Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2

Percentage Accurate: 61.1% → 91.8%

Time: 9.7s

Alternatives: 19

Speedup: 1.6×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) * z) / sqrt(((z * z) - (t * a)));
}

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, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((x * y) * z) / sqrt(((z * z) - (t * a)))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) * z) / Math.sqrt(((z * z) - (t * a)));
}

Python

def code(x, y, z, t, a):
	return ((x * y) * z) / math.sqrt(((z * z) - (t * a)))

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) * z) / sqrt(Float64(Float64(z * z) - Float64(t * a))))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) * z) / sqrt(((z * z) - (t * a)));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 61.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) * z) / sqrt(((z * z) - (t * a)));
}

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, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((x * y) * z) / sqrt(((z * z) - (t * a)))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) * z) / Math.sqrt(((z * z) - (t * a)));
}

Python

def code(x, y, z, t, a):
	return ((x * y) * z) / math.sqrt(((z * z) - (t * a)))

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) * z) / sqrt(Float64(Float64(z * z) - Float64(t * a))))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) * z) / sqrt(((z * z) - (t * a)));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 91.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2 \cdot 10^{+130}:\\ \;\;\;\;\left(\frac{z\_m}{\sqrt{z\_m \cdot z\_m - a \cdot t}} \cdot y\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m \cdot x\_m}{\mathsf{fma}\left(-0.5 \cdot a, \frac{t}{z\_m}, z\_m\right)} \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 2e+130)
      (* (* (/ z_m (sqrt (- (* z_m z_m) (* a t)))) y_m) x_m)
      (* (/ (* z_m x_m) (fma (* -0.5 a) (/ t z_m) z_m)) y_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2e+130) {
		tmp = ((z_m / sqrt(((z_m * z_m) - (a * t)))) * y_m) * x_m;
	} else {
		tmp = ((z_m * x_m) / fma((-0.5 * a), (t / z_m), z_m)) * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2e+130)
		tmp = Float64(Float64(Float64(z_m / sqrt(Float64(Float64(z_m * z_m) - Float64(a * t)))) * y_m) * x_m);
	else
		tmp = Float64(Float64(Float64(z_m * x_m) / fma(Float64(-0.5 * a), Float64(t / z_m), z_m)) * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2e+130], N[(N[(N[(z$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(z$95$m * x$95$m), $MachinePrecision] / N[(N[(-0.5 * a), $MachinePrecision] * N[(t / z$95$m), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.0000000000000001e130

   1. Initial program 61.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. mult-flip N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. mult-flip-rev N/A

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

      14. lower-/.f64 59.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 59.3

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

   3. Applied rewrites 59.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. mult-flip-rev N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 65.7%

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

   if 2.0000000000000001e130 < z

   1. Initial program 61.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 64.8

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{z \cdot \left(1 + -0.5 \cdot \frac{a \cdot t}{{z}^{\color{blue}{2}}}\right)} \]

   4. Applied rewrites 64.8%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z \cdot \left(1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\frac{z \cdot x}{\mathsf{fma}\left(-0.5 \cdot a, \frac{t}{z}, z\right)} \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 91.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.6 \cdot 10^{+109}:\\ \;\;\;\;\left(\frac{z\_m}{\sqrt{z\_m \cdot z\_m - a \cdot t}} \cdot y\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot z\_m}{1 \cdot z\_m} \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 2.6e+109)
      (* (* (/ z_m (sqrt (- (* z_m z_m) (* a t)))) y_m) x_m)
      (* (/ (* x_m z_m) (* 1.0 z_m)) y_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.6e+109) {
		tmp = ((z_m / sqrt(((z_m * z_m) - (a * t)))) * y_m) * x_m;
	} else {
		tmp = ((x_m * z_m) / (1.0 * z_m)) * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
NOTE: x_m, y_m, z_m, t, and a 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(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 2.6d+109) then
        tmp = ((z_m / sqrt(((z_m * z_m) - (a * t)))) * y_m) * x_m
    else
        tmp = ((x_m * z_m) / (1.0d0 * z_m)) * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.6e+109) {
		tmp = ((z_m / Math.sqrt(((z_m * z_m) - (a * t)))) * y_m) * x_m;
	} else {
		tmp = ((x_m * z_m) / (1.0 * z_m)) * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 2.6e+109:
		tmp = ((z_m / math.sqrt(((z_m * z_m) - (a * t)))) * y_m) * x_m
	else:
		tmp = ((x_m * z_m) / (1.0 * z_m)) * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.6e+109)
		tmp = Float64(Float64(Float64(z_m / sqrt(Float64(Float64(z_m * z_m) - Float64(a * t)))) * y_m) * x_m);
	else
		tmp = Float64(Float64(Float64(x_m * z_m) / Float64(1.0 * z_m)) * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2.6e+109)
		tmp = ((z_m / sqrt(((z_m * z_m) - (a * t)))) * y_m) * x_m;
	else
		tmp = ((x_m * z_m) / (1.0 * z_m)) * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2.6e+109], N[(N[(N[(z$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(x$95$m * z$95$m), $MachinePrecision] / N[(1.0 * z$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.5999999999999998e109

   1. Initial program 61.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. mult-flip N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. mult-flip-rev N/A

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

      14. lower-/.f64 59.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 59.3

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

   3. Applied rewrites 59.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. mult-flip-rev N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 65.7%

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

   if 2.5999999999999998e109 < z

   1. Initial program 61.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 64.8

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{z \cdot \left(1 + -0.5 \cdot \frac{a \cdot t}{{z}^{\color{blue}{2}}}\right)} \]

   4. Applied rewrites 64.8%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z \cdot \left(1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}\right)}} \]

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 65.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   9. Applied rewrites 74.5%

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

Alternative 3: 90.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.7 \cdot 10^{-140}:\\ \;\;\;\;\frac{\left(x\_m \cdot y\_m\right) \cdot z\_m}{\sqrt{-t} \cdot \sqrt{a}}\\ \mathbf{elif}\;z\_m \leq 4.5 \cdot 10^{+58}:\\ \;\;\;\;\left(\frac{y\_m}{\sqrt{z\_m \cdot z\_m - a \cdot t}} \cdot x\_m\right) \cdot z\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot z\_m}{1 \cdot z\_m} \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1.7e-140)
      (/ (* (* x_m y_m) z_m) (* (sqrt (- t)) (sqrt a)))
      (if (<= z_m 4.5e+58)
        (* (* (/ y_m (sqrt (- (* z_m z_m) (* a t)))) x_m) z_m)
        (* (/ (* x_m z_m) (* 1.0 z_m)) y_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.7e-140) {
		tmp = ((x_m * y_m) * z_m) / (sqrt(-t) * sqrt(a));
	} else if (z_m <= 4.5e+58) {
		tmp = ((y_m / sqrt(((z_m * z_m) - (a * t)))) * x_m) * z_m;
	} else {
		tmp = ((x_m * z_m) / (1.0 * z_m)) * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
NOTE: x_m, y_m, z_m, t, and a 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(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.7d-140) then
        tmp = ((x_m * y_m) * z_m) / (sqrt(-t) * sqrt(a))
    else if (z_m <= 4.5d+58) then
        tmp = ((y_m / sqrt(((z_m * z_m) - (a * t)))) * x_m) * z_m
    else
        tmp = ((x_m * z_m) / (1.0d0 * z_m)) * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.7e-140) {
		tmp = ((x_m * y_m) * z_m) / (Math.sqrt(-t) * Math.sqrt(a));
	} else if (z_m <= 4.5e+58) {
		tmp = ((y_m / Math.sqrt(((z_m * z_m) - (a * t)))) * x_m) * z_m;
	} else {
		tmp = ((x_m * z_m) / (1.0 * z_m)) * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.7e-140:
		tmp = ((x_m * y_m) * z_m) / (math.sqrt(-t) * math.sqrt(a))
	elif z_m <= 4.5e+58:
		tmp = ((y_m / math.sqrt(((z_m * z_m) - (a * t)))) * x_m) * z_m
	else:
		tmp = ((x_m * z_m) / (1.0 * z_m)) * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.7e-140)
		tmp = Float64(Float64(Float64(x_m * y_m) * z_m) / Float64(sqrt(Float64(-t)) * sqrt(a)));
	elseif (z_m <= 4.5e+58)
		tmp = Float64(Float64(Float64(y_m / sqrt(Float64(Float64(z_m * z_m) - Float64(a * t)))) * x_m) * z_m);
	else
		tmp = Float64(Float64(Float64(x_m * z_m) / Float64(1.0 * z_m)) * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.7e-140)
		tmp = ((x_m * y_m) * z_m) / (sqrt(-t) * sqrt(a));
	elseif (z_m <= 4.5e+58)
		tmp = ((y_m / sqrt(((z_m * z_m) - (a * t)))) * x_m) * z_m;
	else
		tmp = ((x_m * z_m) / (1.0 * z_m)) * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.7e-140], N[(N[(N[(x$95$m * y$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] / N[(N[Sqrt[(-t)], $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 4.5e+58], N[(N[(N[(y$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] * z$95$m), $MachinePrecision], N[(N[(N[(x$95$m * z$95$m), $MachinePrecision] / N[(1.0 * z$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 1.70000000000000004e-140

   1. Initial program 61.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

   2. Taylor expanded in z around 0

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

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\mathsf{neg}\left(a \cdot t\right)}} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{-a \cdot t}} \]

      3. lower-*.f64 33.7

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{-a \cdot t}} \]

   4. Applied rewrites 33.7%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{-a \cdot t}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\mathsf{neg}\left(a \cdot t\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\mathsf{neg}\left(a \cdot t\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\mathsf{neg}\left(t \cdot a\right)}} \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\left(\mathsf{neg}\left(t\right)\right) \cdot a}} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\mathsf{neg}\left(t\right)} \cdot \color{blue}{\sqrt{a}}} \]

      7. lower-special-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\mathsf{neg}\left(t\right)} \cdot \color{blue}{\sqrt{a}}} \]

      8. lower-special-sqrt.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\mathsf{neg}\left(t\right)} \cdot \sqrt{\color{blue}{a}}} \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{-t} \cdot \sqrt{a}} \]

      10. lower-special-sqrt.f64 18.6

          \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{-t} \cdot \sqrt{a}} \]

   6. Applied rewrites 18.6%

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

   if 1.70000000000000004e-140 < z < 4.4999999999999998e58

   1. Initial program 61.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot z} \]

      4. mult-flip-rev N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 61.1

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 61.1

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

   3. Applied rewrites 61.1%

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

   if 4.4999999999999998e58 < z

   1. Initial program 61.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 64.8

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{z \cdot \left(1 + -0.5 \cdot \frac{a \cdot t}{{z}^{\color{blue}{2}}}\right)} \]

   4. Applied rewrites 64.8%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z \cdot \left(1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}\right)}} \]

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 65.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   9. Applied rewrites 74.5%

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

Alternative 4: 83.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.6 \cdot 10^{+109}:\\ \;\;\;\;\left(\frac{y\_m}{\sqrt{z\_m \cdot z\_m - a \cdot t}} \cdot z\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot z\_m}{1 \cdot z\_m} \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 2.6e+109)
      (* (* (/ y_m (sqrt (- (* z_m z_m) (* a t)))) z_m) x_m)
      (* (/ (* x_m z_m) (* 1.0 z_m)) y_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.6e+109) {
		tmp = ((y_m / sqrt(((z_m * z_m) - (a * t)))) * z_m) * x_m;
	} else {
		tmp = ((x_m * z_m) / (1.0 * z_m)) * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
NOTE: x_m, y_m, z_m, t, and a 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(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 2.6d+109) then
        tmp = ((y_m / sqrt(((z_m * z_m) - (a * t)))) * z_m) * x_m
    else
        tmp = ((x_m * z_m) / (1.0d0 * z_m)) * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.6e+109) {
		tmp = ((y_m / Math.sqrt(((z_m * z_m) - (a * t)))) * z_m) * x_m;
	} else {
		tmp = ((x_m * z_m) / (1.0 * z_m)) * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 2.6e+109:
		tmp = ((y_m / math.sqrt(((z_m * z_m) - (a * t)))) * z_m) * x_m
	else:
		tmp = ((x_m * z_m) / (1.0 * z_m)) * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.6e+109)
		tmp = Float64(Float64(Float64(y_m / sqrt(Float64(Float64(z_m * z_m) - Float64(a * t)))) * z_m) * x_m);
	else
		tmp = Float64(Float64(Float64(x_m * z_m) / Float64(1.0 * z_m)) * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2.6e+109)
		tmp = ((y_m / sqrt(((z_m * z_m) - (a * t)))) * z_m) * x_m;
	else
		tmp = ((x_m * z_m) / (1.0 * z_m)) * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2.6e+109], N[(N[(N[(y$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(x$95$m * z$95$m), $MachinePrecision] / N[(1.0 * z$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.5999999999999998e109

   1. Initial program 61.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. mult-flip-rev N/A

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

      14. lower-/.f64 64.5

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 64.5

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

   3. Applied rewrites 64.5%

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

   if 2.5999999999999998e109 < z

   1. Initial program 61.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 64.8

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{z \cdot \left(1 + -0.5 \cdot \frac{a \cdot t}{{z}^{\color{blue}{2}}}\right)} \]

   4. Applied rewrites 64.8%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z \cdot \left(1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}\right)}} \]

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 65.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   9. Applied rewrites 74.5%

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

Alternative 5: 82.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.65 \cdot 10^{-140}:\\ \;\;\;\;\frac{\left(x\_m \cdot y\_m\right) \cdot z\_m}{\sqrt{-t} \cdot \sqrt{a}}\\ \mathbf{elif}\;z\_m \leq 2.3 \cdot 10^{+16}:\\ \;\;\;\;\left(\frac{x\_m}{\sqrt{z\_m \cdot z\_m - a \cdot t}} \cdot z\_m\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot z\_m}{1 \cdot z\_m} \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1.65e-140)
      (/ (* (* x_m y_m) z_m) (* (sqrt (- t)) (sqrt a)))
      (if (<= z_m 2.3e+16)
        (* (* (/ x_m (sqrt (- (* z_m z_m) (* a t)))) z_m) y_m)
        (* (/ (* x_m z_m) (* 1.0 z_m)) y_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.65e-140) {
		tmp = ((x_m * y_m) * z_m) / (sqrt(-t) * sqrt(a));
	} else if (z_m <= 2.3e+16) {
		tmp = ((x_m / sqrt(((z_m * z_m) - (a * t)))) * z_m) * y_m;
	} else {
		tmp = ((x_m * z_m) / (1.0 * z_m)) * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
NOTE: x_m, y_m, z_m, t, and a 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(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.65d-140) then
        tmp = ((x_m * y_m) * z_m) / (sqrt(-t) * sqrt(a))
    else if (z_m <= 2.3d+16) then
        tmp = ((x_m / sqrt(((z_m * z_m) - (a * t)))) * z_m) * y_m
    else
        tmp = ((x_m * z_m) / (1.0d0 * z_m)) * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.65e-140) {
		tmp = ((x_m * y_m) * z_m) / (Math.sqrt(-t) * Math.sqrt(a));
	} else if (z_m <= 2.3e+16) {
		tmp = ((x_m / Math.sqrt(((z_m * z_m) - (a * t)))) * z_m) * y_m;
	} else {
		tmp = ((x_m * z_m) / (1.0 * z_m)) * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.65e-140:
		tmp = ((x_m * y_m) * z_m) / (math.sqrt(-t) * math.sqrt(a))
	elif z_m <= 2.3e+16:
		tmp = ((x_m / math.sqrt(((z_m * z_m) - (a * t)))) * z_m) * y_m
	else:
		tmp = ((x_m * z_m) / (1.0 * z_m)) * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.65e-140)
		tmp = Float64(Float64(Float64(x_m * y_m) * z_m) / Float64(sqrt(Float64(-t)) * sqrt(a)));
	elseif (z_m <= 2.3e+16)
		tmp = Float64(Float64(Float64(x_m / sqrt(Float64(Float64(z_m * z_m) - Float64(a * t)))) * z_m) * y_m);
	else
		tmp = Float64(Float64(Float64(x_m * z_m) / Float64(1.0 * z_m)) * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.65e-140)
		tmp = ((x_m * y_m) * z_m) / (sqrt(-t) * sqrt(a));
	elseif (z_m <= 2.3e+16)
		tmp = ((x_m / sqrt(((z_m * z_m) - (a * t)))) * z_m) * y_m;
	else
		tmp = ((x_m * z_m) / (1.0 * z_m)) * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.65e-140], N[(N[(N[(x$95$m * y$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] / N[(N[Sqrt[(-t)], $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 2.3e+16], N[(N[(N[(x$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(x$95$m * z$95$m), $MachinePrecision] / N[(1.0 * z$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 1.64999999999999994e-140

   1. Initial program 61.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

   2. Taylor expanded in z around 0

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

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\mathsf{neg}\left(a \cdot t\right)}} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{-a \cdot t}} \]

      3. lower-*.f64 33.7

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{-a \cdot t}} \]

   4. Applied rewrites 33.7%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{-a \cdot t}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\mathsf{neg}\left(a \cdot t\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\mathsf{neg}\left(a \cdot t\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\mathsf{neg}\left(t \cdot a\right)}} \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\left(\mathsf{neg}\left(t\right)\right) \cdot a}} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\mathsf{neg}\left(t\right)} \cdot \color{blue}{\sqrt{a}}} \]

      7. lower-special-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\mathsf{neg}\left(t\right)} \cdot \color{blue}{\sqrt{a}}} \]

      8. lower-special-sqrt.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\mathsf{neg}\left(t\right)} \cdot \sqrt{\color{blue}{a}}} \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{-t} \cdot \sqrt{a}} \]

      10. lower-special-sqrt.f64 18.6

          \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{-t} \cdot \sqrt{a}} \]

   6. Applied rewrites 18.6%

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

   if 1.64999999999999994e-140 < z < 2.3e16

   1. Initial program 61.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. mult-flip N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. mult-flip-rev N/A

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

      14. lower-/.f64 59.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 59.3

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

   3. Applied rewrites 59.3%

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

   if 2.3e16 < z

   1. Initial program 61.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 64.8

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{z \cdot \left(1 + -0.5 \cdot \frac{a \cdot t}{{z}^{\color{blue}{2}}}\right)} \]

   4. Applied rewrites 64.8%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z \cdot \left(1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}\right)}} \]

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 65.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   9. Applied rewrites 74.5%

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

Alternative 6: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.25 \cdot 10^{-49}:\\ \;\;\;\;\frac{\left(x\_m \cdot y\_m\right) \cdot z\_m}{\sqrt{-t} \cdot \sqrt{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot z\_m}{1 \cdot z\_m} \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1.25e-49)
      (/ (* (* x_m y_m) z_m) (* (sqrt (- t)) (sqrt a)))
      (* (/ (* x_m z_m) (* 1.0 z_m)) y_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.25e-49) {
		tmp = ((x_m * y_m) * z_m) / (sqrt(-t) * sqrt(a));
	} else {
		tmp = ((x_m * z_m) / (1.0 * z_m)) * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
NOTE: x_m, y_m, z_m, t, and a 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(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.25d-49) then
        tmp = ((x_m * y_m) * z_m) / (sqrt(-t) * sqrt(a))
    else
        tmp = ((x_m * z_m) / (1.0d0 * z_m)) * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.25e-49) {
		tmp = ((x_m * y_m) * z_m) / (Math.sqrt(-t) * Math.sqrt(a));
	} else {
		tmp = ((x_m * z_m) / (1.0 * z_m)) * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.25e-49:
		tmp = ((x_m * y_m) * z_m) / (math.sqrt(-t) * math.sqrt(a))
	else:
		tmp = ((x_m * z_m) / (1.0 * z_m)) * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.25e-49)
		tmp = Float64(Float64(Float64(x_m * y_m) * z_m) / Float64(sqrt(Float64(-t)) * sqrt(a)));
	else
		tmp = Float64(Float64(Float64(x_m * z_m) / Float64(1.0 * z_m)) * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.25e-49)
		tmp = ((x_m * y_m) * z_m) / (sqrt(-t) * sqrt(a));
	else
		tmp = ((x_m * z_m) / (1.0 * z_m)) * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.25e-49], N[(N[(N[(x$95$m * y$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] / N[(N[Sqrt[(-t)], $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m * z$95$m), $MachinePrecision] / N[(1.0 * z$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.25e-49

   1. Initial program 61.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

   2. Taylor expanded in z around 0

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

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\mathsf{neg}\left(a \cdot t\right)}} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{-a \cdot t}} \]

      3. lower-*.f64 33.7

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{-a \cdot t}} \]

   4. Applied rewrites 33.7%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{-a \cdot t}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\mathsf{neg}\left(a \cdot t\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\mathsf{neg}\left(a \cdot t\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\mathsf{neg}\left(t \cdot a\right)}} \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\left(\mathsf{neg}\left(t\right)\right) \cdot a}} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\mathsf{neg}\left(t\right)} \cdot \color{blue}{\sqrt{a}}} \]

      7. lower-special-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\mathsf{neg}\left(t\right)} \cdot \color{blue}{\sqrt{a}}} \]

      8. lower-special-sqrt.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\mathsf{neg}\left(t\right)} \cdot \sqrt{\color{blue}{a}}} \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{-t} \cdot \sqrt{a}} \]

      10. lower-special-sqrt.f64 18.6

          \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{-t} \cdot \sqrt{a}} \]

   6. Applied rewrites 18.6%

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

   if 1.25e-49 < z

   1. Initial program 61.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 64.8

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{z \cdot \left(1 + -0.5 \cdot \frac{a \cdot t}{{z}^{\color{blue}{2}}}\right)} \]

   4. Applied rewrites 64.8%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z \cdot \left(1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}\right)}} \]

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 65.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   9. Applied rewrites 74.5%

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

Alternative 7: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.55 \cdot 10^{-49}:\\ \;\;\;\;\frac{y\_m \cdot z\_m}{\sqrt{-a \cdot t}} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot z\_m}{1 \cdot z\_m} \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1.55e-49)
      (* (/ (* y_m z_m) (sqrt (- (* a t)))) x_m)
      (* (/ (* x_m z_m) (* 1.0 z_m)) y_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.55e-49) {
		tmp = ((y_m * z_m) / sqrt(-(a * t))) * x_m;
	} else {
		tmp = ((x_m * z_m) / (1.0 * z_m)) * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
NOTE: x_m, y_m, z_m, t, and a 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(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.55d-49) then
        tmp = ((y_m * z_m) / sqrt(-(a * t))) * x_m
    else
        tmp = ((x_m * z_m) / (1.0d0 * z_m)) * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.55e-49) {
		tmp = ((y_m * z_m) / Math.sqrt(-(a * t))) * x_m;
	} else {
		tmp = ((x_m * z_m) / (1.0 * z_m)) * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.55e-49:
		tmp = ((y_m * z_m) / math.sqrt(-(a * t))) * x_m
	else:
		tmp = ((x_m * z_m) / (1.0 * z_m)) * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.55e-49)
		tmp = Float64(Float64(Float64(y_m * z_m) / sqrt(Float64(-Float64(a * t)))) * x_m);
	else
		tmp = Float64(Float64(Float64(x_m * z_m) / Float64(1.0 * z_m)) * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.55e-49)
		tmp = ((y_m * z_m) / sqrt(-(a * t))) * x_m;
	else
		tmp = ((x_m * z_m) / (1.0 * z_m)) * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.55e-49], N[(N[(N[(y$95$m * z$95$m), $MachinePrecision] / N[Sqrt[(-N[(a * t), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(x$95$m * z$95$m), $MachinePrecision] / N[(1.0 * z$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.55e-49

   1. Initial program 61.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. mult-flip N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. mult-flip-rev N/A

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

      14. lower-/.f64 59.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 59.3

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

   3. Applied rewrites 59.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. mult-flip-rev N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 65.7%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\sqrt{\mathsf{neg}\left(a \cdot t\right)}}} \cdot x \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot z}{\sqrt{\color{blue}{\mathsf{neg}\left(a \cdot t\right)}}} \cdot x \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{y \cdot z}{\sqrt{\mathsf{neg}\left(a \cdot t\right)}} \cdot x \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{y \cdot z}{\sqrt{-a \cdot t}} \cdot x \]

      5. lower-*.f64 34.6

         \[\leadsto \frac{y \cdot z}{\sqrt{-a \cdot t}} \cdot x \]

   8. Applied rewrites 34.6%

      \[\leadsto \color{blue}{\frac{y \cdot z}{\sqrt{-a \cdot t}}} \cdot x \]

   if 1.55e-49 < z

   1. Initial program 61.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 64.8

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{z \cdot \left(1 + -0.5 \cdot \frac{a \cdot t}{{z}^{\color{blue}{2}}}\right)} \]

   4. Applied rewrites 64.8%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z \cdot \left(1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}\right)}} \]

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 65.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   9. Applied rewrites 74.5%

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

Alternative 8: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.55 \cdot 10^{-49}:\\ \;\;\;\;\frac{x\_m \cdot \left(y\_m \cdot z\_m\right)}{\sqrt{-a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot z\_m}{1 \cdot z\_m} \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1.55e-49)
      (/ (* x_m (* y_m z_m)) (sqrt (- (* a t))))
      (* (/ (* x_m z_m) (* 1.0 z_m)) y_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.55e-49) {
		tmp = (x_m * (y_m * z_m)) / sqrt(-(a * t));
	} else {
		tmp = ((x_m * z_m) / (1.0 * z_m)) * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
NOTE: x_m, y_m, z_m, t, and a 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(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.55d-49) then
        tmp = (x_m * (y_m * z_m)) / sqrt(-(a * t))
    else
        tmp = ((x_m * z_m) / (1.0d0 * z_m)) * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.55e-49) {
		tmp = (x_m * (y_m * z_m)) / Math.sqrt(-(a * t));
	} else {
		tmp = ((x_m * z_m) / (1.0 * z_m)) * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.55e-49:
		tmp = (x_m * (y_m * z_m)) / math.sqrt(-(a * t))
	else:
		tmp = ((x_m * z_m) / (1.0 * z_m)) * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.55e-49)
		tmp = Float64(Float64(x_m * Float64(y_m * z_m)) / sqrt(Float64(-Float64(a * t))));
	else
		tmp = Float64(Float64(Float64(x_m * z_m) / Float64(1.0 * z_m)) * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.55e-49)
		tmp = (x_m * (y_m * z_m)) / sqrt(-(a * t));
	else
		tmp = ((x_m * z_m) / (1.0 * z_m)) * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.55e-49], N[(N[(x$95$m * N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-N[(a * t), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m * z$95$m), $MachinePrecision] / N[(1.0 * z$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.55e-49

   1. Initial program 61.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot \left(y \cdot z\right)}{\color{blue}{\sqrt{\mathsf{neg}\left(a \cdot t\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot \left(y \cdot z\right)}{\sqrt{\color{blue}{\mathsf{neg}\left(a \cdot t\right)}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot \left(y \cdot z\right)}{\sqrt{\mathsf{neg}\left(a \cdot t\right)}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{x \cdot \left(y \cdot z\right)}{\sqrt{\mathsf{neg}\left(a \cdot t\right)}} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{x \cdot \left(y \cdot z\right)}{\sqrt{-a \cdot t}} \]

      6. lower-*.f64 33.7

         \[\leadsto \frac{x \cdot \left(y \cdot z\right)}{\sqrt{-a \cdot t}} \]

   4. Applied rewrites 33.7%

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

   if 1.55e-49 < z

   1. Initial program 61.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 64.8

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{z \cdot \left(1 + -0.5 \cdot \frac{a \cdot t}{{z}^{\color{blue}{2}}}\right)} \]

   4. Applied rewrites 64.8%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z \cdot \left(1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}\right)}} \]

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 65.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   9. Applied rewrites 74.5%

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

Alternative 9: 73.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot y\_m \leq 5 \cdot 10^{+28}:\\ \;\;\;\;\frac{\left(x\_m \cdot y\_m\right) \cdot z\_m}{z\_m \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;z\_m \cdot \left(x\_m \cdot \frac{y\_m}{1 \cdot z\_m}\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= (* x_m y_m) 5e+28)
      (/ (* (* x_m y_m) z_m) (* z_m 1.0))
      (* z_m (* x_m (/ y_m (* 1.0 z_m)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if ((x_m * y_m) <= 5e+28) {
		tmp = ((x_m * y_m) * z_m) / (z_m * 1.0);
	} else {
		tmp = z_m * (x_m * (y_m / (1.0 * z_m)));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
NOTE: x_m, y_m, z_m, t, and a 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(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((x_m * y_m) <= 5d+28) then
        tmp = ((x_m * y_m) * z_m) / (z_m * 1.0d0)
    else
        tmp = z_m * (x_m * (y_m / (1.0d0 * z_m)))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if ((x_m * y_m) <= 5e+28) {
		tmp = ((x_m * y_m) * z_m) / (z_m * 1.0);
	} else {
		tmp = z_m * (x_m * (y_m / (1.0 * z_m)));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if (x_m * y_m) <= 5e+28:
		tmp = ((x_m * y_m) * z_m) / (z_m * 1.0)
	else:
		tmp = z_m * (x_m * (y_m / (1.0 * z_m)))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (Float64(x_m * y_m) <= 5e+28)
		tmp = Float64(Float64(Float64(x_m * y_m) * z_m) / Float64(z_m * 1.0));
	else
		tmp = Float64(z_m * Float64(x_m * Float64(y_m / Float64(1.0 * z_m))));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if ((x_m * y_m) <= 5e+28)
		tmp = ((x_m * y_m) * z_m) / (z_m * 1.0);
	else
		tmp = z_m * (x_m * (y_m / (1.0 * z_m)));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[N[(x$95$m * y$95$m), $MachinePrecision], 5e+28], N[(N[(N[(x$95$m * y$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] / N[(z$95$m * 1.0), $MachinePrecision]), $MachinePrecision], N[(z$95$m * N[(x$95$m * N[(y$95$m / N[(1.0 * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 4.99999999999999957e28

   1. Initial program 61.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 64.8

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{z \cdot \left(1 + -0.5 \cdot \frac{a \cdot t}{{z}^{\color{blue}{2}}}\right)} \]

   4. Applied rewrites 64.8%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z \cdot \left(1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}\right)}} \]

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 65.6%

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

   if 4.99999999999999957e28 < (*.f64 x y)

   1. Initial program 61.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 64.8

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{z \cdot \left(1 + -0.5 \cdot \frac{a \cdot t}{{z}^{\color{blue}{2}}}\right)} \]

   4. Applied rewrites 64.8%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z \cdot \left(1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}\right)}} \]

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 65.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   9. Applied rewrites 74.5%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. mult-flip N/A

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

       4. associate-*l* N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*l* N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 N/A

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

   11. Applied rewrites 61.6%

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

Alternative 10: 71.3% accurate, 1.6× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (* z_s (* y_s (* x_s (* (/ (* x_m z_m) (* 1.0 z_m)) y_m)))))

C

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

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
NOTE: x_m, y_m, z_m, t, and a 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(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = z_s * (y_s * (x_s * (((x_m * z_m) / (1.0d0 * z_m)) * y_m)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.1%

   \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

2. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 64.8

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{z \cdot \left(1 + -0.5 \cdot \frac{a \cdot t}{{z}^{\color{blue}{2}}}\right)} \]

4. Applied rewrites 64.8%

   \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z \cdot \left(1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}\right)}} \]

5. Taylor expanded in z around inf

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

7. Applied rewrites 65.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

9. Applied rewrites 74.5%

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

Alternative 11: 71.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 7.5 \cdot 10^{-135}:\\ \;\;\;\;\frac{\left(-y\_m\right) \cdot \left(z\_m \cdot x\_m\right)}{-\left(-z\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot z\_m\right) \cdot \frac{y\_m}{1 \cdot z\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 7.5e-135)
      (/ (* (- y_m) (* z_m x_m)) (- (- z_m)))
      (* (* x_m z_m) (/ y_m (* 1.0 z_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 7.5e-135) {
		tmp = (-y_m * (z_m * x_m)) / -(-z_m);
	} else {
		tmp = (x_m * z_m) * (y_m / (1.0 * z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
NOTE: x_m, y_m, z_m, t, and a 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(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 7.5d-135) then
        tmp = (-y_m * (z_m * x_m)) / -(-z_m)
    else
        tmp = (x_m * z_m) * (y_m / (1.0d0 * z_m))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 7.5e-135) {
		tmp = (-y_m * (z_m * x_m)) / -(-z_m);
	} else {
		tmp = (x_m * z_m) * (y_m / (1.0 * z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 7.5e-135:
		tmp = (-y_m * (z_m * x_m)) / -(-z_m)
	else:
		tmp = (x_m * z_m) * (y_m / (1.0 * z_m))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 7.5e-135)
		tmp = Float64(Float64(Float64(-y_m) * Float64(z_m * x_m)) / Float64(-Float64(-z_m)));
	else
		tmp = Float64(Float64(x_m * z_m) * Float64(y_m / Float64(1.0 * z_m)));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 7.5e-135)
		tmp = (-y_m * (z_m * x_m)) / -(-z_m);
	else
		tmp = (x_m * z_m) * (y_m / (1.0 * z_m));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 7.5e-135], N[(N[((-y$95$m) * N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / (-(-z$95$m))), $MachinePrecision], N[(N[(x$95$m * z$95$m), $MachinePrecision] * N[(y$95$m / N[(1.0 * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 7.5e-135

   1. Initial program 61.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

   2. Taylor expanded in z around -inf

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

      1. lower-*.f64 17.6

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

   4. Applied rewrites 17.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-neg-out N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 19.0

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

      19. lift-*.f64 N/A

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

      20. mul-1-neg N/A

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

      21. lower-neg.f64 19.0

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

   6. Applied rewrites 19.0%

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

   if 7.5e-135 < z

   1. Initial program 61.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 64.8

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{z \cdot \left(1 + -0.5 \cdot \frac{a \cdot t}{{z}^{\color{blue}{2}}}\right)} \]

   4. Applied rewrites 64.8%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z \cdot \left(1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}\right)}} \]

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 65.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 70.4

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 70.4

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

   9. Applied rewrites 70.4%

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

Alternative 12: 64.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.7 \cdot 10^{-134}:\\ \;\;\;\;\frac{\left(-y\_m\right) \cdot \left(z\_m \cdot x\_m\right)}{-\left(-z\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;z\_m \cdot \left(x\_m \cdot \frac{y\_m}{1 \cdot z\_m}\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 2.7e-134)
      (/ (* (- y_m) (* z_m x_m)) (- (- z_m)))
      (* z_m (* x_m (/ y_m (* 1.0 z_m)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.7e-134) {
		tmp = (-y_m * (z_m * x_m)) / -(-z_m);
	} else {
		tmp = z_m * (x_m * (y_m / (1.0 * z_m)));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
NOTE: x_m, y_m, z_m, t, and a 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(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 2.7d-134) then
        tmp = (-y_m * (z_m * x_m)) / -(-z_m)
    else
        tmp = z_m * (x_m * (y_m / (1.0d0 * z_m)))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.7e-134) {
		tmp = (-y_m * (z_m * x_m)) / -(-z_m);
	} else {
		tmp = z_m * (x_m * (y_m / (1.0 * z_m)));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 2.7e-134:
		tmp = (-y_m * (z_m * x_m)) / -(-z_m)
	else:
		tmp = z_m * (x_m * (y_m / (1.0 * z_m)))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.7e-134)
		tmp = Float64(Float64(Float64(-y_m) * Float64(z_m * x_m)) / Float64(-Float64(-z_m)));
	else
		tmp = Float64(z_m * Float64(x_m * Float64(y_m / Float64(1.0 * z_m))));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2.7e-134)
		tmp = (-y_m * (z_m * x_m)) / -(-z_m);
	else
		tmp = z_m * (x_m * (y_m / (1.0 * z_m)));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2.7e-134], N[(N[((-y$95$m) * N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / (-(-z$95$m))), $MachinePrecision], N[(z$95$m * N[(x$95$m * N[(y$95$m / N[(1.0 * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.6999999999999998e-134

   1. Initial program 61.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

   2. Taylor expanded in z around -inf

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

      1. lower-*.f64 17.6

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

   4. Applied rewrites 17.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-neg-out N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 19.0

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

      19. lift-*.f64 N/A

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

      20. mul-1-neg N/A

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

      21. lower-neg.f64 19.0

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

   6. Applied rewrites 19.0%

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

   if 2.6999999999999998e-134 < z

   1. Initial program 61.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 64.8

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{z \cdot \left(1 + -0.5 \cdot \frac{a \cdot t}{{z}^{\color{blue}{2}}}\right)} \]

   4. Applied rewrites 64.8%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z \cdot \left(1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}\right)}} \]

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 65.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   9. Applied rewrites 74.5%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. mult-flip N/A

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

       4. associate-*l* N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*l* N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 N/A

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

   11. Applied rewrites 61.6%

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

Alternative 13: 56.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.2 \cdot 10^{-136}:\\ \;\;\;\;\frac{\left(-y\_m\right) \cdot \left(z\_m \cdot x\_m\right)}{-\left(-z\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x\_m}{z\_m} \cdot z\_m\right) \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1.2e-136)
      (/ (* (- y_m) (* z_m x_m)) (- (- z_m)))
      (* (* (/ x_m z_m) z_m) y_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.2e-136) {
		tmp = (-y_m * (z_m * x_m)) / -(-z_m);
	} else {
		tmp = ((x_m / z_m) * z_m) * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
NOTE: x_m, y_m, z_m, t, and a 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(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.2d-136) then
        tmp = (-y_m * (z_m * x_m)) / -(-z_m)
    else
        tmp = ((x_m / z_m) * z_m) * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.2e-136) {
		tmp = (-y_m * (z_m * x_m)) / -(-z_m);
	} else {
		tmp = ((x_m / z_m) * z_m) * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.2e-136:
		tmp = (-y_m * (z_m * x_m)) / -(-z_m)
	else:
		tmp = ((x_m / z_m) * z_m) * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.2e-136)
		tmp = Float64(Float64(Float64(-y_m) * Float64(z_m * x_m)) / Float64(-Float64(-z_m)));
	else
		tmp = Float64(Float64(Float64(x_m / z_m) * z_m) * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.2e-136)
		tmp = (-y_m * (z_m * x_m)) / -(-z_m);
	else
		tmp = ((x_m / z_m) * z_m) * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.2e-136], N[(N[((-y$95$m) * N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / (-(-z$95$m))), $MachinePrecision], N[(N[(N[(x$95$m / z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.1999999999999999e-136

   1. Initial program 61.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

   2. Taylor expanded in z around -inf

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

      1. lower-*.f64 17.6

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

   4. Applied rewrites 17.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-neg-out N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 19.0

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

      19. lift-*.f64 N/A

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

      20. mul-1-neg N/A

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

      21. lower-neg.f64 19.0

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

   6. Applied rewrites 19.0%

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

   if 1.1999999999999999e-136 < z

   1. Initial program 61.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. mult-flip N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. mult-flip-rev N/A

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

      14. lower-/.f64 59.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 59.3

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

   3. Applied rewrites 59.3%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 54.9

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

   6. Applied rewrites 54.9%

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

Alternative 14: 56.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.2 \cdot 10^{-136}:\\ \;\;\;\;\frac{\left(y\_m \cdot x\_m\right) \cdot z\_m}{-z\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x\_m}{z\_m} \cdot z\_m\right) \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1.2e-136)
      (/ (* (* y_m x_m) z_m) (- z_m))
      (* (* (/ x_m z_m) z_m) y_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.2e-136) {
		tmp = ((y_m * x_m) * z_m) / -z_m;
	} else {
		tmp = ((x_m / z_m) * z_m) * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
NOTE: x_m, y_m, z_m, t, and a 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(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.2d-136) then
        tmp = ((y_m * x_m) * z_m) / -z_m
    else
        tmp = ((x_m / z_m) * z_m) * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.2e-136) {
		tmp = ((y_m * x_m) * z_m) / -z_m;
	} else {
		tmp = ((x_m / z_m) * z_m) * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.2e-136:
		tmp = ((y_m * x_m) * z_m) / -z_m
	else:
		tmp = ((x_m / z_m) * z_m) * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.2e-136)
		tmp = Float64(Float64(Float64(y_m * x_m) * z_m) / Float64(-z_m));
	else
		tmp = Float64(Float64(Float64(x_m / z_m) * z_m) * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.2e-136)
		tmp = ((y_m * x_m) * z_m) / -z_m;
	else
		tmp = ((x_m / z_m) * z_m) * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.2e-136], N[(N[(N[(y$95$m * x$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] / (-z$95$m)), $MachinePrecision], N[(N[(N[(x$95$m / z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.1999999999999999e-136

   1. Initial program 61.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

   2. Taylor expanded in z around -inf

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

      1. lower-*.f64 17.6

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

   4. Applied rewrites 17.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 17.6

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 17.6

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

   6. Applied rewrites 17.6%

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

   if 1.1999999999999999e-136 < z

   1. Initial program 61.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. mult-flip N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. mult-flip-rev N/A

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

      14. lower-/.f64 59.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 59.3

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

   3. Applied rewrites 59.3%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 54.9

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

   6. Applied rewrites 54.9%

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

Alternative 15: 55.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.35 \cdot 10^{-155}:\\ \;\;\;\;\frac{z\_m \cdot x\_m}{-z\_m} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x\_m}{z\_m} \cdot z\_m\right) \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1.35e-155)
      (* (/ (* z_m x_m) (- z_m)) y_m)
      (* (* (/ x_m z_m) z_m) y_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.35e-155) {
		tmp = ((z_m * x_m) / -z_m) * y_m;
	} else {
		tmp = ((x_m / z_m) * z_m) * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
NOTE: x_m, y_m, z_m, t, and a 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(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.35d-155) then
        tmp = ((z_m * x_m) / -z_m) * y_m
    else
        tmp = ((x_m / z_m) * z_m) * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.35e-155) {
		tmp = ((z_m * x_m) / -z_m) * y_m;
	} else {
		tmp = ((x_m / z_m) * z_m) * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.35e-155:
		tmp = ((z_m * x_m) / -z_m) * y_m
	else:
		tmp = ((x_m / z_m) * z_m) * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.35e-155)
		tmp = Float64(Float64(Float64(z_m * x_m) / Float64(-z_m)) * y_m);
	else
		tmp = Float64(Float64(Float64(x_m / z_m) * z_m) * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.35e-155)
		tmp = ((z_m * x_m) / -z_m) * y_m;
	else
		tmp = ((x_m / z_m) * z_m) * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.35e-155], N[(N[(N[(z$95$m * x$95$m), $MachinePrecision] / (-z$95$m)), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(x$95$m / z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.34999999999999991e-155

   1. Initial program 61.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

   2. Taylor expanded in z around -inf

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

      1. lower-*.f64 17.6

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

   4. Applied rewrites 17.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 18.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 18.6

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

      15. lift-*.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 18.6

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

   6. Applied rewrites 18.6%

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

   if 1.34999999999999991e-155 < z

   1. Initial program 61.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. mult-flip N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. mult-flip-rev N/A

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

      14. lower-/.f64 59.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 59.3

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

   3. Applied rewrites 59.3%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 54.9

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

   6. Applied rewrites 54.9%

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

Alternative 16: 54.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
NOTE: x_m, y_m, z_m, t, and a 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(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = z_s * (y_s * (x_s * (((x_m / z_m) * z_m) * y_m)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. mult-flip N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

   13. mult-flip-rev N/A

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

   14. lower-/.f64 59.3

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 59.3

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

3. Applied rewrites 59.3%

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

4. Taylor expanded in z around inf

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

   1. lower-/.f64 54.9

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

6. Applied rewrites 54.9%

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

Alternative 17: 54.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3.4 \cdot 10^{-44}:\\ \;\;\;\;\left(y\_m \cdot z\_m\right) \cdot \frac{x\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x\_m}{z\_m} \cdot y\_m\right) \cdot z\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= y_m 3.4e-44)
      (* (* y_m z_m) (/ x_m z_m))
      (* (* (/ x_m z_m) y_m) z_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (y_m <= 3.4e-44) {
		tmp = (y_m * z_m) * (x_m / z_m);
	} else {
		tmp = ((x_m / z_m) * y_m) * z_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
NOTE: x_m, y_m, z_m, t, and a 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(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (y_m <= 3.4d-44) then
        tmp = (y_m * z_m) * (x_m / z_m)
    else
        tmp = ((x_m / z_m) * y_m) * z_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (y_m <= 3.4e-44) {
		tmp = (y_m * z_m) * (x_m / z_m);
	} else {
		tmp = ((x_m / z_m) * y_m) * z_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if y_m <= 3.4e-44:
		tmp = (y_m * z_m) * (x_m / z_m)
	else:
		tmp = ((x_m / z_m) * y_m) * z_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (y_m <= 3.4e-44)
		tmp = Float64(Float64(y_m * z_m) * Float64(x_m / z_m));
	else
		tmp = Float64(Float64(Float64(x_m / z_m) * y_m) * z_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (y_m <= 3.4e-44)
		tmp = (y_m * z_m) * (x_m / z_m);
	else
		tmp = ((x_m / z_m) * y_m) * z_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 3.4e-44], N[(N[(y$95$m * z$95$m), $MachinePrecision] * N[(x$95$m / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m / z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] * z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.40000000000000016e-44

   1. Initial program 61.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. mult-flip N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. mult-flip-rev N/A

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

      14. lower-/.f64 59.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 59.3

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

   3. Applied rewrites 59.3%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 54.9

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

   6. Applied rewrites 54.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 46.6

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

   8. Applied rewrites 46.6%

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

   if 3.40000000000000016e-44 < y

   1. Initial program 61.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. mult-flip N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. mult-flip-rev N/A

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

      14. lower-/.f64 59.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 59.3

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

   3. Applied rewrites 59.3%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 54.9

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

   6. Applied rewrites 54.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 53.1

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

   8. Applied rewrites 53.1%

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

Alternative 18: 46.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
NOTE: x_m, y_m, z_m, t, and a 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(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = z_s * (y_s * (x_s * ((y_m * z_m) * (x_m / z_m))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. mult-flip N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

   13. mult-flip-rev N/A

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

   14. lower-/.f64 59.3

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 59.3

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

3. Applied rewrites 59.3%

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

4. Taylor expanded in z around inf

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

   1. lower-/.f64 54.9

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

6. Applied rewrites 54.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 46.6

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

8. Applied rewrites 46.6%

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

Alternative 19: 14.1% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
NOTE: x_m, y_m, z_m, t, and a 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(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = z_s * (y_s * (x_s * (-y_m * x_m)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.1%

   \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]

2. Taylor expanded in z around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 14.1

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

4. Applied rewrites 14.1%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(x \cdot y\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(x \cdot y\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(y \cdot x\right) \]

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

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

   6. lower-*.f64 N/A

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

   7. lower-neg.f64 14.1

      \[\leadsto \left(-y\right) \cdot x \]

6. Applied rewrites 14.1%

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

Reproduce

herbie shell --seed 2025154 
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"
  :precision binary64
  (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))