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

Percentage Accurate: 61.2% → 91.7%

Time: 6.9s

Alternatives: 7

Speedup: 7.5×

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: 61.2% 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: 91.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x, y_m, z_m, t, a] = \mathsf{sort}([x, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.45 \cdot 10^{-151}:\\ \;\;\;\;\left(z\_m \cdot y\_m\right) \cdot \frac{x}{\sqrt{\left(-a\right) \cdot t}}\\ \mathbf{elif}\;z\_m \leq 2 \cdot 10^{+123}:\\ \;\;\;\;y\_m \cdot \left(x \cdot \frac{z\_m}{\sqrt{z\_m \cdot z\_m - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (if (<= z_m 2.45e-151)
     (* (* z_m y_m) (/ x (sqrt (* (- a) t))))
     (if (<= z_m 2e+123)
       (* y_m (* x (/ z_m (sqrt (- (* z_m z_m) (* a t))))))
       (* y_m x))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.45e-151) {
		tmp = (z_m * y_m) * (x / sqrt((-a * t)));
	} else if (z_m <= 2e+123) {
		tmp = y_m * (x * (z_m / sqrt(((z_m * z_m) - (a * t)))));
	} else {
		tmp = y_m * x;
	}
	return z_s * (y_s * tmp);
}

Fortran

y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z_s, y_s, x, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 2.45d-151) then
        tmp = (z_m * y_m) * (x / sqrt((-a * t)))
    else if (z_m <= 2d+123) then
        tmp = y_m * (x * (z_m / sqrt(((z_m * z_m) - (a * t)))))
    else
        tmp = y_m * x
    end if
    code = z_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.45e-151) {
		tmp = (z_m * y_m) * (x / Math.sqrt((-a * t)));
	} else if (z_m <= 2e+123) {
		tmp = y_m * (x * (z_m / Math.sqrt(((z_m * z_m) - (a * t)))));
	} else {
		tmp = y_m * x;
	}
	return z_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x, y_m, z_m, t, a] = sort([x, y_m, z_m, t, a])
def code(z_s, y_s, x, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 2.45e-151:
		tmp = (z_m * y_m) * (x / math.sqrt((-a * t)))
	elif z_m <= 2e+123:
		tmp = y_m * (x * (z_m / math.sqrt(((z_m * z_m) - (a * t)))))
	else:
		tmp = y_m * x
	return z_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x, y_m, z_m, t, a = sort([x, y_m, z_m, t, a])
function code(z_s, y_s, x, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.45e-151)
		tmp = Float64(Float64(z_m * y_m) * Float64(x / sqrt(Float64(Float64(-a) * t))));
	elseif (z_m <= 2e+123)
		tmp = Float64(y_m * Float64(x * Float64(z_m / sqrt(Float64(Float64(z_m * z_m) - Float64(a * t))))));
	else
		tmp = Float64(y_m * x);
	end
	return Float64(z_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x, y_m, z_m, t, a = num2cell(sort([x, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2.45e-151)
		tmp = (z_m * y_m) * (x / sqrt((-a * t)));
	elseif (z_m <= 2e+123)
		tmp = y_m * (x * (z_m / sqrt(((z_m * z_m) - (a * t)))));
	else
		tmp = y_m * x;
	end
	tmp_2 = z_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * If[LessEqual[z$95$m, 2.45e-151], N[(N[(z$95$m * y$95$m), $MachinePrecision] * N[(x / N[Sqrt[N[((-a) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 2e+123], N[(y$95$m * N[(x * N[(z$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 2.44999999999999983e-151

   1. Initial program 63.1%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 39.1

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

   5. Applied rewrites 39.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 38.5

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

      9. *-commutative 38.5

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

      10. lift-neg.f64 N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. mul-1-neg N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. sqrt-pow2 N/A

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

      19. *-commutative N/A

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

   7. Applied rewrites 38.5%

      \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot \frac{z}{\sqrt{\left(-t\right) \cdot a}}} \]
   8. 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

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. *-commutative N/A

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

      14. mul-1-neg N/A

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

      15. *-commutative N/A

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

      16. mul-1-neg N/A

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

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

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

      18. lift-*.f64 N/A

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

      19. lift-neg.f64 N/A

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

   9. Applied rewrites 36.6%

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

   if 2.44999999999999983e-151 < z < 1.99999999999999996e123

   1. Initial program 98.2%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Add Preprocessing
   3. 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

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

      7. lift-*.f64 N/A

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

      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

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

      15. lift--.f64 N/A

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

      16. lift-sqrt.f64 99.9

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

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

      7. lift-*.f64 N/A

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

      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

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

      13. lift--.f64 N/A

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

      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 98.6

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

   6. Applied rewrites 98.6%

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

   if 1.99999999999999996e123 < z

   1. Initial program 25.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 96.1

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

   5. Applied rewrites 96.1%

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

Alternative 2: 75.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x, y_m, z_m, t, a] = \mathsf{sort}([x, y_m, z_m, t, a])\\ \\ \begin{array}{l} t_1 := \left(x \cdot y\_m\right) \cdot z\_m\\ z\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{z\_m \cdot z\_m - t \cdot a}} \leq 2 \cdot 10^{-308}:\\ \;\;\;\;\frac{t\_1}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x y_m z_m t a)
 :precision binary64
 (let* ((t_1 (* (* x y_m) z_m)))
   (*
    z_s
    (*
     y_s
     (if (<= (/ t_1 (sqrt (- (* z_m z_m) (* t a)))) 2e-308)
       (/ t_1 z_m)
       (* y_m x))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x, double y_m, double z_m, double t, double a) {
	double t_1 = (x * y_m) * z_m;
	double tmp;
	if ((t_1 / sqrt(((z_m * z_m) - (t * a)))) <= 2e-308) {
		tmp = t_1 / z_m;
	} else {
		tmp = y_m * x;
	}
	return z_s * (y_s * tmp);
}

Fortran

y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z_s, y_s, x, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    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 * y_m) * z_m
    if ((t_1 / sqrt(((z_m * z_m) - (t * a)))) <= 2d-308) then
        tmp = t_1 / z_m
    else
        tmp = y_m * x
    end if
    code = z_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x, double y_m, double z_m, double t, double a) {
	double t_1 = (x * y_m) * z_m;
	double tmp;
	if ((t_1 / Math.sqrt(((z_m * z_m) - (t * a)))) <= 2e-308) {
		tmp = t_1 / z_m;
	} else {
		tmp = y_m * x;
	}
	return z_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x, y_m, z_m, t, a] = sort([x, y_m, z_m, t, a])
def code(z_s, y_s, x, y_m, z_m, t, a):
	t_1 = (x * y_m) * z_m
	tmp = 0
	if (t_1 / math.sqrt(((z_m * z_m) - (t * a)))) <= 2e-308:
		tmp = t_1 / z_m
	else:
		tmp = y_m * x
	return z_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x, y_m, z_m, t, a = sort([x, y_m, z_m, t, a])
function code(z_s, y_s, x, y_m, z_m, t, a)
	t_1 = Float64(Float64(x * y_m) * z_m)
	tmp = 0.0
	if (Float64(t_1 / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))) <= 2e-308)
		tmp = Float64(t_1 / z_m);
	else
		tmp = Float64(y_m * x);
	end
	return Float64(z_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x, y_m, z_m, t, a = num2cell(sort([x, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x, y_m, z_m, t, a)
	t_1 = (x * y_m) * z_m;
	tmp = 0.0;
	if ((t_1 / sqrt(((z_m * z_m) - (t * a)))) <= 2e-308)
		tmp = t_1 / z_m;
	else
		tmp = y_m * x;
	end
	tmp_2 = z_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x_, y$95$m_, z$95$m_, t_, a_] := Block[{t$95$1 = N[(N[(x * y$95$m), $MachinePrecision] * z$95$m), $MachinePrecision]}, N[(z$95$s * N[(y$95$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], 2e-308], N[(t$95$1 / z$95$m), $MachinePrecision], N[(y$95$m * x), $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)))) < 1.9999999999999998e-308

   1. Initial program 73.0%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 51.5%

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

   if 1.9999999999999998e-308 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))))

   1. Initial program 50.3%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 43.4

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

   5. Applied rewrites 43.4%

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

Alternative 3: 91.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x, y_m, z_m, t, a] = \mathsf{sort}([x, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2 \cdot 10^{+123}:\\ \;\;\;\;\left(y\_m \cdot x\right) \cdot \frac{z\_m}{\sqrt{z\_m \cdot z\_m - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (if (<= z_m 2e+123)
     (* (* y_m x) (/ z_m (sqrt (- (* z_m z_m) (* a t)))))
     (* y_m x)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2e+123) {
		tmp = (y_m * x) * (z_m / sqrt(((z_m * z_m) - (a * t))));
	} else {
		tmp = y_m * x;
	}
	return z_s * (y_s * tmp);
}

Fortran

y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z_s, y_s, x, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 2d+123) then
        tmp = (y_m * x) * (z_m / sqrt(((z_m * z_m) - (a * t))))
    else
        tmp = y_m * x
    end if
    code = z_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2e+123) {
		tmp = (y_m * x) * (z_m / Math.sqrt(((z_m * z_m) - (a * t))));
	} else {
		tmp = y_m * x;
	}
	return z_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x, y_m, z_m, t, a] = sort([x, y_m, z_m, t, a])
def code(z_s, y_s, x, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 2e+123:
		tmp = (y_m * x) * (z_m / math.sqrt(((z_m * z_m) - (a * t))))
	else:
		tmp = y_m * x
	return z_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x, y_m, z_m, t, a = sort([x, y_m, z_m, t, a])
function code(z_s, y_s, x, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2e+123)
		tmp = Float64(Float64(y_m * x) * Float64(z_m / sqrt(Float64(Float64(z_m * z_m) - Float64(a * t)))));
	else
		tmp = Float64(y_m * x);
	end
	return Float64(z_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x, y_m, z_m, t, a = num2cell(sort([x, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2e+123)
		tmp = (y_m * x) * (z_m / sqrt(((z_m * z_m) - (a * t))));
	else
		tmp = y_m * x;
	end
	tmp_2 = z_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * If[LessEqual[z$95$m, 2e+123], N[(N[(y$95$m * x), $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[(y$95$m * x), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.99999999999999996e123

   1. Initial program 72.8%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Add Preprocessing
   3. 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

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

      7. lift-*.f64 N/A

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

      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

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

      15. lift--.f64 N/A

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

      16. lift-sqrt.f64 75.6

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

      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 75.6

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

   4. Applied rewrites 75.6%

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

   if 1.99999999999999996e123 < z

   1. Initial program 25.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 96.1

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

   5. Applied rewrites 96.1%

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

Alternative 4: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x, y_m, z_m, t, a] = \mathsf{sort}([x, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3 \cdot 10^{-147}:\\ \;\;\;\;\left(z\_m \cdot y\_m\right) \cdot \frac{x}{\sqrt{\left(-a\right) \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (if (<= z_m 3e-147) (* (* z_m y_m) (/ x (sqrt (* (- a) t)))) (* y_m x)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3e-147) {
		tmp = (z_m * y_m) * (x / sqrt((-a * t)));
	} else {
		tmp = y_m * x;
	}
	return z_s * (y_s * tmp);
}

Fortran

y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z_s, y_s, x, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 3d-147) then
        tmp = (z_m * y_m) * (x / sqrt((-a * t)))
    else
        tmp = y_m * x
    end if
    code = z_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3e-147) {
		tmp = (z_m * y_m) * (x / Math.sqrt((-a * t)));
	} else {
		tmp = y_m * x;
	}
	return z_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x, y_m, z_m, t, a] = sort([x, y_m, z_m, t, a])
def code(z_s, y_s, x, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 3e-147:
		tmp = (z_m * y_m) * (x / math.sqrt((-a * t)))
	else:
		tmp = y_m * x
	return z_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x, y_m, z_m, t, a = sort([x, y_m, z_m, t, a])
function code(z_s, y_s, x, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 3e-147)
		tmp = Float64(Float64(z_m * y_m) * Float64(x / sqrt(Float64(Float64(-a) * t))));
	else
		tmp = Float64(y_m * x);
	end
	return Float64(z_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x, y_m, z_m, t, a = num2cell(sort([x, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 3e-147)
		tmp = (z_m * y_m) * (x / sqrt((-a * t)));
	else
		tmp = y_m * x;
	end
	tmp_2 = z_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * If[LessEqual[z$95$m, 3e-147], N[(N[(z$95$m * y$95$m), $MachinePrecision] * N[(x / N[Sqrt[N[((-a) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.0000000000000002e-147

   1. Initial program 63.8%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 39.7

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

   5. Applied rewrites 39.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 39.2

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

      9. *-commutative 39.2

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

      10. lift-neg.f64 N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. mul-1-neg N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. sqrt-pow2 N/A

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

      19. *-commutative N/A

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

   7. Applied rewrites 39.2%

      \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot \frac{z}{\sqrt{\left(-t\right) \cdot a}}} \]
   8. 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

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. *-commutative N/A

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

      14. mul-1-neg N/A

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

      15. *-commutative N/A

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

      16. mul-1-neg N/A

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

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

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

      18. lift-*.f64 N/A

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

      19. lift-neg.f64 N/A

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

   9. Applied rewrites 37.3%

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

   if 3.0000000000000002e-147 < z

   1. Initial program 65.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 86.9

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

   5. Applied rewrites 86.9%

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

Alternative 5: 82.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x, y_m, z_m, t, a] = \mathsf{sort}([x, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.35 \cdot 10^{-147}:\\ \;\;\;\;z\_m \cdot \left(y\_m \cdot \frac{x}{\sqrt{\left(-t\right) \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (if (<= z_m 2.35e-147)
     (* z_m (* y_m (/ x (sqrt (* (- t) a)))))
     (* y_m x)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.35e-147) {
		tmp = z_m * (y_m * (x / sqrt((-t * a))));
	} else {
		tmp = y_m * x;
	}
	return z_s * (y_s * tmp);
}

Fortran

y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z_s, y_s, x, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 2.35d-147) then
        tmp = z_m * (y_m * (x / sqrt((-t * a))))
    else
        tmp = y_m * x
    end if
    code = z_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.35e-147) {
		tmp = z_m * (y_m * (x / Math.sqrt((-t * a))));
	} else {
		tmp = y_m * x;
	}
	return z_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x, y_m, z_m, t, a] = sort([x, y_m, z_m, t, a])
def code(z_s, y_s, x, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 2.35e-147:
		tmp = z_m * (y_m * (x / math.sqrt((-t * a))))
	else:
		tmp = y_m * x
	return z_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x, y_m, z_m, t, a = sort([x, y_m, z_m, t, a])
function code(z_s, y_s, x, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.35e-147)
		tmp = Float64(z_m * Float64(y_m * Float64(x / sqrt(Float64(Float64(-t) * a)))));
	else
		tmp = Float64(y_m * x);
	end
	return Float64(z_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x, y_m, z_m, t, a = num2cell(sort([x, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2.35e-147)
		tmp = z_m * (y_m * (x / sqrt((-t * a))));
	else
		tmp = y_m * x;
	end
	tmp_2 = z_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * If[LessEqual[z$95$m, 2.35e-147], N[(z$95$m * N[(y$95$m * N[(x / N[Sqrt[N[((-t) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.34999999999999994e-147

   1. Initial program 63.8%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 39.7

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

   5. Applied rewrites 39.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 39.2

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

      9. *-commutative 39.2

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

      10. lift-neg.f64 N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. mul-1-neg N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. sqrt-pow2 N/A

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

      19. *-commutative N/A

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

   7. Applied rewrites 39.2%

      \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot \frac{z}{\sqrt{\left(-t\right) \cdot a}}} \]
   8. 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

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. *-commutative N/A

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

      14. mul-1-neg N/A

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

      15. *-commutative N/A

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

      16. mul-1-neg N/A

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

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

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

      18. lift-*.f64 N/A

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

      19. lift-neg.f64 N/A

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

   9. Applied rewrites 37.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 38.5

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

       6. lift-neg.f64 N/A

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

       7. lift-*.f64 N/A

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

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

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

       9. mul-1-neg N/A

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

       10. *-commutative N/A

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

       11. associate-*r* N/A

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

       12. metadata-eval N/A

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

       13. metadata-eval N/A

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

       14. sqrt-pow2 N/A

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

       15. *-commutative N/A

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

       16. sqrt-pow2 N/A

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

       17. metadata-eval N/A

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

       18. metadata-eval N/A

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

       19. *-commutative N/A

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

       20. mul-1-neg N/A

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

       21. lower-*.f64 N/A

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

   11. Applied rewrites 38.5%

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

   if 2.34999999999999994e-147 < z

   1. Initial program 65.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 86.9

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

   5. Applied rewrites 86.9%

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

Alternative 6: 81.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x, y_m, z_m, t, a] = \mathsf{sort}([x, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.6 \cdot 10^{-151}:\\ \;\;\;\;y\_m \cdot \left(x \cdot \frac{z\_m}{\sqrt{\left(-a\right) \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (if (<= z_m 1.6e-151) (* y_m (* x (/ z_m (sqrt (* (- a) t))))) (* y_m x)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.6e-151) {
		tmp = y_m * (x * (z_m / sqrt((-a * t))));
	} else {
		tmp = y_m * x;
	}
	return z_s * (y_s * tmp);
}

Fortran

y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z_s, y_s, x, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.6d-151) then
        tmp = y_m * (x * (z_m / sqrt((-a * t))))
    else
        tmp = y_m * x
    end if
    code = z_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.6e-151) {
		tmp = y_m * (x * (z_m / Math.sqrt((-a * t))));
	} else {
		tmp = y_m * x;
	}
	return z_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x, y_m, z_m, t, a] = sort([x, y_m, z_m, t, a])
def code(z_s, y_s, x, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.6e-151:
		tmp = y_m * (x * (z_m / math.sqrt((-a * t))))
	else:
		tmp = y_m * x
	return z_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x, y_m, z_m, t, a = sort([x, y_m, z_m, t, a])
function code(z_s, y_s, x, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.6e-151)
		tmp = Float64(y_m * Float64(x * Float64(z_m / sqrt(Float64(Float64(-a) * t)))));
	else
		tmp = Float64(y_m * x);
	end
	return Float64(z_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x, y_m, z_m, t, a = num2cell(sort([x, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.6e-151)
		tmp = y_m * (x * (z_m / sqrt((-a * t))));
	else
		tmp = y_m * x;
	end
	tmp_2 = z_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * If[LessEqual[z$95$m, 1.6e-151], N[(y$95$m * N[(x * N[(z$95$m / N[Sqrt[N[((-a) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.60000000000000011e-151

   1. Initial program 63.1%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 39.1

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

   5. Applied rewrites 39.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 38.5

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

      9. *-commutative 38.5

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

      10. lift-neg.f64 N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. mul-1-neg N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. sqrt-pow2 N/A

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

      19. *-commutative N/A

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

   7. Applied rewrites 38.5%

      \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot \frac{z}{\sqrt{\left(-t\right) \cdot a}}} \]
   8. 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 36.1

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. mul-1-neg N/A

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

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

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 36.1

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

   9. Applied rewrites 36.1%

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

   if 1.60000000000000011e-151 < z

   1. Initial program 66.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 85.7

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

   5. Applied rewrites 85.7%

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

Alternative 7: 73.2% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z_s, y_s, x, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = z_s * (y_s * (y_m * x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 45.4

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

5. Applied rewrites 45.4%

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

Reproduce

herbie shell --seed 2025058 
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -31921305903852764000000000000000000000000000000) (- (* y x)) (if (< z 5976268120920894000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y)) (* y x))))

  (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))