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

Percentage Accurate: 61.1% → 93.8%

Time: 14.5s

Alternatives: 11

Speedup: 37.7×

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: 61.1% 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: 93.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 6.8 \cdot 10^{-127}:\\ \;\;\;\;\left(z\_m \cdot y\_m\right) \cdot \frac{x\_m}{\sqrt{0 - t} \cdot \sqrt{a}}\\ \mathbf{elif}\;z\_m \leq 10^{+113}:\\ \;\;\;\;\frac{y\_m \cdot x\_m}{\frac{\sqrt{z\_m \cdot z\_m - t \cdot a}}{z\_m}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{1 + \frac{a \cdot -0.5}{\frac{z\_m}{\frac{t}{z\_m}}}}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 6.8e-127)
      (* (* z_m y_m) (/ x_m (* (sqrt (- 0.0 t)) (sqrt a))))
      (if (<= z_m 1e+113)
        (/ (* y_m x_m) (/ (sqrt (- (* z_m z_m) (* t a))) z_m))
        (* y_m (/ x_m (+ 1.0 (/ (* a -0.5) (/ z_m (/ t z_m))))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 6.8e-127) {
		tmp = (z_m * y_m) * (x_m / (sqrt((0.0 - t)) * sqrt(a)));
	} else if (z_m <= 1e+113) {
		tmp = (y_m * x_m) / (sqrt(((z_m * z_m) - (t * a))) / z_m);
	} else {
		tmp = y_m * (x_m / (1.0 + ((a * -0.5) / (z_m / (t / z_m)))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_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 <= 6.8d-127) then
        tmp = (z_m * y_m) * (x_m / (sqrt((0.0d0 - t)) * sqrt(a)))
    else if (z_m <= 1d+113) then
        tmp = (y_m * x_m) / (sqrt(((z_m * z_m) - (t * a))) / z_m)
    else
        tmp = y_m * (x_m / (1.0d0 + ((a * (-0.5d0)) / (z_m / (t / z_m)))))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 6.8e-127) {
		tmp = (z_m * y_m) * (x_m / (Math.sqrt((0.0 - t)) * Math.sqrt(a)));
	} else if (z_m <= 1e+113) {
		tmp = (y_m * x_m) / (Math.sqrt(((z_m * z_m) - (t * a))) / z_m);
	} else {
		tmp = y_m * (x_m / (1.0 + ((a * -0.5) / (z_m / (t / z_m)))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 6.8e-127:
		tmp = (z_m * y_m) * (x_m / (math.sqrt((0.0 - t)) * math.sqrt(a)))
	elif z_m <= 1e+113:
		tmp = (y_m * x_m) / (math.sqrt(((z_m * z_m) - (t * a))) / z_m)
	else:
		tmp = y_m * (x_m / (1.0 + ((a * -0.5) / (z_m / (t / z_m)))))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 6.8e-127)
		tmp = Float64(Float64(z_m * y_m) * Float64(x_m / Float64(sqrt(Float64(0.0 - t)) * sqrt(a))));
	elseif (z_m <= 1e+113)
		tmp = Float64(Float64(y_m * x_m) / Float64(sqrt(Float64(Float64(z_m * z_m) - Float64(t * a))) / z_m));
	else
		tmp = Float64(y_m * Float64(x_m / Float64(1.0 + Float64(Float64(a * -0.5) / Float64(z_m / Float64(t / z_m))))));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 6.8e-127)
		tmp = (z_m * y_m) * (x_m / (sqrt((0.0 - t)) * sqrt(a)));
	elseif (z_m <= 1e+113)
		tmp = (y_m * x_m) / (sqrt(((z_m * z_m) - (t * a))) / z_m);
	else
		tmp = y_m * (x_m / (1.0 + ((a * -0.5) / (z_m / (t / z_m)))));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 6.8e-127], N[(N[(z$95$m * y$95$m), $MachinePrecision] * N[(x$95$m / N[(N[Sqrt[N[(0.0 - t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 1e+113], N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m / N[(1.0 + N[(N[(a * -0.5), $MachinePrecision] / N[(z$95$m / N[(t / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 6.7999999999999997e-127

   1. Initial program 62.4%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot z\right), \color{blue}{\left(\frac{x}{\sqrt{z \cdot z - t \cdot a}}\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \left(\frac{\color{blue}{x}}{\sqrt{z \cdot z - t \cdot a}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \color{blue}{\left(\sqrt{z \cdot z - t \cdot a}\right)}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 62.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 62.8%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot t\right)\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(a \cdot t\right)\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\left(0 - a \cdot t\right)\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot t\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 38.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, t\right)\right)\right)\right)\right) \]

   7. Simplified 38.2%

      \[\leadsto \left(y \cdot z\right) \cdot \frac{x}{\sqrt{\color{blue}{0 - a \cdot t}}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \left(\sqrt{\mathsf{neg}\left(a \cdot t\right)}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \left(\sqrt{\mathsf{neg}\left(t \cdot a\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \left(\sqrt{\left(\mathsf{neg}\left(t\right)\right) \cdot a}\right)\right)\right) \]

      4. sqrt-prod N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \left(\sqrt{\mathsf{neg}\left(t\right)} \cdot \color{blue}{\sqrt{a}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(\sqrt{\mathsf{neg}\left(t\right)}\right), \color{blue}{\left(\sqrt{a}\right)}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(t\right)\right)\right), \left(\sqrt{\color{blue}{a}}\right)\right)\right)\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(0 - t\right)\right), \left(\sqrt{a}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, t\right)\right), \left(\sqrt{a}\right)\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 19.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, t\right)\right), \mathsf{sqrt.f64}\left(a\right)\right)\right)\right) \]

   9. Applied egg-rr 19.3%

      \[\leadsto \left(y \cdot z\right) \cdot \frac{x}{\color{blue}{\sqrt{0 - t} \cdot \sqrt{a}}} \]

   if 6.7999999999999997e-127 < z < 1e113

   1. Initial program 94.5%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\frac{\sqrt{z \cdot z - t \cdot a}}{z}\right)}\right) \]

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{/.f64}\left(\left(\sqrt{z \cdot z - t \cdot a}\right), \color{blue}{z}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right), z\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right), z\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right), z\right)\right) \]

      10. *-lowering-*.f64 94.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right), z\right)\right) \]

   4. Applied egg-rr 94.7%

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

   if 1e113 < z

   1. Initial program 32.4%

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

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \color{blue}{\left(z \cdot \left(1 + \frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \color{blue}{\left(1 + \frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{a \cdot t}{{z}^{2}}\right)}\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(a \cdot \color{blue}{\frac{t}{{z}^{2}}}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{t}{{z}^{2}}\right)}\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \color{blue}{\left({z}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left(z \cdot \color{blue}{z}\right)\right)\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 73.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 73.8%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z \cdot \left(1 + -0.5 \cdot \left(a \cdot \frac{t}{z \cdot z}\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

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

      2. associate-*l* N/A

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

      3. associate-/r* N/A

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

      4. *-inverses N/A

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

      5. un-div-inv N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{1 + \frac{-1}{2} \cdot \left(a \cdot \frac{t}{z \cdot z}\right)}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(1 + \frac{-1}{2} \cdot \left(a \cdot \frac{t}{z \cdot z}\right)\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot \left(a \cdot \frac{t}{z \cdot z}\right)\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot \left(\frac{t}{z \cdot z} \cdot \color{blue}{a}\right)\right)\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{2} \cdot \frac{t}{z \cdot z}\right) \cdot \color{blue}{a}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{-1}{2} \cdot \frac{t}{z \cdot z}\right), \color{blue}{a}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{t}{z \cdot z}\right)\right), a\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(t, \left(z \cdot z\right)\right)\right), a\right)\right)\right)\right) \]

      14. *-lowering-*.f64 97.8%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, z\right)\right)\right), a\right)\right)\right)\right) \]

   7. Applied egg-rr 97.8%

      \[\leadsto \color{blue}{x \cdot \frac{y}{1 + \left(-0.5 \cdot \frac{t}{z \cdot z}\right) \cdot a}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{y}{1 + \left(\frac{-1}{2} \cdot \frac{t}{z \cdot z}\right) \cdot a}\right), \color{blue}{x}\right) \]

   9. Applied egg-rr 97.8%

      \[\leadsto \color{blue}{\frac{y}{1 + \frac{-0.5}{\frac{\frac{z}{\frac{t}{z}}}{a}}} \cdot x} \]
   10. Step-by-step derivation

       1. associate-*l/ N/A

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

       2. associate-/l* N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{x}{1 + \frac{\frac{-1}{2}}{\frac{\frac{z}{\frac{t}{z}}}{a}}}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \color{blue}{\left(1 + \frac{\frac{-1}{2}}{\frac{\frac{z}{\frac{t}{z}}}{a}}\right)}\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2}}{\frac{\frac{z}{\frac{t}{z}}}{a}}\right)}\right)\right)\right) \]

       6. associate-/r/ N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{2}}{\frac{z}{\frac{t}{z}}} \cdot \color{blue}{a}\right)\right)\right)\right) \]

       7. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{2} \cdot a}{\color{blue}{\frac{z}{\frac{t}{z}}}}\right)\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot a\right), \color{blue}{\left(\frac{z}{\frac{t}{z}}\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, a\right), \left(\frac{\color{blue}{z}}{\frac{t}{z}}\right)\right)\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, a\right), \mathsf{/.f64}\left(z, \color{blue}{\left(\frac{t}{z}\right)}\right)\right)\right)\right)\right) \]

       11. /-lowering-/.f64 97.8%

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, a\right), \mathsf{/.f64}\left(z, \mathsf{/.f64}\left(t, \color{blue}{z}\right)\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 97.8%

       \[\leadsto \color{blue}{y \cdot \frac{x}{1 + \frac{-0.5 \cdot a}{\frac{z}{\frac{t}{z}}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.8 \cdot 10^{-127}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{x}{\sqrt{0 - t} \cdot \sqrt{a}}\\ \mathbf{elif}\;z \leq 10^{+113}:\\ \;\;\;\;\frac{y \cdot x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{1 + \frac{a \cdot -0.5}{\frac{z}{\frac{t}{z}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 10^{+113}:\\ \;\;\;\;\frac{x\_m}{\frac{\frac{\sqrt{z\_m \cdot z\_m - t \cdot a}}{z\_m}}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{1 + \frac{a \cdot -0.5}{\frac{z\_m}{\frac{t}{z\_m}}}}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1e+113)
      (/ x_m (/ (/ (sqrt (- (* z_m z_m) (* t a))) z_m) y_m))
      (* y_m (/ x_m (+ 1.0 (/ (* a -0.5) (/ z_m (/ t z_m)))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1e+113) {
		tmp = x_m / ((sqrt(((z_m * z_m) - (t * a))) / z_m) / y_m);
	} else {
		tmp = y_m * (x_m / (1.0 + ((a * -0.5) / (z_m / (t / z_m)))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1d+113) then
        tmp = x_m / ((sqrt(((z_m * z_m) - (t * a))) / z_m) / y_m)
    else
        tmp = y_m * (x_m / (1.0d0 + ((a * (-0.5d0)) / (z_m / (t / z_m)))))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1e+113) {
		tmp = x_m / ((Math.sqrt(((z_m * z_m) - (t * a))) / z_m) / y_m);
	} else {
		tmp = y_m * (x_m / (1.0 + ((a * -0.5) / (z_m / (t / z_m)))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1e+113:
		tmp = x_m / ((math.sqrt(((z_m * z_m) - (t * a))) / z_m) / y_m)
	else:
		tmp = y_m * (x_m / (1.0 + ((a * -0.5) / (z_m / (t / z_m)))))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1e+113)
		tmp = Float64(x_m / Float64(Float64(sqrt(Float64(Float64(z_m * z_m) - Float64(t * a))) / z_m) / y_m));
	else
		tmp = Float64(y_m * Float64(x_m / Float64(1.0 + Float64(Float64(a * -0.5) / Float64(z_m / Float64(t / z_m))))));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1e+113)
		tmp = x_m / ((sqrt(((z_m * z_m) - (t * a))) / z_m) / y_m);
	else
		tmp = y_m * (x_m / (1.0 + ((a * -0.5) / (z_m / (t / z_m)))));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1e+113], N[(x$95$m / N[(N[(N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m / N[(1.0 + N[(N[(a * -0.5), $MachinePrecision] / N[(z$95$m / N[(t / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1e113

   1. Initial program 70.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot z\right), \color{blue}{\left(\frac{x}{\sqrt{z \cdot z - t \cdot a}}\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \left(\frac{\color{blue}{x}}{\sqrt{z \cdot z - t \cdot a}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \color{blue}{\left(\sqrt{z \cdot z - t \cdot a}\right)}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 69.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 69.5%

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. associate-*l/ N/A

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

      5. associate-/r/ N/A

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

      6. associate-/l* N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{y}\right)}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{\sqrt{z \cdot z - t \cdot a}}{z}\right), \color{blue}{y}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{z \cdot z - t \cdot a}\right), z\right), y\right)\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right), z\right), y\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right), z\right), y\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right), z\right), y\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(a \cdot t\right)\right)\right), z\right), y\right)\right) \]

      16. *-lowering-*.f64 73.5%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(a, t\right)\right)\right), z\right), y\right)\right) \]

   6. Applied egg-rr 73.5%

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

   if 1e113 < z

   1. Initial program 32.4%

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

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \color{blue}{\left(z \cdot \left(1 + \frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \color{blue}{\left(1 + \frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{a \cdot t}{{z}^{2}}\right)}\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(a \cdot \color{blue}{\frac{t}{{z}^{2}}}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{t}{{z}^{2}}\right)}\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \color{blue}{\left({z}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left(z \cdot \color{blue}{z}\right)\right)\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 73.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 73.8%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z \cdot \left(1 + -0.5 \cdot \left(a \cdot \frac{t}{z \cdot z}\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

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

      2. associate-*l* N/A

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

      3. associate-/r* N/A

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

      4. *-inverses N/A

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

      5. un-div-inv N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{1 + \frac{-1}{2} \cdot \left(a \cdot \frac{t}{z \cdot z}\right)}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(1 + \frac{-1}{2} \cdot \left(a \cdot \frac{t}{z \cdot z}\right)\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot \left(a \cdot \frac{t}{z \cdot z}\right)\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot \left(\frac{t}{z \cdot z} \cdot \color{blue}{a}\right)\right)\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{2} \cdot \frac{t}{z \cdot z}\right) \cdot \color{blue}{a}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{-1}{2} \cdot \frac{t}{z \cdot z}\right), \color{blue}{a}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{t}{z \cdot z}\right)\right), a\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(t, \left(z \cdot z\right)\right)\right), a\right)\right)\right)\right) \]

      14. *-lowering-*.f64 97.8%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, z\right)\right)\right), a\right)\right)\right)\right) \]

   7. Applied egg-rr 97.8%

      \[\leadsto \color{blue}{x \cdot \frac{y}{1 + \left(-0.5 \cdot \frac{t}{z \cdot z}\right) \cdot a}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{y}{1 + \left(\frac{-1}{2} \cdot \frac{t}{z \cdot z}\right) \cdot a}\right), \color{blue}{x}\right) \]

   9. Applied egg-rr 97.8%

      \[\leadsto \color{blue}{\frac{y}{1 + \frac{-0.5}{\frac{\frac{z}{\frac{t}{z}}}{a}}} \cdot x} \]
   10. Step-by-step derivation

       1. associate-*l/ N/A

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

       2. associate-/l* N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{x}{1 + \frac{\frac{-1}{2}}{\frac{\frac{z}{\frac{t}{z}}}{a}}}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \color{blue}{\left(1 + \frac{\frac{-1}{2}}{\frac{\frac{z}{\frac{t}{z}}}{a}}\right)}\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2}}{\frac{\frac{z}{\frac{t}{z}}}{a}}\right)}\right)\right)\right) \]

       6. associate-/r/ N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{2}}{\frac{z}{\frac{t}{z}}} \cdot \color{blue}{a}\right)\right)\right)\right) \]

       7. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{2} \cdot a}{\color{blue}{\frac{z}{\frac{t}{z}}}}\right)\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot a\right), \color{blue}{\left(\frac{z}{\frac{t}{z}}\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, a\right), \left(\frac{\color{blue}{z}}{\frac{t}{z}}\right)\right)\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, a\right), \mathsf{/.f64}\left(z, \color{blue}{\left(\frac{t}{z}\right)}\right)\right)\right)\right)\right) \]

       11. /-lowering-/.f64 97.8%

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, a\right), \mathsf{/.f64}\left(z, \mathsf{/.f64}\left(t, \color{blue}{z}\right)\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 97.8%

       \[\leadsto \color{blue}{y \cdot \frac{x}{1 + \frac{-0.5 \cdot a}{\frac{z}{\frac{t}{z}}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{+113}:\\ \;\;\;\;\frac{x}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{1 + \frac{a \cdot -0.5}{\frac{z}{\frac{t}{z}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.95 \cdot 10^{+71}:\\ \;\;\;\;\left(z\_m \cdot y\_m\right) \cdot \frac{x\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{1 + \frac{t}{z\_m} \cdot \left(a \cdot \frac{-0.5}{z\_m}\right)}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1.95e+71)
      (* (* z_m y_m) (/ x_m (sqrt (- (* z_m z_m) (* t a)))))
      (* x_m (/ y_m (+ 1.0 (* (/ t z_m) (* a (/ -0.5 z_m)))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.95e+71) {
		tmp = (z_m * y_m) * (x_m / sqrt(((z_m * z_m) - (t * a))));
	} else {
		tmp = x_m * (y_m / (1.0 + ((t / z_m) * (a * (-0.5 / z_m)))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_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+71) then
        tmp = (z_m * y_m) * (x_m / sqrt(((z_m * z_m) - (t * a))))
    else
        tmp = x_m * (y_m / (1.0d0 + ((t / z_m) * (a * ((-0.5d0) / z_m)))))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.95e+71) {
		tmp = (z_m * y_m) * (x_m / Math.sqrt(((z_m * z_m) - (t * a))));
	} else {
		tmp = x_m * (y_m / (1.0 + ((t / z_m) * (a * (-0.5 / z_m)))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.95e+71:
		tmp = (z_m * y_m) * (x_m / math.sqrt(((z_m * z_m) - (t * a))))
	else:
		tmp = x_m * (y_m / (1.0 + ((t / z_m) * (a * (-0.5 / z_m)))))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.95e+71)
		tmp = Float64(Float64(z_m * y_m) * Float64(x_m / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))));
	else
		tmp = Float64(x_m * Float64(y_m / Float64(1.0 + Float64(Float64(t / z_m) * Float64(a * Float64(-0.5 / z_m))))));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.95e+71)
		tmp = (z_m * y_m) * (x_m / sqrt(((z_m * z_m) - (t * a))));
	else
		tmp = x_m * (y_m / (1.0 + ((t / z_m) * (a * (-0.5 / z_m)))));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.95e+71], N[(N[(z$95$m * y$95$m), $MachinePrecision] * N[(x$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(y$95$m / N[(1.0 + N[(N[(t / z$95$m), $MachinePrecision] * N[(a * N[(-0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.9500000000000001e71

   1. Initial program 69.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot z\right), \color{blue}{\left(\frac{x}{\sqrt{z \cdot z - t \cdot a}}\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \left(\frac{\color{blue}{x}}{\sqrt{z \cdot z - t \cdot a}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \color{blue}{\left(\sqrt{z \cdot z - t \cdot a}\right)}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 69.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 69.4%

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

   if 1.9500000000000001e71 < z

   1. Initial program 43.4%

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

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \color{blue}{\left(z \cdot \left(1 + \frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \color{blue}{\left(1 + \frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{a \cdot t}{{z}^{2}}\right)}\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(a \cdot \color{blue}{\frac{t}{{z}^{2}}}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{t}{{z}^{2}}\right)}\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \color{blue}{\left({z}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left(z \cdot \color{blue}{z}\right)\right)\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 76.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 76.9%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z \cdot \left(1 + -0.5 \cdot \left(a \cdot \frac{t}{z \cdot z}\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

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

      2. associate-*l* N/A

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

      3. associate-/r* N/A

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

      4. *-inverses N/A

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

      5. un-div-inv N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{1 + \frac{-1}{2} \cdot \left(a \cdot \frac{t}{z \cdot z}\right)}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(1 + \frac{-1}{2} \cdot \left(a \cdot \frac{t}{z \cdot z}\right)\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot \left(a \cdot \frac{t}{z \cdot z}\right)\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot \left(\frac{t}{z \cdot z} \cdot \color{blue}{a}\right)\right)\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{2} \cdot \frac{t}{z \cdot z}\right) \cdot \color{blue}{a}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{-1}{2} \cdot \frac{t}{z \cdot z}\right), \color{blue}{a}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{t}{z \cdot z}\right)\right), a\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(t, \left(z \cdot z\right)\right)\right), a\right)\right)\right)\right) \]

      14. *-lowering-*.f64 96.5%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, z\right)\right)\right), a\right)\right)\right)\right) \]

   7. Applied egg-rr 96.5%

      \[\leadsto \color{blue}{x \cdot \frac{y}{1 + \left(-0.5 \cdot \frac{t}{z \cdot z}\right) \cdot a}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(a \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{t}{z \cdot z}\right)}\right)\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(a \cdot \frac{\frac{-1}{2} \cdot t}{\color{blue}{z \cdot z}}\right)\right)\right)\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(a \cdot \left(\frac{\frac{-1}{2}}{z} \cdot \color{blue}{\frac{t}{z}}\right)\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(a \cdot \frac{\frac{-1}{2}}{z}\right) \cdot \color{blue}{\frac{t}{z}}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(a \cdot \frac{\frac{-1}{2}}{z}\right), \color{blue}{\left(\frac{t}{z}\right)}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{\frac{-1}{2}}{z}\right)\right), \left(\frac{\color{blue}{t}}{z}\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\frac{-1}{2}, z\right)\right), \left(\frac{t}{z}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 96.5%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\frac{-1}{2}, z\right)\right), \mathsf{/.f64}\left(t, \color{blue}{z}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 96.5%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.95 \cdot 10^{+71}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{1 + \frac{t}{z} \cdot \left(a \cdot \frac{-0.5}{z}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.85 \cdot 10^{-65}:\\ \;\;\;\;\frac{z\_m \cdot y\_m}{\frac{\sqrt{0 - t \cdot a}}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{1 + \frac{t}{z\_m} \cdot \left(a \cdot \frac{-0.5}{z\_m}\right)}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1.85e-65)
      (/ (* z_m y_m) (/ (sqrt (- 0.0 (* t a))) x_m))
      (* x_m (/ y_m (+ 1.0 (* (/ t z_m) (* a (/ -0.5 z_m)))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.85e-65) {
		tmp = (z_m * y_m) / (sqrt((0.0 - (t * a))) / x_m);
	} else {
		tmp = x_m * (y_m / (1.0 + ((t / z_m) * (a * (-0.5 / z_m)))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_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 <= 1.85d-65) then
        tmp = (z_m * y_m) / (sqrt((0.0d0 - (t * a))) / x_m)
    else
        tmp = x_m * (y_m / (1.0d0 + ((t / z_m) * (a * ((-0.5d0) / z_m)))))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.85e-65) {
		tmp = (z_m * y_m) / (Math.sqrt((0.0 - (t * a))) / x_m);
	} else {
		tmp = x_m * (y_m / (1.0 + ((t / z_m) * (a * (-0.5 / z_m)))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.85e-65:
		tmp = (z_m * y_m) / (math.sqrt((0.0 - (t * a))) / x_m)
	else:
		tmp = x_m * (y_m / (1.0 + ((t / z_m) * (a * (-0.5 / z_m)))))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.85e-65)
		tmp = Float64(Float64(z_m * y_m) / Float64(sqrt(Float64(0.0 - Float64(t * a))) / x_m));
	else
		tmp = Float64(x_m * Float64(y_m / Float64(1.0 + Float64(Float64(t / z_m) * Float64(a * Float64(-0.5 / z_m))))));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.85e-65)
		tmp = (z_m * y_m) / (sqrt((0.0 - (t * a))) / x_m);
	else
		tmp = x_m * (y_m / (1.0 + ((t / z_m) * (a * (-0.5 / z_m)))));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.85e-65], N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[(N[Sqrt[N[(0.0 - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(y$95$m / N[(1.0 + N[(N[(t / z$95$m), $MachinePrecision] * N[(a * N[(-0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.85e-65

   1. Initial program 64.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot z\right), \color{blue}{\left(\frac{x}{\sqrt{z \cdot z - t \cdot a}}\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \left(\frac{\color{blue}{x}}{\sqrt{z \cdot z - t \cdot a}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \color{blue}{\left(\sqrt{z \cdot z - t \cdot a}\right)}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 64.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 64.5%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot t\right)\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(a \cdot t\right)\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\left(0 - a \cdot t\right)\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot t\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 40.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, t\right)\right)\right)\right)\right) \]

   7. Simplified 40.1%

      \[\leadsto \left(y \cdot z\right) \cdot \frac{x}{\sqrt{\color{blue}{0 - a \cdot t}}} \]
   8. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \left(y \cdot z\right) \cdot \frac{1}{\color{blue}{\frac{\sqrt{0 - a \cdot t}}{x}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{\frac{\sqrt{0 - a \cdot t}}{x}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot z\right), \color{blue}{\left(\frac{\sqrt{0 - a \cdot t}}{x}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \left(\frac{\color{blue}{\sqrt{0 - a \cdot t}}}{x}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(\left(\sqrt{0 - a \cdot t}\right), \color{blue}{x}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(0 - a \cdot t\right)\right), x\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot t\right)\right)\right), x\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(t \cdot a\right)\right)\right), x\right)\right) \]

      9. *-lowering-*.f64 40.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(t, a\right)\right)\right), x\right)\right) \]

   9. Applied egg-rr 40.1%

      \[\leadsto \color{blue}{\frac{y \cdot z}{\frac{\sqrt{0 - t \cdot a}}{x}}} \]

   if 1.85e-65 < z

   1. Initial program 63.9%

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

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \color{blue}{\left(z \cdot \left(1 + \frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \color{blue}{\left(1 + \frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{a \cdot t}{{z}^{2}}\right)}\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(a \cdot \color{blue}{\frac{t}{{z}^{2}}}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{t}{{z}^{2}}\right)}\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \color{blue}{\left({z}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left(z \cdot \color{blue}{z}\right)\right)\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 76.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 76.6%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z \cdot \left(1 + -0.5 \cdot \left(a \cdot \frac{t}{z \cdot z}\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

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

      2. associate-*l* N/A

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

      3. associate-/r* N/A

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

      4. *-inverses N/A

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

      5. un-div-inv N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{1 + \frac{-1}{2} \cdot \left(a \cdot \frac{t}{z \cdot z}\right)}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(1 + \frac{-1}{2} \cdot \left(a \cdot \frac{t}{z \cdot z}\right)\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot \left(a \cdot \frac{t}{z \cdot z}\right)\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot \left(\frac{t}{z \cdot z} \cdot \color{blue}{a}\right)\right)\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{2} \cdot \frac{t}{z \cdot z}\right) \cdot \color{blue}{a}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{-1}{2} \cdot \frac{t}{z \cdot z}\right), \color{blue}{a}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{t}{z \cdot z}\right)\right), a\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(t, \left(z \cdot z\right)\right)\right), a\right)\right)\right)\right) \]

      14. *-lowering-*.f64 88.4%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, z\right)\right)\right), a\right)\right)\right)\right) \]

   7. Applied egg-rr 88.4%

      \[\leadsto \color{blue}{x \cdot \frac{y}{1 + \left(-0.5 \cdot \frac{t}{z \cdot z}\right) \cdot a}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(a \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{t}{z \cdot z}\right)}\right)\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(a \cdot \frac{\frac{-1}{2} \cdot t}{\color{blue}{z \cdot z}}\right)\right)\right)\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(a \cdot \left(\frac{\frac{-1}{2}}{z} \cdot \color{blue}{\frac{t}{z}}\right)\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(a \cdot \frac{\frac{-1}{2}}{z}\right) \cdot \color{blue}{\frac{t}{z}}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(a \cdot \frac{\frac{-1}{2}}{z}\right), \color{blue}{\left(\frac{t}{z}\right)}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{\frac{-1}{2}}{z}\right)\right), \left(\frac{\color{blue}{t}}{z}\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\frac{-1}{2}, z\right)\right), \left(\frac{t}{z}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 88.4%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\frac{-1}{2}, z\right)\right), \mathsf{/.f64}\left(t, \color{blue}{z}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 88.4%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.85 \cdot 10^{-65}:\\ \;\;\;\;\frac{z \cdot y}{\frac{\sqrt{0 - t \cdot a}}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{1 + \frac{t}{z} \cdot \left(a \cdot \frac{-0.5}{z}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.1 \cdot 10^{-70}:\\ \;\;\;\;\left(z\_m \cdot y\_m\right) \cdot \frac{x\_m}{\sqrt{0 - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{1 + \frac{t}{z\_m} \cdot \left(a \cdot \frac{-0.5}{z\_m}\right)}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1.1e-70)
      (* (* z_m y_m) (/ x_m (sqrt (- 0.0 (* t a)))))
      (* x_m (/ y_m (+ 1.0 (* (/ t z_m) (* a (/ -0.5 z_m)))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.1e-70) {
		tmp = (z_m * y_m) * (x_m / sqrt((0.0 - (t * a))));
	} else {
		tmp = x_m * (y_m / (1.0 + ((t / z_m) * (a * (-0.5 / z_m)))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.1d-70) then
        tmp = (z_m * y_m) * (x_m / sqrt((0.0d0 - (t * a))))
    else
        tmp = x_m * (y_m / (1.0d0 + ((t / z_m) * (a * ((-0.5d0) / z_m)))))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.1e-70) {
		tmp = (z_m * y_m) * (x_m / Math.sqrt((0.0 - (t * a))));
	} else {
		tmp = x_m * (y_m / (1.0 + ((t / z_m) * (a * (-0.5 / z_m)))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.1e-70:
		tmp = (z_m * y_m) * (x_m / math.sqrt((0.0 - (t * a))))
	else:
		tmp = x_m * (y_m / (1.0 + ((t / z_m) * (a * (-0.5 / z_m)))))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.1e-70)
		tmp = Float64(Float64(z_m * y_m) * Float64(x_m / sqrt(Float64(0.0 - Float64(t * a)))));
	else
		tmp = Float64(x_m * Float64(y_m / Float64(1.0 + Float64(Float64(t / z_m) * Float64(a * Float64(-0.5 / z_m))))));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.1e-70)
		tmp = (z_m * y_m) * (x_m / sqrt((0.0 - (t * a))));
	else
		tmp = x_m * (y_m / (1.0 + ((t / z_m) * (a * (-0.5 / z_m)))));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.1e-70], N[(N[(z$95$m * y$95$m), $MachinePrecision] * N[(x$95$m / N[Sqrt[N[(0.0 - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(y$95$m / N[(1.0 + N[(N[(t / z$95$m), $MachinePrecision] * N[(a * N[(-0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.0999999999999999e-70

   1. Initial program 64.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot z\right), \color{blue}{\left(\frac{x}{\sqrt{z \cdot z - t \cdot a}}\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \left(\frac{\color{blue}{x}}{\sqrt{z \cdot z - t \cdot a}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \color{blue}{\left(\sqrt{z \cdot z - t \cdot a}\right)}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 64.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 64.5%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot t\right)\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(a \cdot t\right)\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\left(0 - a \cdot t\right)\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot t\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 40.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, t\right)\right)\right)\right)\right) \]

   7. Simplified 40.1%

      \[\leadsto \left(y \cdot z\right) \cdot \frac{x}{\sqrt{\color{blue}{0 - a \cdot t}}} \]

   if 1.0999999999999999e-70 < z

   1. Initial program 63.9%

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

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \color{blue}{\left(z \cdot \left(1 + \frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \color{blue}{\left(1 + \frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{a \cdot t}{{z}^{2}}\right)}\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(a \cdot \color{blue}{\frac{t}{{z}^{2}}}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{t}{{z}^{2}}\right)}\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \color{blue}{\left({z}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left(z \cdot \color{blue}{z}\right)\right)\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 76.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 76.6%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z \cdot \left(1 + -0.5 \cdot \left(a \cdot \frac{t}{z \cdot z}\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

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

      2. associate-*l* N/A

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

      3. associate-/r* N/A

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

      4. *-inverses N/A

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

      5. un-div-inv N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{1 + \frac{-1}{2} \cdot \left(a \cdot \frac{t}{z \cdot z}\right)}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(1 + \frac{-1}{2} \cdot \left(a \cdot \frac{t}{z \cdot z}\right)\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot \left(a \cdot \frac{t}{z \cdot z}\right)\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot \left(\frac{t}{z \cdot z} \cdot \color{blue}{a}\right)\right)\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{2} \cdot \frac{t}{z \cdot z}\right) \cdot \color{blue}{a}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{-1}{2} \cdot \frac{t}{z \cdot z}\right), \color{blue}{a}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{t}{z \cdot z}\right)\right), a\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(t, \left(z \cdot z\right)\right)\right), a\right)\right)\right)\right) \]

      14. *-lowering-*.f64 88.4%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, z\right)\right)\right), a\right)\right)\right)\right) \]

   7. Applied egg-rr 88.4%

      \[\leadsto \color{blue}{x \cdot \frac{y}{1 + \left(-0.5 \cdot \frac{t}{z \cdot z}\right) \cdot a}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(a \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{t}{z \cdot z}\right)}\right)\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(a \cdot \frac{\frac{-1}{2} \cdot t}{\color{blue}{z \cdot z}}\right)\right)\right)\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(a \cdot \left(\frac{\frac{-1}{2}}{z} \cdot \color{blue}{\frac{t}{z}}\right)\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(a \cdot \frac{\frac{-1}{2}}{z}\right) \cdot \color{blue}{\frac{t}{z}}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(a \cdot \frac{\frac{-1}{2}}{z}\right), \color{blue}{\left(\frac{t}{z}\right)}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{\frac{-1}{2}}{z}\right)\right), \left(\frac{\color{blue}{t}}{z}\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\frac{-1}{2}, z\right)\right), \left(\frac{t}{z}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 88.4%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\frac{-1}{2}, z\right)\right), \mathsf{/.f64}\left(t, \color{blue}{z}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 88.4%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.1 \cdot 10^{-70}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{x}{\sqrt{0 - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{1 + \frac{t}{z} \cdot \left(a \cdot \frac{-0.5}{z}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.2% accurate, 5.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.05 \cdot 10^{-52}:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{1 + a \cdot \left(-0.5 \cdot \frac{t}{z\_m \cdot z\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= x_m 1.05e-52)
      (* x_m (/ y_m (+ 1.0 (* a (* -0.5 (/ t (* z_m z_m)))))))
      (* y_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (x_m <= 1.05e-52) {
		tmp = x_m * (y_m / (1.0 + (a * (-0.5 * (t / (z_m * z_m))))));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_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 (x_m <= 1.05d-52) then
        tmp = x_m * (y_m / (1.0d0 + (a * ((-0.5d0) * (t / (z_m * z_m))))))
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (x_m <= 1.05e-52) {
		tmp = x_m * (y_m / (1.0 + (a * (-0.5 * (t / (z_m * z_m))))));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if x_m <= 1.05e-52:
		tmp = x_m * (y_m / (1.0 + (a * (-0.5 * (t / (z_m * z_m))))))
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (x_m <= 1.05e-52)
		tmp = Float64(x_m * Float64(y_m / Float64(1.0 + Float64(a * Float64(-0.5 * Float64(t / Float64(z_m * z_m)))))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (x_m <= 1.05e-52)
		tmp = x_m * (y_m / (1.0 + (a * (-0.5 * (t / (z_m * z_m))))));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 1.05e-52], N[(x$95$m * N[(y$95$m / N[(1.0 + N[(a * N[(-0.5 * N[(t / N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.0499999999999999e-52

   1. Initial program 68.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \color{blue}{\left(z \cdot \left(1 + \frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \color{blue}{\left(1 + \frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{a \cdot t}{{z}^{2}}\right)}\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(a \cdot \color{blue}{\frac{t}{{z}^{2}}}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{t}{{z}^{2}}\right)}\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \color{blue}{\left({z}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left(z \cdot \color{blue}{z}\right)\right)\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 46.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 46.3%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z \cdot \left(1 + -0.5 \cdot \left(a \cdot \frac{t}{z \cdot z}\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

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

      2. associate-*l* N/A

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

      3. associate-/r* N/A

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

      4. *-inverses N/A

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

      5. un-div-inv N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{1 + \frac{-1}{2} \cdot \left(a \cdot \frac{t}{z \cdot z}\right)}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(1 + \frac{-1}{2} \cdot \left(a \cdot \frac{t}{z \cdot z}\right)\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot \left(a \cdot \frac{t}{z \cdot z}\right)\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot \left(\frac{t}{z \cdot z} \cdot \color{blue}{a}\right)\right)\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{2} \cdot \frac{t}{z \cdot z}\right) \cdot \color{blue}{a}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{-1}{2} \cdot \frac{t}{z \cdot z}\right), \color{blue}{a}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{t}{z \cdot z}\right)\right), a\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(t, \left(z \cdot z\right)\right)\right), a\right)\right)\right)\right) \]

      14. *-lowering-*.f64 49.4%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, z\right)\right)\right), a\right)\right)\right)\right) \]

   7. Applied egg-rr 49.4%

      \[\leadsto \color{blue}{x \cdot \frac{y}{1 + \left(-0.5 \cdot \frac{t}{z \cdot z}\right) \cdot a}} \]

   if 1.0499999999999999e-52 < x

   1. Initial program 53.7%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 42.0%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   5. Simplified 42.0%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{-52}:\\ \;\;\;\;x \cdot \frac{y}{1 + a \cdot \left(-0.5 \cdot \frac{t}{z \cdot z}\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.6% accurate, 7.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \left(x\_m \cdot \frac{y\_m}{1 + \frac{-0.5}{\frac{\frac{z\_m}{\frac{t}{z\_m}}}{a}}}\right)\right)\right) \end{array} \]

FPCore

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

C

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

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_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 = z_s * (y_s * (x_s * (x_m * (y_m / (1.0d0 + ((-0.5d0) / ((z_m / (t / z_m)) / a)))))))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	return z_s * (y_s * (x_s * (x_m * (y_m / (1.0 + (-0.5 / ((z_m / (t / z_m)) / a)))))));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	return z_s * (y_s * (x_s * (x_m * (y_m / (1.0 + (-0.5 / ((z_m / (t / z_m)) / a)))))))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	return Float64(z_s * Float64(y_s * Float64(x_s * Float64(x_m * Float64(y_m / Float64(1.0 + Float64(-0.5 / Float64(Float64(z_m / Float64(t / z_m)) / a))))))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = z_s * (y_s * (x_s * (x_m * (y_m / (1.0 + (-0.5 / ((z_m / (t / z_m)) / a)))))));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * N[(x$95$m * N[(y$95$m / N[(1.0 + N[(-0.5 / N[(N[(z$95$m / N[(t / z$95$m), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.4%

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

3. Taylor expanded in z around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \color{blue}{\left(z \cdot \left(1 + \frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}\right)\right)}\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \color{blue}{\left(1 + \frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}\right)}\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}\right)}\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{a \cdot t}{{z}^{2}}\right)}\right)\right)\right)\right) \]

   4. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(a \cdot \color{blue}{\frac{t}{{z}^{2}}}\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{t}{{z}^{2}}\right)}\right)\right)\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \color{blue}{\left({z}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left(z \cdot \color{blue}{z}\right)\right)\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 44.3%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 44.3%

   \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z \cdot \left(1 + -0.5 \cdot \left(a \cdot \frac{t}{z \cdot z}\right)\right)}} \]
6. Step-by-step derivation

   1. associate-/l* N/A

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

   2. associate-*l* N/A

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

   3. associate-/r* N/A

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

   4. *-inverses N/A

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

   5. un-div-inv N/A

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

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{1 + \frac{-1}{2} \cdot \left(a \cdot \frac{t}{z \cdot z}\right)}\right)}\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(1 + \frac{-1}{2} \cdot \left(a \cdot \frac{t}{z \cdot z}\right)\right)}\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot \left(a \cdot \frac{t}{z \cdot z}\right)\right)}\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot \left(\frac{t}{z \cdot z} \cdot \color{blue}{a}\right)\right)\right)\right)\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{2} \cdot \frac{t}{z \cdot z}\right) \cdot \color{blue}{a}\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{-1}{2} \cdot \frac{t}{z \cdot z}\right), \color{blue}{a}\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{t}{z \cdot z}\right)\right), a\right)\right)\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(t, \left(z \cdot z\right)\right)\right), a\right)\right)\right)\right) \]

   14. *-lowering-*.f64 48.1%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, z\right)\right)\right), a\right)\right)\right)\right) \]

7. Applied egg-rr 48.1%

   \[\leadsto \color{blue}{x \cdot \frac{y}{1 + \left(-0.5 \cdot \frac{t}{z \cdot z}\right) \cdot a}} \]
8. Step-by-step derivation

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{y}{1 + \left(\frac{-1}{2} \cdot \frac{t}{z \cdot z}\right) \cdot a}\right), \color{blue}{x}\right) \]

9. Applied egg-rr 48.8%

   \[\leadsto \color{blue}{\frac{y}{1 + \frac{-0.5}{\frac{\frac{z}{\frac{t}{z}}}{a}}} \cdot x} \]

10. Final simplification 48.8%

    \[\leadsto x \cdot \frac{y}{1 + \frac{-0.5}{\frac{\frac{z}{\frac{t}{z}}}{a}}} \]
11. Add Preprocessing

Alternative 8: 79.7% accurate, 7.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \left(x\_m \cdot \frac{y\_m}{1 + \frac{t}{z\_m} \cdot \left(a \cdot \frac{-0.5}{z\_m}\right)}\right)\right)\right) \end{array} \]

FPCore

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

C

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

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_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 = z_s * (y_s * (x_s * (x_m * (y_m / (1.0d0 + ((t / z_m) * (a * ((-0.5d0) / z_m))))))))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	return z_s * (y_s * (x_s * (x_m * (y_m / (1.0 + ((t / z_m) * (a * (-0.5 / z_m))))))));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	return z_s * (y_s * (x_s * (x_m * (y_m / (1.0 + ((t / z_m) * (a * (-0.5 / z_m))))))))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	return Float64(z_s * Float64(y_s * Float64(x_s * Float64(x_m * Float64(y_m / Float64(1.0 + Float64(Float64(t / z_m) * Float64(a * Float64(-0.5 / z_m)))))))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = z_s * (y_s * (x_s * (x_m * (y_m / (1.0 + ((t / z_m) * (a * (-0.5 / z_m))))))));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * N[(x$95$m * N[(y$95$m / N[(1.0 + N[(N[(t / z$95$m), $MachinePrecision] * N[(a * N[(-0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.4%

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

3. Taylor expanded in z around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \color{blue}{\left(z \cdot \left(1 + \frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}\right)\right)}\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \color{blue}{\left(1 + \frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}\right)}\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}\right)}\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{a \cdot t}{{z}^{2}}\right)}\right)\right)\right)\right) \]

   4. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(a \cdot \color{blue}{\frac{t}{{z}^{2}}}\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{t}{{z}^{2}}\right)}\right)\right)\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \color{blue}{\left({z}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left(z \cdot \color{blue}{z}\right)\right)\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 44.3%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 44.3%

   \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z \cdot \left(1 + -0.5 \cdot \left(a \cdot \frac{t}{z \cdot z}\right)\right)}} \]
6. Step-by-step derivation

   1. associate-/l* N/A

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

   2. associate-*l* N/A

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

   3. associate-/r* N/A

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

   4. *-inverses N/A

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

   5. un-div-inv N/A

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

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{1 + \frac{-1}{2} \cdot \left(a \cdot \frac{t}{z \cdot z}\right)}\right)}\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(1 + \frac{-1}{2} \cdot \left(a \cdot \frac{t}{z \cdot z}\right)\right)}\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot \left(a \cdot \frac{t}{z \cdot z}\right)\right)}\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot \left(\frac{t}{z \cdot z} \cdot \color{blue}{a}\right)\right)\right)\right)\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{2} \cdot \frac{t}{z \cdot z}\right) \cdot \color{blue}{a}\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{-1}{2} \cdot \frac{t}{z \cdot z}\right), \color{blue}{a}\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{t}{z \cdot z}\right)\right), a\right)\right)\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(t, \left(z \cdot z\right)\right)\right), a\right)\right)\right)\right) \]

   14. *-lowering-*.f64 48.1%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, z\right)\right)\right), a\right)\right)\right)\right) \]

7. Applied egg-rr 48.1%

   \[\leadsto \color{blue}{x \cdot \frac{y}{1 + \left(-0.5 \cdot \frac{t}{z \cdot z}\right) \cdot a}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(a \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{t}{z \cdot z}\right)}\right)\right)\right)\right) \]

   2. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(a \cdot \frac{\frac{-1}{2} \cdot t}{\color{blue}{z \cdot z}}\right)\right)\right)\right) \]

   3. times-frac N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(a \cdot \left(\frac{\frac{-1}{2}}{z} \cdot \color{blue}{\frac{t}{z}}\right)\right)\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(a \cdot \frac{\frac{-1}{2}}{z}\right) \cdot \color{blue}{\frac{t}{z}}\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(a \cdot \frac{\frac{-1}{2}}{z}\right), \color{blue}{\left(\frac{t}{z}\right)}\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{\frac{-1}{2}}{z}\right)\right), \left(\frac{\color{blue}{t}}{z}\right)\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\frac{-1}{2}, z\right)\right), \left(\frac{t}{z}\right)\right)\right)\right)\right) \]

   8. /-lowering-/.f64 48.8%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\frac{-1}{2}, z\right)\right), \mathsf{/.f64}\left(t, \color{blue}{z}\right)\right)\right)\right)\right) \]

9. Applied egg-rr 48.8%

   \[\leadsto x \cdot \frac{y}{1 + \color{blue}{\left(a \cdot \frac{-0.5}{z}\right) \cdot \frac{t}{z}}} \]

10. Final simplification 48.8%

    \[\leadsto x \cdot \frac{y}{1 + \frac{t}{z} \cdot \left(a \cdot \frac{-0.5}{z}\right)} \]
11. Add Preprocessing

Alternative 9: 77.3% accurate, 8.1× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (* x_s (if (<= (* t a) -1e-206) (* y_m (/ (* z_m x_m) z_m)) (* y_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if ((t * a) <= -1e-206) {
		tmp = y_m * ((z_m * x_m) / z_m);
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_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 ((t * a) <= (-1d-206)) then
        tmp = y_m * ((z_m * x_m) / z_m)
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if ((t * a) <= -1e-206) {
		tmp = y_m * ((z_m * x_m) / z_m);
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if (t * a) <= -1e-206:
		tmp = y_m * ((z_m * x_m) / z_m)
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (Float64(t * a) <= -1e-206)
		tmp = Float64(y_m * Float64(Float64(z_m * x_m) / z_m));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if ((t * a) <= -1e-206)
		tmp = y_m * ((z_m * x_m) / z_m);
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[N[(t * a), $MachinePrecision], -1e-206], N[(y$95$m * N[(N[(z$95$m * x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t a) < -1.00000000000000003e-206

   1. Initial program 68.3%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 33.3%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   5. Simplified 33.3%

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

      1. *-rgt-identity N/A

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

      2. *-inverses N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-rgt-identity N/A

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

      7. times-frac N/A

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

      8. /-rgt-identity N/A

         \[\leadsto \frac{z \cdot x}{z} \cdot y \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{z \cdot x}{z}\right), \color{blue}{y}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(z \cdot x\right), z\right), y\right) \]

      11. *-lowering-*.f64 31.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(z, x\right), z\right), y\right) \]

   7. Applied egg-rr 31.8%

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

   if -1.00000000000000003e-206 < (*.f64 t a)

   1. Initial program 60.3%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 54.6%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   5. Simplified 54.6%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot a \leq -1 \cdot 10^{-206}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.5% accurate, 9.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.8 \cdot 10^{-188}:\\ \;\;\;\;\left(z\_m \cdot x\_m\right) \cdot \frac{y\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (* x_s (if (<= z_m 3.8e-188) (* (* z_m x_m) (/ y_m z_m)) (* y_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.8e-188) {
		tmp = (z_m * x_m) * (y_m / z_m);
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 3.8d-188) then
        tmp = (z_m * x_m) * (y_m / z_m)
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.8e-188) {
		tmp = (z_m * x_m) * (y_m / z_m);
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 3.8e-188:
		tmp = (z_m * x_m) * (y_m / z_m)
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 3.8e-188)
		tmp = Float64(Float64(z_m * x_m) * Float64(y_m / z_m));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 3.8e-188)
		tmp = (z_m * x_m) * (y_m / z_m);
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 3.8e-188], N[(N[(z$95$m * x$95$m), $MachinePrecision] * N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.8e-188

   1. Initial program 62.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot z\right), \color{blue}{\left(\frac{x}{\sqrt{z \cdot z - t \cdot a}}\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \left(\frac{\color{blue}{x}}{\sqrt{z \cdot z - t \cdot a}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \color{blue}{\left(\sqrt{z \cdot z - t \cdot a}\right)}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 63.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 63.3%

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

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \color{blue}{\left(\frac{x}{z}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 18.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{/.f64}\left(x, \color{blue}{z}\right)\right) \]

   7. Simplified 18.5%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

         \[\leadsto z \cdot \frac{x \cdot y}{z} \]

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(z \cdot x\right), \color{blue}{\left(\frac{y}{z}\right)}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, x\right), \left(\frac{\color{blue}{y}}{z}\right)\right) \]

      9. /-lowering-/.f64 15.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, x\right), \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right) \]

   9. Applied egg-rr 15.3%

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

   if 3.8e-188 < z

   1. Initial program 67.4%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 84.1%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   5. Simplified 84.1%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.8 \cdot 10^{-188}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.5% accurate, 37.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.4%

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

3. Taylor expanded in z around inf

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

   1. *-lowering-*.f64 43.6%

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

5. Simplified 43.6%

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

6. Final simplification 43.6%

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

Developer Target 1: 87.4% accurate, 0.9× 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 2024152 
(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)))))