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

Percentage Accurate: 61.5% → 90.6%

Time: 4.0s

Alternatives: 7

Speedup: 7.5×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2 \cdot 10^{+103}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z\_m}{\sqrt{\mathsf{fma}\left(z\_m, z\_m, \left(-t\right) \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t a)
 :precision binary64
 (*
  z_s
  (if (<= z_m 2e+103)
    (* (* y x) (/ z_m (sqrt (fma z_m z_m (* (- t) a)))))
    (* y x))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2e+103) {
		tmp = (y * x) * (z_m / sqrt(fma(z_m, z_m, (-t * a))));
	} else {
		tmp = y * x;
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2e+103)
		tmp = Float64(Float64(y * x) * Float64(z_m / sqrt(fma(z_m, z_m, Float64(Float64(-t) * a)))));
	else
		tmp = Float64(y * x);
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * If[LessEqual[z$95$m, 2e+103], N[(N[(y * x), $MachinePrecision] * N[(z$95$m / N[Sqrt[N[(z$95$m * z$95$m + N[((-t) * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2e103

   1. Initial program 84.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 83.1

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

   3. Applied rewrites 83.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. pow2 N/A

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

      18. *-commutative N/A

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

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

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

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

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

      21. mul-1-neg N/A

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

      22. +-commutative N/A

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

      23. *-commutative N/A

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

   5. Applied rewrites 86.0%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. pow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. *-commutative N/A

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

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

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

      16. lift-neg.f64 N/A

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

      17. lift-*.f64 86.2

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

   7. Applied rewrites 86.2%

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

   if 2e103 < z

   1. Initial program 30.0%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 96.6

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

   4. Applied rewrites 96.6%

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

Alternative 2: 74.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \left(x \cdot y\right) \cdot z\_m\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{z\_m \cdot z\_m - t \cdot a}} \leq 5 \cdot 10^{-306}:\\ \;\;\;\;\frac{t\_1}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t a)
 :precision binary64
 (let* ((t_1 (* (* x y) z_m)))
   (*
    z_s
    (if (<= (/ t_1 (sqrt (- (* z_m z_m) (* t a)))) 5e-306)
      (/ t_1 z_m)
      (* y x)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	double t_1 = (x * y) * z_m;
	double tmp;
	if ((t_1 / sqrt(((z_m * z_m) - (t * a)))) <= 5e-306) {
		tmp = t_1 / z_m;
	} else {
		tmp = y * x;
	}
	return z_s * tmp;
}

Fortran

z\_m =     private
z\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z_s, x, y, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * y) * z_m
    if ((t_1 / sqrt(((z_m * z_m) - (t * a)))) <= 5d-306) then
        tmp = t_1 / z_m
    else
        tmp = y * x
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t, double a) {
	double t_1 = (x * y) * z_m;
	double tmp;
	if ((t_1 / Math.sqrt(((z_m * z_m) - (t * a)))) <= 5e-306) {
		tmp = t_1 / z_m;
	} else {
		tmp = y * x;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t, a):
	t_1 = (x * y) * z_m
	tmp = 0
	if (t_1 / math.sqrt(((z_m * z_m) - (t * a)))) <= 5e-306:
		tmp = t_1 / z_m
	else:
		tmp = y * x
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t, a)
	t_1 = Float64(Float64(x * y) * z_m)
	tmp = 0.0
	if (Float64(t_1 / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))) <= 5e-306)
		tmp = Float64(t_1 / z_m);
	else
		tmp = Float64(y * x);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t, a)
	t_1 = (x * y) * z_m;
	tmp = 0.0;
	if ((t_1 / sqrt(((z_m * z_m) - (t * a)))) <= 5e-306)
		tmp = t_1 / z_m;
	else
		tmp = y * x;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] * z$95$m), $MachinePrecision]}, N[(z$95$s * If[LessEqual[N[(t$95$1 / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e-306], N[(t$95$1 / z$95$m), $MachinePrecision], N[(y * x), $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)))) < 4.99999999999999998e-306

   1. Initial program 66.7%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 73.6%

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

   if 4.99999999999999998e-306 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))))

   1. Initial program 53.4%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 75.3

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

   4. Applied rewrites 75.3%

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

Alternative 3: 88.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 9 \cdot 10^{+49}:\\ \;\;\;\;z\_m \cdot \left(y \cdot \frac{x}{\sqrt{\mathsf{fma}\left(a, -t, z\_m \cdot z\_m\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t a)
 :precision binary64
 (*
  z_s
  (if (<= z_m 9e+49)
    (* z_m (* y (/ x (sqrt (fma a (- t) (* z_m z_m))))))
    (* y x))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 9e+49) {
		tmp = z_m * (y * (x / sqrt(fma(a, -t, (z_m * z_m)))));
	} else {
		tmp = y * x;
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 9e+49)
		tmp = Float64(z_m * Float64(y * Float64(x / sqrt(fma(a, Float64(-t), Float64(z_m * z_m))))));
	else
		tmp = Float64(y * x);
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * If[LessEqual[z$95$m, 9e+49], N[(z$95$m * N[(y * N[(x / N[Sqrt[N[(a * (-t) + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 8.99999999999999965e49

   1. Initial program 83.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 82.9

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

   3. Applied rewrites 82.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. pow2 N/A

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

      18. *-commutative N/A

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

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

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

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

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

      21. mul-1-neg N/A

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

      22. +-commutative N/A

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

      23. *-commutative N/A

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

   5. Applied rewrites 84.6%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. pow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. *-commutative N/A

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

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

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

      16. lift-neg.f64 N/A

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

      17. lift-*.f64 84.8

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

   7. Applied rewrites 84.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. pow2 N/A

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

   9. Applied rewrites 81.8%

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

   if 8.99999999999999965e49 < z

   1. Initial program 40.9%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 94.6

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

   4. Applied rewrites 94.6%

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

Alternative 4: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.9 \cdot 10^{-149}:\\ \;\;\;\;\left(z\_m \cdot y\right) \cdot \frac{x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t a)
 :precision binary64
 (* z_s (if (<= z_m 1.9e-149) (* (* z_m y) (/ x (sqrt (* a (- t))))) (* y x))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.9e-149) {
		tmp = (z_m * y) * (x / sqrt((a * -t)));
	} else {
		tmp = y * x;
	}
	return z_s * tmp;
}

Fortran

z\_m =     private
z\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z_s, x, y, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.9d-149) then
        tmp = (z_m * y) * (x / sqrt((a * -t)))
    else
        tmp = y * x
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.9e-149) {
		tmp = (z_m * y) * (x / Math.sqrt((a * -t)));
	} else {
		tmp = y * x;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t, a):
	tmp = 0
	if z_m <= 1.9e-149:
		tmp = (z_m * y) * (x / math.sqrt((a * -t)))
	else:
		tmp = y * x
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.9e-149)
		tmp = Float64(Float64(z_m * y) * Float64(x / sqrt(Float64(a * Float64(-t)))));
	else
		tmp = Float64(y * x);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.9e-149)
		tmp = (z_m * y) * (x / sqrt((a * -t)));
	else
		tmp = y * x;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * If[LessEqual[z$95$m, 1.9e-149], N[(N[(z$95$m * y), $MachinePrecision] * N[(x / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.90000000000000003e-149

   1. Initial program 73.4%

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

   2. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 71.5

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

   4. Applied rewrites 71.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 72.6

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

   6. Applied rewrites 72.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 71.6

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

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. lift-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 71.6

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

   8. Applied rewrites 71.6%

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

   if 1.90000000000000003e-149 < z

   1. Initial program 58.8%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 82.6

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

   4. Applied rewrites 82.6%

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

Alternative 5: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.9 \cdot 10^{-149}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z\_m}{\sqrt{\left(-t\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t a)
 :precision binary64
 (* z_s (if (<= z_m 1.9e-149) (* (* y x) (/ z_m (sqrt (* (- t) a)))) (* y x))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.9e-149) {
		tmp = (y * x) * (z_m / sqrt((-t * a)));
	} else {
		tmp = y * x;
	}
	return z_s * tmp;
}

Fortran

z\_m =     private
z\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z_s, x, y, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.9d-149) then
        tmp = (y * x) * (z_m / sqrt((-t * a)))
    else
        tmp = y * x
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.9e-149) {
		tmp = (y * x) * (z_m / Math.sqrt((-t * a)));
	} else {
		tmp = y * x;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t, a):
	tmp = 0
	if z_m <= 1.9e-149:
		tmp = (y * x) * (z_m / math.sqrt((-t * a)))
	else:
		tmp = y * x
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.9e-149)
		tmp = Float64(Float64(y * x) * Float64(z_m / sqrt(Float64(Float64(-t) * a))));
	else
		tmp = Float64(y * x);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.9e-149)
		tmp = (y * x) * (z_m / sqrt((-t * a)));
	else
		tmp = y * x;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * If[LessEqual[z$95$m, 1.9e-149], N[(N[(y * x), $MachinePrecision] * N[(z$95$m / N[Sqrt[N[((-t) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.90000000000000003e-149

   1. Initial program 73.4%

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

   2. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 71.5

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

   4. Applied rewrites 71.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 70.8

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

      9. lift-neg.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-pow2 N/A

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

      18. *-commutative N/A

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

      19. sqrt-pow2 N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. *-commutative N/A

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

      23. mul-1-neg N/A

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

   6. Applied rewrites 70.8%

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

   if 1.90000000000000003e-149 < z

   1. Initial program 58.8%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 82.6

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

   4. Applied rewrites 82.6%

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

Alternative 6: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.9 \cdot 10^{-149}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z\_m}{\sqrt{\left(-t\right) \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t a)
 :precision binary64
 (* z_s (if (<= z_m 1.9e-149) (* y (* x (/ z_m (sqrt (* (- t) a))))) (* y x))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.9e-149) {
		tmp = y * (x * (z_m / sqrt((-t * a))));
	} else {
		tmp = y * x;
	}
	return z_s * tmp;
}

Fortran

z\_m =     private
z\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z_s, x, y, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.9d-149) then
        tmp = y * (x * (z_m / sqrt((-t * a))))
    else
        tmp = y * x
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.9e-149) {
		tmp = y * (x * (z_m / Math.sqrt((-t * a))));
	} else {
		tmp = y * x;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t, a):
	tmp = 0
	if z_m <= 1.9e-149:
		tmp = y * (x * (z_m / math.sqrt((-t * a))))
	else:
		tmp = y * x
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.9e-149)
		tmp = Float64(y * Float64(x * Float64(z_m / sqrt(Float64(Float64(-t) * a)))));
	else
		tmp = Float64(y * x);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.9e-149)
		tmp = y * (x * (z_m / sqrt((-t * a))));
	else
		tmp = y * x;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * If[LessEqual[z$95$m, 1.9e-149], N[(y * N[(x * N[(z$95$m / N[Sqrt[N[((-t) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.90000000000000003e-149

   1. Initial program 73.4%

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

   2. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 71.5

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

   4. Applied rewrites 71.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 70.8

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

      9. lift-neg.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-pow2 N/A

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

      18. *-commutative N/A

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

      19. sqrt-pow2 N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. *-commutative N/A

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

      23. mul-1-neg N/A

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

   6. Applied rewrites 70.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 70.1

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

      6. pow2 70.1

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

      7. fp-cancel-sub-sign-inv 70.1

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

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-neg.f64 70.1

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

      13. pow2 70.1

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

   8. Applied rewrites 70.1%

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

   if 1.90000000000000003e-149 < z

   1. Initial program 58.8%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 82.6

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

   4. Applied rewrites 82.6%

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

Alternative 7: 72.1% accurate, 7.5× speedup

Math

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

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t a) :precision binary64 (* z_s (* y x)))

C

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

Fortran

z\_m =     private
z\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp = code(z_s, x, y, z_m, t, a)
	tmp = z_s * (y * x);
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(y * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.5%

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

2. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 72.1

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

4. Applied rewrites 72.1%

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

Developer Target 1: 88.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025093 
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"
  :precision binary64

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

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