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

Percentage Accurate: 59.6% → 92.2%

Time: 11.4s

Alternatives: 13

Speedup: 2.6×

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: 59.6% 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: 92.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4 \cdot 10^{+25}:\\ \;\;\;\;x\_m \cdot \frac{y\_m \cdot z\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot y\_m}{\frac{\mathsf{fma}\left(\frac{a}{z\_m}, -0.5 \cdot t, z\_m\right)}{z\_m}}\\ \end{array}\right)\right) \end{array} \]

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 4e+25)
      (* x_m (/ (* y_m z_m) (sqrt (fma (- a) t (* z_m z_m)))))
      (/ (* x_m y_m) (/ (fma (/ a z_m) (* -0.5 t) z_m) z_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 <= 4e+25) {
		tmp = x_m * ((y_m * z_m) / sqrt(fma(-a, t, (z_m * z_m))));
	} else {
		tmp = (x_m * y_m) / (fma((a / z_m), (-0.5 * t), z_m) / z_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 <= 4e+25)
		tmp = Float64(x_m * Float64(Float64(y_m * z_m) / sqrt(fma(Float64(-a), t, Float64(z_m * z_m)))));
	else
		tmp = Float64(Float64(x_m * y_m) / Float64(fma(Float64(a / z_m), Float64(-0.5 * t), z_m) / z_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, 4e+25], N[(x$95$m * N[(N[(y$95$m * z$95$m), $MachinePrecision] / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[(N[(N[(a / z$95$m), $MachinePrecision] * N[(-0.5 * t), $MachinePrecision] + z$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.00000000000000036e25

   1. Initial program 71.0%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 73.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 71.9

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

   6. Applied rewrites 71.9%

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

   if 4.00000000000000036e25 < z

   1. Initial program 37.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 80.7

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

   5. Applied rewrites 80.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 98.5

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

   7. Applied rewrites 98.5%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{+25}:\\ \;\;\;\;x \cdot \frac{y \cdot z}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\mathsf{fma}\left(\frac{a}{z}, -0.5 \cdot t, z\right)}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 7 \cdot 10^{+25}:\\ \;\;\;\;x\_m \cdot \frac{y\_m \cdot z\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z\_m}{\mathsf{fma}\left(\frac{a}{z\_m}, -0.5 \cdot t, z\_m\right)} \cdot y\_m\right) \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

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 7e+25)
      (* x_m (/ (* y_m z_m) (sqrt (fma (- a) t (* z_m z_m)))))
      (* (* (/ z_m (fma (/ a z_m) (* -0.5 t) 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 <= 7e+25) {
		tmp = x_m * ((y_m * z_m) / sqrt(fma(-a, t, (z_m * z_m))));
	} else {
		tmp = ((z_m / fma((a / z_m), (-0.5 * t), 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 <= 7e+25)
		tmp = Float64(x_m * Float64(Float64(y_m * z_m) / sqrt(fma(Float64(-a), t, Float64(z_m * z_m)))));
	else
		tmp = Float64(Float64(Float64(z_m / fma(Float64(a / z_m), Float64(-0.5 * t), 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, 7e+25], N[(x$95$m * N[(N[(y$95$m * z$95$m), $MachinePrecision] / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z$95$m / N[(N[(a / z$95$m), $MachinePrecision] * N[(-0.5 * t), $MachinePrecision] + 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 < 6.99999999999999999e25

   1. Initial program 71.0%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 73.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 71.9

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

   6. Applied rewrites 71.9%

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

   if 6.99999999999999999e25 < z

   1. Initial program 37.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 80.7

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

   5. Applied rewrites 80.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 98.5%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7 \cdot 10^{+25}:\\ \;\;\;\;x \cdot \frac{y \cdot z}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{\mathsf{fma}\left(\frac{a}{z}, -0.5 \cdot t, z\right)} \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 6.8 \cdot 10^{+25}:\\ \;\;\;\;\frac{x\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}} \cdot \left(y\_m \cdot z\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z\_m}{\mathsf{fma}\left(\frac{a}{z\_m}, -0.5 \cdot t, z\_m\right)} \cdot y\_m\right) \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

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+25)
      (* (/ x_m (sqrt (fma (- a) t (* z_m z_m)))) (* y_m z_m))
      (* (* (/ z_m (fma (/ a z_m) (* -0.5 t) 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+25) {
		tmp = (x_m / sqrt(fma(-a, t, (z_m * z_m)))) * (y_m * z_m);
	} else {
		tmp = ((z_m / fma((a / z_m), (-0.5 * t), 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+25)
		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(a / z_m), Float64(-0.5 * t), 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+25], 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[(a / z$95$m), $MachinePrecision] * N[(-0.5 * t), $MachinePrecision] + 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 < 6.79999999999999967e25

   1. Initial program 71.0%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 73.0

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

      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 73.0

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

   4. Applied rewrites 73.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 70.2

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

   6. Applied rewrites 70.2%

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

   if 6.79999999999999967e25 < z

   1. Initial program 37.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 80.7

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

   5. Applied rewrites 80.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 98.5%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.8 \cdot 10^{+25}:\\ \;\;\;\;\frac{x}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}} \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{\mathsf{fma}\left(\frac{a}{z}, -0.5 \cdot t, z\right)} \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.65 \cdot 10^{+107}:\\ \;\;\;\;\left(\frac{y\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}} \cdot z\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z\_m}{\mathsf{fma}\left(\frac{a}{z\_m}, -0.5 \cdot t, z\_m\right)} \cdot y\_m\right) \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

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 2.65e+107)
      (* (* (/ y_m (sqrt (fma (- a) t (* z_m z_m)))) z_m) x_m)
      (* (* (/ z_m (fma (/ a z_m) (* -0.5 t) 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 <= 2.65e+107) {
		tmp = ((y_m / sqrt(fma(-a, t, (z_m * z_m)))) * z_m) * x_m;
	} else {
		tmp = ((z_m / fma((a / z_m), (-0.5 * t), 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 <= 2.65e+107)
		tmp = Float64(Float64(Float64(y_m / sqrt(fma(Float64(-a), t, Float64(z_m * z_m)))) * z_m) * x_m);
	else
		tmp = Float64(Float64(Float64(z_m / fma(Float64(a / z_m), Float64(-0.5 * t), 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, 2.65e+107], N[(N[(N[(y$95$m / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(z$95$m / N[(N[(a / z$95$m), $MachinePrecision] * N[(-0.5 * t), $MachinePrecision] + 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 < 2.65e107

   1. Initial program 71.8%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 73.1

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

      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 73.1

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

   4. Applied rewrites 73.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 73.1

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

   6. Applied rewrites 73.1%

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

   if 2.65e107 < z

   1. Initial program 27.1%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 79.9

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

   5. Applied rewrites 79.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 100.0%

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

Alternative 5: 88.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5.4 \cdot 10^{+35}:\\ \;\;\;\;\left(\frac{y\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}} \cdot x\_m\right) \cdot z\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z\_m}{\mathsf{fma}\left(\frac{a}{z\_m}, -0.5 \cdot t, z\_m\right)} \cdot y\_m\right) \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

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 5.4e+35)
      (* (* (/ y_m (sqrt (fma (- a) t (* z_m z_m)))) x_m) z_m)
      (* (* (/ z_m (fma (/ a z_m) (* -0.5 t) 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 <= 5.4e+35) {
		tmp = ((y_m / sqrt(fma(-a, t, (z_m * z_m)))) * x_m) * z_m;
	} else {
		tmp = ((z_m / fma((a / z_m), (-0.5 * t), 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 <= 5.4e+35)
		tmp = Float64(Float64(Float64(y_m / sqrt(fma(Float64(-a), t, Float64(z_m * z_m)))) * x_m) * z_m);
	else
		tmp = Float64(Float64(Float64(z_m / fma(Float64(a / z_m), Float64(-0.5 * t), 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, 5.4e+35], N[(N[(N[(y$95$m / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] * z$95$m), $MachinePrecision], N[(N[(N[(z$95$m / N[(N[(a / z$95$m), $MachinePrecision] * N[(-0.5 * t), $MachinePrecision] + 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 < 5.40000000000000005e35

   1. Initial program 71.0%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 73.0

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

      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 73.0

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

   4. Applied rewrites 73.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 70.3

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 70.6

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

   6. Applied rewrites 70.6%

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

   if 5.40000000000000005e35 < z

   1. Initial program 37.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 80.7

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

   5. Applied rewrites 80.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 98.5%

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

Alternative 6: 88.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5.4 \cdot 10^{+35}:\\ \;\;\;\;\left(\frac{x\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}} \cdot z\_m\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z\_m}{\mathsf{fma}\left(\frac{a}{z\_m}, -0.5 \cdot t, z\_m\right)} \cdot y\_m\right) \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

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 5.4e+35)
      (* (* (/ x_m (sqrt (fma (- a) t (* z_m z_m)))) z_m) y_m)
      (* (* (/ z_m (fma (/ a z_m) (* -0.5 t) 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 <= 5.4e+35) {
		tmp = ((x_m / sqrt(fma(-a, t, (z_m * z_m)))) * z_m) * y_m;
	} else {
		tmp = ((z_m / fma((a / z_m), (-0.5 * t), 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 <= 5.4e+35)
		tmp = Float64(Float64(Float64(x_m / sqrt(fma(Float64(-a), t, Float64(z_m * z_m)))) * z_m) * y_m);
	else
		tmp = Float64(Float64(Float64(z_m / fma(Float64(a / z_m), Float64(-0.5 * t), 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, 5.4e+35], N[(N[(N[(x$95$m / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(z$95$m / N[(N[(a / z$95$m), $MachinePrecision] * N[(-0.5 * t), $MachinePrecision] + 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 < 5.40000000000000005e35

   1. Initial program 71.0%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 73.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. associate-*r* N/A

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

      10. div-inv N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 71.8

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

   6. Applied rewrites 71.8%

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

   if 5.40000000000000005e35 < z

   1. Initial program 37.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 80.7

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

   5. Applied rewrites 80.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 98.5%

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

Alternative 7: 87.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.65 \cdot 10^{+107}:\\ \;\;\;\;\left(x\_m \cdot z\_m\right) \cdot \frac{y\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z\_m}{\mathsf{fma}\left(\frac{a}{z\_m}, -0.5 \cdot t, z\_m\right)} \cdot y\_m\right) \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

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 2.65e+107)
      (* (* x_m z_m) (/ y_m (sqrt (fma (- a) t (* z_m z_m)))))
      (* (* (/ z_m (fma (/ a z_m) (* -0.5 t) 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 <= 2.65e+107) {
		tmp = (x_m * z_m) * (y_m / sqrt(fma(-a, t, (z_m * z_m))));
	} else {
		tmp = ((z_m / fma((a / z_m), (-0.5 * t), 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 <= 2.65e+107)
		tmp = Float64(Float64(x_m * z_m) * Float64(y_m / sqrt(fma(Float64(-a), t, Float64(z_m * z_m)))));
	else
		tmp = Float64(Float64(Float64(z_m / fma(Float64(a / z_m), Float64(-0.5 * t), 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, 2.65e+107], N[(N[(x$95$m * z$95$m), $MachinePrecision] * N[(y$95$m / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z$95$m / N[(N[(a / z$95$m), $MachinePrecision] * N[(-0.5 * t), $MachinePrecision] + 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 < 2.65e107

   1. Initial program 71.8%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 73.1

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

      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 73.1

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

   4. Applied rewrites 73.1%

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

   if 2.65e107 < z

   1. Initial program 27.1%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 79.9

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

   5. Applied rewrites 79.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 100.0%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.65 \cdot 10^{+107}:\\ \;\;\;\;\left(x \cdot z\right) \cdot \frac{y}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{\mathsf{fma}\left(\frac{a}{z}, -0.5 \cdot t, z\right)} \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.7 \cdot 10^{-88}:\\ \;\;\;\;\frac{x\_m \cdot y\_m}{\sqrt{\left(-t\right) \cdot a}} \cdot z\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z\_m}{\mathsf{fma}\left(\frac{a}{z\_m}, -0.5 \cdot t, z\_m\right)} \cdot y\_m\right) \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

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 1.7e-88)
      (* (/ (* x_m y_m) (sqrt (* (- t) a))) z_m)
      (* (* (/ z_m (fma (/ a z_m) (* -0.5 t) 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 <= 1.7e-88) {
		tmp = ((x_m * y_m) / sqrt((-t * a))) * z_m;
	} else {
		tmp = ((z_m / fma((a / z_m), (-0.5 * t), 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 <= 1.7e-88)
		tmp = Float64(Float64(Float64(x_m * y_m) / sqrt(Float64(Float64(-t) * a))) * z_m);
	else
		tmp = Float64(Float64(Float64(z_m / fma(Float64(a / z_m), Float64(-0.5 * t), 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, 1.7e-88], N[(N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[Sqrt[N[((-t) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision], N[(N[(N[(z$95$m / N[(N[(a / z$95$m), $MachinePrecision] * N[(-0.5 * t), $MachinePrecision] + 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 < 1.69999999999999987e-88

   1. Initial program 67.6%

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

   3. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 38.5

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

   5. Applied rewrites 38.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 37.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 37.9

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

   7. Applied rewrites 37.9%

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

   8. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 47.9

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

   10. Applied rewrites 47.9%

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

   if 1.69999999999999987e-88 < z

   1. Initial program 53.2%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 81.7

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

   5. Applied rewrites 81.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 94.7%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.7 \cdot 10^{-88}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{\left(-t\right) \cdot a}} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{\mathsf{fma}\left(\frac{a}{z}, -0.5 \cdot t, z\right)} \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.55 \cdot 10^{-87}:\\ \;\;\;\;\frac{x\_m \cdot y\_m}{\sqrt{\left(-t\right) \cdot a}} \cdot z\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot y\_m}{1}\\ \end{array}\right)\right) \end{array} \]

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 2.55e-87)
      (* (/ (* x_m y_m) (sqrt (* (- t) a))) z_m)
      (/ (* x_m y_m) 1.0))))))

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 <= 2.55e-87) {
		tmp = ((x_m * y_m) / sqrt((-t * a))) * z_m;
	} else {
		tmp = (x_m * y_m) / 1.0;
	}
	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 <= 2.55d-87) then
        tmp = ((x_m * y_m) / sqrt((-t * a))) * z_m
    else
        tmp = (x_m * y_m) / 1.0d0
    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 <= 2.55e-87) {
		tmp = ((x_m * y_m) / Math.sqrt((-t * a))) * z_m;
	} else {
		tmp = (x_m * y_m) / 1.0;
	}
	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 <= 2.55e-87:
		tmp = ((x_m * y_m) / math.sqrt((-t * a))) * z_m
	else:
		tmp = (x_m * y_m) / 1.0
	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 <= 2.55e-87)
		tmp = Float64(Float64(Float64(x_m * y_m) / sqrt(Float64(Float64(-t) * a))) * z_m);
	else
		tmp = Float64(Float64(x_m * y_m) / 1.0);
	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 <= 2.55e-87)
		tmp = ((x_m * y_m) / sqrt((-t * a))) * z_m;
	else
		tmp = (x_m * y_m) / 1.0;
	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, 2.55e-87], N[(N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[Sqrt[N[((-t) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision], N[(N[(x$95$m * y$95$m), $MachinePrecision] / 1.0), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.55000000000000012e-87

   1. Initial program 67.6%

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

   3. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 38.5

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

   5. Applied rewrites 38.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 37.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 37.9

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

   7. Applied rewrites 37.9%

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

   8. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 47.9

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

   10. Applied rewrites 47.9%

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

   if 2.55000000000000012e-87 < z

   1. Initial program 53.2%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 81.7

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

   5. Applied rewrites 81.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 94.7

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

   7. Applied rewrites 94.7%

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

   8. Taylor expanded in z around inf

      \[\leadsto \frac{y \cdot x}{\color{blue}{1}} \]
   9. Step-by-step derivation

   10. Applied rewrites 92.8%

       \[\leadsto \frac{y \cdot x}{\color{blue}{1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.55 \cdot 10^{-87}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{\left(-t\right) \cdot a}} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.6 \cdot 10^{-114}:\\ \;\;\;\;\frac{\frac{x\_m \cdot x\_m}{x\_m} \cdot y\_m}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot y\_m}{1}\\ \end{array}\right)\right) \end{array} \]

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 3.6e-114)
      (/ (* (/ (* x_m x_m) x_m) y_m) 1.0)
      (/ (* x_m y_m) 1.0))))))

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 <= 3.6e-114) {
		tmp = (((x_m * x_m) / x_m) * y_m) / 1.0;
	} else {
		tmp = (x_m * y_m) / 1.0;
	}
	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 <= 3.6d-114) then
        tmp = (((x_m * x_m) / x_m) * y_m) / 1.0d0
    else
        tmp = (x_m * y_m) / 1.0d0
    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 <= 3.6e-114) {
		tmp = (((x_m * x_m) / x_m) * y_m) / 1.0;
	} else {
		tmp = (x_m * y_m) / 1.0;
	}
	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 <= 3.6e-114:
		tmp = (((x_m * x_m) / x_m) * y_m) / 1.0
	else:
		tmp = (x_m * y_m) / 1.0
	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 <= 3.6e-114)
		tmp = Float64(Float64(Float64(Float64(x_m * x_m) / x_m) * y_m) / 1.0);
	else
		tmp = Float64(Float64(x_m * y_m) / 1.0);
	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 <= 3.6e-114)
		tmp = (((x_m * x_m) / x_m) * y_m) / 1.0;
	else
		tmp = (x_m * y_m) / 1.0;
	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, 3.6e-114], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / 1.0), $MachinePrecision], N[(N[(x$95$m * y$95$m), $MachinePrecision] / 1.0), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.60000000000000018e-114

   1. Initial program 67.2%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 30.7

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

   5. Applied rewrites 30.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 31.3

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

   7. Applied rewrites 31.3%

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

   8. Taylor expanded in z around inf

      \[\leadsto \frac{y \cdot x}{\color{blue}{1}} \]
   9. Step-by-step derivation

   10. Applied rewrites 21.7%

       \[\leadsto \frac{y \cdot x}{\color{blue}{1}} \]
   11. Step-by-step derivation

       1. *-rgt-identity N/A

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

       2. *-inverses N/A

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

       3. associate-/l* N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. swap-sqr N/A

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

       8. times-frac N/A

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

       9. frac-2neg N/A

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

       10. neg-sub0 N/A

           \[\leadsto \frac{\frac{\color{blue}{0 - y \cdot y}}{\mathsf{neg}\left(y\right)} \cdot \frac{x \cdot x}{x}}{1} \]

       11. metadata-eval N/A

           \[\leadsto \frac{\frac{\color{blue}{0 \cdot 0} - y \cdot y}{\mathsf{neg}\left(y\right)} \cdot \frac{x \cdot x}{x}}{1} \]

       12. neg-sub0 N/A

           \[\leadsto \frac{\frac{0 \cdot 0 - y \cdot y}{\color{blue}{0 - y}} \cdot \frac{x \cdot x}{x}}{1} \]

       13. flip-+ N/A

           \[\leadsto \frac{\color{blue}{\left(0 + y\right)} \cdot \frac{x \cdot x}{x}}{1} \]

       14. +-lft-identity N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 N/A

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

       17. lower-*.f64 25.3

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

   12. Applied rewrites 25.3%

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

   if 3.60000000000000018e-114 < z

   1. Initial program 54.3%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 81.0

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

   5. Applied rewrites 81.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 93.8

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

   7. Applied rewrites 93.8%

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

   8. Taylor expanded in z around inf

      \[\leadsto \frac{y \cdot x}{\color{blue}{1}} \]
   9. Step-by-step derivation

   10. Applied rewrites 91.9%

       \[\leadsto \frac{y \cdot x}{\color{blue}{1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.6 \cdot 10^{-114}:\\ \;\;\;\;\frac{\frac{x \cdot x}{x} \cdot y}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.2 \cdot 10^{-193}:\\ \;\;\;\;\frac{\left(x\_m \cdot y\_m\right) \cdot z\_m}{-z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot y\_m}{1}\\ \end{array}\right)\right) \end{array} \]

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 2.2e-193)
      (/ (* (* x_m y_m) z_m) (- z_m))
      (/ (* x_m y_m) 1.0))))))

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 <= 2.2e-193) {
		tmp = ((x_m * y_m) * z_m) / -z_m;
	} else {
		tmp = (x_m * y_m) / 1.0;
	}
	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 <= 2.2d-193) then
        tmp = ((x_m * y_m) * z_m) / -z_m
    else
        tmp = (x_m * y_m) / 1.0d0
    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 <= 2.2e-193) {
		tmp = ((x_m * y_m) * z_m) / -z_m;
	} else {
		tmp = (x_m * y_m) / 1.0;
	}
	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 <= 2.2e-193:
		tmp = ((x_m * y_m) * z_m) / -z_m
	else:
		tmp = (x_m * y_m) / 1.0
	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 <= 2.2e-193)
		tmp = Float64(Float64(Float64(x_m * y_m) * z_m) / Float64(-z_m));
	else
		tmp = Float64(Float64(x_m * y_m) / 1.0);
	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 <= 2.2e-193)
		tmp = ((x_m * y_m) * z_m) / -z_m;
	else
		tmp = (x_m * y_m) / 1.0;
	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, 2.2e-193], N[(N[(N[(x$95$m * y$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] / (-z$95$m)), $MachinePrecision], N[(N[(x$95$m * y$95$m), $MachinePrecision] / 1.0), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.19999999999999977e-193

   1. Initial program 65.9%

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

   3. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 64.0

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

   5. Applied rewrites 64.0%

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

   if 2.19999999999999977e-193 < z

   1. Initial program 57.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 78.6

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

   5. Applied rewrites 78.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 89.8

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

   7. Applied rewrites 89.8%

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

   8. Taylor expanded in z around inf

      \[\leadsto \frac{y \cdot x}{\color{blue}{1}} \]
   9. Step-by-step derivation

   10. Applied rewrites 88.2%

       \[\leadsto \frac{y \cdot x}{\color{blue}{1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.2 \cdot 10^{-193}:\\ \;\;\;\;\frac{\left(x \cdot y\right) \cdot z}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.0% accurate, 2.6× speedup

Math

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

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) 1.0)))))

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) / 1.0)));
}

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) / 1.0d0)))
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) / 1.0)));
}

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) / 1.0)))

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(Float64(x_m * y_m) / 1.0))))
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) / 1.0)));
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[(N[(x$95$m * y$95$m), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 t around 0

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. associate-*r* N/A

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

   4. associate-*l/ N/A

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

   5. associate-*r/ N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 48.2

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

5. Applied rewrites 48.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lower-/.f64 53.0

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

7. Applied rewrites 53.0%

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

8. Taylor expanded in z around inf

   \[\leadsto \frac{y \cdot x}{\color{blue}{1}} \]
9. Step-by-step derivation

10. Applied rewrites 46.1%

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

11. Final simplification 46.1%

    \[\leadsto \frac{x \cdot y}{1} \]
12. Add Preprocessing

Alternative 13: 13.1% accurate, 5.6× speedup

Math

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

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 (* (- 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) {
	return x_s * (y_s * (z_s * (-y_m * x_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 * (-y_m * x_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 * (-y_m * x_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 * (-y_m * x_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(Float64(-y_m) * x_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 * (-y_m * x_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[((-y$95$m) * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 z around -inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. neg-mul-1 N/A

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

   5. lower-neg.f64 44.8

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

5. Applied rewrites 44.8%

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

Developer Target 1: 87.4% 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 2024332 
(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)))))