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

Percentage Accurate: 61.1% → 95.0%

Time: 6.6s

Alternatives: 13

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.1% accurate, 1.0× speedup

Math

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

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: 95.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_2 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \left(\mathsf{copysign}\left(1, z\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z\right| \leq 2.5 \cdot 10^{-159}:\\ \;\;\;\;\frac{t\_2}{\sqrt{-\mathsf{min}\left(t, a\right)} \cdot \sqrt{\mathsf{max}\left(t, a\right)}} \cdot \left(\left|z\right| \cdot t\_1\right)\\ \mathbf{elif}\;\left|z\right| \leq 10^{+120}:\\ \;\;\;\;\left(\frac{\left|z\right|}{\sqrt{\left|z\right| \cdot \left|z\right| - \mathsf{max}\left(t, a\right) \cdot \mathsf{min}\left(t, a\right)}} \cdot t\_1\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|z\right|}{\left|z\right| - \mathsf{min}\left(t, a\right) \cdot \left(\mathsf{max}\left(t, a\right) \cdot \frac{0.5}{\left|z\right|}\right)} \cdot \left(t\_1 \cdot t\_2\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fmax (fabs x) (fabs y))) (t_2 (fmin (fabs x) (fabs y))))
  (*
   (copysign 1.0 x)
   (*
    (copysign 1.0 y)
    (*
     (copysign 1.0 z)
     (if (<= (fabs z) 2.5e-159)
       (*
        (/ t_2 (* (sqrt (- (fmin t a))) (sqrt (fmax t a))))
        (* (fabs z) t_1))
       (if (<= (fabs z) 1e+120)
         (*
          (*
           (/
            (fabs z)
            (sqrt
             (- (* (fabs z) (fabs z)) (* (fmax t a) (fmin t a)))))
           t_1)
          t_2)
         (*
          (/
           (fabs z)
           (-
            (fabs z)
            (* (fmin t a) (* (fmax t a) (/ 0.5 (fabs z))))))
          (* t_1 t_2)))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fmax(fabs(x), fabs(y));
	double t_2 = fmin(fabs(x), fabs(y));
	double tmp;
	if (fabs(z) <= 2.5e-159) {
		tmp = (t_2 / (sqrt(-fmin(t, a)) * sqrt(fmax(t, a)))) * (fabs(z) * t_1);
	} else if (fabs(z) <= 1e+120) {
		tmp = ((fabs(z) / sqrt(((fabs(z) * fabs(z)) - (fmax(t, a) * fmin(t, a))))) * t_1) * t_2;
	} else {
		tmp = (fabs(z) / (fabs(z) - (fmin(t, a) * (fmax(t, a) * (0.5 / fabs(z)))))) * (t_1 * t_2);
	}
	return copysign(1.0, x) * (copysign(1.0, y) * (copysign(1.0, z) * tmp));
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = fmax(Math.abs(x), Math.abs(y));
	double t_2 = fmin(Math.abs(x), Math.abs(y));
	double tmp;
	if (Math.abs(z) <= 2.5e-159) {
		tmp = (t_2 / (Math.sqrt(-fmin(t, a)) * Math.sqrt(fmax(t, a)))) * (Math.abs(z) * t_1);
	} else if (Math.abs(z) <= 1e+120) {
		tmp = ((Math.abs(z) / Math.sqrt(((Math.abs(z) * Math.abs(z)) - (fmax(t, a) * fmin(t, a))))) * t_1) * t_2;
	} else {
		tmp = (Math.abs(z) / (Math.abs(z) - (fmin(t, a) * (fmax(t, a) * (0.5 / Math.abs(z)))))) * (t_1 * t_2);
	}
	return Math.copySign(1.0, x) * (Math.copySign(1.0, y) * (Math.copySign(1.0, z) * tmp));
}

Python

def code(x, y, z, t, a):
	t_1 = fmax(math.fabs(x), math.fabs(y))
	t_2 = fmin(math.fabs(x), math.fabs(y))
	tmp = 0
	if math.fabs(z) <= 2.5e-159:
		tmp = (t_2 / (math.sqrt(-fmin(t, a)) * math.sqrt(fmax(t, a)))) * (math.fabs(z) * t_1)
	elif math.fabs(z) <= 1e+120:
		tmp = ((math.fabs(z) / math.sqrt(((math.fabs(z) * math.fabs(z)) - (fmax(t, a) * fmin(t, a))))) * t_1) * t_2
	else:
		tmp = (math.fabs(z) / (math.fabs(z) - (fmin(t, a) * (fmax(t, a) * (0.5 / math.fabs(z)))))) * (t_1 * t_2)
	return math.copysign(1.0, x) * (math.copysign(1.0, y) * (math.copysign(1.0, z) * tmp))

Julia

function code(x, y, z, t, a)
	t_1 = fmax(abs(x), abs(y))
	t_2 = fmin(abs(x), abs(y))
	tmp = 0.0
	if (abs(z) <= 2.5e-159)
		tmp = Float64(Float64(t_2 / Float64(sqrt(Float64(-fmin(t, a))) * sqrt(fmax(t, a)))) * Float64(abs(z) * t_1));
	elseif (abs(z) <= 1e+120)
		tmp = Float64(Float64(Float64(abs(z) / sqrt(Float64(Float64(abs(z) * abs(z)) - Float64(fmax(t, a) * fmin(t, a))))) * t_1) * t_2);
	else
		tmp = Float64(Float64(abs(z) / Float64(abs(z) - Float64(fmin(t, a) * Float64(fmax(t, a) * Float64(0.5 / abs(z)))))) * Float64(t_1 * t_2));
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * Float64(copysign(1.0, z) * tmp)))
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = max(abs(x), abs(y));
	t_2 = min(abs(x), abs(y));
	tmp = 0.0;
	if (abs(z) <= 2.5e-159)
		tmp = (t_2 / (sqrt(-min(t, a)) * sqrt(max(t, a)))) * (abs(z) * t_1);
	elseif (abs(z) <= 1e+120)
		tmp = ((abs(z) / sqrt(((abs(z) * abs(z)) - (max(t, a) * min(t, a))))) * t_1) * t_2;
	else
		tmp = (abs(z) / (abs(z) - (min(t, a) * (max(t, a) * (0.5 / abs(z)))))) * (t_1 * t_2);
	end
	tmp_2 = (sign(x) * abs(1.0)) * ((sign(y) * abs(1.0)) * ((sign(z) * abs(1.0)) * tmp));
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z], $MachinePrecision], 2.5e-159], N[(N[(t$95$2 / N[(N[Sqrt[(-N[Min[t, a], $MachinePrecision])], $MachinePrecision] * N[Sqrt[N[Max[t, a], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[z], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[z], $MachinePrecision], 1e+120], N[(N[(N[(N[Abs[z], $MachinePrecision] / N[Sqrt[N[(N[(N[Abs[z], $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision] - N[(N[Max[t, a], $MachinePrecision] * N[Min[t, a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(N[Abs[z], $MachinePrecision] / N[(N[Abs[z], $MachinePrecision] - N[(N[Min[t, a], $MachinePrecision] * N[(N[Max[t, a], $MachinePrecision] * N[(0.5 / N[Abs[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 2.5000000000000002e-159

   1. Initial program 61.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. mult-flip-rev N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 59.1%

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

   3. Applied rewrites 59.1%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 31.8%

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

   6. Applied rewrites 31.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. sqrt-prod N/A

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

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

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

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

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

      9. lower-neg.f64 N/A

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

      10. lower-unsound-sqrt.f64 18.3%

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

   8. Applied rewrites 18.3%

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

   if 2.5000000000000002e-159 < z < 9.9999999999999998e119

   1. Initial program 61.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. inv-pow N/A

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

      7. lift-sqrt.f64 N/A

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

      8. sqrt-fabs-rev N/A

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

      9. lift-sqrt.f64 N/A

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

      10. rem-sqrt-square-rev N/A

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

      11. sqrt-pow2 N/A

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

      12. sqr-neg-rev N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. unpow-prod-down N/A

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

      16. pow-add N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

   3. Applied rewrites 61.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-/.f64 64.0%

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

   5. Applied rewrites 64.0%

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

   if 9.9999999999999998e119 < z

   1. Initial program 61.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 41.8%

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

   4. Applied rewrites 41.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 47.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 49.2%

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

   8. Applied rewrites 49.2%

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

Alternative 2: 92.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_2 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \left(\mathsf{copysign}\left(1, z\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z\right| \leq 1.4 \cdot 10^{+52}:\\ \;\;\;\;\frac{\left|z\right| \cdot t\_1}{\sqrt{\left|z\right| \cdot \left|z\right| - a \cdot t}} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|z\right|}{\left|z\right| - t \cdot \left(a \cdot \frac{0.5}{\left|z\right|}\right)} \cdot \left(t\_1 \cdot t\_2\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fmax (fabs x) (fabs y))) (t_2 (fmin (fabs x) (fabs y))))
  (*
   (copysign 1.0 x)
   (*
    (copysign 1.0 y)
    (*
     (copysign 1.0 z)
     (if (<= (fabs z) 1.4e+52)
       (*
        (/ (* (fabs z) t_1) (sqrt (- (* (fabs z) (fabs z)) (* a t))))
        t_2)
       (*
        (/ (fabs z) (- (fabs z) (* t (* a (/ 0.5 (fabs z))))))
        (* t_1 t_2))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fmax(fabs(x), fabs(y));
	double t_2 = fmin(fabs(x), fabs(y));
	double tmp;
	if (fabs(z) <= 1.4e+52) {
		tmp = ((fabs(z) * t_1) / sqrt(((fabs(z) * fabs(z)) - (a * t)))) * t_2;
	} else {
		tmp = (fabs(z) / (fabs(z) - (t * (a * (0.5 / fabs(z)))))) * (t_1 * t_2);
	}
	return copysign(1.0, x) * (copysign(1.0, y) * (copysign(1.0, z) * tmp));
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = fmax(Math.abs(x), Math.abs(y));
	double t_2 = fmin(Math.abs(x), Math.abs(y));
	double tmp;
	if (Math.abs(z) <= 1.4e+52) {
		tmp = ((Math.abs(z) * t_1) / Math.sqrt(((Math.abs(z) * Math.abs(z)) - (a * t)))) * t_2;
	} else {
		tmp = (Math.abs(z) / (Math.abs(z) - (t * (a * (0.5 / Math.abs(z)))))) * (t_1 * t_2);
	}
	return Math.copySign(1.0, x) * (Math.copySign(1.0, y) * (Math.copySign(1.0, z) * tmp));
}

Python

def code(x, y, z, t, a):
	t_1 = fmax(math.fabs(x), math.fabs(y))
	t_2 = fmin(math.fabs(x), math.fabs(y))
	tmp = 0
	if math.fabs(z) <= 1.4e+52:
		tmp = ((math.fabs(z) * t_1) / math.sqrt(((math.fabs(z) * math.fabs(z)) - (a * t)))) * t_2
	else:
		tmp = (math.fabs(z) / (math.fabs(z) - (t * (a * (0.5 / math.fabs(z)))))) * (t_1 * t_2)
	return math.copysign(1.0, x) * (math.copysign(1.0, y) * (math.copysign(1.0, z) * tmp))

Julia

function code(x, y, z, t, a)
	t_1 = fmax(abs(x), abs(y))
	t_2 = fmin(abs(x), abs(y))
	tmp = 0.0
	if (abs(z) <= 1.4e+52)
		tmp = Float64(Float64(Float64(abs(z) * t_1) / sqrt(Float64(Float64(abs(z) * abs(z)) - Float64(a * t)))) * t_2);
	else
		tmp = Float64(Float64(abs(z) / Float64(abs(z) - Float64(t * Float64(a * Float64(0.5 / abs(z)))))) * Float64(t_1 * t_2));
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * Float64(copysign(1.0, z) * tmp)))
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = max(abs(x), abs(y));
	t_2 = min(abs(x), abs(y));
	tmp = 0.0;
	if (abs(z) <= 1.4e+52)
		tmp = ((abs(z) * t_1) / sqrt(((abs(z) * abs(z)) - (a * t)))) * t_2;
	else
		tmp = (abs(z) / (abs(z) - (t * (a * (0.5 / abs(z)))))) * (t_1 * t_2);
	end
	tmp_2 = (sign(x) * abs(1.0)) * ((sign(y) * abs(1.0)) * ((sign(z) * abs(1.0)) * tmp));
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z], $MachinePrecision], 1.4e+52], N[(N[(N[(N[Abs[z], $MachinePrecision] * t$95$1), $MachinePrecision] / N[Sqrt[N[(N[(N[Abs[z], $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(N[Abs[z], $MachinePrecision] / N[(N[Abs[z], $MachinePrecision] - N[(t * N[(a * N[(0.5 / N[Abs[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 1.4e52

   1. Initial program 61.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. inv-pow N/A

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

      7. lift-sqrt.f64 N/A

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

      8. sqrt-fabs-rev N/A

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

      9. lift-sqrt.f64 N/A

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

      10. rem-sqrt-square-rev N/A

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

      11. sqrt-pow2 N/A

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

      12. sqr-neg-rev N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. unpow-prod-down N/A

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

      16. pow-add N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

   3. Applied rewrites 61.0%

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

   if 1.4e52 < z

   1. Initial program 61.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 41.8%

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

   4. Applied rewrites 41.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 47.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 49.2%

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

   8. Applied rewrites 49.2%

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

Alternative 3: 92.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_2 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \left(\mathsf{copysign}\left(1, z\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z\right| \leq 1.4 \cdot 10^{+52}:\\ \;\;\;\;\frac{\left|z\right| \cdot t\_1}{\sqrt{\left|z\right| \cdot \left|z\right| - a \cdot t}} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(t\_1 \cdot t\_2\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fmax (fabs x) (fabs y))) (t_2 (fmin (fabs x) (fabs y))))
  (*
   (copysign 1.0 x)
   (*
    (copysign 1.0 y)
    (*
     (copysign 1.0 z)
     (if (<= (fabs z) 1.4e+52)
       (*
        (/ (* (fabs z) t_1) (sqrt (- (* (fabs z) (fabs z)) (* a t))))
        t_2)
       (* 1.0 (* t_1 t_2))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fmax(fabs(x), fabs(y));
	double t_2 = fmin(fabs(x), fabs(y));
	double tmp;
	if (fabs(z) <= 1.4e+52) {
		tmp = ((fabs(z) * t_1) / sqrt(((fabs(z) * fabs(z)) - (a * t)))) * t_2;
	} else {
		tmp = 1.0 * (t_1 * t_2);
	}
	return copysign(1.0, x) * (copysign(1.0, y) * (copysign(1.0, z) * tmp));
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = fmax(Math.abs(x), Math.abs(y));
	double t_2 = fmin(Math.abs(x), Math.abs(y));
	double tmp;
	if (Math.abs(z) <= 1.4e+52) {
		tmp = ((Math.abs(z) * t_1) / Math.sqrt(((Math.abs(z) * Math.abs(z)) - (a * t)))) * t_2;
	} else {
		tmp = 1.0 * (t_1 * t_2);
	}
	return Math.copySign(1.0, x) * (Math.copySign(1.0, y) * (Math.copySign(1.0, z) * tmp));
}

Python

def code(x, y, z, t, a):
	t_1 = fmax(math.fabs(x), math.fabs(y))
	t_2 = fmin(math.fabs(x), math.fabs(y))
	tmp = 0
	if math.fabs(z) <= 1.4e+52:
		tmp = ((math.fabs(z) * t_1) / math.sqrt(((math.fabs(z) * math.fabs(z)) - (a * t)))) * t_2
	else:
		tmp = 1.0 * (t_1 * t_2)
	return math.copysign(1.0, x) * (math.copysign(1.0, y) * (math.copysign(1.0, z) * tmp))

Julia

function code(x, y, z, t, a)
	t_1 = fmax(abs(x), abs(y))
	t_2 = fmin(abs(x), abs(y))
	tmp = 0.0
	if (abs(z) <= 1.4e+52)
		tmp = Float64(Float64(Float64(abs(z) * t_1) / sqrt(Float64(Float64(abs(z) * abs(z)) - Float64(a * t)))) * t_2);
	else
		tmp = Float64(1.0 * Float64(t_1 * t_2));
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * Float64(copysign(1.0, z) * tmp)))
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = max(abs(x), abs(y));
	t_2 = min(abs(x), abs(y));
	tmp = 0.0;
	if (abs(z) <= 1.4e+52)
		tmp = ((abs(z) * t_1) / sqrt(((abs(z) * abs(z)) - (a * t)))) * t_2;
	else
		tmp = 1.0 * (t_1 * t_2);
	end
	tmp_2 = (sign(x) * abs(1.0)) * ((sign(y) * abs(1.0)) * ((sign(z) * abs(1.0)) * tmp));
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z], $MachinePrecision], 1.4e+52], N[(N[(N[(N[Abs[z], $MachinePrecision] * t$95$1), $MachinePrecision] / N[Sqrt[N[(N[(N[Abs[z], $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(1.0 * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 1.4e52

   1. Initial program 61.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. inv-pow N/A

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

      7. lift-sqrt.f64 N/A

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

      8. sqrt-fabs-rev N/A

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

      9. lift-sqrt.f64 N/A

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

      10. rem-sqrt-square-rev N/A

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

      11. sqrt-pow2 N/A

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

      12. sqr-neg-rev N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. unpow-prod-down N/A

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

      16. pow-add N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

   3. Applied rewrites 61.0%

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

   if 1.4e52 < z

   1. Initial program 61.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 41.8%

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

   4. Applied rewrites 41.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 47.8%

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

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 42.7%

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

Alternative 4: 92.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_2 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \left(\mathsf{copysign}\left(1, z\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z\right| \leq 10^{+120}:\\ \;\;\;\;\left(\frac{\left|z\right|}{\sqrt{\left|z\right| \cdot \left|z\right| - a \cdot t}} \cdot t\_1\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(t\_1 \cdot t\_2\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fmax (fabs x) (fabs y))) (t_2 (fmin (fabs x) (fabs y))))
  (*
   (copysign 1.0 x)
   (*
    (copysign 1.0 y)
    (*
     (copysign 1.0 z)
     (if (<= (fabs z) 1e+120)
       (*
        (* (/ (fabs z) (sqrt (- (* (fabs z) (fabs z)) (* a t)))) t_1)
        t_2)
       (* 1.0 (* t_1 t_2))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fmax(fabs(x), fabs(y));
	double t_2 = fmin(fabs(x), fabs(y));
	double tmp;
	if (fabs(z) <= 1e+120) {
		tmp = ((fabs(z) / sqrt(((fabs(z) * fabs(z)) - (a * t)))) * t_1) * t_2;
	} else {
		tmp = 1.0 * (t_1 * t_2);
	}
	return copysign(1.0, x) * (copysign(1.0, y) * (copysign(1.0, z) * tmp));
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = fmax(Math.abs(x), Math.abs(y));
	double t_2 = fmin(Math.abs(x), Math.abs(y));
	double tmp;
	if (Math.abs(z) <= 1e+120) {
		tmp = ((Math.abs(z) / Math.sqrt(((Math.abs(z) * Math.abs(z)) - (a * t)))) * t_1) * t_2;
	} else {
		tmp = 1.0 * (t_1 * t_2);
	}
	return Math.copySign(1.0, x) * (Math.copySign(1.0, y) * (Math.copySign(1.0, z) * tmp));
}

Python

def code(x, y, z, t, a):
	t_1 = fmax(math.fabs(x), math.fabs(y))
	t_2 = fmin(math.fabs(x), math.fabs(y))
	tmp = 0
	if math.fabs(z) <= 1e+120:
		tmp = ((math.fabs(z) / math.sqrt(((math.fabs(z) * math.fabs(z)) - (a * t)))) * t_1) * t_2
	else:
		tmp = 1.0 * (t_1 * t_2)
	return math.copysign(1.0, x) * (math.copysign(1.0, y) * (math.copysign(1.0, z) * tmp))

Julia

function code(x, y, z, t, a)
	t_1 = fmax(abs(x), abs(y))
	t_2 = fmin(abs(x), abs(y))
	tmp = 0.0
	if (abs(z) <= 1e+120)
		tmp = Float64(Float64(Float64(abs(z) / sqrt(Float64(Float64(abs(z) * abs(z)) - Float64(a * t)))) * t_1) * t_2);
	else
		tmp = Float64(1.0 * Float64(t_1 * t_2));
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * Float64(copysign(1.0, z) * tmp)))
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = max(abs(x), abs(y));
	t_2 = min(abs(x), abs(y));
	tmp = 0.0;
	if (abs(z) <= 1e+120)
		tmp = ((abs(z) / sqrt(((abs(z) * abs(z)) - (a * t)))) * t_1) * t_2;
	else
		tmp = 1.0 * (t_1 * t_2);
	end
	tmp_2 = (sign(x) * abs(1.0)) * ((sign(y) * abs(1.0)) * ((sign(z) * abs(1.0)) * tmp));
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z], $MachinePrecision], 1e+120], N[(N[(N[(N[Abs[z], $MachinePrecision] / N[Sqrt[N[(N[(N[Abs[z], $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], N[(1.0 * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 9.9999999999999998e119

   1. Initial program 61.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. inv-pow N/A

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

      7. lift-sqrt.f64 N/A

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

      8. sqrt-fabs-rev N/A

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

      9. lift-sqrt.f64 N/A

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

      10. rem-sqrt-square-rev N/A

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

      11. sqrt-pow2 N/A

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

      12. sqr-neg-rev N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. unpow-prod-down N/A

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

      16. pow-add N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

   3. Applied rewrites 61.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-/.f64 64.0%

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

   5. Applied rewrites 64.0%

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

   if 9.9999999999999998e119 < z

   1. Initial program 61.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 41.8%

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

   4. Applied rewrites 41.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 47.8%

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

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 42.7%

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

Alternative 5: 90.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ t_2 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \left(\mathsf{copysign}\left(1, z\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z\right| \leq 5 \cdot 10^{-155}:\\ \;\;\;\;\frac{t\_1}{\sqrt{-a \cdot t}} \cdot \left(\left|z\right| \cdot t\_2\right)\\ \mathbf{elif}\;\left|z\right| \leq 1.4 \cdot 10^{+52}:\\ \;\;\;\;\left(\frac{t\_2}{\sqrt{\left|z\right| \cdot \left|z\right| - a \cdot t}} \cdot t\_1\right) \cdot \left|z\right|\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(t\_2 \cdot t\_1\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fmin (fabs x) (fabs y))) (t_2 (fmax (fabs x) (fabs y))))
  (*
   (copysign 1.0 x)
   (*
    (copysign 1.0 y)
    (*
     (copysign 1.0 z)
     (if (<= (fabs z) 5e-155)
       (* (/ t_1 (sqrt (- (* a t)))) (* (fabs z) t_2))
       (if (<= (fabs z) 1.4e+52)
         (*
          (* (/ t_2 (sqrt (- (* (fabs z) (fabs z)) (* a t)))) t_1)
          (fabs z))
         (* 1.0 (* t_2 t_1)))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fmin(fabs(x), fabs(y));
	double t_2 = fmax(fabs(x), fabs(y));
	double tmp;
	if (fabs(z) <= 5e-155) {
		tmp = (t_1 / sqrt(-(a * t))) * (fabs(z) * t_2);
	} else if (fabs(z) <= 1.4e+52) {
		tmp = ((t_2 / sqrt(((fabs(z) * fabs(z)) - (a * t)))) * t_1) * fabs(z);
	} else {
		tmp = 1.0 * (t_2 * t_1);
	}
	return copysign(1.0, x) * (copysign(1.0, y) * (copysign(1.0, z) * tmp));
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = fmin(Math.abs(x), Math.abs(y));
	double t_2 = fmax(Math.abs(x), Math.abs(y));
	double tmp;
	if (Math.abs(z) <= 5e-155) {
		tmp = (t_1 / Math.sqrt(-(a * t))) * (Math.abs(z) * t_2);
	} else if (Math.abs(z) <= 1.4e+52) {
		tmp = ((t_2 / Math.sqrt(((Math.abs(z) * Math.abs(z)) - (a * t)))) * t_1) * Math.abs(z);
	} else {
		tmp = 1.0 * (t_2 * t_1);
	}
	return Math.copySign(1.0, x) * (Math.copySign(1.0, y) * (Math.copySign(1.0, z) * tmp));
}

Python

def code(x, y, z, t, a):
	t_1 = fmin(math.fabs(x), math.fabs(y))
	t_2 = fmax(math.fabs(x), math.fabs(y))
	tmp = 0
	if math.fabs(z) <= 5e-155:
		tmp = (t_1 / math.sqrt(-(a * t))) * (math.fabs(z) * t_2)
	elif math.fabs(z) <= 1.4e+52:
		tmp = ((t_2 / math.sqrt(((math.fabs(z) * math.fabs(z)) - (a * t)))) * t_1) * math.fabs(z)
	else:
		tmp = 1.0 * (t_2 * t_1)
	return math.copysign(1.0, x) * (math.copysign(1.0, y) * (math.copysign(1.0, z) * tmp))

Julia

function code(x, y, z, t, a)
	t_1 = fmin(abs(x), abs(y))
	t_2 = fmax(abs(x), abs(y))
	tmp = 0.0
	if (abs(z) <= 5e-155)
		tmp = Float64(Float64(t_1 / sqrt(Float64(-Float64(a * t)))) * Float64(abs(z) * t_2));
	elseif (abs(z) <= 1.4e+52)
		tmp = Float64(Float64(Float64(t_2 / sqrt(Float64(Float64(abs(z) * abs(z)) - Float64(a * t)))) * t_1) * abs(z));
	else
		tmp = Float64(1.0 * Float64(t_2 * t_1));
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * Float64(copysign(1.0, z) * tmp)))
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = min(abs(x), abs(y));
	t_2 = max(abs(x), abs(y));
	tmp = 0.0;
	if (abs(z) <= 5e-155)
		tmp = (t_1 / sqrt(-(a * t))) * (abs(z) * t_2);
	elseif (abs(z) <= 1.4e+52)
		tmp = ((t_2 / sqrt(((abs(z) * abs(z)) - (a * t)))) * t_1) * abs(z);
	else
		tmp = 1.0 * (t_2 * t_1);
	end
	tmp_2 = (sign(x) * abs(1.0)) * ((sign(y) * abs(1.0)) * ((sign(z) * abs(1.0)) * tmp));
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z], $MachinePrecision], 5e-155], N[(N[(t$95$1 / N[Sqrt[(-N[(a * t), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[z], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[z], $MachinePrecision], 1.4e+52], N[(N[(N[(t$95$2 / N[Sqrt[N[(N[(N[Abs[z], $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 4.9999999999999999e-155

   1. Initial program 61.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. mult-flip-rev N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 59.1%

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

   3. Applied rewrites 59.1%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 31.8%

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

   6. Applied rewrites 31.8%

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

   if 4.9999999999999999e-155 < z < 1.4e52

   1. Initial program 61.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 59.7%

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

   3. Applied rewrites 59.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lower-*.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 59.6%

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

   5. Applied rewrites 59.6%

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

   if 1.4e52 < z

   1. Initial program 61.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 41.8%

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

   4. Applied rewrites 41.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 47.8%

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

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 42.7%

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

Alternative 6: 83.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_2 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \left(\mathsf{copysign}\left(1, z\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z\right| \leq 1.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{t\_1 \cdot \left|z\right|}{\sqrt{-a \cdot t}} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(t\_1 \cdot t\_2\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fmax (fabs x) (fabs y))) (t_2 (fmin (fabs x) (fabs y))))
  (*
   (copysign 1.0 x)
   (*
    (copysign 1.0 y)
    (*
     (copysign 1.0 z)
     (if (<= (fabs z) 1.8e-21)
       (* (/ (* t_1 (fabs z)) (sqrt (- (* a t)))) t_2)
       (* 1.0 (* t_1 t_2))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fmax(fabs(x), fabs(y));
	double t_2 = fmin(fabs(x), fabs(y));
	double tmp;
	if (fabs(z) <= 1.8e-21) {
		tmp = ((t_1 * fabs(z)) / sqrt(-(a * t))) * t_2;
	} else {
		tmp = 1.0 * (t_1 * t_2);
	}
	return copysign(1.0, x) * (copysign(1.0, y) * (copysign(1.0, z) * tmp));
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = fmax(Math.abs(x), Math.abs(y));
	double t_2 = fmin(Math.abs(x), Math.abs(y));
	double tmp;
	if (Math.abs(z) <= 1.8e-21) {
		tmp = ((t_1 * Math.abs(z)) / Math.sqrt(-(a * t))) * t_2;
	} else {
		tmp = 1.0 * (t_1 * t_2);
	}
	return Math.copySign(1.0, x) * (Math.copySign(1.0, y) * (Math.copySign(1.0, z) * tmp));
}

Python

def code(x, y, z, t, a):
	t_1 = fmax(math.fabs(x), math.fabs(y))
	t_2 = fmin(math.fabs(x), math.fabs(y))
	tmp = 0
	if math.fabs(z) <= 1.8e-21:
		tmp = ((t_1 * math.fabs(z)) / math.sqrt(-(a * t))) * t_2
	else:
		tmp = 1.0 * (t_1 * t_2)
	return math.copysign(1.0, x) * (math.copysign(1.0, y) * (math.copysign(1.0, z) * tmp))

Julia

function code(x, y, z, t, a)
	t_1 = fmax(abs(x), abs(y))
	t_2 = fmin(abs(x), abs(y))
	tmp = 0.0
	if (abs(z) <= 1.8e-21)
		tmp = Float64(Float64(Float64(t_1 * abs(z)) / sqrt(Float64(-Float64(a * t)))) * t_2);
	else
		tmp = Float64(1.0 * Float64(t_1 * t_2));
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * Float64(copysign(1.0, z) * tmp)))
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = max(abs(x), abs(y));
	t_2 = min(abs(x), abs(y));
	tmp = 0.0;
	if (abs(z) <= 1.8e-21)
		tmp = ((t_1 * abs(z)) / sqrt(-(a * t))) * t_2;
	else
		tmp = 1.0 * (t_1 * t_2);
	end
	tmp_2 = (sign(x) * abs(1.0)) * ((sign(y) * abs(1.0)) * ((sign(z) * abs(1.0)) * tmp));
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z], $MachinePrecision], 1.8e-21], N[(N[(N[(t$95$1 * N[Abs[z], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-N[(a * t), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(1.0 * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 1.7999999999999999e-21

   1. Initial program 61.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. inv-pow N/A

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

      7. lift-sqrt.f64 N/A

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

      8. sqrt-fabs-rev N/A

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

      9. lift-sqrt.f64 N/A

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

      10. rem-sqrt-square-rev N/A

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

      11. sqrt-pow2 N/A

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

      12. sqr-neg-rev N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. unpow-prod-down N/A

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

      16. pow-add N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

   3. Applied rewrites 61.0%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 31.9%

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

   6. Applied rewrites 31.9%

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

   if 1.7999999999999999e-21 < z

   1. Initial program 61.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 41.8%

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

   4. Applied rewrites 41.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 47.8%

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

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 42.7%

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

Alternative 7: 83.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ t_2 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \left(\mathsf{copysign}\left(1, z\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z\right| \leq 1.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{t\_1}{\sqrt{-a \cdot t}} \cdot \left(\left|z\right| \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(t\_2 \cdot t\_1\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fmin (fabs x) (fabs y))) (t_2 (fmax (fabs x) (fabs y))))
  (*
   (copysign 1.0 x)
   (*
    (copysign 1.0 y)
    (*
     (copysign 1.0 z)
     (if (<= (fabs z) 1.8e-21)
       (* (/ t_1 (sqrt (- (* a t)))) (* (fabs z) t_2))
       (* 1.0 (* t_2 t_1))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fmin(fabs(x), fabs(y));
	double t_2 = fmax(fabs(x), fabs(y));
	double tmp;
	if (fabs(z) <= 1.8e-21) {
		tmp = (t_1 / sqrt(-(a * t))) * (fabs(z) * t_2);
	} else {
		tmp = 1.0 * (t_2 * t_1);
	}
	return copysign(1.0, x) * (copysign(1.0, y) * (copysign(1.0, z) * tmp));
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = fmin(Math.abs(x), Math.abs(y));
	double t_2 = fmax(Math.abs(x), Math.abs(y));
	double tmp;
	if (Math.abs(z) <= 1.8e-21) {
		tmp = (t_1 / Math.sqrt(-(a * t))) * (Math.abs(z) * t_2);
	} else {
		tmp = 1.0 * (t_2 * t_1);
	}
	return Math.copySign(1.0, x) * (Math.copySign(1.0, y) * (Math.copySign(1.0, z) * tmp));
}

Python

def code(x, y, z, t, a):
	t_1 = fmin(math.fabs(x), math.fabs(y))
	t_2 = fmax(math.fabs(x), math.fabs(y))
	tmp = 0
	if math.fabs(z) <= 1.8e-21:
		tmp = (t_1 / math.sqrt(-(a * t))) * (math.fabs(z) * t_2)
	else:
		tmp = 1.0 * (t_2 * t_1)
	return math.copysign(1.0, x) * (math.copysign(1.0, y) * (math.copysign(1.0, z) * tmp))

Julia

function code(x, y, z, t, a)
	t_1 = fmin(abs(x), abs(y))
	t_2 = fmax(abs(x), abs(y))
	tmp = 0.0
	if (abs(z) <= 1.8e-21)
		tmp = Float64(Float64(t_1 / sqrt(Float64(-Float64(a * t)))) * Float64(abs(z) * t_2));
	else
		tmp = Float64(1.0 * Float64(t_2 * t_1));
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * Float64(copysign(1.0, z) * tmp)))
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = min(abs(x), abs(y));
	t_2 = max(abs(x), abs(y));
	tmp = 0.0;
	if (abs(z) <= 1.8e-21)
		tmp = (t_1 / sqrt(-(a * t))) * (abs(z) * t_2);
	else
		tmp = 1.0 * (t_2 * t_1);
	end
	tmp_2 = (sign(x) * abs(1.0)) * ((sign(y) * abs(1.0)) * ((sign(z) * abs(1.0)) * tmp));
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z], $MachinePrecision], 1.8e-21], N[(N[(t$95$1 / N[Sqrt[(-N[(a * t), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[z], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 1.7999999999999999e-21

   1. Initial program 61.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. mult-flip-rev N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 59.1%

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

   3. Applied rewrites 59.1%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 31.8%

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

   6. Applied rewrites 31.8%

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

   if 1.7999999999999999e-21 < z

   1. Initial program 61.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 41.8%

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

   4. Applied rewrites 41.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 47.8%

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

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 42.7%

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

Alternative 8: 82.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ t_2 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \left(\mathsf{copysign}\left(1, z\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z\right| \leq 1.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{t\_1 \cdot \left(t\_2 \cdot \left|z\right|\right)}{\sqrt{-a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(t\_2 \cdot t\_1\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fmin (fabs x) (fabs y))) (t_2 (fmax (fabs x) (fabs y))))
  (*
   (copysign 1.0 x)
   (*
    (copysign 1.0 y)
    (*
     (copysign 1.0 z)
     (if (<= (fabs z) 1.8e-21)
       (/ (* t_1 (* t_2 (fabs z))) (sqrt (- (* a t))))
       (* 1.0 (* t_2 t_1))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fmin(fabs(x), fabs(y));
	double t_2 = fmax(fabs(x), fabs(y));
	double tmp;
	if (fabs(z) <= 1.8e-21) {
		tmp = (t_1 * (t_2 * fabs(z))) / sqrt(-(a * t));
	} else {
		tmp = 1.0 * (t_2 * t_1);
	}
	return copysign(1.0, x) * (copysign(1.0, y) * (copysign(1.0, z) * tmp));
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = fmin(Math.abs(x), Math.abs(y));
	double t_2 = fmax(Math.abs(x), Math.abs(y));
	double tmp;
	if (Math.abs(z) <= 1.8e-21) {
		tmp = (t_1 * (t_2 * Math.abs(z))) / Math.sqrt(-(a * t));
	} else {
		tmp = 1.0 * (t_2 * t_1);
	}
	return Math.copySign(1.0, x) * (Math.copySign(1.0, y) * (Math.copySign(1.0, z) * tmp));
}

Python

def code(x, y, z, t, a):
	t_1 = fmin(math.fabs(x), math.fabs(y))
	t_2 = fmax(math.fabs(x), math.fabs(y))
	tmp = 0
	if math.fabs(z) <= 1.8e-21:
		tmp = (t_1 * (t_2 * math.fabs(z))) / math.sqrt(-(a * t))
	else:
		tmp = 1.0 * (t_2 * t_1)
	return math.copysign(1.0, x) * (math.copysign(1.0, y) * (math.copysign(1.0, z) * tmp))

Julia

function code(x, y, z, t, a)
	t_1 = fmin(abs(x), abs(y))
	t_2 = fmax(abs(x), abs(y))
	tmp = 0.0
	if (abs(z) <= 1.8e-21)
		tmp = Float64(Float64(t_1 * Float64(t_2 * abs(z))) / sqrt(Float64(-Float64(a * t))));
	else
		tmp = Float64(1.0 * Float64(t_2 * t_1));
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * Float64(copysign(1.0, z) * tmp)))
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = min(abs(x), abs(y));
	t_2 = max(abs(x), abs(y));
	tmp = 0.0;
	if (abs(z) <= 1.8e-21)
		tmp = (t_1 * (t_2 * abs(z))) / sqrt(-(a * t));
	else
		tmp = 1.0 * (t_2 * t_1);
	end
	tmp_2 = (sign(x) * abs(1.0)) * ((sign(y) * abs(1.0)) * ((sign(z) * abs(1.0)) * tmp));
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z], $MachinePrecision], 1.8e-21], N[(N[(t$95$1 * N[(t$95$2 * N[Abs[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-N[(a * t), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 1.7999999999999999e-21

   1. Initial program 61.1%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 32.1%

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

   4. Applied rewrites 32.1%

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

   if 1.7999999999999999e-21 < z

   1. Initial program 61.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 41.8%

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

   4. Applied rewrites 41.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 47.8%

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

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 42.7%

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

Alternative 9: 76.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ t_2 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \left(\mathsf{copysign}\left(1, z\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(t\_1 \cdot t\_2\right) \cdot \left|z\right|}{\sqrt{\left|z\right| \cdot \left|z\right| - t \cdot a}} \leq 5 \cdot 10^{-202}:\\ \;\;\;\;\frac{\left(t\_1 \cdot \left|z\right|\right) \cdot t\_2}{\left|z\right| \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(t\_2 \cdot t\_1\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fmin (fabs x) (fabs y))) (t_2 (fmax (fabs x) (fabs y))))
  (*
   (copysign 1.0 x)
   (*
    (copysign 1.0 y)
    (*
     (copysign 1.0 z)
     (if (<=
          (/
           (* (* t_1 t_2) (fabs z))
           (sqrt (- (* (fabs z) (fabs z)) (* t a))))
          5e-202)
       (/ (* (* t_1 (fabs z)) t_2) (* (fabs z) 1.0))
       (* 1.0 (* t_2 t_1))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fmin(fabs(x), fabs(y));
	double t_2 = fmax(fabs(x), fabs(y));
	double tmp;
	if ((((t_1 * t_2) * fabs(z)) / sqrt(((fabs(z) * fabs(z)) - (t * a)))) <= 5e-202) {
		tmp = ((t_1 * fabs(z)) * t_2) / (fabs(z) * 1.0);
	} else {
		tmp = 1.0 * (t_2 * t_1);
	}
	return copysign(1.0, x) * (copysign(1.0, y) * (copysign(1.0, z) * tmp));
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = fmin(Math.abs(x), Math.abs(y));
	double t_2 = fmax(Math.abs(x), Math.abs(y));
	double tmp;
	if ((((t_1 * t_2) * Math.abs(z)) / Math.sqrt(((Math.abs(z) * Math.abs(z)) - (t * a)))) <= 5e-202) {
		tmp = ((t_1 * Math.abs(z)) * t_2) / (Math.abs(z) * 1.0);
	} else {
		tmp = 1.0 * (t_2 * t_1);
	}
	return Math.copySign(1.0, x) * (Math.copySign(1.0, y) * (Math.copySign(1.0, z) * tmp));
}

Python

def code(x, y, z, t, a):
	t_1 = fmin(math.fabs(x), math.fabs(y))
	t_2 = fmax(math.fabs(x), math.fabs(y))
	tmp = 0
	if (((t_1 * t_2) * math.fabs(z)) / math.sqrt(((math.fabs(z) * math.fabs(z)) - (t * a)))) <= 5e-202:
		tmp = ((t_1 * math.fabs(z)) * t_2) / (math.fabs(z) * 1.0)
	else:
		tmp = 1.0 * (t_2 * t_1)
	return math.copysign(1.0, x) * (math.copysign(1.0, y) * (math.copysign(1.0, z) * tmp))

Julia

function code(x, y, z, t, a)
	t_1 = fmin(abs(x), abs(y))
	t_2 = fmax(abs(x), abs(y))
	tmp = 0.0
	if (Float64(Float64(Float64(t_1 * t_2) * abs(z)) / sqrt(Float64(Float64(abs(z) * abs(z)) - Float64(t * a)))) <= 5e-202)
		tmp = Float64(Float64(Float64(t_1 * abs(z)) * t_2) / Float64(abs(z) * 1.0));
	else
		tmp = Float64(1.0 * Float64(t_2 * t_1));
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * Float64(copysign(1.0, z) * tmp)))
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = min(abs(x), abs(y));
	t_2 = max(abs(x), abs(y));
	tmp = 0.0;
	if ((((t_1 * t_2) * abs(z)) / sqrt(((abs(z) * abs(z)) - (t * a)))) <= 5e-202)
		tmp = ((t_1 * abs(z)) * t_2) / (abs(z) * 1.0);
	else
		tmp = 1.0 * (t_2 * t_1);
	end
	tmp_2 = (sign(x) * abs(1.0)) * ((sign(y) * abs(1.0)) * ((sign(z) * abs(1.0)) * tmp));
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[(t$95$1 * t$95$2), $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(N[Abs[z], $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e-202], N[(N[(N[(t$95$1 * N[Abs[z], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] / N[(N[Abs[z], $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(t$95$2 * t$95$1), $MachinePrecision]), $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)))) < 4.9999999999999997e-202

   1. Initial program 61.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 41.8%

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

   4. Applied rewrites 41.8%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 41.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 38.9%

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

   9. Applied rewrites 38.9%

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

   if 4.9999999999999997e-202 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))))

   1. Initial program 61.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 41.8%

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

   4. Applied rewrites 41.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 47.8%

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

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 42.7%

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

Alternative 10: 76.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ t_2 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \left(\mathsf{copysign}\left(1, z\right) \cdot \begin{array}{l} \mathbf{if}\;t \cdot a \leq -1.7 \cdot 10^{-120}:\\ \;\;\;\;\frac{t\_1 \cdot \left|z\right|}{1 \cdot \left|z\right|} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(t\_2 \cdot t\_1\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fmin (fabs x) (fabs y))) (t_2 (fmax (fabs x) (fabs y))))
  (*
   (copysign 1.0 x)
   (*
    (copysign 1.0 y)
    (*
     (copysign 1.0 z)
     (if (<= (* t a) -1.7e-120)
       (* (/ (* t_1 (fabs z)) (* 1.0 (fabs z))) t_2)
       (* 1.0 (* t_2 t_1))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fmin(fabs(x), fabs(y));
	double t_2 = fmax(fabs(x), fabs(y));
	double tmp;
	if ((t * a) <= -1.7e-120) {
		tmp = ((t_1 * fabs(z)) / (1.0 * fabs(z))) * t_2;
	} else {
		tmp = 1.0 * (t_2 * t_1);
	}
	return copysign(1.0, x) * (copysign(1.0, y) * (copysign(1.0, z) * tmp));
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = fmin(Math.abs(x), Math.abs(y));
	double t_2 = fmax(Math.abs(x), Math.abs(y));
	double tmp;
	if ((t * a) <= -1.7e-120) {
		tmp = ((t_1 * Math.abs(z)) / (1.0 * Math.abs(z))) * t_2;
	} else {
		tmp = 1.0 * (t_2 * t_1);
	}
	return Math.copySign(1.0, x) * (Math.copySign(1.0, y) * (Math.copySign(1.0, z) * tmp));
}

Python

def code(x, y, z, t, a):
	t_1 = fmin(math.fabs(x), math.fabs(y))
	t_2 = fmax(math.fabs(x), math.fabs(y))
	tmp = 0
	if (t * a) <= -1.7e-120:
		tmp = ((t_1 * math.fabs(z)) / (1.0 * math.fabs(z))) * t_2
	else:
		tmp = 1.0 * (t_2 * t_1)
	return math.copysign(1.0, x) * (math.copysign(1.0, y) * (math.copysign(1.0, z) * tmp))

Julia

function code(x, y, z, t, a)
	t_1 = fmin(abs(x), abs(y))
	t_2 = fmax(abs(x), abs(y))
	tmp = 0.0
	if (Float64(t * a) <= -1.7e-120)
		tmp = Float64(Float64(Float64(t_1 * abs(z)) / Float64(1.0 * abs(z))) * t_2);
	else
		tmp = Float64(1.0 * Float64(t_2 * t_1));
	end
	return Float64(copysign(1.0, x) * Float64(copysign(1.0, y) * Float64(copysign(1.0, z) * tmp)))
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = min(abs(x), abs(y));
	t_2 = max(abs(x), abs(y));
	tmp = 0.0;
	if ((t * a) <= -1.7e-120)
		tmp = ((t_1 * abs(z)) / (1.0 * abs(z))) * t_2;
	else
		tmp = 1.0 * (t_2 * t_1);
	end
	tmp_2 = (sign(x) * abs(1.0)) * ((sign(y) * abs(1.0)) * ((sign(z) * abs(1.0)) * tmp));
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Min[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(t * a), $MachinePrecision], -1.7e-120], N[(N[(N[(t$95$1 * N[Abs[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 * N[Abs[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(1.0 * N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t a) < -1.7e-120

   1. Initial program 61.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 41.8%

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

   4. Applied rewrites 41.8%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 41.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   9. Applied rewrites 39.9%

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

   if -1.7e-120 < (*.f64 t a)

   1. Initial program 61.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 41.8%

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

   4. Applied rewrites 41.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 47.8%

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

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 42.7%

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

Alternative 11: 73.6% accurate, 0.3× speedup

Math

\[\mathsf{copysign}\left(1, z\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z\right| \leq 1.02 \cdot 10^{-192}:\\ \;\;\;\;\frac{y}{\left|z\right|} \cdot \left(\left|z\right| \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(y \cdot x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (*
 (copysign 1.0 z)
 (if (<= (fabs z) 1.02e-192)
   (* (/ y (fabs z)) (* (fabs z) x))
   (* 1.0 (* y x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (fabs(z) <= 1.02e-192) {
		tmp = (y / fabs(z)) * (fabs(z) * x);
	} else {
		tmp = 1.0 * (y * x);
	}
	return copysign(1.0, z) * tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (Math.abs(z) <= 1.02e-192) {
		tmp = (y / Math.abs(z)) * (Math.abs(z) * x);
	} else {
		tmp = 1.0 * (y * x);
	}
	return Math.copySign(1.0, z) * tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if math.fabs(z) <= 1.02e-192:
		tmp = (y / math.fabs(z)) * (math.fabs(z) * x)
	else:
		tmp = 1.0 * (y * x)
	return math.copysign(1.0, z) * tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (abs(z) <= 1.02e-192)
		tmp = Float64(Float64(y / abs(z)) * Float64(abs(z) * x));
	else
		tmp = Float64(1.0 * Float64(y * x));
	end
	return Float64(copysign(1.0, z) * tmp)
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (abs(z) <= 1.02e-192)
		tmp = (y / abs(z)) * (abs(z) * x);
	else
		tmp = 1.0 * (y * x);
	end
	tmp_2 = (sign(z) * abs(1.0)) * tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z], $MachinePrecision], 1.02e-192], N[(N[(y / N[Abs[z], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[z], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(y * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.02e-192

   1. Initial program 61.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 59.7%

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

   3. Applied rewrites 59.7%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 36.6%

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

   6. Applied rewrites 36.6%

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

   if 1.02e-192 < z

   1. Initial program 61.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 41.8%

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

   4. Applied rewrites 41.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 47.8%

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

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 42.7%

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

Alternative 12: 72.8% accurate, 0.4× speedup

Math

\[\mathsf{copysign}\left(1, z\right) \cdot \left(1 \cdot \left(y \cdot x\right)\right) \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (* (copysign 1.0 z) (* 1.0 (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	return copysign(1.0, z) * (1.0 * (y * x));
}

Java

public static double code(double x, double y, double z, double t, double a) {
	return Math.copySign(1.0, z) * (1.0 * (y * x));
}

Python

def code(x, y, z, t, a):
	return math.copysign(1.0, z) * (1.0 * (y * x))

Julia

function code(x, y, z, t, a)
	return Float64(copysign(1.0, z) * Float64(1.0 * Float64(y * x)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = (sign(z) * abs(1.0)) * (1.0 * (y * x));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(1.0 * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.1%

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

2. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 41.8%

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

4. Applied rewrites 41.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

6. Applied rewrites 47.8%

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

7. Taylor expanded in z around inf

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

9. Applied rewrites 42.7%

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

Alternative 13: 43.0% accurate, 5.6× speedup

Math

\[\left(-y\right) \cdot x \]

FPCore

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

C

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

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 = -y * x
end function

Java

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

Python

def code(x, y, z, t, a):
	return -y * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.1%

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

2. Taylor expanded in z around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 43.0%

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

4. Applied rewrites 43.0%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

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

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

   6. lower-*.f64 N/A

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

   7. lower-neg.f64 43.0%

      \[\leadsto \left(-y\right) \cdot x \]

6. Applied rewrites 43.0%

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

Reproduce

herbie shell --seed 2025258 
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"
  :precision binary64
  (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))