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

Percentage Accurate: 61.4% → 92.5%

Time: 6.9s

Alternatives: 9

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: 61.4% 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: 92.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.9 \cdot 10^{-161}:\\ \;\;\;\;\left(z\_m \cdot y\_m\right) \cdot \frac{x\_m}{\sqrt{\left(-t\right) \cdot a}}\\ \mathbf{elif}\;z\_m \leq 2.4 \cdot 10^{+102}:\\ \;\;\;\;\left(y\_m \cdot x\_m\right) \cdot \frac{z\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1.9e-161)
      (* (* z_m y_m) (/ x_m (sqrt (* (- t) a))))
      (if (<= z_m 2.4e+102)
        (* (* y_m x_m) (/ z_m (sqrt (fma (- a) t (* z_m z_m)))))
        (* y_m x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.9e-161) {
		tmp = (z_m * y_m) * (x_m / sqrt((-t * a)));
	} else if (z_m <= 2.4e+102) {
		tmp = (y_m * x_m) * (z_m / sqrt(fma(-a, t, (z_m * z_m))));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.9e-161)
		tmp = Float64(Float64(z_m * y_m) * Float64(x_m / sqrt(Float64(Float64(-t) * a))));
	elseif (z_m <= 2.4e+102)
		tmp = Float64(Float64(y_m * x_m) * Float64(z_m / sqrt(fma(Float64(-a), t, Float64(z_m * z_m)))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.9e-161], N[(N[(z$95$m * y$95$m), $MachinePrecision] * N[(x$95$m / N[Sqrt[N[((-t) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 2.4e+102], N[(N[(y$95$m * x$95$m), $MachinePrecision] * N[(z$95$m / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 1.9000000000000001e-161

   1. Initial program 72.2%

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

   3. Taylor expanded in z around 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 71.5

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

   5. Applied rewrites 71.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 79.3

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

   7. Applied rewrites 79.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 82.4

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. sqrt-pow2 N/A

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

      15. *-commutative N/A

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

      16. sqrt-pow2 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. *-commutative N/A

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

      20. mul-1-neg N/A

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

      21. lower-*.f64 N/A

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

      22. lower-neg.f64 82.4

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

   9. Applied rewrites 82.4%

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

   if 1.9000000000000001e-161 < z < 2.39999999999999994e102

   1. Initial program 88.6%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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. lower-sqrt.f64 N/A

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

      14. pow2 N/A

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

      15. *-commutative N/A

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

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

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

      17. mul-1-neg N/A

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

      18. associate-*r* N/A

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

      19. +-commutative N/A

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

      20. associate-*r* N/A

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

      21. mul-1-neg N/A

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

      22. lower-fma.f64 N/A

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

      23. lower-neg.f64 N/A

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

      24. pow2 N/A

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

      25. lift-*.f64 92.2

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

   4. Applied rewrites 92.2%

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

   if 2.39999999999999994e102 < z

   1. Initial program 30.6%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 96.7

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

   5. Applied rewrites 96.7%

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

Alternative 2: 75.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ \begin{array}{l} t_1 := \left(x\_m \cdot y\_m\right) \cdot z\_m\\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{z\_m \cdot z\_m - t \cdot a}} \leq 0:\\ \;\;\;\;\frac{t\_1}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (let* ((t_1 (* (* x_m y_m) z_m)))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= (/ t_1 (sqrt (- (* z_m z_m) (* t a)))) 0.0)
        (/ t_1 z_m)
        (* y_m x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double t_1 = (x_m * y_m) * z_m;
	double tmp;
	if ((t_1 / sqrt(((z_m * z_m) - (t * a)))) <= 0.0) {
		tmp = t_1 / z_m;
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

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

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

real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x_m * y_m) * z_m
    if ((t_1 / sqrt(((z_m * z_m) - (t * a)))) <= 0.0d0) then
        tmp = t_1 / z_m
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double t_1 = (x_m * y_m) * z_m;
	double tmp;
	if ((t_1 / Math.sqrt(((z_m * z_m) - (t * a)))) <= 0.0) {
		tmp = t_1 / z_m;
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	t_1 = (x_m * y_m) * z_m
	tmp = 0
	if (t_1 / math.sqrt(((z_m * z_m) - (t * a)))) <= 0.0:
		tmp = t_1 / z_m
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

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

MATLAB

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

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := Block[{t$95$1 = N[(N[(x$95$m * y$95$m), $MachinePrecision] * z$95$m), $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[N[(t$95$1 / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[(t$95$1 / z$95$m), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $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)))) < 0.0

   1. Initial program 51.6%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 80.8%

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

   if 0.0 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))))

   1. Initial program 66.9%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 72.4

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

   5. Applied rewrites 72.4%

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

Alternative 3: 85.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2 \cdot 10^{-129}:\\ \;\;\;\;\left(z\_m \cdot y\_m\right) \cdot \frac{x\_m}{\sqrt{\left(-t\right) \cdot a}}\\ \mathbf{elif}\;z\_m \leq 1.9 \cdot 10^{+102}:\\ \;\;\;\;y\_m \cdot \left(x\_m \cdot \frac{z\_m}{\mathsf{fma}\left(\frac{a \cdot t}{z\_m}, -0.5, z\_m\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 2e-129)
      (* (* z_m y_m) (/ x_m (sqrt (* (- t) a))))
      (if (<= z_m 1.9e+102)
        (* y_m (* x_m (/ z_m (fma (/ (* a t) z_m) -0.5 z_m))))
        (* y_m x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2e-129) {
		tmp = (z_m * y_m) * (x_m / sqrt((-t * a)));
	} else if (z_m <= 1.9e+102) {
		tmp = y_m * (x_m * (z_m / fma(((a * t) / z_m), -0.5, z_m)));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2e-129)
		tmp = Float64(Float64(z_m * y_m) * Float64(x_m / sqrt(Float64(Float64(-t) * a))));
	elseif (z_m <= 1.9e+102)
		tmp = Float64(y_m * Float64(x_m * Float64(z_m / fma(Float64(Float64(a * t) / z_m), -0.5, z_m))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2e-129], N[(N[(z$95$m * y$95$m), $MachinePrecision] * N[(x$95$m / N[Sqrt[N[((-t) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 1.9e+102], N[(y$95$m * N[(x$95$m * N[(z$95$m / N[(N[(N[(a * t), $MachinePrecision] / z$95$m), $MachinePrecision] * -0.5 + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 1.9999999999999999e-129

   1. Initial program 74.6%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 69.7

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

   5. Applied rewrites 69.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 76.6

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

   7. Applied rewrites 76.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 79.5

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. sqrt-pow2 N/A

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

      15. *-commutative N/A

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

      16. sqrt-pow2 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. *-commutative N/A

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

      20. mul-1-neg N/A

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

      21. lower-*.f64 N/A

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

      22. lower-neg.f64 79.5

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

   9. Applied rewrites 79.5%

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

   if 1.9999999999999999e-129 < z < 1.89999999999999989e102

   1. Initial program 89.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 72.8

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

   5. Applied rewrites 72.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 75.6

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

   7. Applied rewrites 75.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 74.7

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

   9. Applied rewrites 74.7%

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

   if 1.89999999999999989e102 < z

   1. Initial program 30.6%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 96.7

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

   5. Applied rewrites 96.7%

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

Alternative 4: 90.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.1 \cdot 10^{+85}:\\ \;\;\;\;\left(z\_m \cdot y\_m\right) \cdot \frac{x\_m}{\sqrt{\mathsf{fma}\left(-t, a, z\_m \cdot z\_m\right)}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 2.1e+85)
      (* (* z_m y_m) (/ x_m (sqrt (fma (- t) a (* z_m z_m)))))
      (* y_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.1e+85) {
		tmp = (z_m * y_m) * (x_m / sqrt(fma(-t, a, (z_m * z_m))));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.1e+85)
		tmp = Float64(Float64(z_m * y_m) * Float64(x_m / sqrt(fma(Float64(-t), a, Float64(z_m * z_m)))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2.1e+85], N[(N[(z$95$m * y$95$m), $MachinePrecision] * N[(x$95$m / N[Sqrt[N[((-t) * a + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.1000000000000001e85

   1. Initial program 83.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 85.4

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

   4. Applied rewrites 85.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. pow2 N/A

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

      16. *-commutative N/A

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

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

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

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

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

      19. mul-1-neg N/A

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

      20. +-commutative N/A

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

   6. Applied rewrites 86.4%

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

   if 2.1000000000000001e85 < z

   1. Initial program 34.4%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 96.2

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

   5. Applied rewrites 96.2%

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

Alternative 5: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.45 \cdot 10^{-140}:\\ \;\;\;\;\left(z\_m \cdot y\_m\right) \cdot \frac{x\_m}{\sqrt{\left(-t\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\left(y\_m \cdot x\_m\right) \cdot \frac{z\_m}{\mathsf{fma}\left(a \cdot \frac{t}{z\_m}, -0.5, z\_m\right)}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1.45e-140)
      (* (* z_m y_m) (/ x_m (sqrt (* (- t) a))))
      (* (* y_m x_m) (/ z_m (fma (* a (/ t z_m)) -0.5 z_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.45e-140) {
		tmp = (z_m * y_m) * (x_m / sqrt((-t * a)));
	} else {
		tmp = (y_m * x_m) * (z_m / fma((a * (t / z_m)), -0.5, z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.45e-140)
		tmp = Float64(Float64(z_m * y_m) * Float64(x_m / sqrt(Float64(Float64(-t) * a))));
	else
		tmp = Float64(Float64(y_m * x_m) * Float64(z_m / fma(Float64(a * Float64(t / z_m)), -0.5, z_m)));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.45e-140], N[(N[(z$95$m * y$95$m), $MachinePrecision] * N[(x$95$m / N[Sqrt[N[((-t) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * x$95$m), $MachinePrecision] * N[(z$95$m / N[(N[(a * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision] * -0.5 + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.44999999999999999e-140

   1. Initial program 73.6%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 70.4

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

   5. Applied rewrites 70.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 77.8

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

   7. Applied rewrites 77.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 80.8

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. sqrt-pow2 N/A

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

      15. *-commutative N/A

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

      16. sqrt-pow2 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. *-commutative N/A

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

      20. mul-1-neg N/A

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

      21. lower-*.f64 N/A

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

      22. lower-neg.f64 80.8

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

   9. Applied rewrites 80.8%

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

   if 1.44999999999999999e-140 < z

   1. Initial program 58.5%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 71.8

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

   5. Applied rewrites 71.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 83.5

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

   7. Applied rewrites 83.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 86.5

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

   9. Applied rewrites 86.5%

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

Alternative 6: 84.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.4 \cdot 10^{-105}:\\ \;\;\;\;\left(z\_m \cdot y\_m\right) \cdot \frac{x\_m}{\sqrt{\left(-t\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1.4e-105)
      (* (* z_m y_m) (/ x_m (sqrt (* (- t) a))))
      (* y_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.4e-105) {
		tmp = (z_m * y_m) * (x_m / sqrt((-t * a)));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

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

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

real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.4d-105) then
        tmp = (z_m * y_m) * (x_m / sqrt((-t * a)))
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.4e-105) {
		tmp = (z_m * y_m) * (x_m / Math.sqrt((-t * a)));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.4e-105:
		tmp = (z_m * y_m) * (x_m / math.sqrt((-t * a)))
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.4e-105)
		tmp = Float64(Float64(z_m * y_m) * Float64(x_m / sqrt(Float64(Float64(-t) * a))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.4e-105)
		tmp = (z_m * y_m) * (x_m / sqrt((-t * a)));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.4e-105], N[(N[(z$95$m * y$95$m), $MachinePrecision] * N[(x$95$m / N[Sqrt[N[((-t) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.4e-105

   1. Initial program 75.9%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 69.0

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

   5. Applied rewrites 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 75.5

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

   7. Applied rewrites 75.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 78.1

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. sqrt-pow2 N/A

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

      15. *-commutative N/A

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

      16. sqrt-pow2 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. *-commutative N/A

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

      20. mul-1-neg N/A

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

      21. lower-*.f64 N/A

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

      22. lower-neg.f64 78.1

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

   9. Applied rewrites 78.1%

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

   if 1.4e-105 < z

   1. Initial program 56.9%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 86.0

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

   5. Applied rewrites 86.0%

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

Alternative 7: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.4 \cdot 10^{-105}:\\ \;\;\;\;\left(y\_m \cdot x\_m\right) \cdot \frac{z\_m}{\sqrt{\left(-t\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1.4e-105)
      (* (* y_m x_m) (/ z_m (sqrt (* (- t) a))))
      (* y_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.4e-105) {
		tmp = (y_m * x_m) * (z_m / sqrt((-t * a)));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

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

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

real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.4d-105) then
        tmp = (y_m * x_m) * (z_m / sqrt((-t * a)))
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.4e-105) {
		tmp = (y_m * x_m) * (z_m / Math.sqrt((-t * a)));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.4e-105:
		tmp = (y_m * x_m) * (z_m / math.sqrt((-t * a)))
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.4e-105)
		tmp = Float64(Float64(y_m * x_m) * Float64(z_m / sqrt(Float64(Float64(-t) * a))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.4e-105)
		tmp = (y_m * x_m) * (z_m / sqrt((-t * a)));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.4e-105], N[(N[(y$95$m * x$95$m), $MachinePrecision] * N[(z$95$m / N[Sqrt[N[((-t) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.4e-105

   1. Initial program 75.9%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 69.0

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

   5. Applied rewrites 69.0%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{\left(-a\right) \cdot t}}} \]
   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 69.3

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

      9. lift-neg.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-pow2 N/A

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

      18. *-commutative N/A

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

      19. sqrt-pow2 N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. *-commutative N/A

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

      23. mul-1-neg N/A

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

   7. Applied rewrites 69.3%

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

   if 1.4e-105 < z

   1. Initial program 56.9%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 86.0

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

   5. Applied rewrites 86.0%

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

Alternative 8: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5 \cdot 10^{-155}:\\ \;\;\;\;y\_m \cdot \left(x\_m \cdot \frac{z\_m}{\sqrt{\left(-a\right) \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 5e-155)
      (* y_m (* x_m (/ z_m (sqrt (* (- a) t)))))
      (* y_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 5e-155) {
		tmp = y_m * (x_m * (z_m / sqrt((-a * t))));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

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

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

real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 5d-155) then
        tmp = y_m * (x_m * (z_m / sqrt((-a * t))))
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 5e-155) {
		tmp = y_m * (x_m * (z_m / Math.sqrt((-a * t))));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 5e-155:
		tmp = y_m * (x_m * (z_m / math.sqrt((-a * t))))
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 5e-155)
		tmp = Float64(y_m * Float64(x_m * Float64(z_m / sqrt(Float64(Float64(-a) * t)))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 5e-155)
		tmp = y_m * (x_m * (z_m / sqrt((-a * t))));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 5e-155], N[(y$95$m * N[(x$95$m * N[(z$95$m / N[Sqrt[N[((-a) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.9999999999999999e-155

   1. Initial program 72.6%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 71.2

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

   5. Applied rewrites 71.2%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{\left(-a\right) \cdot t}}} \]
   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 70.8

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

      9. lift-neg.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-pow2 N/A

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

      18. *-commutative N/A

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

      19. sqrt-pow2 N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. *-commutative N/A

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

      23. mul-1-neg N/A

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

   7. Applied rewrites 70.8%

      \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot \frac{z}{\sqrt{\left(-t\right) \cdot a}}} \]
   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 65.4

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

      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. associate-*r* N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 65.4

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

   9. Applied rewrites 65.4%

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

   if 4.9999999999999999e-155 < z

   1. Initial program 59.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 82.8

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

   5. Applied rewrites 82.8%

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

Alternative 9: 72.6% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.4%

   \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
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 72.6

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

5. Applied rewrites 72.6%

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

Developer Target 1: 87.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z < -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z < 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (< z -3.1921305903852764e+46)
   (- (* y x))
   (if (< z 5.976268120920894e+90)
     (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y))
     (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z < -3.1921305903852764e+46) {
		tmp = -(y * x);
	} else if (z < 5.976268120920894e+90) {
		tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, 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
    real(8) :: tmp
    if (z < (-3.1921305903852764d+46)) then
        tmp = -(y * x)
    else if (z < 5.976268120920894d+90) then
        tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z < -3.1921305903852764e+46) {
		tmp = -(y * x);
	} else if (z < 5.976268120920894e+90) {
		tmp = (x * z) / (Math.sqrt(((z * z) - (a * t))) / y);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z < -3.1921305903852764e+46:
		tmp = -(y * x)
	elif z < 5.976268120920894e+90:
		tmp = (x * z) / (math.sqrt(((z * z) - (a * t))) / y)
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z < -3.1921305903852764e+46)
		tmp = Float64(-Float64(y * x));
	elseif (z < 5.976268120920894e+90)
		tmp = Float64(Float64(x * z) / Float64(sqrt(Float64(Float64(z * z) - Float64(a * t))) / y));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z < -3.1921305903852764e+46)
		tmp = -(y * x);
	elseif (z < 5.976268120920894e+90)
		tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Less[z, -3.1921305903852764e+46], (-N[(y * x), $MachinePrecision]), If[Less[z, 5.976268120920894e+90], N[(N[(x * z), $MachinePrecision] / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Reproduce

herbie shell --seed 2025089 
(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)))))