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

Percentage Accurate: 60.9% → 92.3%

Time: 15.8s

Alternatives: 14

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: 60.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{z_m \cdot \left(y_m \cdot x_m\right)}{\sqrt{z_m \cdot z_m - t \cdot a}}\\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{y_m \cdot x_m}{\frac{z_m + \frac{-0.5}{\frac{\frac{z_m}{t}}{a}}}{z_m}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+207}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot x_m\\ \end{array}\right)\right) \end{array} \end{array} \]

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)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (let* ((t_1 (/ (* z_m (* y_m x_m)) (sqrt (- (* z_m z_m) (* t a))))))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= t_1 0.0)
        (/ (* y_m x_m) (/ (+ z_m (/ -0.5 (/ (/ z_m t) a))) z_m))
        (if (<= t_1 5e+207) t_1 (* y_m x_m))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
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 = (z_m * (y_m * x_m)) / sqrt(((z_m * z_m) - (t * a)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (y_m * x_m) / ((z_m + (-0.5 / ((z_m / t) / a))) / z_m);
	} else if (t_1 <= 5e+207) {
		tmp = t_1;
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
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 = (z_m * (y_m * x_m)) / sqrt(((z_m * z_m) - (t * a)))
    if (t_1 <= 0.0d0) then
        tmp = (y_m * x_m) / ((z_m + ((-0.5d0) / ((z_m / t) / a))) / z_m)
    else if (t_1 <= 5d+207) then
        tmp = t_1
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
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 = (z_m * (y_m * x_m)) / Math.sqrt(((z_m * z_m) - (t * a)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (y_m * x_m) / ((z_m + (-0.5 / ((z_m / t) / a))) / z_m);
	} else if (t_1 <= 5e+207) {
		tmp = t_1;
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	t_1 = (z_m * (y_m * x_m)) / math.sqrt(((z_m * z_m) - (t * a)))
	tmp = 0
	if t_1 <= 0.0:
		tmp = (y_m * x_m) / ((z_m + (-0.5 / ((z_m / t) / a))) / z_m)
	elif t_1 <= 5e+207:
		tmp = t_1
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = Float64(Float64(z_m * Float64(y_m * x_m)) / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(y_m * x_m) / Float64(Float64(z_m + Float64(-0.5 / Float64(Float64(z_m / t) / a))) / z_m));
	elseif (t_1 <= 5e+207)
		tmp = t_1;
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = (z_m * (y_m * x_m)) / sqrt(((z_m * z_m) - (t * a)));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = (y_m * x_m) / ((z_m + (-0.5 / ((z_m / t) / a))) / z_m);
	elseif (t_1 <= 5e+207)
		tmp = t_1;
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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[(z$95$m * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$1, 0.0], N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(z$95$m + N[(-0.5 / N[(N[(z$95$m / t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+207], t$95$1, N[(y$95$m * x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a)))) < 0.0

   1. Initial program 70.5%

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

   2. Taylor expanded in z around inf 56.4%

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

      1. associate-*r/ 56.4%

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

   4. Simplified 56.4%

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

      1. *-un-lft-identity 56.4%

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

      2. associate-/l* 57.6%

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

      3. associate-/l* 57.6%

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

   6. Applied egg-rr 57.6%

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

      1. expm1-log1p-u 36.1%

         \[\leadsto 1 \cdot \frac{x \cdot y}{\frac{z + \frac{-0.5}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{z}{a \cdot t}\right)\right)}}}{z}} \]

      2. expm1-udef 34.1%

         \[\leadsto 1 \cdot \frac{x \cdot y}{\frac{z + \frac{-0.5}{\color{blue}{e^{\mathsf{log1p}\left(\frac{z}{a \cdot t}\right)} - 1}}}{z}} \]

   8. Applied egg-rr 34.1%

      \[\leadsto 1 \cdot \frac{x \cdot y}{\frac{z + \frac{-0.5}{\color{blue}{e^{\mathsf{log1p}\left(\frac{z}{a \cdot t}\right)} - 1}}}{z}} \]
   9. Step-by-step derivation

      1. expm1-def 36.1%

         \[\leadsto 1 \cdot \frac{x \cdot y}{\frac{z + \frac{-0.5}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{z}{a \cdot t}\right)\right)}}}{z}} \]

      2. expm1-log1p 57.6%

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

      3. *-lft-identity 57.6%

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

      4. associate-/l* 57.6%

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

      5. associate-/l* 57.6%

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

      6. associate-/l* 57.6%

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

      7. *-lft-identity 57.6%

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

   10. Simplified 57.6%

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

   if 0.0 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a)))) < 4.9999999999999999e207

   1. Initial program 99.7%

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

   if 4.9999999999999999e207 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))))

   1. Initial program 25.3%

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

      1. associate-*l/ 29.4%

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

      2. *-commutative 29.4%

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

      3. associate-/l* 29.7%

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

      4. associate-/r/ 32.8%

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

      5. associate-/r* 32.1%

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

   3. Simplified 32.1%

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

   4. Taylor expanded in z around inf 49.4%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(y \cdot x\right)}{\sqrt{z \cdot z - t \cdot a}} \leq 0:\\ \;\;\;\;\frac{y \cdot x}{\frac{z + \frac{-0.5}{\frac{\frac{z}{t}}{a}}}{z}}\\ \mathbf{elif}\;\frac{z \cdot \left(y \cdot x\right)}{\sqrt{z \cdot z - t \cdot a}} \leq 5 \cdot 10^{+207}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2: 89.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 1.2 \cdot 10^{+80}:\\ \;\;\;\;\left(z_m \cdot \frac{y_m}{\sqrt{{z_m}^{2} - t \cdot a}}\right) \cdot x_m\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot x_m\\ \end{array}\right)\right) \end{array} \]

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)
(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 1.2e+80)
      (* (* z_m (/ y_m (sqrt (- (pow z_m 2.0) (* t a))))) x_m)
      (* y_m x_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
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 <= 1.2e+80) {
		tmp = (z_m * (y_m / sqrt((pow(z_m, 2.0) - (t * a))))) * x_m;
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
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 <= 1.2d+80) then
        tmp = (z_m * (y_m / sqrt(((z_m ** 2.0d0) - (t * a))))) * x_m
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
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 <= 1.2e+80) {
		tmp = (z_m * (y_m / Math.sqrt((Math.pow(z_m, 2.0) - (t * a))))) * x_m;
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.2e+80:
		tmp = (z_m * (y_m / math.sqrt((math.pow(z_m, 2.0) - (t * a))))) * x_m
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.2e+80)
		tmp = Float64(Float64(z_m * Float64(y_m / sqrt(Float64((z_m ^ 2.0) - Float64(t * a))))) * x_m);
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.2e+80)
		tmp = (z_m * (y_m / sqrt(((z_m ^ 2.0) - (t * a))))) * x_m;
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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, 1.2e+80], N[(N[(z$95$m * N[(y$95$m / N[Sqrt[N[(N[Power[z$95$m, 2.0], $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.1999999999999999e80

   1. Initial program 75.0%

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

      1. associate-*l/ 75.7%

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

      2. *-commutative 75.7%

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

      3. associate-/l* 74.6%

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

      4. associate-/r/ 76.0%

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

      5. associate-/r* 74.9%

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

   3. Simplified 74.9%

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

      1. associate-/r/ 73.1%

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

      2. *-commutative 73.1%

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

      3. associate-*r* 73.4%

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

      4. pow2 73.4%

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

   5. Applied egg-rr 73.4%

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

   if 1.1999999999999999e80 < z

   1. Initial program 30.7%

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

      1. associate-*l/ 32.4%

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

      2. *-commutative 32.4%

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

      3. associate-/l* 29.6%

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

      4. associate-/r/ 31.1%

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

      5. associate-/r* 30.4%

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

   3. Simplified 30.4%

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

   4. Taylor expanded in z around inf 98.3%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{+80}:\\ \;\;\;\;\left(z \cdot \frac{y}{\sqrt{{z}^{2} - t \cdot a}}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3: 86.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 4.6 \cdot 10^{-71}:\\ \;\;\;\;x_m \cdot \left(z_m \cdot \frac{y_m}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{elif}\;z_m \leq 1.16 \cdot 10^{+80}:\\ \;\;\;\;\frac{y_m}{\frac{\sqrt{z_m \cdot z_m - t \cdot a}}{z_m \cdot x_m}}\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot x_m\\ \end{array}\right)\right) \end{array} \]

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)
(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 4.6e-71)
      (* x_m (* z_m (/ y_m (sqrt (* a (- t))))))
      (if (<= z_m 1.16e+80)
        (/ y_m (/ (sqrt (- (* z_m z_m) (* t a))) (* z_m x_m)))
        (* y_m x_m)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
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 <= 4.6e-71) {
		tmp = x_m * (z_m * (y_m / sqrt((a * -t))));
	} else if (z_m <= 1.16e+80) {
		tmp = y_m / (sqrt(((z_m * z_m) - (t * a))) / (z_m * x_m));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
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 <= 4.6d-71) then
        tmp = x_m * (z_m * (y_m / sqrt((a * -t))))
    else if (z_m <= 1.16d+80) then
        tmp = y_m / (sqrt(((z_m * z_m) - (t * a))) / (z_m * x_m))
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
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 <= 4.6e-71) {
		tmp = x_m * (z_m * (y_m / Math.sqrt((a * -t))));
	} else if (z_m <= 1.16e+80) {
		tmp = y_m / (Math.sqrt(((z_m * z_m) - (t * a))) / (z_m * x_m));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 4.6e-71:
		tmp = x_m * (z_m * (y_m / math.sqrt((a * -t))))
	elif z_m <= 1.16e+80:
		tmp = y_m / (math.sqrt(((z_m * z_m) - (t * a))) / (z_m * x_m))
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 4.6e-71)
		tmp = Float64(x_m * Float64(z_m * Float64(y_m / sqrt(Float64(a * Float64(-t))))));
	elseif (z_m <= 1.16e+80)
		tmp = Float64(y_m / Float64(sqrt(Float64(Float64(z_m * z_m) - Float64(t * a))) / Float64(z_m * x_m)));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 4.6e-71)
		tmp = x_m * (z_m * (y_m / sqrt((a * -t))));
	elseif (z_m <= 1.16e+80)
		tmp = y_m / (sqrt(((z_m * z_m) - (t * a))) / (z_m * x_m));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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, 4.6e-71], N[(x$95$m * N[(z$95$m * N[(y$95$m / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 1.16e+80], N[(y$95$m / N[(N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 4.5999999999999997e-71

   1. Initial program 70.8%

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

      1. associate-*l/ 71.7%

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

      2. *-commutative 71.7%

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

      3. associate-/l* 71.1%

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

      4. associate-/r/ 72.8%

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

      5. associate-/r* 70.1%

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

   3. Simplified 70.1%

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

      1. associate-/r/ 68.2%

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

      2. *-commutative 68.2%

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

      3. associate-*r* 68.8%

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

      4. pow2 68.8%

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

   5. Applied egg-rr 68.8%

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

   6. Taylor expanded in z around 0 41.3%

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

      1. neg-mul-1 41.3%

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

      2. distribute-rgt-neg-in 41.3%

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

   8. Simplified 41.3%

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

   if 4.5999999999999997e-71 < z < 1.15999999999999997e80

   1. Initial program 90.9%

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

      1. associate-*l/ 91.0%

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

      2. *-commutative 91.0%

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

      3. associate-/l* 87.9%

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

      4. associate-/r/ 88.0%

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

      5. associate-/r* 92.8%

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

   3. Simplified 92.8%

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

   if 1.15999999999999997e80 < z

   1. Initial program 30.7%

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

      1. associate-*l/ 32.4%

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

      2. *-commutative 32.4%

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

      3. associate-/l* 29.6%

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

      4. associate-/r/ 31.1%

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

      5. associate-/r* 30.4%

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

   3. Simplified 30.4%

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

   4. Taylor expanded in z around inf 98.3%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.6 \cdot 10^{-71}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{elif}\;z \leq 1.16 \cdot 10^{+80}:\\ \;\;\;\;\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 4: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \sqrt{z_m \cdot z_m - t \cdot a}\\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 8.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{x_m \cdot \left(z_m \cdot y_m\right)}{t_1}\\ \mathbf{elif}\;z_m \leq 1.25 \cdot 10^{+80}:\\ \;\;\;\;\frac{y_m}{\frac{t_1}{z_m \cdot x_m}}\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot x_m\\ \end{array}\right)\right) \end{array} \end{array} \]

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)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (let* ((t_1 (sqrt (- (* z_m z_m) (* t a)))))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= z_m 8.8e+29)
        (/ (* x_m (* z_m y_m)) t_1)
        (if (<= z_m 1.25e+80) (/ y_m (/ t_1 (* z_m x_m))) (* y_m x_m))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
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(((z_m * z_m) - (t * a)));
	double tmp;
	if (z_m <= 8.8e+29) {
		tmp = (x_m * (z_m * y_m)) / t_1;
	} else if (z_m <= 1.25e+80) {
		tmp = y_m / (t_1 / (z_m * x_m));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
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(((z_m * z_m) - (t * a)))
    if (z_m <= 8.8d+29) then
        tmp = (x_m * (z_m * y_m)) / t_1
    else if (z_m <= 1.25d+80) then
        tmp = y_m / (t_1 / (z_m * x_m))
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
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(((z_m * z_m) - (t * a)));
	double tmp;
	if (z_m <= 8.8e+29) {
		tmp = (x_m * (z_m * y_m)) / t_1;
	} else if (z_m <= 1.25e+80) {
		tmp = y_m / (t_1 / (z_m * x_m));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	t_1 = math.sqrt(((z_m * z_m) - (t * a)))
	tmp = 0
	if z_m <= 8.8e+29:
		tmp = (x_m * (z_m * y_m)) / t_1
	elif z_m <= 1.25e+80:
		tmp = y_m / (t_1 / (z_m * x_m))
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))
	tmp = 0.0
	if (z_m <= 8.8e+29)
		tmp = Float64(Float64(x_m * Float64(z_m * y_m)) / t_1);
	elseif (z_m <= 1.25e+80)
		tmp = Float64(y_m / Float64(t_1 / Float64(z_m * x_m)));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = sqrt(((z_m * z_m) - (t * a)));
	tmp = 0.0;
	if (z_m <= 8.8e+29)
		tmp = (x_m * (z_m * y_m)) / t_1;
	elseif (z_m <= 1.25e+80)
		tmp = y_m / (t_1 / (z_m * x_m));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 8.8e+29], N[(N[(x$95$m * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z$95$m, 1.25e+80], N[(y$95$m / N[(t$95$1 / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < 8.8000000000000005e29

   1. Initial program 73.9%

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

      1. associate-*l* 68.8%

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

   3. Simplified 68.8%

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

   if 8.8000000000000005e29 < z < 1.2499999999999999e80

   1. Initial program 88.1%

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

      1. associate-*l/ 82.9%

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

      2. *-commutative 82.9%

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

      3. associate-/l* 82.6%

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

      4. associate-/r/ 82.5%

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

      5. associate-/r* 87.4%

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

   3. Simplified 87.4%

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

   if 1.2499999999999999e80 < z

   1. Initial program 30.7%

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

      1. associate-*l/ 32.4%

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

      2. *-commutative 32.4%

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

      3. associate-/l* 29.6%

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

      4. associate-/r/ 31.1%

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

      5. associate-/r* 30.4%

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

   3. Simplified 30.4%

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

   4. Taylor expanded in z around inf 98.3%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+80}:\\ \;\;\;\;\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 5: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 2.7 \cdot 10^{-80}:\\ \;\;\;\;x_m \cdot \left(z_m \cdot \frac{y_m}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot \left(y_m \cdot \frac{z_m}{\mathsf{fma}\left(a \cdot \frac{-0.5}{z_m}, t, z_m\right)}\right)\\ \end{array}\right)\right) \end{array} \]

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)
(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 2.7e-80)
      (* x_m (* z_m (/ y_m (sqrt (* a (- t))))))
      (* x_m (* y_m (/ z_m (fma (* a (/ -0.5 z_m)) t z_m)))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
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 <= 2.7e-80) {
		tmp = x_m * (z_m * (y_m / sqrt((a * -t))));
	} else {
		tmp = x_m * (y_m * (z_m / fma((a * (-0.5 / z_m)), t, z_m)));
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.7e-80)
		tmp = Float64(x_m * Float64(z_m * Float64(y_m / sqrt(Float64(a * Float64(-t))))));
	else
		tmp = Float64(x_m * Float64(y_m * Float64(z_m / fma(Float64(a * Float64(-0.5 / z_m)), t, 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]
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, 2.7e-80], N[(x$95$m * N[(z$95$m * N[(y$95$m / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(y$95$m * N[(z$95$m / N[(N[(a * N[(-0.5 / z$95$m), $MachinePrecision]), $MachinePrecision] * t + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.7000000000000002e-80

   1. Initial program 70.8%

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

      1. associate-*l/ 71.7%

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

      2. *-commutative 71.7%

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

      3. associate-/l* 71.1%

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

      4. associate-/r/ 72.8%

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

      5. associate-/r* 70.1%

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

   3. Simplified 70.1%

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

      1. associate-/r/ 68.2%

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

      2. *-commutative 68.2%

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

      3. associate-*r* 68.8%

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

      4. pow2 68.8%

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

   5. Applied egg-rr 68.8%

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

   6. Taylor expanded in z around 0 41.3%

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

      1. neg-mul-1 41.3%

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

      2. distribute-rgt-neg-in 41.3%

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

   8. Simplified 41.3%

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

   if 2.7000000000000002e-80 < z

   1. Initial program 56.5%

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

      1. associate-*l/ 57.5%

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

      2. *-commutative 57.5%

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

      3. associate-/l* 54.6%

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

      4. associate-/r/ 55.5%

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

      5. associate-/r* 57.1%

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

   3. Simplified 57.1%

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

   4. Taylor expanded in z around inf 74.6%

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

      1. associate-*r/ 74.2%

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

   6. Simplified 74.6%

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

   7. Taylor expanded in a around 0 74.6%

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

      1. associate-/l* 75.9%

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

   9. Simplified 75.9%

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

       1. div-inv 76.1%

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

       2. +-commutative 76.1%

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

       3. fma-def 76.1%

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

       4. associate-/r/ 76.1%

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

       5. *-commutative 76.1%

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

   11. Applied egg-rr 76.1%

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

   12. Taylor expanded in y around 0 69.3%

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

       1. associate-*r* 74.2%

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

       2. associate-*r/ 74.2%

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

       3. associate-*l/ 74.2%

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

       4. +-commutative 74.2%

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

       5. fma-udef 74.2%

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

       6. associate-/l* 86.8%

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

       7. associate-*r/ 86.9%

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

       8. *-rgt-identity 86.9%

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

       9. associate-*r/ 86.9%

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

       10. associate-/r/ 86.6%

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

       11. associate-*l/ 86.9%

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

       12. *-lft-identity 86.9%

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

       13. fma-udef 86.9%

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

       14. associate-*r* 90.9%

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

       15. fma-def 90.9%

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

       16. *-commutative 90.9%

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

   14. Simplified 90.9%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.7 \cdot 10^{-80}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\mathsf{fma}\left(a \cdot \frac{-0.5}{z}, t, z\right)}\right)\\ \end{array} \]

Alternative 6: 83.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 3.3 \cdot 10^{-80}:\\ \;\;\;\;x_m \cdot \left(z_m \cdot \frac{y_m}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m \cdot x_m}{\frac{z_m + \frac{-0.5}{\frac{\frac{z_m}{t}}{a}}}{z_m}}\\ \end{array}\right)\right) \end{array} \]

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)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 3.3e-80)
      (* x_m (* z_m (/ y_m (sqrt (* a (- t))))))
      (/ (* y_m x_m) (/ (+ z_m (/ -0.5 (/ (/ z_m t) a))) 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);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.3e-80) {
		tmp = x_m * (z_m * (y_m / sqrt((a * -t))));
	} else {
		tmp = (y_m * x_m) / ((z_m + (-0.5 / ((z_m / t) / a))) / z_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 3.3d-80) then
        tmp = x_m * (z_m * (y_m / sqrt((a * -t))))
    else
        tmp = (y_m * x_m) / ((z_m + ((-0.5d0) / ((z_m / t) / a))) / z_m)
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.3e-80) {
		tmp = x_m * (z_m * (y_m / Math.sqrt((a * -t))));
	} else {
		tmp = (y_m * x_m) / ((z_m + (-0.5 / ((z_m / t) / a))) / z_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 3.3e-80:
		tmp = x_m * (z_m * (y_m / math.sqrt((a * -t))))
	else:
		tmp = (y_m * x_m) / ((z_m + (-0.5 / ((z_m / t) / a))) / 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)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 3.3e-80)
		tmp = Float64(x_m * Float64(z_m * Float64(y_m / sqrt(Float64(a * Float64(-t))))));
	else
		tmp = Float64(Float64(y_m * x_m) / Float64(Float64(z_m + Float64(-0.5 / Float64(Float64(z_m / t) / a))) / z_m));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 3.3e-80)
		tmp = x_m * (z_m * (y_m / sqrt((a * -t))));
	else
		tmp = (y_m * x_m) / ((z_m + (-0.5 / ((z_m / t) / a))) / z_m);
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 3.3e-80], N[(x$95$m * N[(z$95$m * N[(y$95$m / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(z$95$m + N[(-0.5 / N[(N[(z$95$m / t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.3e-80

   1. Initial program 70.8%

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

      1. associate-*l/ 71.7%

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

      2. *-commutative 71.7%

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

      3. associate-/l* 71.1%

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

      4. associate-/r/ 72.8%

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

      5. associate-/r* 70.1%

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

   3. Simplified 70.1%

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

      1. associate-/r/ 68.2%

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

      2. *-commutative 68.2%

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

      3. associate-*r* 68.8%

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

      4. pow2 68.8%

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

   5. Applied egg-rr 68.8%

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

   6. Taylor expanded in z around 0 41.3%

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

      1. neg-mul-1 41.3%

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

      2. distribute-rgt-neg-in 41.3%

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

   8. Simplified 41.3%

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

   if 3.3e-80 < z

   1. Initial program 56.5%

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

   2. Taylor expanded in z around inf 74.2%

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

      1. associate-*r/ 74.2%

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

   4. Simplified 74.2%

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

      1. *-un-lft-identity 74.2%

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

      2. associate-/l* 86.8%

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

      3. associate-/l* 86.8%

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

   6. Applied egg-rr 86.8%

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

      1. expm1-log1p-u 46.9%

         \[\leadsto 1 \cdot \frac{x \cdot y}{\frac{z + \frac{-0.5}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{z}{a \cdot t}\right)\right)}}}{z}} \]

      2. expm1-udef 39.2%

         \[\leadsto 1 \cdot \frac{x \cdot y}{\frac{z + \frac{-0.5}{\color{blue}{e^{\mathsf{log1p}\left(\frac{z}{a \cdot t}\right)} - 1}}}{z}} \]

   8. Applied egg-rr 39.2%

      \[\leadsto 1 \cdot \frac{x \cdot y}{\frac{z + \frac{-0.5}{\color{blue}{e^{\mathsf{log1p}\left(\frac{z}{a \cdot t}\right)} - 1}}}{z}} \]
   9. Step-by-step derivation

      1. expm1-def 46.9%

         \[\leadsto 1 \cdot \frac{x \cdot y}{\frac{z + \frac{-0.5}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{z}{a \cdot t}\right)\right)}}}{z}} \]

      2. expm1-log1p 86.8%

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

      3. *-lft-identity 86.8%

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

      4. associate-/l* 86.8%

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

      5. associate-/l* 90.9%

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

      6. associate-/l* 90.9%

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

      7. *-lft-identity 90.9%

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

   10. Simplified 90.9%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.3 \cdot 10^{-80}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\frac{z + \frac{-0.5}{\frac{\frac{z}{t}}{a}}}{z}}\\ \end{array} \]

Alternative 7: 76.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 8.4 \cdot 10^{-175}:\\ \;\;\;\;-2 \cdot \frac{x_m}{\frac{a}{y_m} \cdot \left(\frac{1}{z_m} \cdot \frac{t}{z_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot x_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(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 8.4e-175)
      (* -2.0 (/ x_m (* (/ a y_m) (* (/ 1.0 z_m) (/ t z_m)))))
      (* y_m x_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
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 <= 8.4e-175) {
		tmp = -2.0 * (x_m / ((a / y_m) * ((1.0 / z_m) * (t / z_m))));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
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 <= 8.4d-175) then
        tmp = (-2.0d0) * (x_m / ((a / y_m) * ((1.0d0 / z_m) * (t / z_m))))
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
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 <= 8.4e-175) {
		tmp = -2.0 * (x_m / ((a / y_m) * ((1.0 / z_m) * (t / z_m))));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 8.4e-175:
		tmp = -2.0 * (x_m / ((a / y_m) * ((1.0 / z_m) * (t / z_m))))
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 8.4e-175)
		tmp = Float64(-2.0 * Float64(x_m / Float64(Float64(a / y_m) * Float64(Float64(1.0 / z_m) * Float64(t / z_m)))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 8.4e-175)
		tmp = -2.0 * (x_m / ((a / y_m) * ((1.0 / z_m) * (t / z_m))));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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, 8.4e-175], N[(-2.0 * N[(x$95$m / N[(N[(a / y$95$m), $MachinePrecision] * N[(N[(1.0 / z$95$m), $MachinePrecision] * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 8.4e-175

   1. Initial program 68.7%

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

      1. associate-*l/ 69.6%

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

      2. *-commutative 69.6%

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

      3. associate-/l* 68.9%

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

      4. associate-/r/ 70.8%

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

      5. associate-/r* 67.9%

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

   3. Simplified 67.9%

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

   4. Taylor expanded in z around inf 27.4%

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

      1. associate-*r/ 27.4%

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

   6. Simplified 27.4%

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

   7. Taylor expanded in z around 0 22.6%

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

      1. associate-/l* 22.0%

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

      2. times-frac 22.0%

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

   9. Simplified 22.0%

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{\frac{a}{y} \cdot \frac{t}{{z}^{2}}}} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 22.0%

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

       2. unpow2 22.0%

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

       3. times-frac 22.1%

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

   11. Applied egg-rr 22.1%

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

   if 8.4e-175 < z

   1. Initial program 60.9%

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

      1. associate-*l/ 61.7%

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

      2. *-commutative 61.7%

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

      3. associate-/l* 59.1%

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

      4. associate-/r/ 59.9%

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

      5. associate-/r* 61.4%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in z around inf 84.0%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8.4 \cdot 10^{-175}:\\ \;\;\;\;-2 \cdot \frac{x}{\frac{a}{y} \cdot \left(\frac{1}{z} \cdot \frac{t}{z}\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 8: 77.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 5.2 \cdot 10^{+50}:\\ \;\;\;\;z_m \cdot \left(x_m \cdot \frac{y_m}{z_m + \frac{t \cdot \left(a \cdot -0.5\right)}{z_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot x_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(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 5.2e+50)
      (* z_m (* x_m (/ y_m (+ z_m (/ (* t (* a -0.5)) z_m)))))
      (* y_m x_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
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 <= 5.2e+50) {
		tmp = z_m * (x_m * (y_m / (z_m + ((t * (a * -0.5)) / z_m))));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
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 <= 5.2d+50) then
        tmp = z_m * (x_m * (y_m / (z_m + ((t * (a * (-0.5d0))) / z_m))))
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
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 <= 5.2e+50) {
		tmp = z_m * (x_m * (y_m / (z_m + ((t * (a * -0.5)) / z_m))));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 5.2e+50:
		tmp = z_m * (x_m * (y_m / (z_m + ((t * (a * -0.5)) / z_m))))
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 5.2e+50)
		tmp = Float64(z_m * Float64(x_m * Float64(y_m / Float64(z_m + Float64(Float64(t * Float64(a * -0.5)) / z_m)))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 5.2e+50)
		tmp = z_m * (x_m * (y_m / (z_m + ((t * (a * -0.5)) / z_m))));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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, 5.2e+50], N[(z$95$m * N[(x$95$m * N[(y$95$m / N[(z$95$m + N[(N[(t * N[(a * -0.5), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 5.2000000000000004e50

   1. Initial program 74.0%

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

      1. associate-*l/ 75.7%

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

      2. *-commutative 75.7%

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

      3. associate-/l* 74.5%

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

      4. associate-/r/ 75.9%

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

      5. associate-/r* 74.4%

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

   3. Simplified 74.4%

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

   4. Taylor expanded in z around inf 37.8%

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

      1. associate-*r/ 36.9%

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

   6. Simplified 37.8%

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

      1. associate-/r/ 37.9%

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

      2. associate-*r* 38.1%

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

      3. associate-*r* 38.1%

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

   8. Applied egg-rr 38.1%

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

   if 5.2000000000000004e50 < z

   1. Initial program 39.3%

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

      1. associate-*l/ 38.1%

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

      2. *-commutative 38.1%

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

      3. associate-/l* 35.6%

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

      4. associate-/r/ 36.9%

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

      5. associate-/r* 37.5%

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

   3. Simplified 37.5%

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

   4. Taylor expanded in z around inf 97.1%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.2 \cdot 10^{+50}:\\ \;\;\;\;z \cdot \left(x \cdot \frac{y}{z + \frac{t \cdot \left(a \cdot -0.5\right)}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 9: 78.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 1.6 \cdot 10^{+39}:\\ \;\;\;\;\frac{x_m}{\frac{z_m + \frac{t \cdot \left(a \cdot -0.5\right)}{z_m}}{z_m \cdot y_m}}\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot x_m\\ \end{array}\right)\right) \end{array} \]

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)
(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 1.6e+39)
      (/ x_m (/ (+ z_m (/ (* t (* a -0.5)) z_m)) (* z_m y_m)))
      (* y_m x_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
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 <= 1.6e+39) {
		tmp = x_m / ((z_m + ((t * (a * -0.5)) / z_m)) / (z_m * y_m));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
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 <= 1.6d+39) then
        tmp = x_m / ((z_m + ((t * (a * (-0.5d0))) / z_m)) / (z_m * y_m))
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
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 <= 1.6e+39) {
		tmp = x_m / ((z_m + ((t * (a * -0.5)) / z_m)) / (z_m * y_m));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.6e+39:
		tmp = x_m / ((z_m + ((t * (a * -0.5)) / z_m)) / (z_m * y_m))
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.6e+39)
		tmp = Float64(x_m / Float64(Float64(z_m + Float64(Float64(t * Float64(a * -0.5)) / z_m)) / Float64(z_m * y_m)));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.6e+39)
		tmp = x_m / ((z_m + ((t * (a * -0.5)) / z_m)) / (z_m * y_m));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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, 1.6e+39], N[(x$95$m / N[(N[(z$95$m + N[(N[(t * N[(a * -0.5), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] / N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.59999999999999996e39

   1. Initial program 73.9%

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

      1. associate-*l/ 75.1%

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

      2. *-commutative 75.1%

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

      3. associate-/l* 73.9%

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

      4. associate-/r/ 75.4%

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

      5. associate-/r* 73.8%

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

   3. Simplified 73.8%

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

      1. associate-/r/ 71.9%

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

      2. associate-*l/ 71.3%

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

      3. *-commutative 71.3%

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

      4. associate-*l* 68.9%

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

      5. *-commutative 68.9%

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

      6. associate-/l* 70.9%

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

      7. div-inv 71.0%

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

      8. pow2 71.0%

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

      9. *-commutative 71.0%

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

   5. Applied egg-rr 71.0%

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

      1. associate-*r/ 70.9%

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

      2. *-rgt-identity 70.9%

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

      3. *-commutative 70.9%

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

   7. Simplified 70.9%

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

   8. Taylor expanded in z around inf 35.2%

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

      1. associate-*r/ 35.2%

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

      2. associate-*l* 35.2%

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

      3. *-commutative 35.2%

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

   10. Simplified 35.2%

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

   if 1.59999999999999996e39 < z

   1. Initial program 41.5%

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

      1. associate-*l/ 41.7%

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

      2. *-commutative 41.7%

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

      3. associate-/l* 39.3%

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

      4. associate-/r/ 40.6%

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

      5. associate-/r* 41.1%

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

   3. Simplified 41.1%

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

   4. Taylor expanded in z around inf 95.9%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.6 \cdot 10^{+39}:\\ \;\;\;\;\frac{x}{\frac{z + \frac{t \cdot \left(a \cdot -0.5\right)}{z}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 10: 79.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 6.8 \cdot 10^{+79}:\\ \;\;\;\;\frac{y_m \cdot x_m}{\frac{z_m + \frac{-0.5}{\frac{z_m}{t \cdot a}}}{z_m}}\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot x_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 6.8e+79)
      (/ (* y_m x_m) (/ (+ z_m (/ -0.5 (/ z_m (* t a)))) z_m))
      (* y_m x_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 6.8e+79) {
		tmp = (y_m * x_m) / ((z_m + (-0.5 / (z_m / (t * a)))) / z_m);
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 6.8d+79) then
        tmp = (y_m * x_m) / ((z_m + ((-0.5d0) / (z_m / (t * a)))) / z_m)
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 6.8e+79) {
		tmp = (y_m * x_m) / ((z_m + (-0.5 / (z_m / (t * a)))) / z_m);
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 6.8e+79:
		tmp = (y_m * x_m) / ((z_m + (-0.5 / (z_m / (t * a)))) / z_m)
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 6.8e+79)
		tmp = Float64(Float64(y_m * x_m) / Float64(Float64(z_m + Float64(-0.5 / Float64(z_m / Float64(t * a)))) / z_m));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 6.8e+79)
		tmp = (y_m * x_m) / ((z_m + (-0.5 / (z_m / (t * a)))) / z_m);
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 6.8e+79], N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(z$95$m + N[(-0.5 / N[(z$95$m / N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 6.80000000000000063e79

   1. Initial program 75.0%

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

   2. Taylor expanded in z around inf 39.5%

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

      1. associate-*r/ 39.5%

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

   4. Simplified 39.5%

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

      1. *-un-lft-identity 39.5%

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

      2. associate-/l* 40.4%

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

      3. associate-/l* 40.4%

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

   6. Applied egg-rr 40.4%

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

   if 6.80000000000000063e79 < z

   1. Initial program 30.7%

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

      1. associate-*l/ 32.4%

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

      2. *-commutative 32.4%

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

      3. associate-/l* 29.6%

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

      4. associate-/r/ 31.1%

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

      5. associate-/r* 30.4%

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

   3. Simplified 30.4%

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

   4. Taylor expanded in z around inf 98.3%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.8 \cdot 10^{+79}:\\ \;\;\;\;\frac{y \cdot x}{\frac{z + \frac{-0.5}{\frac{z}{t \cdot a}}}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 11: 79.7% accurate, 7.5× speedup

Math

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

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)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (* y_s (* x_s (/ (* y_m x_m) (/ (+ z_m (/ -0.5 (/ (/ z_m t) a))) 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);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	return z_s * (y_s * (x_s * ((y_m * x_m) / ((z_m + (-0.5 / ((z_m / t) / a))) / z_m))));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = z_s * (y_s * (x_s * ((y_m * x_m) / ((z_m + ((-0.5d0) / ((z_m / t) / a))) / z_m))))
end function

Java

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

Python

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

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
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(Float64(y_m * x_m) / Float64(Float64(z_m + Float64(-0.5 / Float64(Float64(z_m / t) / a))) / z_m)))))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = z_s * (y_s * (x_s * ((y_m * x_m) / ((z_m + (-0.5 / ((z_m / t) / a))) / z_m))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(z$95$m + N[(-0.5 / N[(N[(z$95$m / t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.3%

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

2. Taylor expanded in z around inf 45.9%

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

   1. associate-*r/ 45.9%

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

4. Simplified 45.9%

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

   1. *-un-lft-identity 45.9%

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

   2. associate-/l* 50.7%

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

   3. associate-/l* 50.7%

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

6. Applied egg-rr 50.7%

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

   1. expm1-log1p-u 31.7%

      \[\leadsto 1 \cdot \frac{x \cdot y}{\frac{z + \frac{-0.5}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{z}{a \cdot t}\right)\right)}}}{z}} \]

   2. expm1-udef 28.7%

      \[\leadsto 1 \cdot \frac{x \cdot y}{\frac{z + \frac{-0.5}{\color{blue}{e^{\mathsf{log1p}\left(\frac{z}{a \cdot t}\right)} - 1}}}{z}} \]

8. Applied egg-rr 28.7%

   \[\leadsto 1 \cdot \frac{x \cdot y}{\frac{z + \frac{-0.5}{\color{blue}{e^{\mathsf{log1p}\left(\frac{z}{a \cdot t}\right)} - 1}}}{z}} \]
9. Step-by-step derivation

   1. expm1-def 31.7%

      \[\leadsto 1 \cdot \frac{x \cdot y}{\frac{z + \frac{-0.5}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{z}{a \cdot t}\right)\right)}}}{z}} \]

   2. expm1-log1p 50.7%

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

   3. *-lft-identity 50.7%

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

   4. associate-/l* 50.7%

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

   5. associate-/l* 52.3%

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

   6. associate-/l* 52.3%

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

   7. *-lft-identity 52.3%

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

10. Simplified 52.3%

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

11. Final simplification 52.3%

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

Alternative 12: 76.6% accurate, 12.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 1.4 \cdot 10^{-135}:\\ \;\;\;\;\frac{x_m \cdot \left(z_m \cdot y_m\right)}{z_m}\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot x_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(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 1.4e-135) (/ (* x_m (* z_m y_m)) z_m) (* y_m x_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
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 <= 1.4e-135) {
		tmp = (x_m * (z_m * y_m)) / z_m;
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
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 <= 1.4d-135) then
        tmp = (x_m * (z_m * y_m)) / z_m
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
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 <= 1.4e-135) {
		tmp = (x_m * (z_m * y_m)) / z_m;
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.4e-135:
		tmp = (x_m * (z_m * y_m)) / z_m
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.4e-135)
		tmp = Float64(Float64(x_m * Float64(z_m * y_m)) / z_m);
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.4e-135)
		tmp = (x_m * (z_m * y_m)) / z_m;
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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, 1.4e-135], N[(N[(x$95$m * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.40000000000000012e-135

   1. Initial program 69.3%

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

      1. associate-*l* 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in z around inf 22.7%

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

   if 1.40000000000000012e-135 < z

   1. Initial program 59.8%

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

      1. associate-*l/ 60.7%

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

      2. *-commutative 60.7%

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

      3. associate-/l* 58.0%

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

      4. associate-/r/ 58.8%

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

      5. associate-/r* 60.3%

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

   3. Simplified 60.3%

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

   4. Taylor expanded in z around inf 86.2%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.4 \cdot 10^{-135}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 13: 64.9% accurate, 16.1× speedup

Math

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

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)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (* z_s (* y_s (* x_s (/ y_m (/ z_m (* z_m x_m)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	return z_s * (y_s * (x_s * (y_m / (z_m / (z_m * x_m)))));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = z_s * (y_s * (x_s * (y_m / (z_m / (z_m * x_m)))))
end function

Java

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

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	return z_s * (y_s * (x_s * (y_m / (z_m / (z_m * x_m)))))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	return Float64(z_s * Float64(y_s * Float64(x_s * Float64(y_m / Float64(z_m / Float64(z_m * x_m))))))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = z_s * (y_s * (x_s * (y_m / (z_m / (z_m * x_m)))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * N[(y$95$m / N[(z$95$m / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.3%

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

   1. associate-*l/ 66.3%

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

   2. *-commutative 66.3%

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

   3. associate-/l* 64.8%

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

   4. associate-/r/ 66.1%

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

   5. associate-/r* 65.1%

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

3. Simplified 65.1%

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

4. Taylor expanded in z around inf 41.4%

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

5. Final simplification 41.4%

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

Alternative 14: 73.8% accurate, 37.7× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (* z_s (* y_s (* x_s (* y_m x_m)))))

C

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

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = z_s * (y_s * (x_s * (y_m * x_m)))
end function

Java

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

Python

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

Julia

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

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = z_s * (y_s * (x_s * (y_m * x_m)));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.3%

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

   1. associate-*l/ 66.3%

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

   2. *-commutative 66.3%

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

   3. associate-/l* 64.8%

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

   4. associate-/r/ 66.1%

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

   5. associate-/r* 65.1%

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

3. Simplified 65.1%

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

4. Taylor expanded in z around inf 44.9%

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

5. Final simplification 44.9%

   \[\leadsto y \cdot x \]

Developer target: 89.3% accurate, 1.0× 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 2023322 
(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)))))