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

Percentage Accurate: 61.0% → 90.5%

Time: 16.3s

Alternatives: 12

Speedup: 37.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.3 \cdot 10^{+75}:\\ \;\;\;\;\left(z\_m \cdot \frac{y}{\sqrt{{z\_m}^{2} - t \cdot a}}\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\_m\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x_s x_m y z_m t a)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 3.3e+75)
     (* (* z_m (/ y (sqrt (- (pow z_m 2.0) (* t a))))) x_m)
     (* y x_m)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
z_m = fabs(z);
z_s = copysign(1.0, z);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.3e+75) {
		tmp = (z_m * (y / sqrt((pow(z_m, 2.0) - (t * a))))) * x_m;
	} else {
		tmp = y * x_m;
	}
	return z_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
z_m = abs(z)
z_s = copysign(1.0d0, z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, x_s, x_m, y, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 3.3d+75) then
        tmp = (z_m * (y / sqrt(((z_m ** 2.0d0) - (t * a))))) * x_m
    else
        tmp = y * x_m
    end if
    code = z_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
assert x_m < y && y < z_m && z_m < t && t < a;
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.3e+75) {
		tmp = (z_m * (y / Math.sqrt((Math.pow(z_m, 2.0) - (t * a))))) * x_m;
	} else {
		tmp = y * x_m;
	}
	return z_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
[x_m, y, z_m, t, a] = sort([x_m, y, z_m, t, a])
def code(z_s, x_s, x_m, y, z_m, t, a):
	tmp = 0
	if z_m <= 3.3e+75:
		tmp = (z_m * (y / math.sqrt((math.pow(z_m, 2.0) - (t * a))))) * x_m
	else:
		tmp = y * x_m
	return z_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
z_m = abs(z)
z_s = copysign(1.0, z)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 3.3e+75)
		tmp = Float64(Float64(z_m * Float64(y / sqrt(Float64((z_m ^ 2.0) - Float64(t * a))))) * x_m);
	else
		tmp = Float64(y * x_m);
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
x_m, y, z_m, t, a = num2cell(sort([x_m, y, z_m, t, a])){:}
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 3.3e+75)
		tmp = (z_m * (y / sqrt(((z_m ^ 2.0) - (t * a))))) * x_m;
	else
		tmp = y * x_m;
	end
	tmp_2 = z_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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 3.3e+75], N[(N[(z$95$m * N[(y / 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 * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.29999999999999998e75

   1. Initial program 64.7%

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

      1. associate-*l* 64.8%

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

      2. *-commutative 64.8%

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

      3. associate-*l* 64.4%

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

      4. *-commutative 64.4%

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

      5. associate-/l* 65.0%

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

   3. Simplified 65.0%

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

      1. associate-/r/ 64.0%

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

      2. *-commutative 64.0%

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

      3. associate-*r* 64.5%

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

      4. pow2 64.5%

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

   6. Applied egg-rr 64.5%

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

   if 3.29999999999999998e75 < z

   1. Initial program 43.8%

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

      1. associate-/l* 49.0%

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

      2. associate-*l/ 49.1%

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

      3. *-commutative 49.1%

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

      4. associate-/l* 44.0%

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

   3. Simplified 44.0%

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

   5. Taylor expanded in z around inf 100.0%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.3 \cdot 10^{+75}:\\ \;\;\;\;\left(z \cdot \frac{y}{\sqrt{{z}^{2} - t \cdot a}}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5.1 \cdot 10^{-102}:\\ \;\;\;\;\frac{x\_m \cdot \left(z\_m \cdot y\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{elif}\;z\_m \leq 2.9 \cdot 10^{+76}:\\ \;\;\;\;y \cdot \frac{z\_m \cdot x\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\_m\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x_s x_m y z_m t a)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 5.1e-102)
     (/ (* x_m (* z_m y)) (sqrt (* t (- a))))
     (if (<= z_m 2.9e+76)
       (* y (/ (* z_m x_m) (sqrt (- (* z_m z_m) (* t a)))))
       (* y x_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
z_m = fabs(z);
z_s = copysign(1.0, z);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 5.1e-102) {
		tmp = (x_m * (z_m * y)) / sqrt((t * -a));
	} else if (z_m <= 2.9e+76) {
		tmp = y * ((z_m * x_m) / sqrt(((z_m * z_m) - (t * a))));
	} else {
		tmp = y * x_m;
	}
	return z_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
z_m = abs(z)
z_s = copysign(1.0d0, z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, x_s, x_m, y, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 5.1d-102) then
        tmp = (x_m * (z_m * y)) / sqrt((t * -a))
    else if (z_m <= 2.9d+76) then
        tmp = y * ((z_m * x_m) / sqrt(((z_m * z_m) - (t * a))))
    else
        tmp = y * x_m
    end if
    code = z_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
assert x_m < y && y < z_m && z_m < t && t < a;
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 5.1e-102) {
		tmp = (x_m * (z_m * y)) / Math.sqrt((t * -a));
	} else if (z_m <= 2.9e+76) {
		tmp = y * ((z_m * x_m) / Math.sqrt(((z_m * z_m) - (t * a))));
	} else {
		tmp = y * x_m;
	}
	return z_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
[x_m, y, z_m, t, a] = sort([x_m, y, z_m, t, a])
def code(z_s, x_s, x_m, y, z_m, t, a):
	tmp = 0
	if z_m <= 5.1e-102:
		tmp = (x_m * (z_m * y)) / math.sqrt((t * -a))
	elif z_m <= 2.9e+76:
		tmp = y * ((z_m * x_m) / math.sqrt(((z_m * z_m) - (t * a))))
	else:
		tmp = y * x_m
	return z_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
z_m = abs(z)
z_s = copysign(1.0, z)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 5.1e-102)
		tmp = Float64(Float64(x_m * Float64(z_m * y)) / sqrt(Float64(t * Float64(-a))));
	elseif (z_m <= 2.9e+76)
		tmp = Float64(y * Float64(Float64(z_m * x_m) / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))));
	else
		tmp = Float64(y * x_m);
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
x_m, y, z_m, t, a = num2cell(sort([x_m, y, z_m, t, a])){:}
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 5.1e-102)
		tmp = (x_m * (z_m * y)) / sqrt((t * -a));
	elseif (z_m <= 2.9e+76)
		tmp = y * ((z_m * x_m) / sqrt(((z_m * z_m) - (t * a))));
	else
		tmp = y * x_m;
	end
	tmp_2 = z_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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 5.1e-102], N[(N[(x$95$m * N[(z$95$m * y), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 2.9e+76], N[(y * N[(N[(z$95$m * x$95$m), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 5.09999999999999999e-102

   1. Initial program 60.8%

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

      1. associate-*l* 60.4%

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

   3. Simplified 60.4%

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

   5. Taylor expanded in z around 0 43.5%

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

      1. mul-1-neg 44.3%

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

      2. *-commutative 44.3%

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

      3. distribute-rgt-neg-in 44.3%

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

   7. Simplified 43.5%

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

   if 5.09999999999999999e-102 < z < 2.9000000000000002e76

   1. Initial program 81.6%

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

      1. associate-/l* 84.1%

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

      2. associate-*l/ 89.3%

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

      3. *-commutative 89.3%

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

      4. associate-/l* 84.3%

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

   3. Simplified 84.3%

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

   if 2.9000000000000002e76 < z

   1. Initial program 43.8%

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

      1. associate-/l* 49.0%

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

      2. associate-*l/ 49.1%

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

      3. *-commutative 49.1%

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

      4. associate-/l* 44.0%

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

   3. Simplified 44.0%

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

   5. Taylor expanded in z around inf 100.0%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.1 \cdot 10^{-102}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+76}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.45 \cdot 10^{+35}:\\ \;\;\;\;\frac{x\_m \cdot \left(z\_m \cdot y\right)}{\sqrt{z\_m \cdot z\_m - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\_m\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x_s x_m y z_m t a)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 2.45e+35)
     (/ (* x_m (* z_m y)) (sqrt (- (* z_m z_m) (* t a))))
     (* y x_m)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
z_m = fabs(z);
z_s = copysign(1.0, z);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.45e+35) {
		tmp = (x_m * (z_m * y)) / sqrt(((z_m * z_m) - (t * a)));
	} else {
		tmp = y * x_m;
	}
	return z_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
z_m = abs(z)
z_s = copysign(1.0d0, z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, x_s, x_m, y, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 2.45d+35) then
        tmp = (x_m * (z_m * y)) / sqrt(((z_m * z_m) - (t * a)))
    else
        tmp = y * x_m
    end if
    code = z_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
assert x_m < y && y < z_m && z_m < t && t < a;
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.45e+35) {
		tmp = (x_m * (z_m * y)) / Math.sqrt(((z_m * z_m) - (t * a)));
	} else {
		tmp = y * x_m;
	}
	return z_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
[x_m, y, z_m, t, a] = sort([x_m, y, z_m, t, a])
def code(z_s, x_s, x_m, y, z_m, t, a):
	tmp = 0
	if z_m <= 2.45e+35:
		tmp = (x_m * (z_m * y)) / math.sqrt(((z_m * z_m) - (t * a)))
	else:
		tmp = y * x_m
	return z_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
z_m = abs(z)
z_s = copysign(1.0, z)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.45e+35)
		tmp = Float64(Float64(x_m * Float64(z_m * y)) / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a))));
	else
		tmp = Float64(y * x_m);
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
x_m, y, z_m, t, a = num2cell(sort([x_m, y, z_m, t, a])){:}
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2.45e+35)
		tmp = (x_m * (z_m * y)) / sqrt(((z_m * z_m) - (t * a)));
	else
		tmp = y * x_m;
	end
	tmp_2 = z_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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2.45e+35], N[(N[(x$95$m * N[(z$95$m * y), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.45000000000000013e35

   1. Initial program 63.5%

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

      1. associate-*l* 63.6%

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

   3. Simplified 63.6%

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

   if 2.45000000000000013e35 < z

   1. Initial program 50.7%

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

      1. associate-/l* 56.3%

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

      2. associate-*l/ 56.4%

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

      3. *-commutative 56.4%

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

      4. associate-/l* 50.8%

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

   3. Simplified 50.8%

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

   5. Taylor expanded in z around inf 97.2%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.45 \cdot 10^{+35}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 0.0085:\\ \;\;\;\;y \cdot \frac{z\_m \cdot x\_m}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\_m\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x_s x_m y z_m t a)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 0.0085) (* y (/ (* z_m x_m) (sqrt (* t (- a))))) (* y x_m)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
z_m = fabs(z);
z_s = copysign(1.0, z);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 0.0085) {
		tmp = y * ((z_m * x_m) / sqrt((t * -a)));
	} else {
		tmp = y * x_m;
	}
	return z_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
z_m = abs(z)
z_s = copysign(1.0d0, z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, x_s, x_m, y, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 0.0085d0) then
        tmp = y * ((z_m * x_m) / sqrt((t * -a)))
    else
        tmp = y * x_m
    end if
    code = z_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
assert x_m < y && y < z_m && z_m < t && t < a;
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 0.0085) {
		tmp = y * ((z_m * x_m) / Math.sqrt((t * -a)));
	} else {
		tmp = y * x_m;
	}
	return z_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
[x_m, y, z_m, t, a] = sort([x_m, y, z_m, t, a])
def code(z_s, x_s, x_m, y, z_m, t, a):
	tmp = 0
	if z_m <= 0.0085:
		tmp = y * ((z_m * x_m) / math.sqrt((t * -a)))
	else:
		tmp = y * x_m
	return z_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
z_m = abs(z)
z_s = copysign(1.0, z)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 0.0085)
		tmp = Float64(y * Float64(Float64(z_m * x_m) / sqrt(Float64(t * Float64(-a)))));
	else
		tmp = Float64(y * x_m);
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
x_m, y, z_m, t, a = num2cell(sort([x_m, y, z_m, t, a])){:}
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 0.0085)
		tmp = y * ((z_m * x_m) / sqrt((t * -a)));
	else
		tmp = y * x_m;
	end
	tmp_2 = z_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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 0.0085], N[(y * N[(N[(z$95$m * x$95$m), $MachinePrecision] / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.0085000000000000006

   1. Initial program 63.0%

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

      1. associate-/l* 63.8%

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

      2. associate-*l/ 65.9%

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

      3. *-commutative 65.9%

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

      4. associate-/l* 63.4%

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

   3. Simplified 63.4%

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

   5. Taylor expanded in z around 0 45.1%

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

      1. mul-1-neg 45.1%

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

      2. *-commutative 45.1%

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

      3. distribute-rgt-neg-in 45.1%

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

   7. Simplified 45.1%

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

   if 0.0085000000000000006 < z

   1. Initial program 52.7%

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

      1. associate-/l* 57.9%

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

      2. associate-*l/ 58.1%

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

      3. *-commutative 58.1%

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

      4. associate-/l* 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in z around inf 94.9%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.0085:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 0.0085:\\ \;\;\;\;\frac{x\_m \cdot \left(z\_m \cdot y\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\_m\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x_s x_m y z_m t a)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 0.0085) (/ (* x_m (* z_m y)) (sqrt (* t (- a)))) (* y x_m)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
z_m = fabs(z);
z_s = copysign(1.0, z);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 0.0085) {
		tmp = (x_m * (z_m * y)) / sqrt((t * -a));
	} else {
		tmp = y * x_m;
	}
	return z_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
z_m = abs(z)
z_s = copysign(1.0d0, z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, x_s, x_m, y, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 0.0085d0) then
        tmp = (x_m * (z_m * y)) / sqrt((t * -a))
    else
        tmp = y * x_m
    end if
    code = z_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
assert x_m < y && y < z_m && z_m < t && t < a;
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 0.0085) {
		tmp = (x_m * (z_m * y)) / Math.sqrt((t * -a));
	} else {
		tmp = y * x_m;
	}
	return z_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
[x_m, y, z_m, t, a] = sort([x_m, y, z_m, t, a])
def code(z_s, x_s, x_m, y, z_m, t, a):
	tmp = 0
	if z_m <= 0.0085:
		tmp = (x_m * (z_m * y)) / math.sqrt((t * -a))
	else:
		tmp = y * x_m
	return z_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
z_m = abs(z)
z_s = copysign(1.0, z)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 0.0085)
		tmp = Float64(Float64(x_m * Float64(z_m * y)) / sqrt(Float64(t * Float64(-a))));
	else
		tmp = Float64(y * x_m);
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
x_m, y, z_m, t, a = num2cell(sort([x_m, y, z_m, t, a])){:}
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 0.0085)
		tmp = (x_m * (z_m * y)) / sqrt((t * -a));
	else
		tmp = y * x_m;
	end
	tmp_2 = z_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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 0.0085], N[(N[(x$95$m * N[(z$95$m * y), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.0085000000000000006

   1. Initial program 63.0%

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

      1. associate-*l* 63.2%

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

   3. Simplified 63.2%

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

   5. Taylor expanded in z around 0 43.9%

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

      1. mul-1-neg 45.1%

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

      2. *-commutative 45.1%

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

      3. distribute-rgt-neg-in 45.1%

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

   7. Simplified 43.9%

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

   if 0.0085000000000000006 < z

   1. Initial program 52.7%

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

      1. associate-/l* 57.9%

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

      2. associate-*l/ 58.1%

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

      3. *-commutative 58.1%

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

      4. associate-/l* 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in z around inf 94.9%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.0085:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.6% accurate, 5.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \cdot x\_m \leq 2 \cdot 10^{-64}:\\ \;\;\;\;y \cdot \frac{z\_m \cdot x\_m}{z\_m + -0.5 \cdot \left(a \cdot \frac{t}{z\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\_m\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x_s x_m y z_m t a)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= (* y x_m) 2e-64)
     (* y (/ (* z_m x_m) (+ z_m (* -0.5 (* a (/ t z_m))))))
     (* y x_m)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
z_m = fabs(z);
z_s = copysign(1.0, z);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if ((y * x_m) <= 2e-64) {
		tmp = y * ((z_m * x_m) / (z_m + (-0.5 * (a * (t / z_m)))));
	} else {
		tmp = y * x_m;
	}
	return z_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
z_m = abs(z)
z_s = copysign(1.0d0, z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, x_s, x_m, y, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((y * x_m) <= 2d-64) then
        tmp = y * ((z_m * x_m) / (z_m + ((-0.5d0) * (a * (t / z_m)))))
    else
        tmp = y * x_m
    end if
    code = z_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
assert x_m < y && y < z_m && z_m < t && t < a;
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if ((y * x_m) <= 2e-64) {
		tmp = y * ((z_m * x_m) / (z_m + (-0.5 * (a * (t / z_m)))));
	} else {
		tmp = y * x_m;
	}
	return z_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
[x_m, y, z_m, t, a] = sort([x_m, y, z_m, t, a])
def code(z_s, x_s, x_m, y, z_m, t, a):
	tmp = 0
	if (y * x_m) <= 2e-64:
		tmp = y * ((z_m * x_m) / (z_m + (-0.5 * (a * (t / z_m)))))
	else:
		tmp = y * x_m
	return z_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
z_m = abs(z)
z_s = copysign(1.0, z)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0
	if (Float64(y * x_m) <= 2e-64)
		tmp = Float64(y * Float64(Float64(z_m * x_m) / Float64(z_m + Float64(-0.5 * Float64(a * Float64(t / z_m))))));
	else
		tmp = Float64(y * x_m);
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
x_m, y, z_m, t, a = num2cell(sort([x_m, y, z_m, t, a])){:}
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0;
	if ((y * x_m) <= 2e-64)
		tmp = y * ((z_m * x_m) / (z_m + (-0.5 * (a * (t / z_m)))));
	else
		tmp = y * x_m;
	end
	tmp_2 = z_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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x$95$s * If[LessEqual[N[(y * x$95$m), $MachinePrecision], 2e-64], N[(y * N[(N[(z$95$m * x$95$m), $MachinePrecision] / N[(z$95$m + N[(-0.5 * N[(a * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 1.99999999999999993e-64

   1. Initial program 66.4%

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

      1. associate-/l* 68.3%

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

      2. associate-*l/ 68.8%

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

      3. *-commutative 68.8%

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

      4. associate-/l* 64.7%

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

   3. Simplified 64.7%

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

   5. Taylor expanded in z around inf 49.7%

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

      1. associate-/l* 50.3%

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

   7. Simplified 50.3%

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

      1. clear-num 50.3%

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

      2. associate-/r/ 50.3%

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

      3. clear-num 50.3%

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

   9. Applied egg-rr 50.3%

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

   if 1.99999999999999993e-64 < (*.f64 x y)

   1. Initial program 45.0%

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

      1. associate-/l* 47.7%

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

      2. associate-*l/ 51.5%

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

      3. *-commutative 51.5%

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

      4. associate-/l* 50.0%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in z around inf 27.4%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq 2 \cdot 10^{-64}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z + -0.5 \cdot \left(a \cdot \frac{t}{z}\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 9500000:\\ \;\;\;\;\frac{x\_m \cdot \left(z\_m \cdot y\right)}{z\_m + -0.5 \cdot \frac{t \cdot a}{z\_m}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\_m\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x_s x_m y z_m t a)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 9500000.0)
     (/ (* x_m (* z_m y)) (+ z_m (* -0.5 (/ (* t a) z_m))))
     (* y x_m)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
z_m = fabs(z);
z_s = copysign(1.0, z);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 9500000.0) {
		tmp = (x_m * (z_m * y)) / (z_m + (-0.5 * ((t * a) / z_m)));
	} else {
		tmp = y * x_m;
	}
	return z_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
z_m = abs(z)
z_s = copysign(1.0d0, z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, x_s, x_m, y, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 9500000.0d0) then
        tmp = (x_m * (z_m * y)) / (z_m + ((-0.5d0) * ((t * a) / z_m)))
    else
        tmp = y * x_m
    end if
    code = z_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
assert x_m < y && y < z_m && z_m < t && t < a;
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 9500000.0) {
		tmp = (x_m * (z_m * y)) / (z_m + (-0.5 * ((t * a) / z_m)));
	} else {
		tmp = y * x_m;
	}
	return z_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
[x_m, y, z_m, t, a] = sort([x_m, y, z_m, t, a])
def code(z_s, x_s, x_m, y, z_m, t, a):
	tmp = 0
	if z_m <= 9500000.0:
		tmp = (x_m * (z_m * y)) / (z_m + (-0.5 * ((t * a) / z_m)))
	else:
		tmp = y * x_m
	return z_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
z_m = abs(z)
z_s = copysign(1.0, z)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 9500000.0)
		tmp = Float64(Float64(x_m * Float64(z_m * y)) / Float64(z_m + Float64(-0.5 * Float64(Float64(t * a) / z_m))));
	else
		tmp = Float64(y * x_m);
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
x_m, y, z_m, t, a = num2cell(sort([x_m, y, z_m, t, a])){:}
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 9500000.0)
		tmp = (x_m * (z_m * y)) / (z_m + (-0.5 * ((t * a) / z_m)));
	else
		tmp = y * x_m;
	end
	tmp_2 = z_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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 9500000.0], N[(N[(x$95$m * N[(z$95$m * y), $MachinePrecision]), $MachinePrecision] / N[(z$95$m + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 9.5e6

   1. Initial program 63.0%

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

      1. associate-*l* 63.2%

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

   3. Simplified 63.2%

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

   5. Taylor expanded in z around inf 26.9%

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

   if 9.5e6 < z

   1. Initial program 52.7%

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

      1. associate-/l* 57.9%

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

      2. associate-*l/ 58.1%

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

      3. *-commutative 58.1%

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

      4. associate-/l* 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in z around inf 94.9%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9500000:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{z + -0.5 \cdot \frac{t \cdot a}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 15200000:\\ \;\;\;\;\frac{z\_m \cdot \left(y \cdot x\_m\right)}{z\_m + -0.5 \cdot \frac{t \cdot a}{z\_m}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\_m\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x_s x_m y z_m t a)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 15200000.0)
     (/ (* z_m (* y x_m)) (+ z_m (* -0.5 (/ (* t a) z_m))))
     (* y x_m)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
z_m = fabs(z);
z_s = copysign(1.0, z);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 15200000.0) {
		tmp = (z_m * (y * x_m)) / (z_m + (-0.5 * ((t * a) / z_m)));
	} else {
		tmp = y * x_m;
	}
	return z_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
z_m = abs(z)
z_s = copysign(1.0d0, z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, x_s, x_m, y, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 15200000.0d0) then
        tmp = (z_m * (y * x_m)) / (z_m + ((-0.5d0) * ((t * a) / z_m)))
    else
        tmp = y * x_m
    end if
    code = z_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
assert x_m < y && y < z_m && z_m < t && t < a;
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 15200000.0) {
		tmp = (z_m * (y * x_m)) / (z_m + (-0.5 * ((t * a) / z_m)));
	} else {
		tmp = y * x_m;
	}
	return z_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
[x_m, y, z_m, t, a] = sort([x_m, y, z_m, t, a])
def code(z_s, x_s, x_m, y, z_m, t, a):
	tmp = 0
	if z_m <= 15200000.0:
		tmp = (z_m * (y * x_m)) / (z_m + (-0.5 * ((t * a) / z_m)))
	else:
		tmp = y * x_m
	return z_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
z_m = abs(z)
z_s = copysign(1.0, z)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 15200000.0)
		tmp = Float64(Float64(z_m * Float64(y * x_m)) / Float64(z_m + Float64(-0.5 * Float64(Float64(t * a) / z_m))));
	else
		tmp = Float64(y * x_m);
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
x_m, y, z_m, t, a = num2cell(sort([x_m, y, z_m, t, a])){:}
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 15200000.0)
		tmp = (z_m * (y * x_m)) / (z_m + (-0.5 * ((t * a) / z_m)));
	else
		tmp = y * x_m;
	end
	tmp_2 = z_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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 15200000.0], N[(N[(z$95$m * N[(y * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(z$95$m + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.52e7

   1. Initial program 63.0%

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

   3. Taylor expanded in z around inf 27.4%

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

   if 1.52e7 < z

   1. Initial program 52.7%

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

      1. associate-/l* 57.9%

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

      2. associate-*l/ 58.1%

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

      3. *-commutative 58.1%

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

      4. associate-/l* 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in z around inf 94.9%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 15200000:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{z + -0.5 \cdot \frac{t \cdot a}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.7% accurate, 8.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 8.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{y}{z\_m \cdot \frac{1}{z\_m \cdot x\_m}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\_m\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x_s x_m y z_m t a)
 :precision binary64
 (*
  z_s
  (* x_s (if (<= z_m 8.2e-105) (/ y (* z_m (/ 1.0 (* z_m x_m)))) (* y x_m)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
z_m = fabs(z);
z_s = copysign(1.0, z);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 8.2e-105) {
		tmp = y / (z_m * (1.0 / (z_m * x_m)));
	} else {
		tmp = y * x_m;
	}
	return z_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
z_m = abs(z)
z_s = copysign(1.0d0, z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, x_s, x_m, y, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 8.2d-105) then
        tmp = y / (z_m * (1.0d0 / (z_m * x_m)))
    else
        tmp = y * x_m
    end if
    code = z_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
assert x_m < y && y < z_m && z_m < t && t < a;
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 8.2e-105) {
		tmp = y / (z_m * (1.0 / (z_m * x_m)));
	} else {
		tmp = y * x_m;
	}
	return z_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
[x_m, y, z_m, t, a] = sort([x_m, y, z_m, t, a])
def code(z_s, x_s, x_m, y, z_m, t, a):
	tmp = 0
	if z_m <= 8.2e-105:
		tmp = y / (z_m * (1.0 / (z_m * x_m)))
	else:
		tmp = y * x_m
	return z_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
z_m = abs(z)
z_s = copysign(1.0, z)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 8.2e-105)
		tmp = Float64(y / Float64(z_m * Float64(1.0 / Float64(z_m * x_m))));
	else
		tmp = Float64(y * x_m);
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
x_m, y, z_m, t, a = num2cell(sort([x_m, y, z_m, t, a])){:}
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 8.2e-105)
		tmp = y / (z_m * (1.0 / (z_m * x_m)));
	else
		tmp = y * x_m;
	end
	tmp_2 = z_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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 8.2e-105], N[(y / N[(z$95$m * N[(1.0 / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 8.20000000000000061e-105

   1. Initial program 60.6%

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

      1. associate-*l* 60.2%

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

      2. *-commutative 60.2%

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

      3. associate-*l* 60.3%

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

      4. *-commutative 60.3%

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

      5. associate-/l* 60.4%

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

   3. Simplified 60.4%

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

   5. Taylor expanded in z around inf 20.8%

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

      1. div-inv 20.8%

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

      2. *-commutative 20.8%

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

   7. Applied egg-rr 20.8%

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

   if 8.20000000000000061e-105 < z

   1. Initial program 59.0%

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

      1. associate-/l* 63.1%

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

      2. associate-*l/ 65.1%

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

      3. *-commutative 65.1%

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

      4. associate-/l* 60.1%

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

   3. Simplified 60.1%

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

   5. Taylor expanded in z around inf 85.4%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{y}{z \cdot \frac{1}{z \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.15 \cdot 10^{-108}:\\ \;\;\;\;y \cdot \frac{z\_m \cdot x\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\_m\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x_s x_m y z_m t a)
 :precision binary64
 (* z_s (* x_s (if (<= z_m 1.15e-108) (* y (/ (* z_m x_m) z_m)) (* y x_m)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
z_m = fabs(z);
z_s = copysign(1.0, z);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.15e-108) {
		tmp = y * ((z_m * x_m) / z_m);
	} else {
		tmp = y * x_m;
	}
	return z_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
z_m = abs(z)
z_s = copysign(1.0d0, z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, x_s, x_m, y, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.15d-108) then
        tmp = y * ((z_m * x_m) / z_m)
    else
        tmp = y * x_m
    end if
    code = z_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
assert x_m < y && y < z_m && z_m < t && t < a;
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.15e-108) {
		tmp = y * ((z_m * x_m) / z_m);
	} else {
		tmp = y * x_m;
	}
	return z_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
[x_m, y, z_m, t, a] = sort([x_m, y, z_m, t, a])
def code(z_s, x_s, x_m, y, z_m, t, a):
	tmp = 0
	if z_m <= 1.15e-108:
		tmp = y * ((z_m * x_m) / z_m)
	else:
		tmp = y * x_m
	return z_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
z_m = abs(z)
z_s = copysign(1.0, z)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.15e-108)
		tmp = Float64(y * Float64(Float64(z_m * x_m) / z_m));
	else
		tmp = Float64(y * x_m);
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
x_m, y, z_m, t, a = num2cell(sort([x_m, y, z_m, t, a])){:}
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.15e-108)
		tmp = y * ((z_m * x_m) / z_m);
	else
		tmp = y * x_m;
	end
	tmp_2 = z_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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.15e-108], N[(y * N[(N[(z$95$m * x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(y * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.14999999999999998e-108

   1. Initial program 60.6%

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

      1. associate-/l* 61.5%

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

      2. associate-*l/ 62.7%

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

      3. *-commutative 62.7%

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

      4. associate-/l* 60.4%

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

   3. Simplified 60.4%

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

   5. Taylor expanded in z around inf 20.8%

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

   if 1.14999999999999998e-108 < z

   1. Initial program 59.0%

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

      1. associate-/l* 63.1%

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

      2. associate-*l/ 65.1%

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

      3. *-commutative 65.1%

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

      4. associate-/l* 60.1%

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

   3. Simplified 60.1%

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

   5. Taylor expanded in z around inf 85.4%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.15 \cdot 10^{-108}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.9% accurate, 12.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x_s x_m y z_m t a)
 :precision binary64
 (* z_s (* x_s (/ y (* z_m (/ (/ 1.0 z_m) x_m))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
x_m, y, z_m, t, a = num2cell(sort([x_m, y, z_m, t, a])){:}
function tmp = code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = z_s * (x_s * (y / (z_m * ((1.0 / 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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x$95$s * N[(y / N[(z$95$m * N[(N[(1.0 / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.0%

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

   1. associate-*l* 58.9%

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

   2. *-commutative 58.9%

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

   3. associate-*l* 59.4%

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

   4. *-commutative 59.4%

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

   5. associate-/l* 60.2%

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

3. Simplified 60.2%

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

5. Taylor expanded in z around inf 41.2%

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

   1. div-inv 41.2%

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

   2. *-commutative 41.2%

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

7. Applied egg-rr 41.2%

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

   1. *-un-lft-identity 41.2%

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

   2. associate-/r* 41.3%

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

9. Applied egg-rr 41.3%

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

10. Final simplification 41.3%

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

Alternative 12: 72.8% accurate, 37.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
x_m, y, z_m, t, a = num2cell(sort([x_m, y, z_m, t, a])){:}
function tmp = code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = z_s * (x_s * (y * 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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x$95$s * N[(y * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.0%

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

   1. associate-/l* 62.1%

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

   2. associate-*l/ 63.6%

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

   3. *-commutative 63.6%

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

   4. associate-/l* 60.3%

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

3. Simplified 60.3%

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

5. Taylor expanded in z around inf 41.5%

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

6. Final simplification 41.5%

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

Developer target: 88.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z < -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z < 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (< z -3.1921305903852764e+46)
   (- (* y x))
   (if (< z 5.976268120920894e+90)
     (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y))
     (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z < -3.1921305903852764e+46) {
		tmp = -(y * x);
	} else if (z < 5.976268120920894e+90) {
		tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z < (-3.1921305903852764d+46)) then
        tmp = -(y * x)
    else if (z < 5.976268120920894d+90) then
        tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z < -3.1921305903852764e+46) {
		tmp = -(y * x);
	} else if (z < 5.976268120920894e+90) {
		tmp = (x * z) / (Math.sqrt(((z * z) - (a * t))) / y);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z < -3.1921305903852764e+46:
		tmp = -(y * x)
	elif z < 5.976268120920894e+90:
		tmp = (x * z) / (math.sqrt(((z * z) - (a * t))) / y)
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z < -3.1921305903852764e+46)
		tmp = Float64(-Float64(y * x));
	elseif (z < 5.976268120920894e+90)
		tmp = Float64(Float64(x * z) / Float64(sqrt(Float64(Float64(z * z) - Float64(a * t))) / y));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z < -3.1921305903852764e+46)
		tmp = -(y * x);
	elseif (z < 5.976268120920894e+90)
		tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Less[z, -3.1921305903852764e+46], (-N[(y * x), $MachinePrecision]), If[Less[z, 5.976268120920894e+90], N[(N[(x * z), $MachinePrecision] / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024031 
(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)))))