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

Percentage Accurate: 61.1% → 91.8%

Time: 5.1s

Alternatives: 10

Speedup: 3.0×

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, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.12 \cdot 10^{-153}:\\ \;\;\;\;\frac{\left(z\_m \cdot y\right) \cdot x\_m}{\sqrt{\left(-t\right) \cdot a}}\\ \mathbf{elif}\;z\_m \leq 4.9 \cdot 10^{+129}:\\ \;\;\;\;\left(\frac{z\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}} \cdot x\_m\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\_m\right) \cdot 1\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x_s x_m y z_m t a)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 1.12e-153)
     (/ (* (* z_m y) x_m) (sqrt (* (- t) a)))
     (if (<= z_m 4.9e+129)
       (* (* (/ z_m (sqrt (- (* z_m z_m) (* t a)))) x_m) y)
       (* (* y x_m) 1.0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.12e-153) {
		tmp = ((z_m * y) * x_m) / sqrt((-t * a));
	} else if (z_m <= 4.9e+129) {
		tmp = ((z_m / sqrt(((z_m * z_m) - (t * a)))) * x_m) * y;
	} else {
		tmp = (y * x_m) * 1.0;
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m =     private
x\_s =     private
z\_m =     private
z\_s =     private
NOTE: x_m, y, 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, x_s, x_m, y, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.12d-153) then
        tmp = ((z_m * y) * x_m) / sqrt((-t * a))
    else if (z_m <= 4.9d+129) then
        tmp = ((z_m / sqrt(((z_m * z_m) - (t * a)))) * x_m) * y
    else
        tmp = (y * x_m) * 1.0d0
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y && y < z_m && z_m < t && t < a;
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.12e-153) {
		tmp = ((z_m * y) * x_m) / Math.sqrt((-t * a));
	} else if (z_m <= 4.9e+129) {
		tmp = ((z_m / Math.sqrt(((z_m * z_m) - (t * a)))) * x_m) * y;
	} else {
		tmp = (y * x_m) * 1.0;
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y, z_m, t, a] = sort([x_m, y, z_m, t, a])
def code(z_s, x_s, x_m, y, z_m, t, a):
	tmp = 0
	if z_m <= 1.12e-153:
		tmp = ((z_m * y) * x_m) / math.sqrt((-t * a))
	elif z_m <= 4.9e+129:
		tmp = ((z_m / math.sqrt(((z_m * z_m) - (t * a)))) * x_m) * y
	else:
		tmp = (y * x_m) * 1.0
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.12e-153)
		tmp = Float64(Float64(Float64(z_m * y) * x_m) / sqrt(Float64(Float64(-t) * a)));
	elseif (z_m <= 4.9e+129)
		tmp = Float64(Float64(Float64(z_m / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))) * x_m) * y);
	else
		tmp = Float64(Float64(y * x_m) * 1.0);
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y, z_m, t, a = num2cell(sort([x_m, y, z_m, t, a])){:}
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.12e-153)
		tmp = ((z_m * y) * x_m) / sqrt((-t * a));
	elseif (z_m <= 4.9e+129)
		tmp = ((z_m / sqrt(((z_m * z_m) - (t * a)))) * x_m) * y;
	else
		tmp = (y * x_m) * 1.0;
	end
	tmp_2 = z_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]
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, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.12e-153], N[(N[(N[(z$95$m * y), $MachinePrecision] * x$95$m), $MachinePrecision] / N[Sqrt[N[((-t) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 4.9e+129], N[(N[(N[(z$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] * y), $MachinePrecision], N[(N[(y * x$95$m), $MachinePrecision] * 1.0), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 1.12000000000000005e-153

   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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow2 N/A

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

      6. *-commutative N/A

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

      7. sqrt-pow2 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 33.7

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

   4. Applied rewrites 33.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 33.2

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

   6. Applied rewrites 33.2%

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

   if 1.12000000000000005e-153 < z < 4.9e129

   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. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-sqrt.f64 64.0

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 64.0

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

   3. Applied rewrites 64.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-/.f64 63.3

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

   5. Applied rewrites 63.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 63.3%

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

   if 4.9e129 < 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. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-sqrt.f64 64.0

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 64.0

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

   3. Applied rewrites 64.0%

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

   4. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(a \cdot \frac{t}{{z}^{2}}, \frac{1}{2}, 1\right) \]

      5. pow2 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(a \cdot \frac{t}{z \cdot z}, \frac{1}{2}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, \frac{1}{2}, 1\right) \]

      7. pow2 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{{z}^{2}} \cdot a, \frac{1}{2}, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{{z}^{2}} \cdot a, \frac{1}{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, \frac{1}{2}, 1\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, \frac{1}{2}, 1\right) \]

      11. lift-*.f64 67.1

          \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, 0.5, 1\right) \]

   6. Applied rewrites 67.1%

      \[\leadsto \left(y \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, 0.5, 1\right)} \]

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 73.0%

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

Alternative 2: 91.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4 \cdot 10^{+152}:\\ \;\;\;\;\left(y \cdot x\_m\right) \cdot \frac{z\_m}{\sqrt{z\_m \cdot z\_m - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\_m\right) \cdot \mathsf{fma}\left(\frac{\frac{t}{z\_m}}{z\_m} \cdot a, 0.5, 1\right)\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x_s x_m y z_m t a)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 4e+152)
     (* (* y x_m) (/ z_m (sqrt (- (* z_m z_m) (* a t)))))
     (* (* y x_m) (fma (* (/ (/ t z_m) z_m) a) 0.5 1.0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 4e+152) {
		tmp = (y * x_m) * (z_m / sqrt(((z_m * z_m) - (a * t))));
	} else {
		tmp = (y * x_m) * fma((((t / z_m) / z_m) * a), 0.5, 1.0);
	}
	return z_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 4e+152)
		tmp = Float64(Float64(y * x_m) * Float64(z_m / sqrt(Float64(Float64(z_m * z_m) - Float64(a * t)))));
	else
		tmp = Float64(Float64(y * x_m) * fma(Float64(Float64(Float64(t / z_m) / z_m) * a), 0.5, 1.0));
	end
	return Float64(z_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]
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, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 4e+152], N[(N[(y * x$95$m), $MachinePrecision] * N[(z$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x$95$m), $MachinePrecision] * N[(N[(N[(N[(t / z$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] * a), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.0000000000000002e152

   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. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-sqrt.f64 64.0

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 64.0

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

   3. Applied rewrites 64.0%

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

   if 4.0000000000000002e152 < 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. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-sqrt.f64 64.0

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 64.0

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

   3. Applied rewrites 64.0%

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

   4. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(a \cdot \frac{t}{{z}^{2}}, \frac{1}{2}, 1\right) \]

      5. pow2 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(a \cdot \frac{t}{z \cdot z}, \frac{1}{2}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, \frac{1}{2}, 1\right) \]

      7. pow2 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{{z}^{2}} \cdot a, \frac{1}{2}, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{{z}^{2}} \cdot a, \frac{1}{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, \frac{1}{2}, 1\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, \frac{1}{2}, 1\right) \]

      11. lift-*.f64 67.1

          \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, 0.5, 1\right) \]

   6. Applied rewrites 67.1%

      \[\leadsto \left(y \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, 0.5, 1\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, \frac{1}{2}, 1\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, \frac{1}{2}, 1\right) \]

      3. associate-/r* N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{\frac{t}{z}}{z} \cdot a, \frac{1}{2}, 1\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{\frac{t}{z}}{z} \cdot a, \frac{1}{2}, 1\right) \]

      5. lift-/.f64 68.4

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{\frac{t}{z}}{z} \cdot a, 0.5, 1\right) \]

   8. Applied rewrites 68.4%

      \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{\frac{t}{z}}{z} \cdot a, 0.5, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 91.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4.9 \cdot 10^{+129}:\\ \;\;\;\;\left(y \cdot x\_m\right) \cdot \frac{z\_m}{\sqrt{z\_m \cdot z\_m - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\_m\right) \cdot 1\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x_s x_m y z_m t a)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 4.9e+129)
     (* (* y x_m) (/ z_m (sqrt (- (* z_m z_m) (* a t)))))
     (* (* y x_m) 1.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 4.9e+129) {
		tmp = (y * x_m) * (z_m / sqrt(((z_m * z_m) - (a * t))));
	} else {
		tmp = (y * x_m) * 1.0;
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m =     private
x\_s =     private
z\_m =     private
z\_s =     private
NOTE: x_m, y, 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, x_s, x_m, y, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 4.9d+129) then
        tmp = (y * x_m) * (z_m / sqrt(((z_m * z_m) - (a * t))))
    else
        tmp = (y * x_m) * 1.0d0
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y && y < z_m && z_m < t && t < a;
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 4.9e+129) {
		tmp = (y * x_m) * (z_m / Math.sqrt(((z_m * z_m) - (a * t))));
	} else {
		tmp = (y * x_m) * 1.0;
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y, z_m, t, a] = sort([x_m, y, z_m, t, a])
def code(z_s, x_s, x_m, y, z_m, t, a):
	tmp = 0
	if z_m <= 4.9e+129:
		tmp = (y * x_m) * (z_m / math.sqrt(((z_m * z_m) - (a * t))))
	else:
		tmp = (y * x_m) * 1.0
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 4.9e+129)
		tmp = Float64(Float64(y * x_m) * Float64(z_m / sqrt(Float64(Float64(z_m * z_m) - Float64(a * t)))));
	else
		tmp = Float64(Float64(y * x_m) * 1.0);
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y, z_m, t, a = num2cell(sort([x_m, y, z_m, t, a])){:}
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 4.9e+129)
		tmp = (y * x_m) * (z_m / sqrt(((z_m * z_m) - (a * t))));
	else
		tmp = (y * x_m) * 1.0;
	end
	tmp_2 = z_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]
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, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 4.9e+129], N[(N[(y * x$95$m), $MachinePrecision] * N[(z$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x$95$m), $MachinePrecision] * 1.0), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.9e129

   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. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-sqrt.f64 64.0

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 64.0

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

   3. Applied rewrites 64.0%

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

   if 4.9e129 < 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. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-sqrt.f64 64.0

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 64.0

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

   3. Applied rewrites 64.0%

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

   4. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(a \cdot \frac{t}{{z}^{2}}, \frac{1}{2}, 1\right) \]

      5. pow2 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(a \cdot \frac{t}{z \cdot z}, \frac{1}{2}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, \frac{1}{2}, 1\right) \]

      7. pow2 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{{z}^{2}} \cdot a, \frac{1}{2}, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{{z}^{2}} \cdot a, \frac{1}{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, \frac{1}{2}, 1\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, \frac{1}{2}, 1\right) \]

      11. lift-*.f64 67.1

          \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, 0.5, 1\right) \]

   6. Applied rewrites 67.1%

      \[\leadsto \left(y \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, 0.5, 1\right)} \]

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 73.0%

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

Alternative 4: 82.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.25 \cdot 10^{-129}:\\ \;\;\;\;\frac{\left(z\_m \cdot y\right) \cdot x\_m}{\sqrt{\left(-t\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\_m\right) \cdot 1\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x_s x_m y z_m t a)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 2.25e-129)
     (/ (* (* z_m y) x_m) (sqrt (* (- t) a)))
     (* (* y x_m) 1.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.25e-129) {
		tmp = ((z_m * y) * x_m) / sqrt((-t * a));
	} else {
		tmp = (y * x_m) * 1.0;
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m =     private
x\_s =     private
z\_m =     private
z\_s =     private
NOTE: x_m, y, 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, x_s, x_m, y, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 2.25d-129) then
        tmp = ((z_m * y) * x_m) / sqrt((-t * a))
    else
        tmp = (y * x_m) * 1.0d0
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y && y < z_m && z_m < t && t < a;
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.25e-129) {
		tmp = ((z_m * y) * x_m) / Math.sqrt((-t * a));
	} else {
		tmp = (y * x_m) * 1.0;
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y, z_m, t, a] = sort([x_m, y, z_m, t, a])
def code(z_s, x_s, x_m, y, z_m, t, a):
	tmp = 0
	if z_m <= 2.25e-129:
		tmp = ((z_m * y) * x_m) / math.sqrt((-t * a))
	else:
		tmp = (y * x_m) * 1.0
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.25e-129)
		tmp = Float64(Float64(Float64(z_m * y) * x_m) / sqrt(Float64(Float64(-t) * a)));
	else
		tmp = Float64(Float64(y * x_m) * 1.0);
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y, z_m, t, a = num2cell(sort([x_m, y, z_m, t, a])){:}
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2.25e-129)
		tmp = ((z_m * y) * x_m) / sqrt((-t * a));
	else
		tmp = (y * x_m) * 1.0;
	end
	tmp_2 = z_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]
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, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2.25e-129], N[(N[(N[(z$95$m * y), $MachinePrecision] * x$95$m), $MachinePrecision] / N[Sqrt[N[((-t) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(y * x$95$m), $MachinePrecision] * 1.0), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.25000000000000015e-129

   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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow2 N/A

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

      6. *-commutative N/A

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

      7. sqrt-pow2 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 33.7

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

   4. Applied rewrites 33.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 33.2

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

   6. Applied rewrites 33.2%

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

   if 2.25000000000000015e-129 < 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. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-sqrt.f64 64.0

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 64.0

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

   3. Applied rewrites 64.0%

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

   4. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(a \cdot \frac{t}{{z}^{2}}, \frac{1}{2}, 1\right) \]

      5. pow2 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(a \cdot \frac{t}{z \cdot z}, \frac{1}{2}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, \frac{1}{2}, 1\right) \]

      7. pow2 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{{z}^{2}} \cdot a, \frac{1}{2}, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{{z}^{2}} \cdot a, \frac{1}{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, \frac{1}{2}, 1\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, \frac{1}{2}, 1\right) \]

      11. lift-*.f64 67.1

          \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, 0.5, 1\right) \]

   6. Applied rewrites 67.1%

      \[\leadsto \left(y \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, 0.5, 1\right)} \]

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 73.0%

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

Alternative 5: 82.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.25 \cdot 10^{-129}:\\ \;\;\;\;\left(y \cdot x\_m\right) \cdot \frac{z\_m}{\sqrt{\left(-t\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\_m\right) \cdot 1\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x_s x_m y z_m t a)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 2.25e-129)
     (* (* y x_m) (/ z_m (sqrt (* (- t) a))))
     (* (* y x_m) 1.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.25e-129) {
		tmp = (y * x_m) * (z_m / sqrt((-t * a)));
	} else {
		tmp = (y * x_m) * 1.0;
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m =     private
x\_s =     private
z\_m =     private
z\_s =     private
NOTE: x_m, y, 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, x_s, x_m, y, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 2.25d-129) then
        tmp = (y * x_m) * (z_m / sqrt((-t * a)))
    else
        tmp = (y * x_m) * 1.0d0
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y && y < z_m && z_m < t && t < a;
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.25e-129) {
		tmp = (y * x_m) * (z_m / Math.sqrt((-t * a)));
	} else {
		tmp = (y * x_m) * 1.0;
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y, z_m, t, a] = sort([x_m, y, z_m, t, a])
def code(z_s, x_s, x_m, y, z_m, t, a):
	tmp = 0
	if z_m <= 2.25e-129:
		tmp = (y * x_m) * (z_m / math.sqrt((-t * a)))
	else:
		tmp = (y * x_m) * 1.0
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.25e-129)
		tmp = Float64(Float64(y * x_m) * Float64(z_m / sqrt(Float64(Float64(-t) * a))));
	else
		tmp = Float64(Float64(y * x_m) * 1.0);
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y, z_m, t, a = num2cell(sort([x_m, y, z_m, t, a])){:}
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2.25e-129)
		tmp = (y * x_m) * (z_m / sqrt((-t * a)));
	else
		tmp = (y * x_m) * 1.0;
	end
	tmp_2 = z_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]
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, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2.25e-129], N[(N[(y * x$95$m), $MachinePrecision] * N[(z$95$m / N[Sqrt[N[((-t) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x$95$m), $MachinePrecision] * 1.0), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.25000000000000015e-129

   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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow2 N/A

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

      6. *-commutative N/A

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

      7. sqrt-pow2 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 33.7

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

   4. Applied rewrites 33.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 33.7

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

      9. pow2 33.7

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

      10. fp-cancel-sub-sign-inv 33.7

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

      11. lift-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 33.7

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

      16. pow2 33.7

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

   6. Applied rewrites 33.7%

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

   if 2.25000000000000015e-129 < 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. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-sqrt.f64 64.0

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 64.0

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

   3. Applied rewrites 64.0%

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

   4. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(a \cdot \frac{t}{{z}^{2}}, \frac{1}{2}, 1\right) \]

      5. pow2 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(a \cdot \frac{t}{z \cdot z}, \frac{1}{2}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, \frac{1}{2}, 1\right) \]

      7. pow2 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{{z}^{2}} \cdot a, \frac{1}{2}, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{{z}^{2}} \cdot a, \frac{1}{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, \frac{1}{2}, 1\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, \frac{1}{2}, 1\right) \]

      11. lift-*.f64 67.1

          \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, 0.5, 1\right) \]

   6. Applied rewrites 67.1%

      \[\leadsto \left(y \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, 0.5, 1\right)} \]

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 73.0%

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

Alternative 6: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.25 \cdot 10^{-129}:\\ \;\;\;\;y \cdot \left(x\_m \cdot \frac{z\_m}{\sqrt{\left(-t\right) \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\_m\right) \cdot 1\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x_s x_m y z_m t a)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 1.25e-129)
     (* y (* x_m (/ z_m (sqrt (* (- t) a)))))
     (* (* y x_m) 1.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.25e-129) {
		tmp = y * (x_m * (z_m / sqrt((-t * a))));
	} else {
		tmp = (y * x_m) * 1.0;
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m =     private
x\_s =     private
z\_m =     private
z\_s =     private
NOTE: x_m, y, 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, x_s, x_m, y, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.25d-129) then
        tmp = y * (x_m * (z_m / sqrt((-t * a))))
    else
        tmp = (y * x_m) * 1.0d0
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y && y < z_m && z_m < t && t < a;
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.25e-129) {
		tmp = y * (x_m * (z_m / Math.sqrt((-t * a))));
	} else {
		tmp = (y * x_m) * 1.0;
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y, z_m, t, a] = sort([x_m, y, z_m, t, a])
def code(z_s, x_s, x_m, y, z_m, t, a):
	tmp = 0
	if z_m <= 1.25e-129:
		tmp = y * (x_m * (z_m / math.sqrt((-t * a))))
	else:
		tmp = (y * x_m) * 1.0
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.25e-129)
		tmp = Float64(y * Float64(x_m * Float64(z_m / sqrt(Float64(Float64(-t) * a)))));
	else
		tmp = Float64(Float64(y * x_m) * 1.0);
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y, z_m, t, a = num2cell(sort([x_m, y, z_m, t, a])){:}
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.25e-129)
		tmp = y * (x_m * (z_m / sqrt((-t * a))));
	else
		tmp = (y * x_m) * 1.0;
	end
	tmp_2 = z_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]
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, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.25e-129], N[(y * N[(x$95$m * N[(z$95$m / N[Sqrt[N[((-t) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x$95$m), $MachinePrecision] * 1.0), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.25000000000000007e-129

   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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow2 N/A

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

      6. *-commutative N/A

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

      7. sqrt-pow2 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 33.7

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

   4. Applied rewrites 33.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 33.7

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

      9. pow2 33.7

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

      10. fp-cancel-sub-sign-inv 33.7

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

      11. lift-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 33.7

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

      16. pow2 33.7

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

   6. Applied rewrites 33.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 31.4

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

   8. Applied rewrites 31.4%

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

   if 1.25000000000000007e-129 < 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. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-sqrt.f64 64.0

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 64.0

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

   3. Applied rewrites 64.0%

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

   4. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(a \cdot \frac{t}{{z}^{2}}, \frac{1}{2}, 1\right) \]

      5. pow2 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(a \cdot \frac{t}{z \cdot z}, \frac{1}{2}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, \frac{1}{2}, 1\right) \]

      7. pow2 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{{z}^{2}} \cdot a, \frac{1}{2}, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{{z}^{2}} \cdot a, \frac{1}{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, \frac{1}{2}, 1\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, \frac{1}{2}, 1\right) \]

      11. lift-*.f64 67.1

          \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, 0.5, 1\right) \]

   6. Applied rewrites 67.1%

      \[\leadsto \left(y \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, 0.5, 1\right)} \]

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 73.0%

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

Alternative 7: 74.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ \begin{array}{l} t_1 := \left(x\_m \cdot y\right) \cdot z\_m\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{z\_m \cdot z\_m - t \cdot a}} \leq 5 \cdot 10^{-294}:\\ \;\;\;\;\frac{t\_1}{1 \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\_m\right) \cdot 1\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x_s x_m y z_m t a)
 :precision binary64
 (let* ((t_1 (* (* x_m y) z_m)))
   (*
    z_s
    (*
     x_s
     (if (<= (/ t_1 (sqrt (- (* z_m z_m) (* t a)))) 5e-294)
       (/ t_1 (* 1.0 z_m))
       (* (* y x_m) 1.0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double t_1 = (x_m * y) * z_m;
	double tmp;
	if ((t_1 / sqrt(((z_m * z_m) - (t * a)))) <= 5e-294) {
		tmp = t_1 / (1.0 * z_m);
	} else {
		tmp = (y * x_m) * 1.0;
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m =     private
x\_s =     private
z\_m =     private
z\_s =     private
NOTE: x_m, y, 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, x_s, x_m, y, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x_m * y) * z_m
    if ((t_1 / sqrt(((z_m * z_m) - (t * a)))) <= 5d-294) then
        tmp = t_1 / (1.0d0 * z_m)
    else
        tmp = (y * x_m) * 1.0d0
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y && y < z_m && z_m < t && t < a;
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double t_1 = (x_m * y) * z_m;
	double tmp;
	if ((t_1 / Math.sqrt(((z_m * z_m) - (t * a)))) <= 5e-294) {
		tmp = t_1 / (1.0 * z_m);
	} else {
		tmp = (y * x_m) * 1.0;
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y, z_m, t, a] = sort([x_m, y, z_m, t, a])
def code(z_s, x_s, x_m, y, z_m, t, a):
	t_1 = (x_m * y) * z_m
	tmp = 0
	if (t_1 / math.sqrt(((z_m * z_m) - (t * a)))) <= 5e-294:
		tmp = t_1 / (1.0 * z_m)
	else:
		tmp = (y * x_m) * 1.0
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(z_s, x_s, x_m, y, z_m, t, a)
	t_1 = Float64(Float64(x_m * y) * z_m)
	tmp = 0.0
	if (Float64(t_1 / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))) <= 5e-294)
		tmp = Float64(t_1 / Float64(1.0 * z_m));
	else
		tmp = Float64(Float64(y * x_m) * 1.0);
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y, z_m, t, a = num2cell(sort([x_m, y, z_m, t, a])){:}
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t, a)
	t_1 = (x_m * y) * z_m;
	tmp = 0.0;
	if ((t_1 / sqrt(((z_m * z_m) - (t * a)))) <= 5e-294)
		tmp = t_1 / (1.0 * z_m);
	else
		tmp = (y * x_m) * 1.0;
	end
	tmp_2 = z_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]
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, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := Block[{t$95$1 = N[(N[(x$95$m * y), $MachinePrecision] * z$95$m), $MachinePrecision]}, N[(z$95$s * N[(x$95$s * If[LessEqual[N[(t$95$1 / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e-294], N[(t$95$1 / N[(1.0 * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y * x$95$m), $MachinePrecision] * 1.0), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a)))) < 5.0000000000000003e-294

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\mathsf{fma}\left(\frac{a \cdot t}{{z}^{2}}, \frac{-1}{2}, 1\right) \cdot z} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\mathsf{fma}\left(a \cdot \frac{t}{{z}^{2}}, \frac{-1}{2}, 1\right) \cdot z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\mathsf{fma}\left(a \cdot \frac{t}{{z}^{2}}, \frac{-1}{2}, 1\right) \cdot z} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\mathsf{fma}\left(a \cdot \frac{t}{{z}^{2}}, \frac{-1}{2}, 1\right) \cdot z} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\mathsf{fma}\left(a \cdot \frac{t}{z \cdot z}, \frac{-1}{2}, 1\right) \cdot z} \]

      10. lift-*.f64 66.7

          \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\mathsf{fma}\left(a \cdot \frac{t}{z \cdot z}, -0.5, 1\right) \cdot z} \]

   4. Applied rewrites 66.7%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{\mathsf{fma}\left(a \cdot \frac{t}{z \cdot z}, -0.5, 1\right) \cdot z}} \]

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 65.1%

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

   if 5.0000000000000003e-294 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))))

   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. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-sqrt.f64 64.0

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 64.0

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

   3. Applied rewrites 64.0%

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

   4. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(a \cdot \frac{t}{{z}^{2}}, \frac{1}{2}, 1\right) \]

      5. pow2 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(a \cdot \frac{t}{z \cdot z}, \frac{1}{2}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, \frac{1}{2}, 1\right) \]

      7. pow2 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{{z}^{2}} \cdot a, \frac{1}{2}, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{{z}^{2}} \cdot a, \frac{1}{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, \frac{1}{2}, 1\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, \frac{1}{2}, 1\right) \]

      11. lift-*.f64 67.1

          \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, 0.5, 1\right) \]

   6. Applied rewrites 67.1%

      \[\leadsto \left(y \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, 0.5, 1\right)} \]

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 73.0%

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

Alternative 8: 74.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(x\_m \cdot y\right) \cdot z\_m}{-z\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\_m\right) \cdot 1\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x_s x_m y z_m t a)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 3e-171) (/ (* (* x_m y) z_m) (- z_m)) (* (* y x_m) 1.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3e-171) {
		tmp = ((x_m * y) * z_m) / -z_m;
	} else {
		tmp = (y * x_m) * 1.0;
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m =     private
x\_s =     private
z\_m =     private
z\_s =     private
NOTE: x_m, y, 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, x_s, x_m, y, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 3d-171) then
        tmp = ((x_m * y) * z_m) / -z_m
    else
        tmp = (y * x_m) * 1.0d0
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y && y < z_m && z_m < t && t < a;
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3e-171) {
		tmp = ((x_m * y) * z_m) / -z_m;
	} else {
		tmp = (y * x_m) * 1.0;
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y, z_m, t, a] = sort([x_m, y, z_m, t, a])
def code(z_s, x_s, x_m, y, z_m, t, a):
	tmp = 0
	if z_m <= 3e-171:
		tmp = ((x_m * y) * z_m) / -z_m
	else:
		tmp = (y * x_m) * 1.0
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 3e-171)
		tmp = Float64(Float64(Float64(x_m * y) * z_m) / Float64(-z_m));
	else
		tmp = Float64(Float64(y * x_m) * 1.0);
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y, z_m, t, a = num2cell(sort([x_m, y, z_m, t, a])){:}
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 3e-171)
		tmp = ((x_m * y) * z_m) / -z_m;
	else
		tmp = (y * x_m) * 1.0;
	end
	tmp_2 = z_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]
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, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 3e-171], N[(N[(N[(x$95$m * y), $MachinePrecision] * z$95$m), $MachinePrecision] / (-z$95$m)), $MachinePrecision], N[(N[(y * x$95$m), $MachinePrecision] * 1.0), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3e-171

   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. mul-1-neg N/A

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

      2. lower-neg.f64 17.0

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

   4. Applied rewrites 17.0%

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

   if 3e-171 < 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. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-sqrt.f64 64.0

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 64.0

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

   3. Applied rewrites 64.0%

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

   4. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(a \cdot \frac{t}{{z}^{2}}, \frac{1}{2}, 1\right) \]

      5. pow2 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(a \cdot \frac{t}{z \cdot z}, \frac{1}{2}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, \frac{1}{2}, 1\right) \]

      7. pow2 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{{z}^{2}} \cdot a, \frac{1}{2}, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{{z}^{2}} \cdot a, \frac{1}{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, \frac{1}{2}, 1\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, \frac{1}{2}, 1\right) \]

      11. lift-*.f64 67.1

          \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, 0.5, 1\right) \]

   6. Applied rewrites 67.1%

      \[\leadsto \left(y \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, 0.5, 1\right)} \]

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 73.0%

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

Alternative 9: 73.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y, z_m, t, a = num2cell(sort([x_m, y, z_m, t, a])){:}
function tmp = code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = z_s * (x_s * ((y * x_m) * 1.0));
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]
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, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x$95$s * N[(N[(y * x$95$m), $MachinePrecision] * 1.0), $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. lift-*.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift--.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lift-sqrt.f64 64.0

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 64.0

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

3. Applied rewrites 64.0%

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

4. Taylor expanded in z around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. associate-/l* N/A

      \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(a \cdot \frac{t}{{z}^{2}}, \frac{1}{2}, 1\right) \]

   5. pow2 N/A

      \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(a \cdot \frac{t}{z \cdot z}, \frac{1}{2}, 1\right) \]

   6. *-commutative N/A

      \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, \frac{1}{2}, 1\right) \]

   7. pow2 N/A

      \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{{z}^{2}} \cdot a, \frac{1}{2}, 1\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{{z}^{2}} \cdot a, \frac{1}{2}, 1\right) \]

   9. pow2 N/A

      \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, \frac{1}{2}, 1\right) \]

   10. lift-/.f64 N/A

       \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, \frac{1}{2}, 1\right) \]

   11. lift-*.f64 67.1

       \[\leadsto \left(y \cdot x\right) \cdot \mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, 0.5, 1\right) \]

6. Applied rewrites 67.1%

   \[\leadsto \left(y \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot z} \cdot a, 0.5, 1\right)} \]

7. Taylor expanded in z around inf

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

9. Applied rewrites 73.0%

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

Alternative 10: 13.6% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y, z_m, t, a = num2cell(sort([x_m, y, z_m, t, a])){:}
function tmp = code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = z_s * (x_s * -(y * 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]
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, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x$95$s * (-N[(y * x$95$m), $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. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

      \[\leadsto -x \cdot y \]

   3. *-commutative N/A

      \[\leadsto -y \cdot x \]

   4. lower-*.f64 13.6

      \[\leadsto -y \cdot x \]

4. Applied rewrites 13.6%

   \[\leadsto \color{blue}{-y \cdot x} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025127 
(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)))))