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

Percentage Accurate: 61.4% → 89.4%

Time: 7.4s

Alternatives: 9

Speedup: 4.1×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.6 \cdot 10^{+133}:\\ \;\;\;\;\frac{\left(x \cdot y\right) \cdot z\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z\_m}{\mathsf{fma}\left(\frac{t}{z\_m}, -0.5 \cdot a, z\_m\right)}\right)\\ \end{array} \end{array} \]

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 3.6e+133)
    (/ (* (* x y) z_m) (sqrt (- (* z_m z_m) (* t a))))
    (* x (* y (/ z_m (fma (/ t z_m) (* -0.5 a) z_m)))))))

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 <= 3.6e+133) {
		tmp = ((x * y) * z_m) / sqrt(((z_m * z_m) - (t * a)));
	} else {
		tmp = x * (y * (z_m / fma((t / z_m), (-0.5 * a), z_m)));
	}
	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 <= 3.6e+133)
		tmp = Float64(Float64(Float64(x * y) * z_m) / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a))));
	else
		tmp = Float64(x * Float64(y * Float64(z_m / fma(Float64(t / z_m), Float64(-0.5 * a), z_m))));
	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, 3.6e+133], N[(N[(N[(x * y), $MachinePrecision] * z$95$m), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(z$95$m / N[(N[(t / z$95$m), $MachinePrecision] * N[(-0.5 * a), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.59999999999999978e133

   1. Initial program 72.2%

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

   if 3.59999999999999978e133 < z

   1. Initial program 26.3%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 65.8

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

   5. Applied rewrites 65.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

Alternative 2: 83.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.5 \cdot 10^{-148}:\\ \;\;\;\;\left(z\_m \cdot y\right) \cdot \frac{x}{\sqrt{\left(-t\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z\_m}{\mathsf{fma}\left(\frac{t}{z\_m}, -0.5 \cdot a, z\_m\right)}\right)\\ \end{array} \end{array} \]

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 1.5e-148)
    (* (* z_m y) (/ x (sqrt (* (- t) a))))
    (* x (* y (/ z_m (fma (/ t z_m) (* -0.5 a) z_m)))))))

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 <= 1.5e-148) {
		tmp = (z_m * y) * (x / sqrt((-t * a)));
	} else {
		tmp = x * (y * (z_m / fma((t / z_m), (-0.5 * a), z_m)));
	}
	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 <= 1.5e-148)
		tmp = Float64(Float64(z_m * y) * Float64(x / sqrt(Float64(Float64(-t) * a))));
	else
		tmp = Float64(x * Float64(y * Float64(z_m / fma(Float64(t / z_m), Float64(-0.5 * a), z_m))));
	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, 1.5e-148], N[(N[(z$95$m * y), $MachinePrecision] * N[(x / N[Sqrt[N[((-t) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(z$95$m / N[(N[(t / z$95$m), $MachinePrecision] * N[(-0.5 * a), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.49999999999999999e-148

   1. Initial program 64.8%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 42.2

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

   5. Applied rewrites 42.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 41.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 35.0

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

   9. Applied rewrites 35.0%

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

   if 1.49999999999999999e-148 < z

   1. Initial program 67.6%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 75.3

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

   5. Applied rewrites 75.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 88.9

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

   7. Applied rewrites 88.9%

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

Alternative 3: 75.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{\left(z\_m \cdot x\right) \cdot y}{-z\_m}\\ \mathbf{elif}\;z\_m \leq 1.8 \cdot 10^{-105}:\\ \;\;\;\;\frac{\left(x \cdot y\right) \cdot z\_m}{\sqrt{z\_m \cdot z\_m}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

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 1.6e-162)
    (/ (* (* z_m x) y) (- z_m))
    (if (<= z_m 1.8e-105)
      (/ (* (* x y) z_m) (sqrt (* z_m z_m)))
      (* 1.0 (* 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 <= 1.6e-162) {
		tmp = ((z_m * x) * y) / -z_m;
	} else if (z_m <= 1.8e-105) {
		tmp = ((x * y) * z_m) / sqrt((z_m * z_m));
	} else {
		tmp = 1.0 * (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 <= 1.6d-162) then
        tmp = ((z_m * x) * y) / -z_m
    else if (z_m <= 1.8d-105) then
        tmp = ((x * y) * z_m) / sqrt((z_m * z_m))
    else
        tmp = 1.0d0 * (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 <= 1.6e-162) {
		tmp = ((z_m * x) * y) / -z_m;
	} else if (z_m <= 1.8e-105) {
		tmp = ((x * y) * z_m) / Math.sqrt((z_m * z_m));
	} else {
		tmp = 1.0 * (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 <= 1.6e-162:
		tmp = ((z_m * x) * y) / -z_m
	elif z_m <= 1.8e-105:
		tmp = ((x * y) * z_m) / math.sqrt((z_m * z_m))
	else:
		tmp = 1.0 * (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 <= 1.6e-162)
		tmp = Float64(Float64(Float64(z_m * x) * y) / Float64(-z_m));
	elseif (z_m <= 1.8e-105)
		tmp = Float64(Float64(Float64(x * y) * z_m) / sqrt(Float64(z_m * z_m)));
	else
		tmp = Float64(1.0 * 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 <= 1.6e-162)
		tmp = ((z_m * x) * y) / -z_m;
	elseif (z_m <= 1.8e-105)
		tmp = ((x * y) * z_m) / sqrt((z_m * z_m));
	else
		tmp = 1.0 * (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, 1.6e-162], N[(N[(N[(z$95$m * x), $MachinePrecision] * y), $MachinePrecision] / (-z$95$m)), $MachinePrecision], If[LessEqual[z$95$m, 1.8e-105], N[(N[(N[(x * y), $MachinePrecision] * z$95$m), $MachinePrecision] / N[Sqrt[N[(z$95$m * z$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(y * x), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 1.59999999999999988e-162

   1. Initial program 63.7%

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

   3. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 64.5

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

   5. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 61.1

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

   7. Applied rewrites 61.1%

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

   if 1.59999999999999988e-162 < z < 1.79999999999999982e-105

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 91.4

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

   5. Applied rewrites 91.4%

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

   if 1.79999999999999982e-105 < z

   1. Initial program 65.3%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 73.6

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

   5. Applied rewrites 73.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 88.1%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 84.5%

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

Alternative 4: 82.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.1 \cdot 10^{-123}:\\ \;\;\;\;\left(z\_m \cdot y\right) \cdot \frac{x}{\sqrt{\left(-t\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

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 3.1e-123)
    (* (* z_m y) (/ x (sqrt (* (- t) a))))
    (* 1.0 (* 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 <= 3.1e-123) {
		tmp = (z_m * y) * (x / sqrt((-t * a)));
	} else {
		tmp = 1.0 * (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 <= 3.1d-123) then
        tmp = (z_m * y) * (x / sqrt((-t * a)))
    else
        tmp = 1.0d0 * (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 <= 3.1e-123) {
		tmp = (z_m * y) * (x / Math.sqrt((-t * a)));
	} else {
		tmp = 1.0 * (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 <= 3.1e-123:
		tmp = (z_m * y) * (x / math.sqrt((-t * a)))
	else:
		tmp = 1.0 * (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 <= 3.1e-123)
		tmp = Float64(Float64(z_m * y) * Float64(x / sqrt(Float64(Float64(-t) * a))));
	else
		tmp = Float64(1.0 * 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 <= 3.1e-123)
		tmp = (z_m * y) * (x / sqrt((-t * a)));
	else
		tmp = 1.0 * (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, 3.1e-123], N[(N[(z$95$m * y), $MachinePrecision] * N[(x / N[Sqrt[N[((-t) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(y * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.09999999999999998e-123

   1. Initial program 65.0%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 42.6

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

   5. Applied rewrites 42.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 42.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 35.4

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

   9. Applied rewrites 35.4%

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

   if 3.09999999999999998e-123 < z

   1. Initial program 67.3%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 75.0

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

   5. Applied rewrites 75.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 88.7%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 84.3%

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

Alternative 5: 82.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.1 \cdot 10^{-123}:\\ \;\;\;\;x \cdot \frac{z\_m \cdot y}{\sqrt{\left(-t\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

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 3.1e-123)
    (* x (/ (* z_m y) (sqrt (* (- t) a))))
    (* 1.0 (* 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 <= 3.1e-123) {
		tmp = x * ((z_m * y) / sqrt((-t * a)));
	} else {
		tmp = 1.0 * (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 <= 3.1d-123) then
        tmp = x * ((z_m * y) / sqrt((-t * a)))
    else
        tmp = 1.0d0 * (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 <= 3.1e-123) {
		tmp = x * ((z_m * y) / Math.sqrt((-t * a)));
	} else {
		tmp = 1.0 * (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 <= 3.1e-123:
		tmp = x * ((z_m * y) / math.sqrt((-t * a)))
	else:
		tmp = 1.0 * (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 <= 3.1e-123)
		tmp = Float64(x * Float64(Float64(z_m * y) / sqrt(Float64(Float64(-t) * a))));
	else
		tmp = Float64(1.0 * 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 <= 3.1e-123)
		tmp = x * ((z_m * y) / sqrt((-t * a)));
	else
		tmp = 1.0 * (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, 3.1e-123], N[(x * N[(N[(z$95$m * y), $MachinePrecision] / N[Sqrt[N[((-t) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(y * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.09999999999999998e-123

   1. Initial program 65.0%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 42.6

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

   5. Applied rewrites 42.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 37.2

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

   7. Applied rewrites 37.2%

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

   if 3.09999999999999998e-123 < z

   1. Initial program 67.3%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 75.0

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

   5. Applied rewrites 75.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 88.7%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 84.3%

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

Alternative 6: 74.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.22 \cdot 10^{-206}:\\ \;\;\;\;\frac{\left(z\_m \cdot x\right) \cdot y}{-z\_m}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

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 1.22e-206) (/ (* (* z_m x) y) (- z_m)) (* 1.0 (* 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 <= 1.22e-206) {
		tmp = ((z_m * x) * y) / -z_m;
	} else {
		tmp = 1.0 * (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 <= 1.22d-206) then
        tmp = ((z_m * x) * y) / -z_m
    else
        tmp = 1.0d0 * (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 <= 1.22e-206) {
		tmp = ((z_m * x) * y) / -z_m;
	} else {
		tmp = 1.0 * (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 <= 1.22e-206:
		tmp = ((z_m * x) * y) / -z_m
	else:
		tmp = 1.0 * (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 <= 1.22e-206)
		tmp = Float64(Float64(Float64(z_m * x) * y) / Float64(-z_m));
	else
		tmp = Float64(1.0 * 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 <= 1.22e-206)
		tmp = ((z_m * x) * y) / -z_m;
	else
		tmp = 1.0 * (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, 1.22e-206], N[(N[(N[(z$95$m * x), $MachinePrecision] * y), $MachinePrecision] / (-z$95$m)), $MachinePrecision], N[(1.0 * N[(y * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.22000000000000002e-206

   1. Initial program 62.5%

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

   3. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 65.9

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

   5. Applied rewrites 65.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 62.3

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

   7. Applied rewrites 62.3%

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

   if 1.22000000000000002e-206 < z

   1. Initial program 70.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 73.1

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

   5. Applied rewrites 73.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 84.3%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 79.5%

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

Alternative 7: 73.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4.9 \cdot 10^{-138}:\\ \;\;\;\;\left(z\_m \cdot x\right) \cdot \frac{y}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

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 4.9e-138) (* (* z_m x) (/ y z_m)) (* 1.0 (* 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 <= 4.9e-138) {
		tmp = (z_m * x) * (y / z_m);
	} else {
		tmp = 1.0 * (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 <= 4.9d-138) then
        tmp = (z_m * x) * (y / z_m)
    else
        tmp = 1.0d0 * (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 <= 4.9e-138) {
		tmp = (z_m * x) * (y / z_m);
	} else {
		tmp = 1.0 * (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 <= 4.9e-138:
		tmp = (z_m * x) * (y / z_m)
	else:
		tmp = 1.0 * (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 <= 4.9e-138)
		tmp = Float64(Float64(z_m * x) * Float64(y / z_m));
	else
		tmp = Float64(1.0 * 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 <= 4.9e-138)
		tmp = (z_m * x) * (y / z_m);
	else
		tmp = 1.0 * (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, 4.9e-138], N[(N[(z$95$m * x), $MachinePrecision] * N[(y / z$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(y * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.90000000000000016e-138

   1. Initial program 64.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 23.1

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

   5. Applied rewrites 23.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 23.1

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

   7. Applied rewrites 23.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l/ N/A

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

   9. Applied rewrites 23.2%

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

   10. Taylor expanded in z around inf

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

       1. lower-/.f64 16.9

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

   12. Applied rewrites 16.9%

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

   if 4.90000000000000016e-138 < z

   1. Initial program 67.6%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 75.3

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

   5. Applied rewrites 75.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 88.9%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 84.5%

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

Alternative 8: 73.0% accurate, 4.1× speedup

Math

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

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 (* 1.0 (* 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 * (1.0 * (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 * (1.0d0 * (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 * (1.0 * (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 * (1.0 * (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(1.0 * 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 * (1.0 * (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[(1.0 * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 41.7

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

5. Applied rewrites 41.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

7. Applied rewrites 46.5%

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

8. Taylor expanded in z around inf

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

10. Applied rewrites 39.9%

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

Alternative 9: 14.1% accurate, 5.6× speedup

Math

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

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(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 65.8%

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

3. Taylor expanded in z around -inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 47.8

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

5. Applied rewrites 47.8%

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

Developer Target 1: 89.0% 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 2024359 
(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)))))