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

Percentage Accurate: 61.0% → 90.9%

Time: 19.6s

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: 61.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 1.85 \cdot 10^{+56}:\\ \;\;\;\;\frac{x_m \cdot \left(z_m \cdot y_m\right)}{\sqrt{z_m \cdot z_m - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m \cdot y_m}{\frac{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z_m}, z_m\right)}{z_m}}\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
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+56)
      (/ (* x_m (* z_m y_m)) (sqrt (- (* z_m z_m) (* t a))))
      (/ (* x_m y_m) (/ (fma -0.5 (* a (/ t z_m)) z_m) 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);
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+56) {
		tmp = (x_m * (z_m * y_m)) / sqrt(((z_m * z_m) - (t * a)));
	} else {
		tmp = (x_m * y_m) / (fma(-0.5, (a * (t / z_m)), z_m) / 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])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.85e+56)
		tmp = Float64(Float64(x_m * Float64(z_m * y_m)) / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a))));
	else
		tmp = Float64(Float64(x_m * y_m) / Float64(fma(-0.5, Float64(a * Float64(t / z_m)), z_m) / z_m));
	end
	return Float64(z_s * Float64(y_s * Float64(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.
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+56], N[(N[(x$95$m * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[(N[(-0.5 * N[(a * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision] + z$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.84999999999999998e56

   1. Initial program 71.6%

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

      1. associate-*l* 70.2%

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

   3. Simplified 70.2%

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

   if 1.84999999999999998e56 < z

   1. Initial program 34.2%

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

   3. Taylor expanded in z around inf 74.6%

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

      1. associate-/l* 91.0%

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

      2. div-inv 91.0%

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

      3. +-commutative 91.0%

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

      4. fma-def 91.0%

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

      5. associate-/l* 98.6%

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

      6. div-inv 98.6%

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

      7. clear-num 98.6%

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

   5. Applied egg-rr 98.6%

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

      1. associate-*r/ 98.6%

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

      2. *-rgt-identity 98.6%

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

   7. Simplified 98.6%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.85 \cdot 10^{+56}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z}, z\right)}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ \begin{array}{l} t_1 := y_m \cdot \frac{x_m}{\frac{\sqrt{t \cdot \left(-a\right)}}{z_m}}\\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 3.7 \cdot 10^{-114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z_m \leq 5.4 \cdot 10^{-79}:\\ \;\;\;\;x_m \cdot \frac{z_m \cdot y_m}{z_m + -0.5 \cdot \frac{t \cdot a}{z_m}}\\ \mathbf{elif}\;z_m \leq 3.2 \cdot 10^{-62}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot y_m\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
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
 (let* ((t_1 (* y_m (/ x_m (/ (sqrt (* t (- a))) z_m)))))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= z_m 3.7e-114)
        t_1
        (if (<= z_m 5.4e-79)
          (* x_m (/ (* z_m y_m) (+ z_m (* -0.5 (/ (* t a) z_m)))))
          (if (<= z_m 3.2e-62) t_1 (* x_m y_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);
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 t_1 = y_m * (x_m / (sqrt((t * -a)) / z_m));
	double tmp;
	if (z_m <= 3.7e-114) {
		tmp = t_1;
	} else if (z_m <= 5.4e-79) {
		tmp = x_m * ((z_m * y_m) / (z_m + (-0.5 * ((t * a) / z_m))));
	} else if (z_m <= 3.2e-62) {
		tmp = t_1;
	} else {
		tmp = x_m * y_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.
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) :: t_1
    real(8) :: tmp
    t_1 = y_m * (x_m / (sqrt((t * -a)) / z_m))
    if (z_m <= 3.7d-114) then
        tmp = t_1
    else if (z_m <= 5.4d-79) then
        tmp = x_m * ((z_m * y_m) / (z_m + ((-0.5d0) * ((t * a) / z_m))))
    else if (z_m <= 3.2d-62) then
        tmp = t_1
    else
        tmp = x_m * y_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;
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 t_1 = y_m * (x_m / (Math.sqrt((t * -a)) / z_m));
	double tmp;
	if (z_m <= 3.7e-114) {
		tmp = t_1;
	} else if (z_m <= 5.4e-79) {
		tmp = x_m * ((z_m * y_m) / (z_m + (-0.5 * ((t * a) / z_m))));
	} else if (z_m <= 3.2e-62) {
		tmp = t_1;
	} else {
		tmp = x_m * y_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])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	t_1 = y_m * (x_m / (math.sqrt((t * -a)) / z_m))
	tmp = 0
	if z_m <= 3.7e-114:
		tmp = t_1
	elif z_m <= 5.4e-79:
		tmp = x_m * ((z_m * y_m) / (z_m + (-0.5 * ((t * a) / z_m))))
	elif z_m <= 3.2e-62:
		tmp = t_1
	else:
		tmp = x_m * y_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])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = Float64(y_m * Float64(x_m / Float64(sqrt(Float64(t * Float64(-a))) / z_m)))
	tmp = 0.0
	if (z_m <= 3.7e-114)
		tmp = t_1;
	elseif (z_m <= 5.4e-79)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / Float64(z_m + Float64(-0.5 * Float64(Float64(t * a) / z_m)))));
	elseif (z_m <= 3.2e-62)
		tmp = t_1;
	else
		tmp = Float64(x_m * y_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])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = y_m * (x_m / (sqrt((t * -a)) / z_m));
	tmp = 0.0;
	if (z_m <= 3.7e-114)
		tmp = t_1;
	elseif (z_m <= 5.4e-79)
		tmp = x_m * ((z_m * y_m) / (z_m + (-0.5 * ((t * a) / z_m))));
	elseif (z_m <= 3.2e-62)
		tmp = t_1;
	else
		tmp = x_m * y_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.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := Block[{t$95$1 = N[(y$95$m * N[(x$95$m / N[(N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 3.7e-114], t$95$1, If[LessEqual[z$95$m, 5.4e-79], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[(z$95$m + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 3.2e-62], t$95$1, N[(x$95$m * y$95$m), $MachinePrecision]]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < 3.69999999999999965e-114 or 5.4000000000000004e-79 < z < 3.20000000000000021e-62

   1. Initial program 68.5%

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

      1. associate-/l* 68.1%

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

      2. associate-*l/ 66.7%

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

      3. *-commutative 66.7%

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

      4. associate-/l* 67.6%

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

   3. Simplified 67.6%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l* 66.7%

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

      2. div-inv 66.7%

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

      3. pow2 66.7%

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

   6. Applied egg-rr 66.7%

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

      1. associate-*r/ 66.7%

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

      2. *-rgt-identity 66.7%

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

      3. *-commutative 66.7%

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

   8. Simplified 66.7%

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

   9. Taylor expanded in z around 0 38.1%

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

       1. mul-1-neg 38.1%

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

       2. distribute-rgt-neg-out 38.1%

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

   11. Simplified 38.1%

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

   if 3.69999999999999965e-114 < z < 5.4000000000000004e-79

   1. Initial program 76.5%

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

   3. Taylor expanded in z around inf 52.9%

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

      1. associate-/l* 76.4%

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

      2. div-inv 76.4%

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

      3. +-commutative 76.4%

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

      4. fma-def 76.4%

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

      5. associate-/l* 76.4%

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

      6. div-inv 76.4%

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

      7. clear-num 76.4%

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

   5. Applied egg-rr 76.4%

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

      1. associate-*r/ 76.4%

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

      2. *-rgt-identity 76.4%

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

   7. Simplified 76.4%

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

   8. Taylor expanded in x around 0 52.9%

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

      1. associate-*r* 52.9%

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

      2. +-commutative 52.9%

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

      3. associate-*r/ 52.9%

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

      4. *-commutative 52.9%

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

      5. fma-udef 52.9%

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

      6. associate-*l/ 76.4%

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

      7. associate-*r/ 76.1%

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

      8. associate-*l* 76.1%

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

      9. *-commutative 76.1%

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

      10. fma-udef 76.1%

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

      11. *-commutative 76.1%

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

      12. fma-def 76.1%

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

   10. Simplified 76.1%

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

   11. Taylor expanded in y around 0 76.4%

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

   if 3.20000000000000021e-62 < z

   1. Initial program 50.0%

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

      1. associate-/l* 51.3%

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

      2. associate-*l/ 50.4%

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

      3. *-commutative 50.4%

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

      4. associate-/l* 48.7%

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

   3. Simplified 48.7%

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

   5. Taylor expanded in z around inf 90.4%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.7 \cdot 10^{-114}:\\ \;\;\;\;y \cdot \frac{x}{\frac{\sqrt{t \cdot \left(-a\right)}}{z}}\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-79}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z + -0.5 \cdot \frac{t \cdot a}{z}}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-62}:\\ \;\;\;\;y \cdot \frac{x}{\frac{\sqrt{t \cdot \left(-a\right)}}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ \begin{array}{l} t_1 := \sqrt{t \cdot \left(-a\right)}\\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 3.6 \cdot 10^{-113}:\\ \;\;\;\;\frac{y_m}{\frac{t_1}{z_m \cdot x_m}}\\ \mathbf{elif}\;z_m \leq 10^{-79}:\\ \;\;\;\;x_m \cdot \frac{z_m \cdot y_m}{z_m + -0.5 \cdot \frac{t \cdot a}{z_m}}\\ \mathbf{elif}\;z_m \leq 1.7 \cdot 10^{-68}:\\ \;\;\;\;y_m \cdot \frac{x_m}{\frac{t_1}{z_m}}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot y_m\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
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
 (let* ((t_1 (sqrt (* t (- a)))))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= z_m 3.6e-113)
        (/ y_m (/ t_1 (* z_m x_m)))
        (if (<= z_m 1e-79)
          (* x_m (/ (* z_m y_m) (+ z_m (* -0.5 (/ (* t a) z_m)))))
          (if (<= z_m 1.7e-68) (* y_m (/ x_m (/ t_1 z_m))) (* x_m y_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);
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 t_1 = sqrt((t * -a));
	double tmp;
	if (z_m <= 3.6e-113) {
		tmp = y_m / (t_1 / (z_m * x_m));
	} else if (z_m <= 1e-79) {
		tmp = x_m * ((z_m * y_m) / (z_m + (-0.5 * ((t * a) / z_m))));
	} else if (z_m <= 1.7e-68) {
		tmp = y_m * (x_m / (t_1 / z_m));
	} else {
		tmp = x_m * y_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.
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) :: t_1
    real(8) :: tmp
    t_1 = sqrt((t * -a))
    if (z_m <= 3.6d-113) then
        tmp = y_m / (t_1 / (z_m * x_m))
    else if (z_m <= 1d-79) then
        tmp = x_m * ((z_m * y_m) / (z_m + ((-0.5d0) * ((t * a) / z_m))))
    else if (z_m <= 1.7d-68) then
        tmp = y_m * (x_m / (t_1 / z_m))
    else
        tmp = x_m * y_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;
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 t_1 = Math.sqrt((t * -a));
	double tmp;
	if (z_m <= 3.6e-113) {
		tmp = y_m / (t_1 / (z_m * x_m));
	} else if (z_m <= 1e-79) {
		tmp = x_m * ((z_m * y_m) / (z_m + (-0.5 * ((t * a) / z_m))));
	} else if (z_m <= 1.7e-68) {
		tmp = y_m * (x_m / (t_1 / z_m));
	} else {
		tmp = x_m * y_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])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	t_1 = math.sqrt((t * -a))
	tmp = 0
	if z_m <= 3.6e-113:
		tmp = y_m / (t_1 / (z_m * x_m))
	elif z_m <= 1e-79:
		tmp = x_m * ((z_m * y_m) / (z_m + (-0.5 * ((t * a) / z_m))))
	elif z_m <= 1.7e-68:
		tmp = y_m * (x_m / (t_1 / z_m))
	else:
		tmp = x_m * y_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])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = sqrt(Float64(t * Float64(-a)))
	tmp = 0.0
	if (z_m <= 3.6e-113)
		tmp = Float64(y_m / Float64(t_1 / Float64(z_m * x_m)));
	elseif (z_m <= 1e-79)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / Float64(z_m + Float64(-0.5 * Float64(Float64(t * a) / z_m)))));
	elseif (z_m <= 1.7e-68)
		tmp = Float64(y_m * Float64(x_m / Float64(t_1 / z_m)));
	else
		tmp = Float64(x_m * y_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])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = sqrt((t * -a));
	tmp = 0.0;
	if (z_m <= 3.6e-113)
		tmp = y_m / (t_1 / (z_m * x_m));
	elseif (z_m <= 1e-79)
		tmp = x_m * ((z_m * y_m) / (z_m + (-0.5 * ((t * a) / z_m))));
	elseif (z_m <= 1.7e-68)
		tmp = y_m * (x_m / (t_1 / z_m));
	else
		tmp = x_m * y_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.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := Block[{t$95$1 = N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 3.6e-113], N[(y$95$m / N[(t$95$1 / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 1e-79], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[(z$95$m + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 1.7e-68], N[(y$95$m * N[(x$95$m / N[(t$95$1 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if z < 3.59999999999999975e-113

   1. Initial program 68.1%

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

      1. associate-*l* 66.4%

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

      2. *-commutative 66.4%

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

      3. associate-*l* 65.4%

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

      4. *-commutative 65.4%

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

      5. associate-/l* 66.8%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in z around 0 38.1%

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

      1. mul-1-neg 38.1%

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

      2. *-commutative 38.1%

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

      3. distribute-rgt-neg-in 38.1%

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

   7. Simplified 38.1%

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

   if 3.59999999999999975e-113 < z < 1e-79

   1. Initial program 76.5%

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

   3. Taylor expanded in z around inf 52.9%

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

      1. associate-/l* 76.4%

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

      2. div-inv 76.4%

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

      3. +-commutative 76.4%

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

      4. fma-def 76.4%

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

      5. associate-/l* 76.4%

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

      6. div-inv 76.4%

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

      7. clear-num 76.4%

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

   5. Applied egg-rr 76.4%

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

      1. associate-*r/ 76.4%

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

      2. *-rgt-identity 76.4%

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

   7. Simplified 76.4%

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

   8. Taylor expanded in x around 0 52.9%

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

      1. associate-*r* 52.9%

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

      2. +-commutative 52.9%

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

      3. associate-*r/ 52.9%

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

      4. *-commutative 52.9%

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

      5. fma-udef 52.9%

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

      6. associate-*l/ 76.4%

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

      7. associate-*r/ 76.1%

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

      8. associate-*l* 76.1%

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

      9. *-commutative 76.1%

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

      10. fma-udef 76.1%

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

      11. *-commutative 76.1%

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

      12. fma-def 76.1%

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

   10. Simplified 76.1%

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

   11. Taylor expanded in y around 0 76.4%

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

   if 1e-79 < z < 1.70000000000000009e-68

   1. Initial program 99.2%

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

      1. associate-/l* 99.2%

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

      2. associate-*l/ 99.2%

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

      3. *-commutative 99.2%

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

      4. associate-/l* 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l* 99.2%

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

      2. div-inv 100.0%

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

      3. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-*r/ 99.2%

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

      2. *-rgt-identity 99.2%

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

      3. *-commutative 99.2%

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

   8. Simplified 99.2%

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

   9. Taylor expanded in z around 0 99.2%

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

       1. mul-1-neg 99.2%

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

       2. distribute-rgt-neg-out 99.2%

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

   11. Simplified 99.2%

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

   if 1.70000000000000009e-68 < z

   1. Initial program 50.0%

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

      1. associate-/l* 51.3%

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

      2. associate-*l/ 50.4%

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

      3. *-commutative 50.4%

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

      4. associate-/l* 48.7%

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

   3. Simplified 48.7%

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

   5. Taylor expanded in z around inf 90.4%

      \[\leadsto y \cdot \color{blue}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.6 \cdot 10^{-113}:\\ \;\;\;\;\frac{y}{\frac{\sqrt{t \cdot \left(-a\right)}}{z \cdot x}}\\ \mathbf{elif}\;z \leq 10^{-79}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z + -0.5 \cdot \frac{t \cdot a}{z}}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-68}:\\ \;\;\;\;y \cdot \frac{x}{\frac{\sqrt{t \cdot \left(-a\right)}}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ \begin{array}{l} t_1 := \frac{x_m \cdot \left(z_m \cdot y_m\right)}{\sqrt{t \cdot \left(-a\right)}}\\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 1.2 \cdot 10^{-112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z_m \leq 2.1 \cdot 10^{-81}:\\ \;\;\;\;x_m \cdot \frac{z_m \cdot y_m}{z_m + -0.5 \cdot \frac{t \cdot a}{z_m}}\\ \mathbf{elif}\;z_m \leq 6.2 \cdot 10^{-65}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot y_m\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
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
 (let* ((t_1 (/ (* x_m (* z_m y_m)) (sqrt (* t (- a))))))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= z_m 1.2e-112)
        t_1
        (if (<= z_m 2.1e-81)
          (* x_m (/ (* z_m y_m) (+ z_m (* -0.5 (/ (* t a) z_m)))))
          (if (<= z_m 6.2e-65) t_1 (* x_m y_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);
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 t_1 = (x_m * (z_m * y_m)) / sqrt((t * -a));
	double tmp;
	if (z_m <= 1.2e-112) {
		tmp = t_1;
	} else if (z_m <= 2.1e-81) {
		tmp = x_m * ((z_m * y_m) / (z_m + (-0.5 * ((t * a) / z_m))));
	} else if (z_m <= 6.2e-65) {
		tmp = t_1;
	} else {
		tmp = x_m * y_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.
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) :: t_1
    real(8) :: tmp
    t_1 = (x_m * (z_m * y_m)) / sqrt((t * -a))
    if (z_m <= 1.2d-112) then
        tmp = t_1
    else if (z_m <= 2.1d-81) then
        tmp = x_m * ((z_m * y_m) / (z_m + ((-0.5d0) * ((t * a) / z_m))))
    else if (z_m <= 6.2d-65) then
        tmp = t_1
    else
        tmp = x_m * y_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;
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 t_1 = (x_m * (z_m * y_m)) / Math.sqrt((t * -a));
	double tmp;
	if (z_m <= 1.2e-112) {
		tmp = t_1;
	} else if (z_m <= 2.1e-81) {
		tmp = x_m * ((z_m * y_m) / (z_m + (-0.5 * ((t * a) / z_m))));
	} else if (z_m <= 6.2e-65) {
		tmp = t_1;
	} else {
		tmp = x_m * y_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])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	t_1 = (x_m * (z_m * y_m)) / math.sqrt((t * -a))
	tmp = 0
	if z_m <= 1.2e-112:
		tmp = t_1
	elif z_m <= 2.1e-81:
		tmp = x_m * ((z_m * y_m) / (z_m + (-0.5 * ((t * a) / z_m))))
	elif z_m <= 6.2e-65:
		tmp = t_1
	else:
		tmp = x_m * y_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])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = Float64(Float64(x_m * Float64(z_m * y_m)) / sqrt(Float64(t * Float64(-a))))
	tmp = 0.0
	if (z_m <= 1.2e-112)
		tmp = t_1;
	elseif (z_m <= 2.1e-81)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / Float64(z_m + Float64(-0.5 * Float64(Float64(t * a) / z_m)))));
	elseif (z_m <= 6.2e-65)
		tmp = t_1;
	else
		tmp = Float64(x_m * y_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])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = (x_m * (z_m * y_m)) / sqrt((t * -a));
	tmp = 0.0;
	if (z_m <= 1.2e-112)
		tmp = t_1;
	elseif (z_m <= 2.1e-81)
		tmp = x_m * ((z_m * y_m) / (z_m + (-0.5 * ((t * a) / z_m))));
	elseif (z_m <= 6.2e-65)
		tmp = t_1;
	else
		tmp = x_m * y_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.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := Block[{t$95$1 = N[(N[(x$95$m * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.2e-112], t$95$1, If[LessEqual[z$95$m, 2.1e-81], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[(z$95$m + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 6.2e-65], t$95$1, N[(x$95$m * y$95$m), $MachinePrecision]]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < 1.2e-112 or 2.0999999999999999e-81 < z < 6.20000000000000032e-65

   1. Initial program 68.5%

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

      1. associate-*l* 66.8%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in z around 0 42.0%

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

      1. mul-1-neg 38.8%

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

      2. *-commutative 38.8%

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

      3. distribute-rgt-neg-in 38.8%

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

   7. Simplified 42.0%

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

   if 1.2e-112 < z < 2.0999999999999999e-81

   1. Initial program 76.5%

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

   3. Taylor expanded in z around inf 52.9%

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

      1. associate-/l* 76.4%

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

      2. div-inv 76.4%

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

      3. +-commutative 76.4%

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

      4. fma-def 76.4%

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

      5. associate-/l* 76.4%

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

      6. div-inv 76.4%

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

      7. clear-num 76.4%

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

   5. Applied egg-rr 76.4%

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

      1. associate-*r/ 76.4%

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

      2. *-rgt-identity 76.4%

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

   7. Simplified 76.4%

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

   8. Taylor expanded in x around 0 52.9%

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

      1. associate-*r* 52.9%

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

      2. +-commutative 52.9%

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

      3. associate-*r/ 52.9%

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

      4. *-commutative 52.9%

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

      5. fma-udef 52.9%

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

      6. associate-*l/ 76.4%

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

      7. associate-*r/ 76.1%

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

      8. associate-*l* 76.1%

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

      9. *-commutative 76.1%

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

      10. fma-udef 76.1%

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

      11. *-commutative 76.1%

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

      12. fma-def 76.1%

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

   10. Simplified 76.1%

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

   11. Taylor expanded in y around 0 76.4%

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

   if 6.20000000000000032e-65 < z

   1. Initial program 50.0%

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

      1. associate-/l* 51.3%

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

      2. associate-*l/ 50.4%

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

      3. *-commutative 50.4%

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

      4. associate-/l* 48.7%

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

   3. Simplified 48.7%

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

   5. Taylor expanded in z around inf 90.4%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-81}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z + -0.5 \cdot \frac{t \cdot a}{z}}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-65}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 2.1 \cdot 10^{-130}:\\ \;\;\;\;\frac{x_m \cdot \left(z_m \cdot y_m\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{elif}\;z_m \leq 2 \cdot 10^{+95}:\\ \;\;\;\;y_m \cdot \frac{z_m \cdot x_m}{\sqrt{z_m \cdot z_m - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot y_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
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 2.1e-130)
      (/ (* x_m (* z_m y_m)) (sqrt (* t (- a))))
      (if (<= z_m 2e+95)
        (* y_m (/ (* z_m x_m) (sqrt (- (* z_m z_m) (* t a)))))
        (* x_m y_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);
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 <= 2.1e-130) {
		tmp = (x_m * (z_m * y_m)) / sqrt((t * -a));
	} else if (z_m <= 2e+95) {
		tmp = y_m * ((z_m * x_m) / sqrt(((z_m * z_m) - (t * a))));
	} else {
		tmp = x_m * y_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.
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 <= 2.1d-130) then
        tmp = (x_m * (z_m * y_m)) / sqrt((t * -a))
    else if (z_m <= 2d+95) then
        tmp = y_m * ((z_m * x_m) / sqrt(((z_m * z_m) - (t * a))))
    else
        tmp = x_m * y_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;
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 <= 2.1e-130) {
		tmp = (x_m * (z_m * y_m)) / Math.sqrt((t * -a));
	} else if (z_m <= 2e+95) {
		tmp = y_m * ((z_m * x_m) / Math.sqrt(((z_m * z_m) - (t * a))));
	} else {
		tmp = x_m * y_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])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 2.1e-130:
		tmp = (x_m * (z_m * y_m)) / math.sqrt((t * -a))
	elif z_m <= 2e+95:
		tmp = y_m * ((z_m * x_m) / math.sqrt(((z_m * z_m) - (t * a))))
	else:
		tmp = x_m * y_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])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.1e-130)
		tmp = Float64(Float64(x_m * Float64(z_m * y_m)) / sqrt(Float64(t * Float64(-a))));
	elseif (z_m <= 2e+95)
		tmp = Float64(y_m * Float64(Float64(z_m * x_m) / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))));
	else
		tmp = Float64(x_m * y_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])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2.1e-130)
		tmp = (x_m * (z_m * y_m)) / sqrt((t * -a));
	elseif (z_m <= 2e+95)
		tmp = y_m * ((z_m * x_m) / sqrt(((z_m * z_m) - (t * a))));
	else
		tmp = x_m * y_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.
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, 2.1e-130], N[(N[(x$95$m * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 2e+95], N[(y$95$m * N[(N[(z$95$m * x$95$m), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 2.10000000000000002e-130

   1. Initial program 67.5%

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

      1. associate-*l* 65.8%

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

   3. Simplified 65.8%

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

   5. Taylor expanded in z around 0 40.2%

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

      1. mul-1-neg 36.9%

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

      2. *-commutative 36.9%

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

      3. distribute-rgt-neg-in 36.9%

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

   7. Simplified 40.2%

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

   if 2.10000000000000002e-130 < z < 2.00000000000000004e95

   1. Initial program 92.7%

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

      1. associate-/l* 95.2%

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

      2. associate-*l/ 92.8%

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

      3. *-commutative 92.8%

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

      4. associate-/l* 92.8%

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

   3. Simplified 92.8%

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

   if 2.00000000000000004e95 < z

   1. Initial program 26.2%

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

      1. associate-/l* 28.2%

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

      2. associate-*l/ 28.5%

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

      3. *-commutative 28.5%

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

      4. associate-/l* 25.9%

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

   3. Simplified 25.9%

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

   5. Taylor expanded in z around inf 98.4%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.1 \cdot 10^{-130}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+95}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 1.26 \cdot 10^{-129}:\\ \;\;\;\;\frac{x_m \cdot \left(z_m \cdot y_m\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{elif}\;z_m \leq 4 \cdot 10^{+95}:\\ \;\;\;\;y_m \cdot \frac{z_m \cdot x_m}{\sqrt{z_m \cdot z_m - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m \cdot y_m}{\frac{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z_m}, z_m\right)}{z_m}}\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
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 1.26e-129)
      (/ (* x_m (* z_m y_m)) (sqrt (* t (- a))))
      (if (<= z_m 4e+95)
        (* y_m (/ (* z_m x_m) (sqrt (- (* z_m z_m) (* t a)))))
        (/ (* x_m y_m) (/ (fma -0.5 (* a (/ t z_m)) z_m) 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);
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 <= 1.26e-129) {
		tmp = (x_m * (z_m * y_m)) / sqrt((t * -a));
	} else if (z_m <= 4e+95) {
		tmp = y_m * ((z_m * x_m) / sqrt(((z_m * z_m) - (t * a))));
	} else {
		tmp = (x_m * y_m) / (fma(-0.5, (a * (t / z_m)), z_m) / 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])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.26e-129)
		tmp = Float64(Float64(x_m * Float64(z_m * y_m)) / sqrt(Float64(t * Float64(-a))));
	elseif (z_m <= 4e+95)
		tmp = Float64(y_m * Float64(Float64(z_m * x_m) / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))));
	else
		tmp = Float64(Float64(x_m * y_m) / Float64(fma(-0.5, Float64(a * Float64(t / z_m)), z_m) / z_m));
	end
	return Float64(z_s * Float64(y_s * Float64(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.
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, 1.26e-129], N[(N[(x$95$m * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 4e+95], N[(y$95$m * N[(N[(z$95$m * x$95$m), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[(N[(-0.5 * N[(a * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision] + z$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 1.2599999999999999e-129

   1. Initial program 67.5%

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

      1. associate-*l* 65.8%

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

   3. Simplified 65.8%

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

   5. Taylor expanded in z around 0 40.2%

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

      1. mul-1-neg 36.9%

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

      2. *-commutative 36.9%

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

      3. distribute-rgt-neg-in 36.9%

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

   7. Simplified 40.2%

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

   if 1.2599999999999999e-129 < z < 4.00000000000000008e95

   1. Initial program 92.7%

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

      1. associate-/l* 95.2%

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

      2. associate-*l/ 92.8%

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

      3. *-commutative 92.8%

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

      4. associate-/l* 92.8%

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

   3. Simplified 92.8%

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

   if 4.00000000000000008e95 < z

   1. Initial program 26.2%

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

   3. Taylor expanded in z around inf 71.5%

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

      1. associate-/l* 89.9%

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

      2. div-inv 89.9%

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

      3. +-commutative 89.9%

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

      4. fma-def 89.9%

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

      5. associate-/l* 98.4%

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

      6. div-inv 98.4%

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

      7. clear-num 98.4%

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

   5. Applied egg-rr 98.4%

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

      1. associate-*r/ 98.4%

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

      2. *-rgt-identity 98.4%

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

   7. Simplified 98.4%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.26 \cdot 10^{-129}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+95}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z}, z\right)}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.5% accurate, 5.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \cdot y_m \leq 4 \cdot 10^{+49}:\\ \;\;\;\;\frac{y_m}{\frac{z_m + -0.5 \cdot \frac{a}{\frac{z_m}{t}}}{z_m \cdot x_m}}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot y_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
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 y_m) 4e+49)
      (/ y_m (/ (+ z_m (* -0.5 (/ a (/ z_m t)))) (* z_m x_m)))
      (* x_m y_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);
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 * y_m) <= 4e+49) {
		tmp = y_m / ((z_m + (-0.5 * (a / (z_m / t)))) / (z_m * x_m));
	} else {
		tmp = x_m * y_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.
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 * y_m) <= 4d+49) then
        tmp = y_m / ((z_m + ((-0.5d0) * (a / (z_m / t)))) / (z_m * x_m))
    else
        tmp = x_m * y_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;
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 * y_m) <= 4e+49) {
		tmp = y_m / ((z_m + (-0.5 * (a / (z_m / t)))) / (z_m * x_m));
	} else {
		tmp = x_m * y_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])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if (x_m * y_m) <= 4e+49:
		tmp = y_m / ((z_m + (-0.5 * (a / (z_m / t)))) / (z_m * x_m))
	else:
		tmp = x_m * y_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])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (Float64(x_m * y_m) <= 4e+49)
		tmp = Float64(y_m / Float64(Float64(z_m + Float64(-0.5 * Float64(a / Float64(z_m / t)))) / Float64(z_m * x_m)));
	else
		tmp = Float64(x_m * y_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])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if ((x_m * y_m) <= 4e+49)
		tmp = y_m / ((z_m + (-0.5 * (a / (z_m / t)))) / (z_m * x_m));
	else
		tmp = x_m * y_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.
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[(x$95$m * y$95$m), $MachinePrecision], 4e+49], N[(y$95$m / N[(N[(z$95$m + N[(-0.5 * N[(a / N[(z$95$m / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 3.99999999999999979e49

   1. Initial program 66.9%

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

      1. associate-*l* 65.0%

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

      2. *-commutative 65.0%

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

      3. associate-*l* 64.7%

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

      4. *-commutative 64.7%

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

      5. associate-/l* 66.0%

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

   3. Simplified 66.0%

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

   5. Taylor expanded in z around inf 46.3%

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

      1. associate-/l* 48.2%

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

   7. Simplified 48.2%

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

   if 3.99999999999999979e49 < (*.f64 x y)

   1. Initial program 42.2%

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

      1. associate-/l* 44.8%

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

      2. associate-*l/ 43.0%

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

      3. *-commutative 43.0%

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

      4. associate-/l* 40.5%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in z around inf 38.3%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq 4 \cdot 10^{+49}:\\ \;\;\;\;\frac{y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 1.5 \cdot 10^{+56}:\\ \;\;\;\;x_m \cdot \frac{z_m \cdot y_m}{z_m + -0.5 \cdot \frac{t \cdot a}{z_m}}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot y_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
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 1.5e+56)
      (* x_m (/ (* z_m y_m) (+ z_m (* -0.5 (/ (* t a) z_m)))))
      (* x_m y_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);
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 <= 1.5e+56) {
		tmp = x_m * ((z_m * y_m) / (z_m + (-0.5 * ((t * a) / z_m))));
	} else {
		tmp = x_m * y_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.
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 <= 1.5d+56) then
        tmp = x_m * ((z_m * y_m) / (z_m + ((-0.5d0) * ((t * a) / z_m))))
    else
        tmp = x_m * y_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;
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 <= 1.5e+56) {
		tmp = x_m * ((z_m * y_m) / (z_m + (-0.5 * ((t * a) / z_m))));
	} else {
		tmp = x_m * y_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])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.5e+56:
		tmp = x_m * ((z_m * y_m) / (z_m + (-0.5 * ((t * a) / z_m))))
	else:
		tmp = x_m * y_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])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.5e+56)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / Float64(z_m + Float64(-0.5 * Float64(Float64(t * a) / z_m)))));
	else
		tmp = Float64(x_m * y_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])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.5e+56)
		tmp = x_m * ((z_m * y_m) / (z_m + (-0.5 * ((t * a) / z_m))));
	else
		tmp = x_m * y_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.
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, 1.5e+56], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[(z$95$m + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.50000000000000003e56

   1. Initial program 71.6%

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

   3. Taylor expanded in z around inf 34.6%

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

      1. associate-/l* 35.2%

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

      2. div-inv 35.2%

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

      3. +-commutative 35.2%

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

      4. fma-def 35.2%

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

      5. associate-/l* 35.2%

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

      6. div-inv 35.2%

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

      7. clear-num 35.2%

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

   5. Applied egg-rr 35.2%

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

      1. associate-*r/ 35.2%

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

      2. *-rgt-identity 35.2%

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

   7. Simplified 35.2%

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

   8. Taylor expanded in x around 0 33.6%

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

      1. associate-*r* 34.6%

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

      2. +-commutative 34.6%

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

      3. associate-*r/ 34.6%

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

      4. *-commutative 34.6%

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

      5. fma-udef 34.6%

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

      6. associate-*l/ 34.8%

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

      7. associate-*r/ 34.8%

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

      8. associate-*l* 34.8%

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

      9. *-commutative 34.8%

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

      10. fma-udef 34.8%

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

      11. *-commutative 34.8%

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

      12. fma-def 34.8%

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

   10. Simplified 34.8%

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

   11. Taylor expanded in y around 0 34.1%

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

   if 1.50000000000000003e56 < z

   1. Initial program 34.2%

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

      1. associate-/l* 36.0%

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

      2. associate-*l/ 36.3%

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

      3. *-commutative 36.3%

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

      4. associate-/l* 34.0%

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

   3. Simplified 34.0%

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

   5. Taylor expanded in z around inf 98.5%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.5 \cdot 10^{+56}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z + -0.5 \cdot \frac{t \cdot a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.3% accurate, 8.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 2.45 \cdot 10^{-92}:\\ \;\;\;\;\frac{y_m}{z_m \cdot \frac{1}{z_m \cdot x_m}}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot y_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
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 2.45e-92) (/ y_m (* z_m (/ 1.0 (* z_m x_m)))) (* x_m y_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);
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 <= 2.45e-92) {
		tmp = y_m / (z_m * (1.0 / (z_m * x_m)));
	} else {
		tmp = x_m * y_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.
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 <= 2.45d-92) then
        tmp = y_m / (z_m * (1.0d0 / (z_m * x_m)))
    else
        tmp = x_m * y_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;
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 <= 2.45e-92) {
		tmp = y_m / (z_m * (1.0 / (z_m * x_m)));
	} else {
		tmp = x_m * y_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])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 2.45e-92:
		tmp = y_m / (z_m * (1.0 / (z_m * x_m)))
	else:
		tmp = x_m * y_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])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.45e-92)
		tmp = Float64(y_m / Float64(z_m * Float64(1.0 / Float64(z_m * x_m))));
	else
		tmp = Float64(x_m * y_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])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2.45e-92)
		tmp = y_m / (z_m * (1.0 / (z_m * x_m)));
	else
		tmp = x_m * y_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.
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, 2.45e-92], N[(y$95$m / N[(z$95$m * N[(1.0 / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.45e-92

   1. Initial program 68.3%

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

      1. associate-*l* 66.7%

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

      2. *-commutative 66.7%

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

      3. associate-*l* 65.7%

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

      4. *-commutative 65.7%

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

      5. associate-/l* 67.6%

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

   3. Simplified 67.6%

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

   5. Taylor expanded in z around inf 24.9%

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

      1. div-inv 24.9%

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

      2. *-commutative 24.9%

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

   7. Applied egg-rr 24.9%

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

   if 2.45e-92 < z

   1. Initial program 51.1%

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

      1. associate-/l* 52.4%

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

      2. associate-*l/ 51.5%

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

      3. *-commutative 51.5%

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

      4. associate-/l* 49.9%

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

   3. Simplified 49.9%

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

   5. Taylor expanded in z around inf 88.5%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.45 \cdot 10^{-92}:\\ \;\;\;\;\frac{y}{z \cdot \frac{1}{z \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.2% accurate, 9.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 2 \cdot 10^{-97}:\\ \;\;\;\;y_m \cdot \frac{z_m \cdot x_m}{z_m}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot y_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
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 2e-97) (* y_m (/ (* z_m x_m) z_m)) (* x_m y_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);
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 <= 2e-97) {
		tmp = y_m * ((z_m * x_m) / z_m);
	} else {
		tmp = x_m * y_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.
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 <= 2d-97) then
        tmp = y_m * ((z_m * x_m) / z_m)
    else
        tmp = x_m * y_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;
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 <= 2e-97) {
		tmp = y_m * ((z_m * x_m) / z_m);
	} else {
		tmp = x_m * y_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])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 2e-97:
		tmp = y_m * ((z_m * x_m) / z_m)
	else:
		tmp = x_m * y_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])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2e-97)
		tmp = Float64(y_m * Float64(Float64(z_m * x_m) / z_m));
	else
		tmp = Float64(x_m * y_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])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2e-97)
		tmp = y_m * ((z_m * x_m) / z_m);
	else
		tmp = x_m * y_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.
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, 2e-97], N[(y$95$m * N[(N[(z$95$m * x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.00000000000000007e-97

   1. Initial program 68.1%

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

      1. associate-/l* 68.3%

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

      2. associate-*l/ 66.9%

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

      3. *-commutative 66.9%

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

      4. associate-/l* 67.8%

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

   3. Simplified 67.8%

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

   5. Taylor expanded in z around inf 25.0%

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

   if 2.00000000000000007e-97 < z

   1. Initial program 51.6%

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

      1. associate-/l* 52.9%

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

      2. associate-*l/ 52.0%

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

      3. *-commutative 52.0%

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

      4. associate-/l* 50.4%

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

   3. Simplified 50.4%

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

   5. Taylor expanded in z around inf 87.6%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-97}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.2% accurate, 37.7× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
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)))))

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);
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)));
}

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.
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)))
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;
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)));
}

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])
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)))

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])
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 * y_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])){:}
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)));
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.
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 * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.2%

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

   1. associate-/l* 62.8%

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

   2. associate-*l/ 61.6%

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

   3. *-commutative 61.6%

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

   4. associate-/l* 61.6%

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

3. Simplified 61.6%

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

5. Taylor expanded in z around inf 41.4%

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

6. Final simplification 41.4%

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

Developer target: 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 2024020 
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z -3.1921305903852764e+46) (- (* y x)) (if (< z 5.976268120920894e+90) (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y)) (* y x)))

  (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))