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

Percentage Accurate: 61.8% → 91.9%

Time: 19.0s

Alternatives: 7

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: 61.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 10^{+119}:\\ \;\;\;\;x\_m \cdot \left(y \cdot \frac{z\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(y \cdot \frac{z\_m}{z\_m + a \cdot \left(t \cdot \frac{-0.5}{z\_m}\right)}\right)\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) 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 1e+119)
     (* x_m (* y (/ z_m (sqrt (- (* z_m z_m) (* t a))))))
     (* x_m (* y (/ z_m (+ z_m (* a (* t (/ -0.5 z_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 <= 1e+119) {
		tmp = x_m * (y * (z_m / sqrt(((z_m * z_m) - (t * a)))));
	} else {
		tmp = x_m * (y * (z_m / (z_m + (a * (t * (-0.5 / z_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 <= 1d+119) then
        tmp = x_m * (y * (z_m / sqrt(((z_m * z_m) - (t * a)))))
    else
        tmp = x_m * (y * (z_m / (z_m + (a * (t * ((-0.5d0) / z_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 <= 1e+119) {
		tmp = x_m * (y * (z_m / Math.sqrt(((z_m * z_m) - (t * a)))));
	} else {
		tmp = x_m * (y * (z_m / (z_m + (a * (t * (-0.5 / z_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 <= 1e+119:
		tmp = x_m * (y * (z_m / math.sqrt(((z_m * z_m) - (t * a)))))
	else:
		tmp = x_m * (y * (z_m / (z_m + (a * (t * (-0.5 / z_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 <= 1e+119)
		tmp = Float64(x_m * Float64(y * Float64(z_m / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a))))));
	else
		tmp = Float64(x_m * Float64(y * Float64(z_m / Float64(z_m + Float64(a * Float64(t * Float64(-0.5 / z_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 <= 1e+119)
		tmp = x_m * (y * (z_m / sqrt(((z_m * z_m) - (t * a)))));
	else
		tmp = x_m * (y * (z_m / (z_m + (a * (t * (-0.5 / z_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, 1e+119], N[(x$95$m * N[(y * N[(z$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(y * N[(z$95$m / N[(z$95$m + N[(a * N[(t * N[(-0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 9.99999999999999944e118

   1. Initial program 70.4%

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

      1. associate-/l* 75.5%

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

      2. associate-*l* 75.2%

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

   3. Simplified 75.2%

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

   if 9.99999999999999944e118 < z

   1. Initial program 20.2%

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

      1. associate-/l* 24.2%

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

      2. associate-*l* 24.4%

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

   3. Simplified 24.4%

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

   5. Taylor expanded in t around 0 86.7%

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

      1. associate-*r/ 86.7%

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

   7. Simplified 86.7%

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

   8. Taylor expanded in a around 0 86.7%

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

      1. associate-*r/ 86.7%

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

      2. *-commutative 86.7%

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

      3. associate-*r/ 86.7%

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

      4. associate-*l* 98.7%

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

   10. Simplified 98.7%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{+119}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + a \cdot \left(t \cdot \frac{-0.5}{z}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 9 \cdot 10^{-111}:\\ \;\;\;\;x\_m \cdot \left(y \cdot \frac{z\_m}{\sqrt{t \cdot \left(-a\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(y \cdot \frac{z\_m}{z\_m + a \cdot \left(t \cdot \frac{-0.5}{z\_m}\right)}\right)\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) 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 9e-111)
     (* x_m (* y (/ z_m (sqrt (* t (- a))))))
     (* x_m (* y (/ z_m (+ z_m (* a (* t (/ -0.5 z_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 <= 9e-111) {
		tmp = x_m * (y * (z_m / sqrt((t * -a))));
	} else {
		tmp = x_m * (y * (z_m / (z_m + (a * (t * (-0.5 / z_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 <= 9d-111) then
        tmp = x_m * (y * (z_m / sqrt((t * -a))))
    else
        tmp = x_m * (y * (z_m / (z_m + (a * (t * ((-0.5d0) / z_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 <= 9e-111) {
		tmp = x_m * (y * (z_m / Math.sqrt((t * -a))));
	} else {
		tmp = x_m * (y * (z_m / (z_m + (a * (t * (-0.5 / z_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 <= 9e-111:
		tmp = x_m * (y * (z_m / math.sqrt((t * -a))))
	else:
		tmp = x_m * (y * (z_m / (z_m + (a * (t * (-0.5 / z_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 <= 9e-111)
		tmp = Float64(x_m * Float64(y * Float64(z_m / sqrt(Float64(t * Float64(-a))))));
	else
		tmp = Float64(x_m * Float64(y * Float64(z_m / Float64(z_m + Float64(a * Float64(t * Float64(-0.5 / z_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 <= 9e-111)
		tmp = x_m * (y * (z_m / sqrt((t * -a))));
	else
		tmp = x_m * (y * (z_m / (z_m + (a * (t * (-0.5 / z_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, 9e-111], N[(x$95$m * N[(y * N[(z$95$m / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(y * N[(z$95$m / N[(z$95$m + N[(a * N[(t * N[(-0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 8.99999999999999987e-111

   1. Initial program 64.8%

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

      1. associate-/l* 69.1%

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

      2. associate-*l* 69.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in z around 0 41.3%

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

      1. associate-*r* 41.3%

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

      2. neg-mul-1 41.3%

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

   7. Simplified 41.3%

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

   if 8.99999999999999987e-111 < z

   1. Initial program 53.5%

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

      1. associate-/l* 59.3%

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

      2. associate-*l* 58.4%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in t around 0 84.2%

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

      1. associate-*r/ 84.2%

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

   7. Simplified 84.2%

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

   8. Taylor expanded in a around 0 84.2%

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

      1. associate-*r/ 84.2%

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

      2. *-commutative 84.2%

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

      3. associate-*r/ 84.2%

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

      4. associate-*l* 90.4%

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

   10. Simplified 90.4%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9 \cdot 10^{-111}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{t \cdot \left(-a\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + a \cdot \left(t \cdot \frac{-0.5}{z}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.8% accurate, 6.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 6.4 \cdot 10^{-187}:\\ \;\;\;\;y \cdot \left(x\_m \cdot \frac{z\_m}{t \cdot \frac{a \cdot -0.5}{z\_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) 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 6.4e-187)
     (* y (* x_m (/ z_m (* t (/ (* a -0.5) z_m)))))
     (* x_m y)))))

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 <= 6.4e-187) {
		tmp = y * (x_m * (z_m / (t * ((a * -0.5) / z_m))));
	} else {
		tmp = x_m * y;
	}
	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 <= 6.4d-187) then
        tmp = y * (x_m * (z_m / (t * ((a * (-0.5d0)) / z_m))))
    else
        tmp = x_m * y
    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 <= 6.4e-187) {
		tmp = y * (x_m * (z_m / (t * ((a * -0.5) / z_m))));
	} else {
		tmp = x_m * y;
	}
	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 <= 6.4e-187:
		tmp = y * (x_m * (z_m / (t * ((a * -0.5) / z_m))))
	else:
		tmp = x_m * y
	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 <= 6.4e-187)
		tmp = Float64(y * Float64(x_m * Float64(z_m / Float64(t * Float64(Float64(a * -0.5) / z_m)))));
	else
		tmp = Float64(x_m * y);
	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 <= 6.4e-187)
		tmp = y * (x_m * (z_m / (t * ((a * -0.5) / z_m))));
	else
		tmp = x_m * y;
	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, 6.4e-187], N[(y * N[(x$95$m * N[(z$95$m / N[(t * N[(N[(a * -0.5), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 6.3999999999999997e-187

   1. Initial program 64.2%

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

   3. Taylor expanded in t around 0 26.3%

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

      1. associate-*r/ 26.5%

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

   5. Simplified 26.3%

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

   6. Taylor expanded in t around inf 26.4%

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

   7. Taylor expanded in t around inf 25.0%

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

      1. associate-*r/ 25.0%

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

      2. associate-*r* 25.0%

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

      3. associate-*l/ 24.3%

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

      4. *-commutative 24.3%

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

      5. associate-*r/ 24.3%

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

   9. Simplified 24.3%

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

       1. associate-/l* 24.1%

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

       2. *-commutative 24.1%

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

       3. associate-*r/ 24.1%

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

   11. Applied egg-rr 24.1%

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

       1. associate-*l* 21.7%

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

   13. Simplified 21.7%

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

   if 6.3999999999999997e-187 < z

   1. Initial program 55.5%

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

      1. associate-/l* 60.7%

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

      2. associate-*l* 61.7%

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

   3. Simplified 61.7%

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

   5. Taylor expanded in z around inf 82.4%

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

      1. *-commutative 82.4%

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

   7. Simplified 82.4%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.4 \cdot 10^{-187}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{t \cdot \frac{a \cdot -0.5}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 6.1 \cdot 10^{-183}:\\ \;\;\;\;z\_m \cdot \frac{\frac{x\_m \cdot \left(z\_m \cdot y\right)}{t}}{a \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) 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 6.1e-183)
     (* z_m (/ (/ (* x_m (* z_m y)) t) (* a -0.5)))
     (* x_m y)))))

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 <= 6.1e-183) {
		tmp = z_m * (((x_m * (z_m * y)) / t) / (a * -0.5));
	} else {
		tmp = x_m * y;
	}
	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 <= 6.1d-183) then
        tmp = z_m * (((x_m * (z_m * y)) / t) / (a * (-0.5d0)))
    else
        tmp = x_m * y
    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 <= 6.1e-183) {
		tmp = z_m * (((x_m * (z_m * y)) / t) / (a * -0.5));
	} else {
		tmp = x_m * y;
	}
	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 <= 6.1e-183:
		tmp = z_m * (((x_m * (z_m * y)) / t) / (a * -0.5))
	else:
		tmp = x_m * y
	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 <= 6.1e-183)
		tmp = Float64(z_m * Float64(Float64(Float64(x_m * Float64(z_m * y)) / t) / Float64(a * -0.5)));
	else
		tmp = Float64(x_m * y);
	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 <= 6.1e-183)
		tmp = z_m * (((x_m * (z_m * y)) / t) / (a * -0.5));
	else
		tmp = x_m * y;
	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, 6.1e-183], N[(z$95$m * N[(N[(N[(x$95$m * N[(z$95$m * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / N[(a * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 6.1000000000000002e-183

   1. Initial program 64.4%

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

   3. Taylor expanded in t around 0 26.1%

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

      1. associate-*r/ 26.3%

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

   5. Simplified 26.1%

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

   6. Taylor expanded in t around inf 26.3%

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

   7. Taylor expanded in t around inf 24.8%

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

      1. associate-*r/ 24.8%

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

      2. associate-*r* 24.8%

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

      3. associate-*l/ 24.2%

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

      4. *-commutative 24.2%

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

      5. associate-*r/ 24.2%

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

   9. Simplified 24.2%

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

       1. associate-/l* 24.0%

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

       2. *-commutative 24.0%

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

       3. associate-*r/ 24.0%

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

   11. Applied egg-rr 24.0%

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

       1. associate-*r/ 24.2%

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

       2. *-commutative 24.2%

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

       3. associate-*r* 22.3%

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

       4. *-commutative 22.3%

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

       5. associate-/l/ 24.7%

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

       6. associate-/r/ 26.1%

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

       7. *-commutative 26.1%

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

       8. *-commutative 26.1%

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

   13. Simplified 26.1%

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

   if 6.1000000000000002e-183 < z

   1. Initial program 55.1%

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

      1. associate-/l* 60.4%

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

      2. associate-*l* 61.3%

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

   3. Simplified 61.3%

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

   5. Taylor expanded in z around inf 83.1%

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

      1. *-commutative 83.1%

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

   7. Simplified 83.1%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.1 \cdot 10^{-183}:\\ \;\;\;\;z \cdot \frac{\frac{x \cdot \left(z \cdot y\right)}{t}}{a \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.8% accurate, 7.5× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) 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 (* x_m (* y (/ z_m (+ z_m (* a (* t (/ -0.5 z_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 * (x_m * (y * (z_m / (z_m + (a * (t * (-0.5 / z_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 * (x_m * (y * (z_m / (z_m + (a * (t * ((-0.5d0) / z_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 * (x_m * (y * (z_m / (z_m + (a * (t * (-0.5 / z_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 * (x_m * (y * (z_m / (z_m + (a * (t * (-0.5 / z_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(x_m * Float64(y * Float64(z_m / Float64(z_m + Float64(a * Float64(t * Float64(-0.5 / z_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 * (x_m * (y * (z_m / (z_m + (a * (t * (-0.5 / z_m))))))));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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[(x$95$m * N[(y * N[(z$95$m / N[(z$95$m + N[(a * N[(t * N[(-0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.4%

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

   1. associate-/l* 65.3%

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

   2. associate-*l* 65.0%

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

3. Simplified 65.0%

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

5. Taylor expanded in t around 0 49.0%

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

   1. associate-*r/ 49.0%

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

7. Simplified 49.0%

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

8. Taylor expanded in a around 0 49.0%

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

   1. associate-*r/ 49.0%

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

   2. *-commutative 49.0%

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

   3. associate-*r/ 49.0%

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

   4. associate-*l* 51.4%

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

10. Simplified 51.4%

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

11. Final simplification 51.4%

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

Alternative 6: 75.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.4 \cdot 10^{-136}:\\ \;\;\;\;y \cdot \frac{z\_m \cdot x\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) 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 3.4e-136) (* y (/ (* z_m x_m) z_m)) (* x_m y)))))

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 <= 3.4e-136) {
		tmp = y * ((z_m * x_m) / z_m);
	} else {
		tmp = x_m * y;
	}
	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 <= 3.4d-136) then
        tmp = y * ((z_m * x_m) / z_m)
    else
        tmp = x_m * y
    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 <= 3.4e-136) {
		tmp = y * ((z_m * x_m) / z_m);
	} else {
		tmp = x_m * y;
	}
	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 <= 3.4e-136:
		tmp = y * ((z_m * x_m) / z_m)
	else:
		tmp = x_m * y
	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 <= 3.4e-136)
		tmp = Float64(y * Float64(Float64(z_m * x_m) / z_m));
	else
		tmp = Float64(x_m * y);
	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 <= 3.4e-136)
		tmp = y * ((z_m * x_m) / z_m);
	else
		tmp = x_m * y;
	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, 3.4e-136], N[(y * N[(N[(z$95$m * x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.4e-136

   1. Initial program 64.4%

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

      1. associate-/l* 68.7%

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

      2. *-commutative 68.7%

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

      3. associate-*l* 69.6%

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

      4. associate-*r/ 65.2%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in z around inf 23.7%

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

   if 3.4e-136 < z

   1. Initial program 54.4%

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

      1. associate-/l* 60.1%

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

      2. associate-*l* 59.3%

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

   3. Simplified 59.3%

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

   5. Taylor expanded in z around inf 87.1%

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

      1. *-commutative 87.1%

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

   7. Simplified 87.1%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.4 \cdot 10^{-136}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.8% accurate, 37.7× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) 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 (* x_m y))))

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 * (x_m * y));
}

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 * (x_m * y))
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 * (x_m * y));
}

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 * (x_m * y))

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(x_m * y)))
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 * (x_m * y));
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[(x$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.4%

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

   1. associate-/l* 65.3%

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

   2. associate-*l* 65.0%

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

3. Simplified 65.0%

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

5. Taylor expanded in z around inf 45.8%

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

   1. *-commutative 45.8%

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

7. Simplified 45.8%

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

8. Final simplification 45.8%

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

Developer target: 88.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024095 
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (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)))))