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

Percentage Accurate: 60.7% → 91.9%

Time: 11.1s

Alternatives: 11

Speedup: 7.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 60.7% accurate, 1.0× speedup

Math

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

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

real(8) function code(x, y, z, t, a)
    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: 91.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5.6 \cdot 10^{+97}:\\ \;\;\;\;x\_m \cdot \left(\left(y\_m \cdot z\_m\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-\log \left(\frac{\mathsf{fma}\left(-0.5 \cdot t, \frac{a}{z\_m}, z\_m\right)}{z\_m}\right)} \cdot y\_m\right) \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 5.6e+97)
      (* x_m (* (* y_m z_m) (sqrt (/ 1.0 (fma (- a) t (* z_m z_m))))))
      (*
       (* (exp (- (log (/ (fma (* -0.5 t) (/ a z_m) z_m) z_m)))) y_m)
       x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 5.6e+97) {
		tmp = x_m * ((y_m * z_m) * sqrt((1.0 / fma(-a, t, (z_m * z_m)))));
	} else {
		tmp = (exp(-log((fma((-0.5 * t), (a / z_m), z_m) / z_m))) * y_m) * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 5.6e+97)
		tmp = Float64(x_m * Float64(Float64(y_m * z_m) * sqrt(Float64(1.0 / fma(Float64(-a), t, Float64(z_m * z_m))))));
	else
		tmp = Float64(Float64(exp(Float64(-log(Float64(fma(Float64(-0.5 * t), Float64(a / z_m), z_m) / z_m)))) * y_m) * x_m);
	end
	return Float64(x_s * Float64(y_s * 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]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 5.6e+97], N[(x$95$m * N[(N[(y$95$m * z$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 / N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[(-N[Log[N[(N[(N[(-0.5 * t), $MachinePrecision] * N[(a / z$95$m), $MachinePrecision] + z$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]], $MachinePrecision])], $MachinePrecision] * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 5.5999999999999998e97

   1. Initial program 65.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 69.6%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.0

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

   7. Applied rewrites 67.0%

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

   if 5.5999999999999998e97 < z

   1. Initial program 26.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 32.7%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.8

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

   7. Applied rewrites 92.8%

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

   9. Applied rewrites 96.5%

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

       1. lift-/.f64 N/A

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

       2. clear-num N/A

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

       3. inv-pow N/A

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

       4. pow-to-exp N/A

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

       5. lower-exp.f64 N/A

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

       6. lower-*.f64 N/A

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

   11. Applied rewrites 96.5%

       \[\leadsto \left(\color{blue}{e^{\log \left(\frac{\mathsf{fma}\left(-0.5 \cdot t, \frac{a}{z}, z\right)}{z}\right) \cdot -1}} \cdot y\right) \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.6 \cdot 10^{+97}:\\ \;\;\;\;x \cdot \left(\left(y \cdot z\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-a, t, z \cdot z\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-\log \left(\frac{\mathsf{fma}\left(-0.5 \cdot t, \frac{a}{z}, z\right)}{z}\right)} \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 6.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{x\_m}{\sqrt{t \cdot \left(-a\right)}} \cdot \left(y\_m \cdot z\_m\right)\\ \mathbf{elif}\;z\_m \leq 6.8 \cdot 10^{+128}:\\ \;\;\;\;\frac{y\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}} \cdot \left(x\_m \cdot z\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z\_m}{\mathsf{fma}\left(\frac{a}{z\_m} \cdot t, -0.5, z\_m\right)} \cdot y\_m\right) \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 6.2e-114)
      (* (/ x_m (sqrt (* t (- a)))) (* y_m z_m))
      (if (<= z_m 6.8e+128)
        (* (/ y_m (sqrt (fma (- a) t (* z_m z_m)))) (* x_m z_m))
        (* (* (/ z_m (fma (* (/ a z_m) t) -0.5 z_m)) y_m) x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 6.2e-114) {
		tmp = (x_m / sqrt((t * -a))) * (y_m * z_m);
	} else if (z_m <= 6.8e+128) {
		tmp = (y_m / sqrt(fma(-a, t, (z_m * z_m)))) * (x_m * z_m);
	} else {
		tmp = ((z_m / fma(((a / z_m) * t), -0.5, z_m)) * y_m) * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 6.2e-114)
		tmp = Float64(Float64(x_m / sqrt(Float64(t * Float64(-a)))) * Float64(y_m * z_m));
	elseif (z_m <= 6.8e+128)
		tmp = Float64(Float64(y_m / sqrt(fma(Float64(-a), t, Float64(z_m * z_m)))) * Float64(x_m * z_m));
	else
		tmp = Float64(Float64(Float64(z_m / fma(Float64(Float64(a / z_m) * t), -0.5, z_m)) * y_m) * x_m);
	end
	return Float64(x_s * Float64(y_s * 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]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 6.2e-114], N[(N[(x$95$m / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 6.8e+128], N[(N[(y$95$m / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z$95$m / N[(N[(N[(a / z$95$m), $MachinePrecision] * t), $MachinePrecision] * -0.5 + z$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 6.2e-114

   1. Initial program 59.6%

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

   3. Taylor expanded in a around inf

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{-1 \cdot \left(a \cdot t\right)}}} \]
   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. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 44.2

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

   5. Applied rewrites 44.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 46.0

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

   7. Applied rewrites 46.0%

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

   if 6.2e-114 < z < 6.7999999999999997e128

   1. Initial program 90.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 90.1

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

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

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 90.1

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

   4. Applied rewrites 90.1%

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

   if 6.7999999999999997e128 < z

   1. Initial program 19.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 24.1%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 91.9

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

   7. Applied rewrites 91.9%

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

   9. Applied rewrites 96.0%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{x}{\sqrt{t \cdot \left(-a\right)}} \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+128}:\\ \;\;\;\;\frac{y}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}} \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{\mathsf{fma}\left(\frac{a}{z} \cdot t, -0.5, z\right)} \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5.6 \cdot 10^{+97}:\\ \;\;\;\;x\_m \cdot \left(\left(y\_m \cdot z\_m\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z\_m}{\mathsf{fma}\left(\frac{a}{z\_m} \cdot t, -0.5, z\_m\right)} \cdot y\_m\right) \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 5.6e+97)
      (* x_m (* (* y_m z_m) (sqrt (/ 1.0 (fma (- a) t (* z_m z_m))))))
      (* (* (/ z_m (fma (* (/ a z_m) t) -0.5 z_m)) y_m) x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 5.6e+97) {
		tmp = x_m * ((y_m * z_m) * sqrt((1.0 / fma(-a, t, (z_m * z_m)))));
	} else {
		tmp = ((z_m / fma(((a / z_m) * t), -0.5, z_m)) * y_m) * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 5.6e+97)
		tmp = Float64(x_m * Float64(Float64(y_m * z_m) * sqrt(Float64(1.0 / fma(Float64(-a), t, Float64(z_m * z_m))))));
	else
		tmp = Float64(Float64(Float64(z_m / fma(Float64(Float64(a / z_m) * t), -0.5, z_m)) * y_m) * x_m);
	end
	return Float64(x_s * Float64(y_s * 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]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 5.6e+97], N[(x$95$m * N[(N[(y$95$m * z$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 / N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z$95$m / N[(N[(N[(a / z$95$m), $MachinePrecision] * t), $MachinePrecision] * -0.5 + z$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 5.5999999999999998e97

   1. Initial program 65.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 69.6%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.0

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

   7. Applied rewrites 67.0%

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

   if 5.5999999999999998e97 < z

   1. Initial program 26.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 32.7%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.8

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

   7. Applied rewrites 92.8%

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

   9. Applied rewrites 96.5%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.6 \cdot 10^{+97}:\\ \;\;\;\;x \cdot \left(\left(y \cdot z\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-a, t, z \cdot z\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{\mathsf{fma}\left(\frac{a}{z} \cdot t, -0.5, z\right)} \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 6.8 \cdot 10^{+128}:\\ \;\;\;\;\left(\frac{z\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}} \cdot y\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z\_m}{\mathsf{fma}\left(\frac{a}{z\_m} \cdot t, -0.5, z\_m\right)} \cdot y\_m\right) \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 6.8e+128)
      (* (* (/ z_m (sqrt (fma (- a) t (* z_m z_m)))) y_m) x_m)
      (* (* (/ z_m (fma (* (/ a z_m) t) -0.5 z_m)) y_m) x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 6.8e+128) {
		tmp = ((z_m / sqrt(fma(-a, t, (z_m * z_m)))) * y_m) * x_m;
	} else {
		tmp = ((z_m / fma(((a / z_m) * t), -0.5, z_m)) * y_m) * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 6.8e+128)
		tmp = Float64(Float64(Float64(z_m / sqrt(fma(Float64(-a), t, Float64(z_m * z_m)))) * y_m) * x_m);
	else
		tmp = Float64(Float64(Float64(z_m / fma(Float64(Float64(a / z_m) * t), -0.5, z_m)) * y_m) * x_m);
	end
	return Float64(x_s * Float64(y_s * 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]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 6.8e+128], N[(N[(N[(z$95$m / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(z$95$m / N[(N[(N[(a / z$95$m), $MachinePrecision] * t), $MachinePrecision] * -0.5 + z$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 6.7999999999999997e128

   1. Initial program 66.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 70.5%

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

   if 6.7999999999999997e128 < z

   1. Initial program 19.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 24.1%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 91.9

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

   7. Applied rewrites 91.9%

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

   9. Applied rewrites 96.0%

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

Alternative 5: 91.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4.2 \cdot 10^{+43}:\\ \;\;\;\;\frac{x\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}} \cdot \left(y\_m \cdot z\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z\_m}{\mathsf{fma}\left(\frac{a}{z\_m} \cdot t, -0.5, z\_m\right)} \cdot y\_m\right) \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 4.2e+43)
      (* (/ x_m (sqrt (fma (- a) t (* z_m z_m)))) (* y_m z_m))
      (* (* (/ z_m (fma (* (/ a z_m) t) -0.5 z_m)) y_m) x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 4.2e+43) {
		tmp = (x_m / sqrt(fma(-a, t, (z_m * z_m)))) * (y_m * z_m);
	} else {
		tmp = ((z_m / fma(((a / z_m) * t), -0.5, z_m)) * y_m) * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 4.2e+43)
		tmp = Float64(Float64(x_m / sqrt(fma(Float64(-a), t, Float64(z_m * z_m)))) * Float64(y_m * z_m));
	else
		tmp = Float64(Float64(Float64(z_m / fma(Float64(Float64(a / z_m) * t), -0.5, z_m)) * y_m) * x_m);
	end
	return Float64(x_s * Float64(y_s * 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]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 4.2e+43], N[(N[(x$95$m / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z$95$m / N[(N[(N[(a / z$95$m), $MachinePrecision] * t), $MachinePrecision] * -0.5 + z$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.20000000000000003e43

   1. Initial program 64.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 67.1

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 67.1

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

   4. Applied rewrites 67.1%

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

   if 4.20000000000000003e43 < z

   1. Initial program 33.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 38.9%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 90.5

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

   7. Applied rewrites 90.5%

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

   9. Applied rewrites 93.7%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.2 \cdot 10^{+43}:\\ \;\;\;\;\frac{x}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}} \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{\mathsf{fma}\left(\frac{a}{z} \cdot t, -0.5, z\right)} \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.3 \cdot 10^{-26}:\\ \;\;\;\;\frac{x\_m}{\sqrt{t \cdot \left(-a\right)}} \cdot \left(y\_m \cdot z\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z\_m}{\mathsf{fma}\left(\frac{a}{z\_m} \cdot t, -0.5, z\_m\right)} \cdot y\_m\right) \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 3.3e-26)
      (* (/ x_m (sqrt (* t (- a)))) (* y_m z_m))
      (* (* (/ z_m (fma (* (/ a z_m) t) -0.5 z_m)) y_m) x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.3e-26) {
		tmp = (x_m / sqrt((t * -a))) * (y_m * z_m);
	} else {
		tmp = ((z_m / fma(((a / z_m) * t), -0.5, z_m)) * y_m) * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 3.3e-26)
		tmp = Float64(Float64(x_m / sqrt(Float64(t * Float64(-a)))) * Float64(y_m * z_m));
	else
		tmp = Float64(Float64(Float64(z_m / fma(Float64(Float64(a / z_m) * t), -0.5, z_m)) * y_m) * x_m);
	end
	return Float64(x_s * Float64(y_s * 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]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 3.3e-26], N[(N[(x$95$m / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z$95$m / N[(N[(N[(a / z$95$m), $MachinePrecision] * t), $MachinePrecision] * -0.5 + z$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.2999999999999998e-26

   1. Initial program 62.7%

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

   3. Taylor expanded in a around inf

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{-1 \cdot \left(a \cdot t\right)}}} \]
   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. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 46.3

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

   5. Applied rewrites 46.3%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{\left(-a\right) \cdot t}}} \]
   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. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 48.4

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

   7. Applied rewrites 48.4%

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

   if 3.2999999999999998e-26 < z

   1. Initial program 43.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 48.4%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 88.0

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

   7. Applied rewrites 88.0%

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

   9. Applied rewrites 90.8%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.3 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{\sqrt{t \cdot \left(-a\right)}} \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{\mathsf{fma}\left(\frac{a}{z} \cdot t, -0.5, z\right)} \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.1 \cdot 10^{-36}:\\ \;\;\;\;\frac{x\_m}{\sqrt{t \cdot \left(-a\right)}} \cdot \left(y\_m \cdot z\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 2.1e-36)
      (* (/ x_m (sqrt (* t (- a)))) (* y_m z_m))
      (* x_m y_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.1e-36) {
		tmp = (x_m / sqrt((t * -a))) * (y_m * z_m);
	} else {
		tmp = x_m * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 2.1d-36) then
        tmp = (x_m / sqrt((t * -a))) * (y_m * z_m)
    else
        tmp = x_m * y_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.1e-36) {
		tmp = (x_m / Math.sqrt((t * -a))) * (y_m * z_m);
	} else {
		tmp = x_m * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 2.1e-36:
		tmp = (x_m / math.sqrt((t * -a))) * (y_m * z_m)
	else:
		tmp = x_m * y_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.1e-36)
		tmp = Float64(Float64(x_m / sqrt(Float64(t * Float64(-a)))) * Float64(y_m * z_m));
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2.1e-36)
		tmp = (x_m / sqrt((t * -a))) * (y_m * z_m);
	else
		tmp = x_m * y_m;
	end
	tmp_2 = x_s * (y_s * (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]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 2.1e-36], N[(N[(x$95$m / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.09999999999999991e-36

   1. Initial program 62.1%

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

   3. Taylor expanded in a around inf

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{-1 \cdot \left(a \cdot t\right)}}} \]
   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. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 46.0

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

   5. Applied rewrites 46.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 48.1

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

   7. Applied rewrites 48.1%

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

   if 2.09999999999999991e-36 < z

   1. Initial program 46.1%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 89.9

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

   5. Applied rewrites 89.9%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.1 \cdot 10^{-36}:\\ \;\;\;\;\frac{x}{\sqrt{t \cdot \left(-a\right)}} \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.1 \cdot 10^{-36}:\\ \;\;\;\;\frac{y\_m}{\sqrt{t \cdot \left(-a\right)}} \cdot \left(x\_m \cdot z\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 2.1e-36)
      (* (/ y_m (sqrt (* t (- a)))) (* x_m z_m))
      (* x_m y_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.1e-36) {
		tmp = (y_m / sqrt((t * -a))) * (x_m * z_m);
	} else {
		tmp = x_m * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 2.1d-36) then
        tmp = (y_m / sqrt((t * -a))) * (x_m * z_m)
    else
        tmp = x_m * y_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.1e-36) {
		tmp = (y_m / Math.sqrt((t * -a))) * (x_m * z_m);
	} else {
		tmp = x_m * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 2.1e-36:
		tmp = (y_m / math.sqrt((t * -a))) * (x_m * z_m)
	else:
		tmp = x_m * y_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.1e-36)
		tmp = Float64(Float64(y_m / sqrt(Float64(t * Float64(-a)))) * Float64(x_m * z_m));
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2.1e-36)
		tmp = (y_m / sqrt((t * -a))) * (x_m * z_m);
	else
		tmp = x_m * y_m;
	end
	tmp_2 = x_s * (y_s * (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]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 2.1e-36], N[(N[(y$95$m / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.09999999999999991e-36

   1. Initial program 62.1%

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

   3. Taylor expanded in a around inf

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{-1 \cdot \left(a \cdot t\right)}}} \]
   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. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 46.0

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

   5. Applied rewrites 46.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 46.9

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

   7. Applied rewrites 46.9%

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

   if 2.09999999999999991e-36 < z

   1. Initial program 46.1%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 89.9

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

   5. Applied rewrites 89.9%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.1 \cdot 10^{-36}:\\ \;\;\;\;\frac{y}{\sqrt{t \cdot \left(-a\right)}} \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 10^{-121}:\\ \;\;\;\;\frac{\left(x\_m \cdot z\_m\right) \cdot y\_m}{-z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (* z_s (if (<= z_m 1e-121) (/ (* (* x_m z_m) y_m) (- z_m)) (* x_m y_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1e-121) {
		tmp = ((x_m * z_m) * y_m) / -z_m;
	} else {
		tmp = x_m * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1d-121) then
        tmp = ((x_m * z_m) * y_m) / -z_m
    else
        tmp = x_m * y_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1e-121) {
		tmp = ((x_m * z_m) * y_m) / -z_m;
	} else {
		tmp = x_m * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1e-121:
		tmp = ((x_m * z_m) * y_m) / -z_m
	else:
		tmp = x_m * y_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1e-121)
		tmp = Float64(Float64(Float64(x_m * z_m) * y_m) / Float64(-z_m));
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1e-121)
		tmp = ((x_m * z_m) * y_m) / -z_m;
	else
		tmp = x_m * y_m;
	end
	tmp_2 = x_s * (y_s * (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]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 1e-121], N[(N[(N[(x$95$m * z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / (-z$95$m)), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 9.9999999999999998e-122

   1. Initial program 59.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 53.7

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

   5. Applied rewrites 53.7%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{-z}} \]
   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. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 51.5

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

   7. Applied rewrites 51.5%

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

   if 9.9999999999999998e-122 < z

   1. Initial program 54.5%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 81.5

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

   5. Applied rewrites 81.5%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{-121}:\\ \;\;\;\;\frac{\left(x \cdot z\right) \cdot y}{-z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.1 \cdot 10^{-121}:\\ \;\;\;\;\frac{y\_m}{z\_m} \cdot \left(x\_m \cdot z\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (* z_s (if (<= z_m 1.1e-121) (* (/ y_m z_m) (* x_m z_m)) (* x_m y_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.1e-121) {
		tmp = (y_m / z_m) * (x_m * z_m);
	} else {
		tmp = x_m * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.1d-121) then
        tmp = (y_m / z_m) * (x_m * z_m)
    else
        tmp = x_m * y_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.1e-121) {
		tmp = (y_m / z_m) * (x_m * z_m);
	} else {
		tmp = x_m * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.1e-121:
		tmp = (y_m / z_m) * (x_m * z_m)
	else:
		tmp = x_m * y_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.1e-121)
		tmp = Float64(Float64(y_m / z_m) * Float64(x_m * z_m));
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.1e-121)
		tmp = (y_m / z_m) * (x_m * z_m);
	else
		tmp = x_m * y_m;
	end
	tmp_2 = x_s * (y_s * (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]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 1.1e-121], N[(N[(y$95$m / z$95$m), $MachinePrecision] * N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.10000000000000011e-121

   1. Initial program 59.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 61.6

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

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

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 61.6

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

   4. Applied rewrites 61.6%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 16.4

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

   7. Applied rewrites 16.4%

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

   if 1.10000000000000011e-121 < z

   1. Initial program 54.5%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 81.5

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

   5. Applied rewrites 81.5%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.1 \cdot 10^{-121}:\\ \;\;\;\;\frac{y}{z} \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.5% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x_s * (y_s * (z_s * (x_m * y_m)))
end function

Java

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

Python

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

Julia

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

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = x_s * (y_s * (z_s * (x_m * y_m)));
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]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.5%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 37.5

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

5. Applied rewrites 37.5%

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

6. Final simplification 37.5%

   \[\leadsto x \cdot y \]
7. Add Preprocessing

Developer Target 1: 87.1% 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

real(8) function code(x, y, z, t, a)
    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 2024278 
(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)))))