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

Percentage Accurate: 60.0% → 91.8%

Time: 11.7s

Alternatives: 10

Speedup: 7.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 60.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ \begin{array}{l} t_1 := {\left(\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)\right)}^{0.25}\\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.35 \cdot 10^{+39}:\\ \;\;\;\;\frac{x\_m}{t\_1} \cdot \frac{y\_m \cdot z\_m}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot y\_m\right) \cdot \frac{z\_m}{\mathsf{fma}\left(\frac{a}{z\_m}, -0.5 \cdot t, z\_m\right)}\\ \end{array}\right)\right) \end{array} \end{array} \]

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
 (let* ((t_1 (pow (fma (- a) t (* z_m z_m)) 0.25)))
   (*
    x_s
    (*
     y_s
     (*
      z_s
      (if (<= z_m 1.35e+39)
        (* (/ x_m t_1) (/ (* y_m z_m) t_1))
        (* (* x_m y_m) (/ z_m (fma (/ a z_m) (* -0.5 t) 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 t_1 = pow(fma(-a, t, (z_m * z_m)), 0.25);
	double tmp;
	if (z_m <= 1.35e+39) {
		tmp = (x_m / t_1) * ((y_m * z_m) / t_1);
	} else {
		tmp = (x_m * y_m) * (z_m / fma((a / z_m), (-0.5 * t), 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)
	t_1 = fma(Float64(-a), t, Float64(z_m * z_m)) ^ 0.25
	tmp = 0.0
	if (z_m <= 1.35e+39)
		tmp = Float64(Float64(x_m / t_1) * Float64(Float64(y_m * z_m) / t_1));
	else
		tmp = Float64(Float64(x_m * y_m) * Float64(z_m / fma(Float64(a / z_m), Float64(-0.5 * t), 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_] := Block[{t$95$1 = N[Power[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]}, N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 1.35e+39], N[(N[(x$95$m / t$95$1), $MachinePrecision] * N[(N[(y$95$m * z$95$m), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * y$95$m), $MachinePrecision] * N[(z$95$m / N[(N[(a / z$95$m), $MachinePrecision] * N[(-0.5 * t), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.35000000000000002e39

   1. Initial program 60.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-sqrt.f64 N/A

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

      7. pow1/2 N/A

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

      8. sqr-pow N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 60.3%

      \[\leadsto \color{blue}{\frac{z \cdot y}{{\left(\mathsf{fma}\left(-a, t, z \cdot z\right)\right)}^{0.25}} \cdot \frac{x}{{\left(\mathsf{fma}\left(-a, t, z \cdot z\right)\right)}^{0.25}}} \]

   if 1.35000000000000002e39 < z

   1. Initial program 47.9%

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

   3. Taylor expanded in a around 0

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z + \frac{-1}{2} \cdot \frac{a \cdot t}{z}}} \]
   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 74.1

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

   5. Applied rewrites 74.1%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{\mathsf{fma}\left(-0.5 \cdot t, \frac{a}{z}, z\right)}} \]
   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. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

         \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{a}{z}, \frac{-1}{2} \cdot t, z\right)\right)\right)} \cdot \color{blue}{\left(y \cdot x\right)} \]

   7. Applied rewrites 97.0%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.35 \cdot 10^{+39}:\\ \;\;\;\;\frac{x}{{\left(\mathsf{fma}\left(-a, t, z \cdot z\right)\right)}^{0.25}} \cdot \frac{y \cdot z}{{\left(\mathsf{fma}\left(-a, t, z \cdot z\right)\right)}^{0.25}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(\frac{a}{z}, -0.5 \cdot t, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.2 \cdot 10^{-89}:\\ \;\;\;\;\frac{x\_m \cdot \left(y\_m \cdot z\_m\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{elif}\;z\_m \leq 2.5 \cdot 10^{+39}:\\ \;\;\;\;\frac{y\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}} \cdot \left(x\_m \cdot z\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot y\_m\right) \cdot \frac{z\_m}{\mathsf{fma}\left(\frac{a}{z\_m}, -0.5 \cdot t, z\_m\right)}\\ \end{array}\right)\right) \end{array} \]

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-89)
      (/ (* x_m (* y_m z_m)) (sqrt (* t (- a))))
      (if (<= z_m 2.5e+39)
        (* (/ y_m (sqrt (fma (- a) t (* z_m z_m)))) (* x_m z_m))
        (* (* x_m y_m) (/ z_m (fma (/ a z_m) (* -0.5 t) 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 <= 2.2e-89) {
		tmp = (x_m * (y_m * z_m)) / sqrt((t * -a));
	} else if (z_m <= 2.5e+39) {
		tmp = (y_m / sqrt(fma(-a, t, (z_m * z_m)))) * (x_m * z_m);
	} else {
		tmp = (x_m * y_m) * (z_m / fma((a / z_m), (-0.5 * t), 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 <= 2.2e-89)
		tmp = Float64(Float64(x_m * Float64(y_m * z_m)) / sqrt(Float64(t * Float64(-a))));
	elseif (z_m <= 2.5e+39)
		tmp = Float64(Float64(y_m / sqrt(fma(Float64(-a), t, Float64(z_m * z_m)))) * Float64(x_m * z_m));
	else
		tmp = Float64(Float64(x_m * y_m) * Float64(z_m / fma(Float64(a / z_m), Float64(-0.5 * t), 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, 2.2e-89], N[(N[(x$95$m * N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 2.5e+39], N[(N[(y$95$m / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * y$95$m), $MachinePrecision] * N[(z$95$m / N[(N[(a / z$95$m), $MachinePrecision] * N[(-0.5 * t), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 2.20000000000000012e-89

   1. Initial program 55.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 56.3

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

   4. Applied rewrites 56.3%

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

   5. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 39.5

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

   7. Applied rewrites 39.5%

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

   if 2.20000000000000012e-89 < z < 2.50000000000000008e39

   1. Initial program 96.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-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 95.7

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

      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 95.7

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

   4. Applied rewrites 95.7%

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

   if 2.50000000000000008e39 < z

   1. Initial program 47.9%

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

   3. Taylor expanded in a around 0

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z + \frac{-1}{2} \cdot \frac{a \cdot t}{z}}} \]
   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 74.1

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

   5. Applied rewrites 74.1%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{\mathsf{fma}\left(-0.5 \cdot t, \frac{a}{z}, z\right)}} \]
   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. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

         \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{a}{z}, \frac{-1}{2} \cdot t, z\right)\right)\right)} \cdot \color{blue}{\left(y \cdot x\right)} \]

   7. Applied rewrites 97.0%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.2 \cdot 10^{-89}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+39}:\\ \;\;\;\;\frac{y}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}} \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(\frac{a}{z}, -0.5 \cdot t, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.35 \cdot 10^{+39}:\\ \;\;\;\;\frac{x\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}} \cdot \left(y\_m \cdot z\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot y\_m\right) \cdot \frac{z\_m}{\mathsf{fma}\left(\frac{a}{z\_m}, -0.5 \cdot t, z\_m\right)}\\ \end{array}\right)\right) \end{array} \]

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 1.35e+39)
      (* (/ x_m (sqrt (fma (- a) t (* z_m z_m)))) (* y_m z_m))
      (* (* x_m y_m) (/ z_m (fma (/ a z_m) (* -0.5 t) 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 <= 1.35e+39) {
		tmp = (x_m / sqrt(fma(-a, t, (z_m * z_m)))) * (y_m * z_m);
	} else {
		tmp = (x_m * y_m) * (z_m / fma((a / z_m), (-0.5 * t), 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 <= 1.35e+39)
		tmp = Float64(Float64(x_m / sqrt(fma(Float64(-a), t, Float64(z_m * z_m)))) * Float64(y_m * z_m));
	else
		tmp = Float64(Float64(x_m * y_m) * Float64(z_m / fma(Float64(a / z_m), Float64(-0.5 * t), 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, 1.35e+39], 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[(x$95$m * y$95$m), $MachinePrecision] * N[(z$95$m / N[(N[(a / z$95$m), $MachinePrecision] * N[(-0.5 * t), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.35000000000000002e39

   1. Initial program 60.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 58.7

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 58.7

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

   4. Applied rewrites 58.7%

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

   if 1.35000000000000002e39 < z

   1. Initial program 47.9%

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

   3. Taylor expanded in a around 0

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z + \frac{-1}{2} \cdot \frac{a \cdot t}{z}}} \]
   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 74.1

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

   5. Applied rewrites 74.1%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{\mathsf{fma}\left(-0.5 \cdot t, \frac{a}{z}, z\right)}} \]
   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. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

         \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{a}{z}, \frac{-1}{2} \cdot t, z\right)\right)\right)} \cdot \color{blue}{\left(y \cdot x\right)} \]

   7. Applied rewrites 97.0%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.35 \cdot 10^{+39}:\\ \;\;\;\;\frac{x}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}} \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(\frac{a}{z}, -0.5 \cdot t, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.15 \cdot 10^{-83}:\\ \;\;\;\;\left(\frac{z\_m}{\sqrt{t \cdot \left(-a\right)}} \cdot y\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot y\_m\right) \cdot \frac{z\_m}{\mathsf{fma}\left(\frac{a}{z\_m}, -0.5 \cdot t, z\_m\right)}\\ \end{array}\right)\right) \end{array} \]

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 1.15e-83)
      (* (* (/ z_m (sqrt (* t (- a)))) y_m) x_m)
      (* (* x_m y_m) (/ z_m (fma (/ a z_m) (* -0.5 t) 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 <= 1.15e-83) {
		tmp = ((z_m / sqrt((t * -a))) * y_m) * x_m;
	} else {
		tmp = (x_m * y_m) * (z_m / fma((a / z_m), (-0.5 * t), 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 <= 1.15e-83)
		tmp = Float64(Float64(Float64(z_m / sqrt(Float64(t * Float64(-a)))) * y_m) * x_m);
	else
		tmp = Float64(Float64(x_m * y_m) * Float64(z_m / fma(Float64(a / z_m), Float64(-0.5 * t), 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, 1.15e-83], N[(N[(N[(z$95$m / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(x$95$m * y$95$m), $MachinePrecision] * N[(z$95$m / N[(N[(a / z$95$m), $MachinePrecision] * N[(-0.5 * t), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.14999999999999995e-83

   1. Initial program 55.4%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 55.3

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 55.3

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 55.3

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 55.3

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

   4. Applied rewrites 55.3%

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

   5. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 38.5

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

   7. Applied rewrites 38.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   9. Applied rewrites 38.7%

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

   if 1.14999999999999995e-83 < z

   1. Initial program 60.9%

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

   3. Taylor expanded in a around 0

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z + \frac{-1}{2} \cdot \frac{a \cdot t}{z}}} \]
   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 74.6

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

   5. Applied rewrites 74.6%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{\mathsf{fma}\left(-0.5 \cdot t, \frac{a}{z}, z\right)}} \]
   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. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

         \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{a}{z}, \frac{-1}{2} \cdot t, z\right)\right)\right)} \cdot \color{blue}{\left(y \cdot x\right)} \]

   7. Applied rewrites 91.3%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.15 \cdot 10^{-83}:\\ \;\;\;\;\left(\frac{z}{\sqrt{t \cdot \left(-a\right)}} \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(\frac{a}{z}, -0.5 \cdot t, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.15 \cdot 10^{-83}:\\ \;\;\;\;\left(\frac{z\_m}{\sqrt{t \cdot \left(-a\right)}} \cdot y\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z\_m}{\mathsf{fma}\left(\frac{a}{z\_m}, -0.5 \cdot t, z\_m\right)} \cdot x\_m\right) \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 1.15e-83)
      (* (* (/ z_m (sqrt (* t (- a)))) y_m) x_m)
      (* (* (/ z_m (fma (/ a z_m) (* -0.5 t) z_m)) x_m) y_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.15e-83) {
		tmp = ((z_m / sqrt((t * -a))) * y_m) * x_m;
	} else {
		tmp = ((z_m / fma((a / z_m), (-0.5 * t), z_m)) * x_m) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.15e-83)
		tmp = Float64(Float64(Float64(z_m / sqrt(Float64(t * Float64(-a)))) * y_m) * x_m);
	else
		tmp = Float64(Float64(Float64(z_m / fma(Float64(a / z_m), Float64(-0.5 * t), z_m)) * x_m) * y_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, 1.15e-83], N[(N[(N[(z$95$m / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$95$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] * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.14999999999999995e-83

   1. Initial program 55.4%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 55.3

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 55.3

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 55.3

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 55.3

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

   4. Applied rewrites 55.3%

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

   5. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 38.5

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

   7. Applied rewrites 38.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   9. Applied rewrites 38.7%

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

   if 1.14999999999999995e-83 < z

   1. Initial program 60.9%

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

   3. Taylor expanded in a around 0

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z + \frac{-1}{2} \cdot \frac{a \cdot t}{z}}} \]
   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 74.6

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

   5. Applied rewrites 74.6%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{\mathsf{fma}\left(-0.5 \cdot t, \frac{a}{z}, z\right)}} \]
   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. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 91.2

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

   7. Applied rewrites 91.2%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.15 \cdot 10^{-83}:\\ \;\;\;\;\left(\frac{z}{\sqrt{t \cdot \left(-a\right)}} \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{\mathsf{fma}\left(\frac{a}{z}, -0.5 \cdot t, z\right)} \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.15 \cdot 10^{-83}:\\ \;\;\;\;\left(\frac{z\_m}{\sqrt{t \cdot \left(-a\right)}} \cdot y\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 1.15e-83)
      (* (* (/ z_m (sqrt (* t (- a)))) y_m) x_m)
      (* x_m y_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.15e-83) {
		tmp = ((z_m / sqrt((t * -a))) * y_m) * x_m;
	} else {
		tmp = x_m * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.15d-83) then
        tmp = ((z_m / sqrt((t * -a))) * y_m) * x_m
    else
        tmp = x_m * y_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.15e-83) {
		tmp = ((z_m / Math.sqrt((t * -a))) * y_m) * x_m;
	} else {
		tmp = x_m * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.15e-83:
		tmp = ((z_m / math.sqrt((t * -a))) * y_m) * x_m
	else:
		tmp = x_m * y_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.15e-83)
		tmp = Float64(Float64(Float64(z_m / sqrt(Float64(t * Float64(-a)))) * y_m) * x_m);
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.15e-83)
		tmp = ((z_m / sqrt((t * -a))) * y_m) * x_m;
	else
		tmp = x_m * y_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 1.15e-83], N[(N[(N[(z$95$m / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.14999999999999995e-83

   1. Initial program 55.4%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 55.3

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 55.3

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 55.3

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 55.3

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

   4. Applied rewrites 55.3%

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

   5. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 38.5

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

   7. Applied rewrites 38.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   9. Applied rewrites 38.7%

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

   if 1.14999999999999995e-83 < z

   1. Initial program 60.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 91.3

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

   5. Applied rewrites 91.3%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.15 \cdot 10^{-83}:\\ \;\;\;\;\left(\frac{z}{\sqrt{t \cdot \left(-a\right)}} \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.15 \cdot 10^{-83}:\\ \;\;\;\;\frac{x\_m}{\sqrt{t \cdot \left(-a\right)}} \cdot \left(y\_m \cdot z\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 1.15e-83)
      (* (/ x_m (sqrt (* t (- a)))) (* y_m z_m))
      (* x_m y_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.15e-83) {
		tmp = (x_m / sqrt((t * -a))) * (y_m * z_m);
	} else {
		tmp = x_m * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.15d-83) then
        tmp = (x_m / sqrt((t * -a))) * (y_m * z_m)
    else
        tmp = x_m * y_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.15e-83) {
		tmp = (x_m / Math.sqrt((t * -a))) * (y_m * z_m);
	} else {
		tmp = x_m * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.15e-83:
		tmp = (x_m / math.sqrt((t * -a))) * (y_m * z_m)
	else:
		tmp = x_m * y_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.15e-83)
		tmp = Float64(Float64(x_m / sqrt(Float64(t * Float64(-a)))) * Float64(y_m * z_m));
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.15e-83)
		tmp = (x_m / sqrt((t * -a))) * (y_m * z_m);
	else
		tmp = x_m * y_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 1.15e-83], N[(N[(x$95$m / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.14999999999999995e-83

   1. Initial program 55.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 55.2

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 55.2

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

   4. Applied rewrites 55.2%

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

   5. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 37.9

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

   7. Applied rewrites 37.9%

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

   if 1.14999999999999995e-83 < z

   1. Initial program 60.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 91.3

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

   5. Applied rewrites 91.3%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.15 \cdot 10^{-83}:\\ \;\;\;\;\frac{x}{\sqrt{t \cdot \left(-a\right)}} \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.95 \cdot 10^{-146}:\\ \;\;\;\;\frac{\left(x\_m \cdot z\_m\right) \cdot y\_m}{-z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 1.95e-146) (/ (* (* x_m z_m) y_m) (- z_m)) (* x_m y_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.95e-146) {
		tmp = ((x_m * z_m) * y_m) / -z_m;
	} else {
		tmp = x_m * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.95d-146) then
        tmp = ((x_m * z_m) * y_m) / -z_m
    else
        tmp = x_m * y_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.95e-146) {
		tmp = ((x_m * z_m) * y_m) / -z_m;
	} else {
		tmp = x_m * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.95e-146:
		tmp = ((x_m * z_m) * y_m) / -z_m
	else:
		tmp = x_m * y_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.95e-146)
		tmp = Float64(Float64(Float64(x_m * z_m) * y_m) / Float64(-z_m));
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.95e-146)
		tmp = ((x_m * z_m) * y_m) / -z_m;
	else
		tmp = x_m * y_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 1.95e-146], N[(N[(N[(x$95$m * z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / (-z$95$m)), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.95000000000000001e-146

   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 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 59.5

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

   5. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 57.0

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

   7. Applied rewrites 57.0%

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

   if 1.95000000000000001e-146 < z

   1. Initial program 63.6%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 83.0

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

   5. Applied rewrites 83.0%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.95 \cdot 10^{-146}:\\ \;\;\;\;\frac{\left(x \cdot z\right) \cdot y}{-z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.95 \cdot 10^{-146}:\\ \;\;\;\;\frac{\left(x\_m \cdot y\_m\right) \cdot z\_m}{-z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 1.95e-146) (/ (* (* x_m y_m) z_m) (- z_m)) (* x_m y_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.95e-146) {
		tmp = ((x_m * y_m) * z_m) / -z_m;
	} else {
		tmp = x_m * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.95d-146) then
        tmp = ((x_m * y_m) * z_m) / -z_m
    else
        tmp = x_m * y_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.95e-146) {
		tmp = ((x_m * y_m) * z_m) / -z_m;
	} else {
		tmp = x_m * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.95e-146:
		tmp = ((x_m * y_m) * z_m) / -z_m
	else:
		tmp = x_m * y_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.95e-146)
		tmp = Float64(Float64(Float64(x_m * y_m) * z_m) / Float64(-z_m));
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.95e-146)
		tmp = ((x_m * y_m) * z_m) / -z_m;
	else
		tmp = x_m * y_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 1.95e-146], N[(N[(N[(x$95$m * y$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] / (-z$95$m)), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.95000000000000001e-146

   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 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 59.5

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

   5. Applied rewrites 59.5%

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

   if 1.95000000000000001e-146 < z

   1. Initial program 63.6%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 83.0

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

   5. Applied rewrites 83.0%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.95 \cdot 10^{-146}:\\ \;\;\;\;\frac{\left(x \cdot y\right) \cdot z}{-z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.0% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = x_s * (y_s * (z_s * (x_m * y_m)));
end

Wolfram

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

Derivation

1. Initial program 57.2%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 39.6

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

5. Applied rewrites 39.6%

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

6. Final simplification 39.6%

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

Developer Target 1: 87.5% 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 2024240 
(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)))))