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

Percentage Accurate: 60.7% → 93.3%

Time: 4.7s

Alternatives: 9

Speedup: 1.7×

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: 60.7% 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: 93.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_2 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \left(\mathsf{copysign}\left(1, z\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z\right| \leq 3 \cdot 10^{-217}:\\ \;\;\;\;\frac{t\_1 \cdot t\_2}{\sqrt{-\mathsf{min}\left(t, a\right)} \cdot \sqrt{\mathsf{max}\left(t, a\right)}} \cdot \left|z\right|\\ \mathbf{elif}\;\left|z\right| \leq 3.35 \cdot 10^{+125}:\\ \;\;\;\;\left(\frac{\left|z\right|}{\sqrt{\left|z\right| \cdot \left|z\right| - \mathsf{max}\left(t, a\right) \cdot \mathsf{min}\left(t, a\right)}} \cdot t\_1\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot t\_1\right) \cdot t\_2\\ \end{array}\right)\right) \end{array} \]

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) 3e-217)
       (*
        (/ (* t_1 t_2) (* (sqrt (- (fmin t a))) (sqrt (fmax t a))))
        (fabs z))
       (if (<= (fabs z) 3.35e+125)
         (*
          (*
           (/
            (fabs z)
            (sqrt
             (- (* (fabs z) (fabs z)) (* (fmax t a) (fmin t a)))))
           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) <= 3e-217) {
		tmp = ((t_1 * t_2) / (sqrt(-fmin(t, a)) * sqrt(fmax(t, a)))) * fabs(z);
	} else if (fabs(z) <= 3.35e+125) {
		tmp = ((fabs(z) / sqrt(((fabs(z) * fabs(z)) - (fmax(t, a) * fmin(t, a))))) * 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) <= 3e-217) {
		tmp = ((t_1 * t_2) / (Math.sqrt(-fmin(t, a)) * Math.sqrt(fmax(t, a)))) * Math.abs(z);
	} else if (Math.abs(z) <= 3.35e+125) {
		tmp = ((Math.abs(z) / Math.sqrt(((Math.abs(z) * Math.abs(z)) - (fmax(t, a) * fmin(t, a))))) * 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) <= 3e-217:
		tmp = ((t_1 * t_2) / (math.sqrt(-fmin(t, a)) * math.sqrt(fmax(t, a)))) * math.fabs(z)
	elif math.fabs(z) <= 3.35e+125:
		tmp = ((math.fabs(z) / math.sqrt(((math.fabs(z) * math.fabs(z)) - (fmax(t, a) * fmin(t, a))))) * 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) <= 3e-217)
		tmp = Float64(Float64(Float64(t_1 * t_2) / Float64(sqrt(Float64(-fmin(t, a))) * sqrt(fmax(t, a)))) * abs(z));
	elseif (abs(z) <= 3.35e+125)
		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(1.0 * 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) <= 3e-217)
		tmp = ((t_1 * t_2) / (sqrt(-min(t, a)) * sqrt(max(t, a)))) * abs(z);
	elseif (abs(z) <= 3.35e+125)
		tmp = ((abs(z) / sqrt(((abs(z) * abs(z)) - (max(t, a) * min(t, a))))) * 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], 3e-217], N[(N[(N[(t$95$1 * t$95$2), $MachinePrecision] / N[(N[Sqrt[(-N[Min[t, a], $MachinePrecision])], $MachinePrecision] * N[Sqrt[N[Max[t, a], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[z], $MachinePrecision], 3.35e+125], 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[(1.0 * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 3e-217

   1. Initial program 60.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. mult-flip-rev N/A

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

      5. lower-*.f64 N/A

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

      6. mult-flip-rev N/A

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

      7. lower-/.f64 61.4%

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 61.4%

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 61.4%

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

   3. Applied rewrites 61.4%

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

   4. Taylor expanded in z around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 33.3%

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

   6. Applied rewrites 33.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. sqrt-prod N/A

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

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

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

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

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

      9. lower-neg.f64 N/A

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

      10. lower-unsound-sqrt.f64 18.8%

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

   8. Applied rewrites 18.8%

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

   if 3e-217 < z < 3.3500000000000002e125

   1. Initial program 60.7%

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

   3. Applied rewrites 63.8%

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

   if 3.3500000000000002e125 < z

   1. Initial program 60.7%

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

   3. Applied rewrites 63.8%

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

   4. Taylor expanded in z around inf

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

   6. Applied rewrites 42.7%

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

Alternative 2: 89.9% accurate, 0.5× speedup

Math

\[\mathsf{copysign}\left(1, z\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z\right| \leq 3.5 \cdot 10^{-216}:\\ \;\;\;\;\frac{y \cdot x}{\sqrt{-\mathsf{min}\left(t, a\right)} \cdot \sqrt{\mathsf{max}\left(t, a\right)}} \cdot \left|z\right|\\ \mathbf{elif}\;\left|z\right| \leq 5.8 \cdot 10^{+89}:\\ \;\;\;\;\left(\frac{x}{\sqrt{\left|z\right| \cdot \left|z\right| - \mathsf{min}\left(t, a\right) \cdot \mathsf{max}\left(t, a\right)}} \cdot y\right) \cdot \left|z\right|\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot y\right) \cdot x\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (*
 (copysign 1.0 z)
 (if (<= (fabs z) 3.5e-216)
   (*
    (/ (* y x) (* (sqrt (- (fmin t a))) (sqrt (fmax t a))))
    (fabs z))
   (if (<= (fabs z) 5.8e+89)
     (*
      (*
       (/
        x
        (sqrt (- (* (fabs z) (fabs z)) (* (fmin t a) (fmax t a)))))
       y)
      (fabs z))
     (* (* 1.0 y) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (fabs(z) <= 3.5e-216) {
		tmp = ((y * x) / (sqrt(-fmin(t, a)) * sqrt(fmax(t, a)))) * fabs(z);
	} else if (fabs(z) <= 5.8e+89) {
		tmp = ((x / sqrt(((fabs(z) * fabs(z)) - (fmin(t, a) * fmax(t, a))))) * y) * fabs(z);
	} 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) <= 3.5e-216) {
		tmp = ((y * x) / (Math.sqrt(-fmin(t, a)) * Math.sqrt(fmax(t, a)))) * Math.abs(z);
	} else if (Math.abs(z) <= 5.8e+89) {
		tmp = ((x / Math.sqrt(((Math.abs(z) * Math.abs(z)) - (fmin(t, a) * fmax(t, a))))) * y) * Math.abs(z);
	} 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) <= 3.5e-216:
		tmp = ((y * x) / (math.sqrt(-fmin(t, a)) * math.sqrt(fmax(t, a)))) * math.fabs(z)
	elif math.fabs(z) <= 5.8e+89:
		tmp = ((x / math.sqrt(((math.fabs(z) * math.fabs(z)) - (fmin(t, a) * fmax(t, a))))) * y) * math.fabs(z)
	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) <= 3.5e-216)
		tmp = Float64(Float64(Float64(y * x) / Float64(sqrt(Float64(-fmin(t, a))) * sqrt(fmax(t, a)))) * abs(z));
	elseif (abs(z) <= 5.8e+89)
		tmp = Float64(Float64(Float64(x / sqrt(Float64(Float64(abs(z) * abs(z)) - Float64(fmin(t, a) * fmax(t, a))))) * y) * abs(z));
	else
		tmp = Float64(Float64(1.0 * 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) <= 3.5e-216)
		tmp = ((y * x) / (sqrt(-min(t, a)) * sqrt(max(t, a)))) * abs(z);
	elseif (abs(z) <= 5.8e+89)
		tmp = ((x / sqrt(((abs(z) * abs(z)) - (min(t, a) * max(t, a))))) * y) * abs(z);
	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], 3.5e-216], N[(N[(N[(y * x), $MachinePrecision] / N[(N[Sqrt[(-N[Min[t, a], $MachinePrecision])], $MachinePrecision] * N[Sqrt[N[Max[t, a], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[z], $MachinePrecision], 5.8e+89], N[(N[(N[(x / N[Sqrt[N[(N[(N[Abs[z], $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision] - N[(N[Min[t, a], $MachinePrecision] * N[Max[t, a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * y), $MachinePrecision] * x), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 3.4999999999999998e-216

   1. Initial program 60.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. mult-flip-rev N/A

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

      5. lower-*.f64 N/A

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

      6. mult-flip-rev N/A

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

      7. lower-/.f64 61.4%

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 61.4%

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 61.4%

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

   3. Applied rewrites 61.4%

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

   4. Taylor expanded in z around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 33.3%

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

   6. Applied rewrites 33.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. sqrt-prod N/A

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

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

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

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

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

      9. lower-neg.f64 N/A

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

      10. lower-unsound-sqrt.f64 18.8%

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

   8. Applied rewrites 18.8%

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

   if 3.4999999999999998e-216 < z < 5.8000000000000005e89

   1. Initial program 60.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. mult-flip-rev N/A

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

      5. lower-*.f64 N/A

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

      6. mult-flip-rev N/A

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

      7. lower-/.f64 61.4%

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 61.4%

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 61.4%

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

   3. Applied rewrites 61.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 59.8%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 59.8%

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

   5. Applied rewrites 59.8%

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

   if 5.8000000000000005e89 < z

   1. Initial program 60.7%

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

   3. Applied rewrites 63.8%

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

   4. Taylor expanded in z around inf

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

   6. Applied rewrites 42.7%

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

Alternative 3: 79.2% accurate, 0.5× speedup

Math

\[\mathsf{copysign}\left(1, z\right) \cdot \begin{array}{l} \mathbf{if}\;\mathsf{min}\left(t, a\right) \cdot \mathsf{max}\left(t, a\right) \leq -2 \cdot 10^{-120}:\\ \;\;\;\;\frac{y \cdot x}{\sqrt{-\mathsf{min}\left(t, a\right)} \cdot \sqrt{\mathsf{max}\left(t, a\right)}} \cdot \left|z\right|\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot y\right) \cdot x\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (*
 (copysign 1.0 z)
 (if (<= (* (fmin t a) (fmax t a)) -2e-120)
   (*
    (/ (* y x) (* (sqrt (- (fmin t a))) (sqrt (fmax t a))))
    (fabs z))
   (* (* 1.0 y) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((fmin(t, a) * fmax(t, a)) <= -2e-120) {
		tmp = ((y * x) / (sqrt(-fmin(t, a)) * sqrt(fmax(t, a)))) * fabs(z);
	} 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 ((fmin(t, a) * fmax(t, a)) <= -2e-120) {
		tmp = ((y * x) / (Math.sqrt(-fmin(t, a)) * Math.sqrt(fmax(t, a)))) * Math.abs(z);
	} else {
		tmp = (1.0 * y) * x;
	}
	return Math.copySign(1.0, z) * tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (fmin(t, a) * fmax(t, a)) <= -2e-120:
		tmp = ((y * x) / (math.sqrt(-fmin(t, a)) * math.sqrt(fmax(t, a)))) * math.fabs(z)
	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 (Float64(fmin(t, a) * fmax(t, a)) <= -2e-120)
		tmp = Float64(Float64(Float64(y * x) / Float64(sqrt(Float64(-fmin(t, a))) * sqrt(fmax(t, a)))) * abs(z));
	else
		tmp = Float64(Float64(1.0 * 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 ((min(t, a) * max(t, a)) <= -2e-120)
		tmp = ((y * x) / (sqrt(-min(t, a)) * sqrt(max(t, a)))) * abs(z);
	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[(N[Min[t, a], $MachinePrecision] * N[Max[t, a], $MachinePrecision]), $MachinePrecision], -2e-120], N[(N[(N[(y * x), $MachinePrecision] / N[(N[Sqrt[(-N[Min[t, a], $MachinePrecision])], $MachinePrecision] * N[Sqrt[N[Max[t, a], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[z], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * y), $MachinePrecision] * x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 60.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. mult-flip-rev N/A

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

      5. lower-*.f64 N/A

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

      6. mult-flip-rev N/A

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

      7. lower-/.f64 61.4%

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 61.4%

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 61.4%

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

   3. Applied rewrites 61.4%

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

   4. Taylor expanded in z around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 33.3%

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

   6. Applied rewrites 33.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. sqrt-prod N/A

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

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

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

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

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

      9. lower-neg.f64 N/A

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

      10. lower-unsound-sqrt.f64 18.8%

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

   8. Applied rewrites 18.8%

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

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

   1. Initial program 60.7%

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

   3. Applied rewrites 63.8%

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

   4. Taylor expanded in z around inf

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

   6. Applied rewrites 42.7%

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

Alternative 4: 76.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_2 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \left(\mathsf{copysign}\left(1, z\right) \cdot \begin{array}{l} \mathbf{if}\;t \cdot a \leq -2 \cdot 10^{-120}:\\ \;\;\;\;\left(\frac{\left|z\right|}{\sqrt{-a \cdot t}} \cdot t\_1\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot t\_1\right) \cdot t\_2\\ \end{array}\right)\right) \end{array} \]

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 (<= (* t a) -2e-120)
       (* (* (/ (fabs z) (sqrt (- (* 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 ((t * a) <= -2e-120) {
		tmp = ((fabs(z) / sqrt(-(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 ((t * a) <= -2e-120) {
		tmp = ((Math.abs(z) / Math.sqrt(-(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 (t * a) <= -2e-120:
		tmp = ((math.fabs(z) / math.sqrt(-(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 (Float64(t * a) <= -2e-120)
		tmp = Float64(Float64(Float64(abs(z) / sqrt(Float64(-Float64(a * t)))) * t_1) * t_2);
	else
		tmp = Float64(Float64(1.0 * 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 ((t * a) <= -2e-120)
		tmp = ((abs(z) / sqrt(-(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[(t * a), $MachinePrecision], -2e-120], N[(N[(N[(N[Abs[z], $MachinePrecision] / N[Sqrt[(-N[(a * t), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(1.0 * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 60.7%

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

   3. Applied rewrites 63.8%

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

   4. Taylor expanded in z around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 31.6%

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

   6. Applied rewrites 31.6%

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

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

   1. Initial program 60.7%

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

   3. Applied rewrites 63.8%

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

   4. Taylor expanded in z around inf

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

   6. Applied rewrites 42.7%

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

Alternative 5: 76.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ t_2 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \left(\mathsf{copysign}\left(1, z\right) \cdot \begin{array}{l} \mathbf{if}\;t \cdot a \leq -2 \cdot 10^{-120}:\\ \;\;\;\;\frac{t\_1 \cdot \left|z\right|}{\sqrt{-a \cdot t}} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot t\_1\right) \cdot t\_2\\ \end{array}\right)\right) \end{array} \]

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 (<= (* t a) -2e-120)
       (* (/ (* 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 ((t * a) <= -2e-120) {
		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 ((t * a) <= -2e-120) {
		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 (t * a) <= -2e-120:
		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 (Float64(t * a) <= -2e-120)
		tmp = Float64(Float64(Float64(t_1 * abs(z)) / sqrt(Float64(-Float64(a * t)))) * t_2);
	else
		tmp = Float64(Float64(1.0 * 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 ((t * a) <= -2e-120)
		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[(t * a), $MachinePrecision], -2e-120], N[(N[(N[(t$95$1 * N[Abs[z], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-N[(a * t), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(1.0 * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 60.7%

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

   3. Applied rewrites 63.8%

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

   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 31.6%

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

   6. Applied rewrites 31.6%

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

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

   1. Initial program 60.7%

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

   3. Applied rewrites 63.8%

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

   4. Taylor expanded in z around inf

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

   6. Applied rewrites 42.7%

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

Alternative 6: 75.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(x, \left|y\right|\right)\\ t_2 := \mathsf{max}\left(x, \left|y\right|\right)\\ \mathsf{copysign}\left(1, y\right) \cdot \left(\mathsf{copysign}\left(1, z\right) \cdot \begin{array}{l} \mathbf{if}\;t \cdot a \leq -2 \cdot 10^{-120}:\\ \;\;\;\;\frac{t\_1 \cdot \left(t\_2 \cdot \left|z\right|\right)}{\sqrt{-a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot t\_2\right) \cdot t\_1\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fmin x (fabs y))) (t_2 (fmax x (fabs y))))
  (*
   (copysign 1.0 y)
   (*
    (copysign 1.0 z)
    (if (<= (* t a) -2e-120)
      (/ (* 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(x, fabs(y));
	double t_2 = fmax(x, fabs(y));
	double tmp;
	if ((t * a) <= -2e-120) {
		tmp = (t_1 * (t_2 * fabs(z))) / sqrt(-(a * t));
	} else {
		tmp = (1.0 * t_2) * t_1;
	}
	return 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(x, Math.abs(y));
	double t_2 = fmax(x, Math.abs(y));
	double tmp;
	if ((t * a) <= -2e-120) {
		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, y) * (Math.copySign(1.0, z) * tmp);
}

Python

def code(x, y, z, t, a):
	t_1 = fmin(x, math.fabs(y))
	t_2 = fmax(x, math.fabs(y))
	tmp = 0
	if (t * a) <= -2e-120:
		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, y) * (math.copysign(1.0, z) * tmp)

Julia

function code(x, y, z, t, a)
	t_1 = fmin(x, abs(y))
	t_2 = fmax(x, abs(y))
	tmp = 0.0
	if (Float64(t * a) <= -2e-120)
		tmp = Float64(Float64(t_1 * Float64(t_2 * abs(z))) / sqrt(Float64(-Float64(a * t))));
	else
		tmp = Float64(Float64(1.0 * t_2) * t_1);
	end
	return 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(x, abs(y));
	t_2 = max(x, abs(y));
	tmp = 0.0;
	if ((t * a) <= -2e-120)
		tmp = (t_1 * (t_2 * abs(z))) / sqrt(-(a * t));
	else
		tmp = (1.0 * t_2) * t_1;
	end
	tmp_2 = (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[x, N[Abs[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Max[x, N[Abs[y], $MachinePrecision]], $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], -2e-120], N[(N[(t$95$1 * N[(t$95$2 * N[Abs[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-N[(a * t), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 60.7%

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

   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 31.6%

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

   4. Applied rewrites 31.6%

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

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

   1. Initial program 60.7%

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

   3. Applied rewrites 63.8%

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

   4. Taylor expanded in z around inf

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

   6. Applied rewrites 42.7%

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

Alternative 7: 75.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{min}\left(\left|x\right|, \left|y\right|\right)\\ t_2 := \mathsf{max}\left(\left|x\right|, \left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \left(\mathsf{copysign}\left(1, y\right) \cdot \left(\mathsf{copysign}\left(1, z\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z\right| \leq 1.55 \cdot 10^{-220}:\\ \;\;\;\;\frac{\left|z\right| \cdot t\_1}{-\left|z\right|} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot t\_2\right) \cdot t\_1\\ \end{array}\right)\right) \end{array} \]

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) 1.55e-220)
       (* (/ (* (fabs z) t_1) (- (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) <= 1.55e-220) {
		tmp = ((fabs(z) * t_1) / -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) <= 1.55e-220) {
		tmp = ((Math.abs(z) * t_1) / -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) <= 1.55e-220:
		tmp = ((math.fabs(z) * t_1) / -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) <= 1.55e-220)
		tmp = Float64(Float64(Float64(abs(z) * t_1) / Float64(-abs(z))) * t_2);
	else
		tmp = Float64(Float64(1.0 * 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) <= 1.55e-220)
		tmp = ((abs(z) * t_1) / -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], 1.55e-220], N[(N[(N[(N[Abs[z], $MachinePrecision] * t$95$1), $MachinePrecision] / (-N[Abs[z], $MachinePrecision])), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(1.0 * t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 1.5500000000000001e-220

   1. Initial program 60.7%

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

   2. Taylor expanded in z around -inf

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

      1. lower-*.f64 42.3%

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

   4. Applied rewrites 42.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 41.4%

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 41.4%

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

      15. lift-*.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 41.4%

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

   6. Applied rewrites 41.4%

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

   if 1.5500000000000001e-220 < z

   1. Initial program 60.7%

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

   3. Applied rewrites 63.8%

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

   4. Taylor expanded in z around inf

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

   6. Applied rewrites 42.7%

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

Alternative 8: 73.1% accurate, 1.7× speedup

Math

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

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(Float64(1.0 * 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[(N[(1.0 * y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.7%

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

3. Applied rewrites 63.8%

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

4. Taylor expanded in z around inf

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

6. Applied rewrites 42.7%

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

Alternative 9: 44.0% accurate, 4.3× 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 60.7%

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

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 44.0%

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

4. Applied rewrites 44.0%

   \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right)} \]
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. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

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

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

   8. lower-*.f64 N/A

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

   9. lower-neg.f64 44.0%

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

6. Applied rewrites 44.0%

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

Reproduce

herbie shell --seed 2025212 
(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)))))