}}{\left(1 + \frac{1}{z}\right) \cdot {z}^{2}} \]

      15. *-commutative N/A

          \[\leadsto \frac{\frac{y \cdot x}{z}}{\color{blue}{{z}^{2} \cdot \left(1 + \frac{1}{z}\right)}} \]

      16. distribute-lft-in N/A

          \[\leadsto \frac{\frac{y \cdot x}{z}}{\color{blue}{{z}^{2} \cdot 1 + {z}^{2} \cdot \frac{1}{z}}} \]

      17. *-rgt-identity N/A

          \[\leadsto \frac{\frac{y \cdot x}{z}}{\color{blue}{{z}^{2}} + {z}^{2} \cdot \frac{1}{z}} \]

      18. unpow2 N/A

          \[\leadsto \frac{\frac{y \cdot x}{z}}{\color{blue}{z \cdot z} + {z}^{2} \cdot \frac{1}{z}} \]

      19. unpow2 N/A

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

      20. associate-*l* N/A

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

      21. rgt-mult-inverse N/A

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

      22. *-rgt-identity N/A

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

      23. lower-fma.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 1.9999999999999999e247 < (*.f64 x y)

   1. Initial program 77.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft1-in N/A

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

      16. lower-fma.f64 91.2

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

   4. Applied rewrites 91.2%

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

Alternative 2: 97.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot y\_m \leq 2 \cdot 10^{-311}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \left(\frac{y\_m}{z} - y\_m\right)\\ \mathbf{elif}\;x\_m \cdot y\_m \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{\frac{y\_m \cdot x\_m}{z}}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot y\_m\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* x_m y_m) 2e-311)
     (* (/ x_m z) (- (/ y_m z) y_m))
     (if (<= (* x_m y_m) 2e+306)
       (/ (/ (* y_m x_m) z) (fma z z z))
       (* (/ (/ x_m (fma z z z)) z) y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((x_m * y_m) <= 2e-311) {
		tmp = (x_m / z) * ((y_m / z) - y_m);
	} else if ((x_m * y_m) <= 2e+306) {
		tmp = ((y_m * x_m) / z) / fma(z, z, z);
	} else {
		tmp = ((x_m / fma(z, z, z)) / z) * y_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(x_m * y_m) <= 2e-311)
		tmp = Float64(Float64(x_m / z) * Float64(Float64(y_m / z) - y_m));
	elseif (Float64(x_m * y_m) <= 2e+306)
		tmp = Float64(Float64(Float64(y_m * x_m) / z) / fma(z, z, z));
	else
		tmp = Float64(Float64(Float64(x_m / fma(z, z, z)) / z) * y_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(x$95$m * y$95$m), $MachinePrecision], 2e-311], N[(N[(x$95$m / z), $MachinePrecision] * N[(N[(y$95$m / z), $MachinePrecision] - y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x$95$m * y$95$m), $MachinePrecision], 2e+306], N[(N[(N[(y$95$m * x$95$m), $MachinePrecision] / z), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < 1.9999999999999e-311

   1. Initial program 71.6%

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

   3. Taylor expanded in z around 0

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

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

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right) - \left(\mathsf{neg}\left(x\right)\right) \cdot y}}{{z}^{2}} \]

      2. div-sub N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right)}{{z}^{2}} - \frac{\left(\mathsf{neg}\left(x\right)\right) \cdot y}{{z}^{2}}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right)}{{z}^{2}} - \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{y}{{z}^{2}}} \]

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

         \[\leadsto \frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right)}{{z}^{2}} - \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{y}{{z}^{2}}\right)\right)} \]

      5. associate-/l* N/A

         \[\leadsto \frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right)}{{z}^{2}} - \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot y}{{z}^{2}}}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right)}{{z}^{2}} - \color{blue}{-1 \cdot \frac{x \cdot y}{{z}^{2}}} \]

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

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right)}{{z}^{2}} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x \cdot y}{{z}^{2}}} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \left(y \cdot z\right)}{{z}^{2}}} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x \cdot y}{{z}^{2}} \]

      9. associate-*r* N/A

         \[\leadsto -1 \cdot \frac{\color{blue}{\left(x \cdot y\right) \cdot z}}{{z}^{2}} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x \cdot y}{{z}^{2}} \]

      10. associate-/l* N/A

          \[\leadsto -1 \cdot \color{blue}{\left(\left(x \cdot y\right) \cdot \frac{z}{{z}^{2}}\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x \cdot y}{{z}^{2}} \]

      11. unpow2 N/A

          \[\leadsto -1 \cdot \left(\left(x \cdot y\right) \cdot \frac{z}{\color{blue}{z \cdot z}}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x \cdot y}{{z}^{2}} \]

      12. associate-/r* N/A

          \[\leadsto -1 \cdot \left(\left(x \cdot y\right) \cdot \color{blue}{\frac{\frac{z}{z}}{z}}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x \cdot y}{{z}^{2}} \]

      13. *-inverses N/A

          \[\leadsto -1 \cdot \left(\left(x \cdot y\right) \cdot \frac{\color{blue}{1}}{z}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x \cdot y}{{z}^{2}} \]

      14. associate-/l* N/A

          \[\leadsto -1 \cdot \color{blue}{\frac{\left(x \cdot y\right) \cdot 1}{z}} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x \cdot y}{{z}^{2}} \]

      15. *-rgt-identity N/A

          \[\leadsto -1 \cdot \frac{\color{blue}{x \cdot y}}{z} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x \cdot y}{{z}^{2}} \]

      16. metadata-eval N/A

          \[\leadsto -1 \cdot \frac{x \cdot y}{z} + \color{blue}{1} \cdot \frac{x \cdot y}{{z}^{2}} \]

      17. *-lft-identity N/A

          \[\leadsto -1 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]

   5. Applied rewrites 72.3%

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

   if 1.9999999999999e-311 < (*.f64 x y) < 2.00000000000000003e306

   1. Initial program 89.3%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. associate-/r* N/A

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

      4. *-rgt-identity N/A

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

      5. rgt-mult-inverse N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

         \[\leadsto \frac{\frac{x \cdot y}{z}}{\color{blue}{{z}^{2}} \cdot \left(1 + \frac{1}{z}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{x \cdot y}{z}}{\color{blue}{\left(1 + \frac{1}{z}\right) \cdot {z}^{2}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{z}}{\left(1 + \frac{1}{z}\right) \cdot {z}^{2}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{z}}}{\left(1 + \frac{1}{z}\right) \cdot {z}^{2}} \]

      13. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{z}}{\left(1 + \frac{1}{z}\right) \cdot {z}^{2}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{z}}{\left(1 + \frac{1}{z}\right) \cdot {z}^{2}} \]

      15. *-commutative N/A

          \[\leadsto \frac{\frac{y \cdot x}{z}}{\color{blue}{{z}^{2} \cdot \left(1 + \frac{1}{z}\right)}} \]

      16. distribute-lft-in N/A

          \[\leadsto \frac{\frac{y \cdot x}{z}}{\color{blue}{{z}^{2} \cdot 1 + {z}^{2} \cdot \frac{1}{z}}} \]

      17. *-rgt-identity N/A

          \[\leadsto \frac{\frac{y \cdot x}{z}}{\color{blue}{{z}^{2}} + {z}^{2} \cdot \frac{1}{z}} \]

      18. unpow2 N/A

          \[\leadsto \frac{\frac{y \cdot x}{z}}{\color{blue}{z \cdot z} + {z}^{2} \cdot \frac{1}{z}} \]

      19. unpow2 N/A

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

      20. associate-*l* N/A

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

      21. rgt-mult-inverse N/A

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

      22. *-rgt-identity N/A

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

      23. lower-fma.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if 2.00000000000000003e306 < (*.f64 x y)

   1. Initial program 76.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. distribute-lft1-in N/A

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

      17. lower-fma.f64 94.3

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

   4. Applied rewrites 94.3%

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

Alternative 3: 96.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot y\_m \leq 2 \cdot 10^{-311}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \left(\frac{y\_m}{z} - y\_m\right)\\ \mathbf{elif}\;x\_m \cdot y\_m \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{\frac{y\_m \cdot x\_m}{z}}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\_m\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* x_m y_m) 2e-311)
     (* (/ x_m z) (- (/ y_m z) y_m))
     (if (<= (* x_m y_m) 2e+306)
       (/ (/ (* y_m x_m) z) (fma z z z))
       (* (/ x_m (* (fma z z z) z)) y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((x_m * y_m) <= 2e-311) {
		tmp = (x_m / z) * ((y_m / z) - y_m);
	} else if ((x_m * y_m) <= 2e+306) {
		tmp = ((y_m * x_m) / z) / fma(z, z, z);
	} else {
		tmp = (x_m / (fma(z, z, z) * z)) * y_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(x_m * y_m) <= 2e-311)
		tmp = Float64(Float64(x_m / z) * Float64(Float64(y_m / z) - y_m));
	elseif (Float64(x_m * y_m) <= 2e+306)
		tmp = Float64(Float64(Float64(y_m * x_m) / z) / fma(z, z, z));
	else
		tmp = Float64(Float64(x_m / Float64(fma(z, z, z) * z)) * y_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(x$95$m * y$95$m), $MachinePrecision], 2e-311], N[(N[(x$95$m / z), $MachinePrecision] * N[(N[(y$95$m / z), $MachinePrecision] - y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x$95$m * y$95$m), $MachinePrecision], 2e+306], N[(N[(N[(y$95$m * x$95$m), $MachinePrecision] / z), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < 1.9999999999999e-311

   1. Initial program 71.6%

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

   3. Taylor expanded in z around 0

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

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

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right) - \left(\mathsf{neg}\left(x\right)\right) \cdot y}}{{z}^{2}} \]

      2. div-sub N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right)}{{z}^{2}} - \frac{\left(\mathsf{neg}\left(x\right)\right) \cdot y}{{z}^{2}}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right)}{{z}^{2}} - \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{y}{{z}^{2}}} \]

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

         \[\leadsto \frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right)}{{z}^{2}} - \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{y}{{z}^{2}}\right)\right)} \]

      5. associate-/l* N/A

         \[\leadsto \frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right)}{{z}^{2}} - \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot y}{{z}^{2}}}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right)}{{z}^{2}} - \color{blue}{-1 \cdot \frac{x \cdot y}{{z}^{2}}} \]

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

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right)}{{z}^{2}} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x \cdot y}{{z}^{2}}} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \left(y \cdot z\right)}{{z}^{2}}} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x \cdot y}{{z}^{2}} \]

      9. associate-*r* N/A

         \[\leadsto -1 \cdot \frac{\color{blue}{\left(x \cdot y\right) \cdot z}}{{z}^{2}} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x \cdot y}{{z}^{2}} \]

      10. associate-/l* N/A

          \[\leadsto -1 \cdot \color{blue}{\left(\left(x \cdot y\right) \cdot \frac{z}{{z}^{2}}\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x \cdot y}{{z}^{2}} \]

      11. unpow2 N/A

          \[\leadsto -1 \cdot \left(\left(x \cdot y\right) \cdot \frac{z}{\color{blue}{z \cdot z}}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x \cdot y}{{z}^{2}} \]

      12. associate-/r* N/A

          \[\leadsto -1 \cdot \left(\left(x \cdot y\right) \cdot \color{blue}{\frac{\frac{z}{z}}{z}}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x \cdot y}{{z}^{2}} \]

      13. *-inverses N/A

          \[\leadsto -1 \cdot \left(\left(x \cdot y\right) \cdot \frac{\color{blue}{1}}{z}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x \cdot y}{{z}^{2}} \]

      14. associate-/l* N/A

          \[\leadsto -1 \cdot \color{blue}{\frac{\left(x \cdot y\right) \cdot 1}{z}} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x \cdot y}{{z}^{2}} \]

      15. *-rgt-identity N/A

          \[\leadsto -1 \cdot \frac{\color{blue}{x \cdot y}}{z} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{x \cdot y}{{z}^{2}} \]

      16. metadata-eval N/A

          \[\leadsto -1 \cdot \frac{x \cdot y}{z} + \color{blue}{1} \cdot \frac{x \cdot y}{{z}^{2}} \]

      17. *-lft-identity N/A

          \[\leadsto -1 \cdot \frac{x \cdot y}{z} + \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]

   5. Applied rewrites 72.3%

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

   if 1.9999999999999e-311 < (*.f64 x y) < 2.00000000000000003e306

   1. Initial program 89.3%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. associate-/r* N/A

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

      4. *-rgt-identity N/A

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

      5. rgt-mult-inverse N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

         \[\leadsto \frac{\frac{x \cdot y}{z}}{\color{blue}{{z}^{2}} \cdot \left(1 + \frac{1}{z}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{x \cdot y}{z}}{\color{blue}{\left(1 + \frac{1}{z}\right) \cdot {z}^{2}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{z}}{\left(1 + \frac{1}{z}\right) \cdot {z}^{2}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{z}}}{\left(1 + \frac{1}{z}\right) \cdot {z}^{2}} \]

      13. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{z}}{\left(1 + \frac{1}{z}\right) \cdot {z}^{2}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{z}}{\left(1 + \frac{1}{z}\right) \cdot {z}^{2}} \]

      15. *-commutative N/A

          \[\leadsto \frac{\frac{y \cdot x}{z}}{\color{blue}{{z}^{2} \cdot \left(1 + \frac{1}{z}\right)}} \]

      16. distribute-lft-in N/A

          \[\leadsto \frac{\frac{y \cdot x}{z}}{\color{blue}{{z}^{2} \cdot 1 + {z}^{2} \cdot \frac{1}{z}}} \]

      17. *-rgt-identity N/A

          \[\leadsto \frac{\frac{y \cdot x}{z}}{\color{blue}{{z}^{2}} + {z}^{2} \cdot \frac{1}{z}} \]

      18. unpow2 N/A

          \[\leadsto \frac{\frac{y \cdot x}{z}}{\color{blue}{z \cdot z} + {z}^{2} \cdot \frac{1}{z}} \]

      19. unpow2 N/A

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

      20. associate-*l* N/A

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

      21. rgt-mult-inverse N/A

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

      22. *-rgt-identity N/A

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

      23. lower-fma.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if 2.00000000000000003e306 < (*.f64 x y)

   1. Initial program 76.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 94.4

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

   4. Applied rewrites 94.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 88.7

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

   6. Applied rewrites 88.7%

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

Alternative 4: 92.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-53} \lor \neg \left(z \leq 1.28 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{y\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (or (<= z -1.25e-53) (not (<= z 1.28e-16)))
     (* (/ y_m (* (fma z z z) z)) x_m)
     (* (/ x_m z) (/ y_m z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z <= -1.25e-53) || !(z <= 1.28e-16)) {
		tmp = (y_m / (fma(z, z, z) * z)) * x_m;
	} else {
		tmp = (x_m / z) * (y_m / z);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if ((z <= -1.25e-53) || !(z <= 1.28e-16))
		tmp = Float64(Float64(y_m / Float64(fma(z, z, z) * z)) * x_m);
	else
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[Or[LessEqual[z, -1.25e-53], N[Not[LessEqual[z, 1.28e-16]], $MachinePrecision]], N[(N[(y$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.25e-53 or 1.2800000000000001e-16 < z

   1. Initial program 78.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft1-in N/A

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

      16. lower-fma.f64 90.3

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

   4. Applied rewrites 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 83.2

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

   6. Applied rewrites 83.2%

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

   if -1.25e-53 < z < 1.2800000000000001e-16

   1. Initial program 78.3%

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

   3. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 98.9

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

   5. Applied rewrites 98.9%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-53} \lor \neg \left(z \leq 1.28 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot y\_m \leq 2 \cdot 10^{-146}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\_m\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* x_m y_m) 2e-146)
     (* (/ x_m z) (/ y_m z))
     (* (/ x_m (* (fma z z z) z)) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((x_m * y_m) <= 2e-146) {
		tmp = (x_m / z) * (y_m / z);
	} else {
		tmp = (x_m / (fma(z, z, z) * z)) * y_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(x_m * y_m) <= 2e-146)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	else
		tmp = Float64(Float64(x_m / Float64(fma(z, z, z) * z)) * y_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(x$95$m * y$95$m), $MachinePrecision], 2e-146], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 2.00000000000000005e-146

   1. Initial program 74.2%

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

   3. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 79.4

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

   5. Applied rewrites 79.4%

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

   if 2.00000000000000005e-146 < (*.f64 x y)

   1. Initial program 86.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 94.9

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

   4. Applied rewrites 94.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 82.5

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

   6. Applied rewrites 82.5%

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

Alternative 6: 80.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot y\_m \leq 2 \cdot 10^{-146}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z \cdot z} \cdot y\_m\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* x_m y_m) 2e-146)
     (* (/ x_m z) (/ y_m z))
     (* (/ x_m (* z z)) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((x_m * y_m) <= 2e-146) {
		tmp = (x_m / z) * (y_m / z);
	} else {
		tmp = (x_m / (z * z)) * y_m;
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m =     private
y\_s =     private
x\_m =     private
x\_s =     private
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
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_s, y_s, x_m, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x_m * y_m) <= 2d-146) then
        tmp = (x_m / z) * (y_m / z)
    else
        tmp = (x_m / (z * z)) * y_m
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((x_m * y_m) <= 2e-146) {
		tmp = (x_m / z) * (y_m / z);
	} else {
		tmp = (x_m / (z * z)) * y_m;
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	tmp = 0
	if (x_m * y_m) <= 2e-146:
		tmp = (x_m / z) * (y_m / z)
	else:
		tmp = (x_m / (z * z)) * y_m
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(x_m * y_m) <= 2e-146)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	else
		tmp = Float64(Float64(x_m / Float64(z * z)) * y_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0;
	if ((x_m * y_m) <= 2e-146)
		tmp = (x_m / z) * (y_m / z);
	else
		tmp = (x_m / (z * z)) * y_m;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(x$95$m * y$95$m), $MachinePrecision], 2e-146], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 2.00000000000000005e-146

   1. Initial program 74.2%

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

   3. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 79.4

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

   5. Applied rewrites 79.4%

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

   if 2.00000000000000005e-146 < (*.f64 x y)

   1. Initial program 86.3%

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

   3. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 69.5

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

   5. Applied rewrites 69.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 68.9

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

   7. Applied rewrites 68.9%

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

Alternative 7: 94.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. times-frac N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-+.f64 N/A

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

   13. distribute-lft1-in N/A

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

   14. lower-fma.f64 95.2

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

4. Applied rewrites 95.2%

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

Alternative 8: 74.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

y\_m =     private
y\_s =     private
x\_m =     private
x\_s =     private
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
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_s, y_s, x_m, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = x_s * (y_s * ((x_m / (z * z)) * y_m))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

3. Taylor expanded in z around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 66.5

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

5. Applied rewrites 66.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 69.7

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

7. Applied rewrites 69.7%

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

Developer Target 1: 97.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< z 249.6182814532307)
   (/ (* y (/ x z)) (+ z (* z z)))
   (/ (* (/ (/ y z) (+ 1.0 z)) x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z < 249.6182814532307) {
		tmp = (y * (x / z)) / (z + (z * z));
	} else {
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z < 249.6182814532307d0) then
        tmp = (y * (x / z)) / (z + (z * z))
    else
        tmp = (((y / z) / (1.0d0 + z)) * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z < 249.6182814532307) {
		tmp = (y * (x / z)) / (z + (z * z));
	} else {
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z < 249.6182814532307:
		tmp = (y * (x / z)) / (z + (z * z))
	else:
		tmp = (((y / z) / (1.0 + z)) * x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z < 249.6182814532307)
		tmp = Float64(Float64(y * Float64(x / z)) / Float64(z + Float64(z * z)));
	else
		tmp = Float64(Float64(Float64(Float64(y / z) / Float64(1.0 + z)) * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z < 249.6182814532307)
		tmp = (y * (x / z)) / (z + (z * z));
	else
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[z, 249.6182814532307], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / N[(z + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y / z), $MachinePrecision] / N[(1.0 + z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision]]

Reproduce

herbie shell --seed 2024352 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z 2496182814532307/10000000000000) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1 z)) x) z)))

  (/ (* x y) (* (* z z) (+ z 1.0))))