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

Percentage Accurate: 61.2% → 92.5%

Time: 20.5s

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: 61.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x, y_m, z_m, t, a] = \mathsf{sort}([x, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.5 \cdot 10^{-208}:\\ \;\;\;\;z\_m \cdot \frac{x \cdot y\_m}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{elif}\;z\_m \leq 3.8 \cdot 10^{+96}:\\ \;\;\;\;x \cdot \left(y\_m \cdot \frac{z\_m}{\sqrt{z\_m \cdot z\_m - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\_m\\ \end{array}\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (if (<= z_m 3.5e-208)
     (* z_m (/ (* x y_m) (sqrt (* a (- t)))))
     (if (<= z_m 3.8e+96)
       (* x (* y_m (/ z_m (sqrt (- (* z_m z_m) (* a t))))))
       (* x y_m))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
assert(x < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.5e-208) {
		tmp = z_m * ((x * y_m) / sqrt((a * -t)));
	} else if (z_m <= 3.8e+96) {
		tmp = x * (y_m * (z_m / sqrt(((z_m * z_m) - (a * t)))));
	} else {
		tmp = x * y_m;
	}
	return z_s * (y_s * tmp);
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 3.5d-208) then
        tmp = z_m * ((x * y_m) / sqrt((a * -t)))
    else if (z_m <= 3.8d+96) then
        tmp = x * (y_m * (z_m / sqrt(((z_m * z_m) - (a * t)))))
    else
        tmp = x * y_m
    end if
    code = z_s * (y_s * tmp)
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
assert x < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.5e-208) {
		tmp = z_m * ((x * y_m) / Math.sqrt((a * -t)));
	} else if (z_m <= 3.8e+96) {
		tmp = x * (y_m * (z_m / Math.sqrt(((z_m * z_m) - (a * t)))));
	} else {
		tmp = x * y_m;
	}
	return z_s * (y_s * tmp);
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
[x, y_m, z_m, t, a] = sort([x, y_m, z_m, t, a])
def code(z_s, y_s, x, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 3.5e-208:
		tmp = z_m * ((x * y_m) / math.sqrt((a * -t)))
	elif z_m <= 3.8e+96:
		tmp = x * (y_m * (z_m / math.sqrt(((z_m * z_m) - (a * t)))))
	else:
		tmp = x * y_m
	return z_s * (y_s * tmp)

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
x, y_m, z_m, t, a = sort([x, y_m, z_m, t, a])
function code(z_s, y_s, x, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 3.5e-208)
		tmp = Float64(z_m * Float64(Float64(x * y_m) / sqrt(Float64(a * Float64(-t)))));
	elseif (z_m <= 3.8e+96)
		tmp = Float64(x * Float64(y_m * Float64(z_m / sqrt(Float64(Float64(z_m * z_m) - Float64(a * t))))));
	else
		tmp = Float64(x * y_m);
	end
	return Float64(z_s * Float64(y_s * tmp))
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
x, y_m, z_m, t, a = num2cell(sort([x, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 3.5e-208)
		tmp = z_m * ((x * y_m) / sqrt((a * -t)));
	elseif (z_m <= 3.8e+96)
		tmp = x * (y_m * (z_m / sqrt(((z_m * z_m) - (a * t)))));
	else
		tmp = x * y_m;
	end
	tmp_2 = z_s * (y_s * tmp);
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * If[LessEqual[z$95$m, 3.5e-208], N[(z$95$m * N[(N[(x * y$95$m), $MachinePrecision] / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 3.8e+96], N[(x * N[(y$95$m * N[(z$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 3.49999999999999991e-208

   1. Initial program 62.2%

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

      1. *-commutative 62.2%

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

      2. associate-/l* 61.6%

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

      3. *-commutative 61.6%

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

      4. pow2 61.6%

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

   4. Applied egg-rr 61.6%

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

   5. Taylor expanded in z around 0 38.1%

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

      1. associate-*r* 38.1%

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

      2. neg-mul-1 38.1%

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

   7. Simplified 38.1%

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

   if 3.49999999999999991e-208 < z < 3.8000000000000002e96

   1. Initial program 89.1%

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

      1. associate-/l* 92.6%

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

      2. associate-*l* 91.0%

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

   3. Simplified 91.0%

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

   if 3.8000000000000002e96 < z

   1. Initial program 40.5%

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

      1. associate-/l* 44.0%

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

      2. associate-*l* 44.0%

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

   3. Simplified 44.0%

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

   5. Taylor expanded in z around inf 96.8%

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

      1. *-commutative 96.8%

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

   7. Simplified 96.8%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{-208}:\\ \;\;\;\;z \cdot \frac{x \cdot y}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+96}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.2% accurate, 0.5× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (if (<= z_m 2e+95)
     (* x (* z_m (/ y_m (sqrt (- (pow z_m 2.0) (* a t))))))
     (* x y_m)))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
assert(x < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2e+95) {
		tmp = x * (z_m * (y_m / sqrt((pow(z_m, 2.0) - (a * t)))));
	} else {
		tmp = x * y_m;
	}
	return z_s * (y_s * tmp);
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 2d+95) then
        tmp = x * (z_m * (y_m / sqrt(((z_m ** 2.0d0) - (a * t)))))
    else
        tmp = x * y_m
    end if
    code = z_s * (y_s * tmp)
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
assert x < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2e+95) {
		tmp = x * (z_m * (y_m / Math.sqrt((Math.pow(z_m, 2.0) - (a * t)))));
	} else {
		tmp = x * y_m;
	}
	return z_s * (y_s * tmp);
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
[x, y_m, z_m, t, a] = sort([x, y_m, z_m, t, a])
def code(z_s, y_s, x, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 2e+95:
		tmp = x * (z_m * (y_m / math.sqrt((math.pow(z_m, 2.0) - (a * t)))))
	else:
		tmp = x * y_m
	return z_s * (y_s * tmp)

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
x, y_m, z_m, t, a = sort([x, y_m, z_m, t, a])
function code(z_s, y_s, x, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2e+95)
		tmp = Float64(x * Float64(z_m * Float64(y_m / sqrt(Float64((z_m ^ 2.0) - Float64(a * t))))));
	else
		tmp = Float64(x * y_m);
	end
	return Float64(z_s * Float64(y_s * tmp))
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
x, y_m, z_m, t, a = num2cell(sort([x, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2e+95)
		tmp = x * (z_m * (y_m / sqrt(((z_m ^ 2.0) - (a * t)))));
	else
		tmp = x * y_m;
	end
	tmp_2 = z_s * (y_s * tmp);
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * If[LessEqual[z$95$m, 2e+95], N[(x * N[(z$95$m * N[(y$95$m / N[Sqrt[N[(N[Power[z$95$m, 2.0], $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.00000000000000004e95

   1. Initial program 69.4%

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

      1. associate-/l* 70.1%

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

      2. associate-*l* 69.5%

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

   3. Simplified 69.5%

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

      1. clear-num 69.0%

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

      2. un-div-inv 69.0%

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

      3. pow2 69.0%

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

   6. Applied egg-rr 69.0%

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

      1. associate-/r/ 69.9%

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

      2. *-commutative 69.9%

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

   8. Simplified 69.9%

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

   if 2.00000000000000004e95 < z

   1. Initial program 40.5%

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

      1. associate-/l* 44.0%

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

      2. associate-*l* 44.0%

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

   3. Simplified 44.0%

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

   5. Taylor expanded in z around inf 96.8%

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

      1. *-commutative 96.8%

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

   7. Simplified 96.8%

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

4. Final simplification 75.9%

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

Alternative 3: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x, y_m, z_m, t, a] = \mathsf{sort}([x, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.8 \cdot 10^{-67}:\\ \;\;\;\;x \cdot \left(y\_m \cdot \frac{z\_m}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y\_m \cdot \frac{z\_m}{z\_m + -0.5 \cdot \left(a \cdot \frac{t}{z\_m}\right)}\right)\\ \end{array}\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (if (<= z_m 3.8e-67)
     (* x (* y_m (/ z_m (sqrt (* a (- t))))))
     (* x (* y_m (/ z_m (+ z_m (* -0.5 (* a (/ t z_m)))))))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
assert(x < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.8e-67) {
		tmp = x * (y_m * (z_m / sqrt((a * -t))));
	} else {
		tmp = x * (y_m * (z_m / (z_m + (-0.5 * (a * (t / z_m))))));
	}
	return z_s * (y_s * tmp);
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 3.8d-67) then
        tmp = x * (y_m * (z_m / sqrt((a * -t))))
    else
        tmp = x * (y_m * (z_m / (z_m + ((-0.5d0) * (a * (t / z_m))))))
    end if
    code = z_s * (y_s * tmp)
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
assert x < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.8e-67) {
		tmp = x * (y_m * (z_m / Math.sqrt((a * -t))));
	} else {
		tmp = x * (y_m * (z_m / (z_m + (-0.5 * (a * (t / z_m))))));
	}
	return z_s * (y_s * tmp);
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
[x, y_m, z_m, t, a] = sort([x, y_m, z_m, t, a])
def code(z_s, y_s, x, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 3.8e-67:
		tmp = x * (y_m * (z_m / math.sqrt((a * -t))))
	else:
		tmp = x * (y_m * (z_m / (z_m + (-0.5 * (a * (t / z_m))))))
	return z_s * (y_s * tmp)

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
x, y_m, z_m, t, a = sort([x, y_m, z_m, t, a])
function code(z_s, y_s, x, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 3.8e-67)
		tmp = Float64(x * Float64(y_m * Float64(z_m / sqrt(Float64(a * Float64(-t))))));
	else
		tmp = Float64(x * Float64(y_m * Float64(z_m / Float64(z_m + Float64(-0.5 * Float64(a * Float64(t / z_m)))))));
	end
	return Float64(z_s * Float64(y_s * tmp))
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
x, y_m, z_m, t, a = num2cell(sort([x, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 3.8e-67)
		tmp = x * (y_m * (z_m / sqrt((a * -t))));
	else
		tmp = x * (y_m * (z_m / (z_m + (-0.5 * (a * (t / z_m))))));
	end
	tmp_2 = z_s * (y_s * tmp);
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * If[LessEqual[z$95$m, 3.8e-67], N[(x * N[(y$95$m * N[(z$95$m / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y$95$m * N[(z$95$m / N[(z$95$m + N[(-0.5 * N[(a * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.79999999999999988e-67

   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* 65.4%

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

      2. associate-*l* 64.6%

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

   3. Simplified 64.6%

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

   5. Taylor expanded in z around 0 36.2%

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

      1. mul-1-neg 36.2%

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

      2. distribute-rgt-neg-out 36.2%

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

   7. Simplified 36.2%

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

   if 3.79999999999999988e-67 < z

   1. Initial program 60.2%

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

      1. associate-/l* 62.3%

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

      2. associate-*l* 62.4%

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

   3. Simplified 62.4%

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

   5. Taylor expanded in z around inf 83.3%

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

      1. associate-/l* 85.5%

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

   7. Simplified 85.5%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.8 \cdot 10^{-67}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + -0.5 \cdot \left(a \cdot \frac{t}{z}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x, y_m, z_m, t, a] = \mathsf{sort}([x, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5.2 \cdot 10^{-70}:\\ \;\;\;\;x \cdot \left(z\_m \cdot \frac{y\_m}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y\_m \cdot \frac{z\_m}{z\_m + -0.5 \cdot \left(a \cdot \frac{t}{z\_m}\right)}\right)\\ \end{array}\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (if (<= z_m 5.2e-70)
     (* x (* z_m (/ y_m (sqrt (* a (- t))))))
     (* x (* y_m (/ z_m (+ z_m (* -0.5 (* a (/ t z_m)))))))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
assert(x < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 5.2e-70) {
		tmp = x * (z_m * (y_m / sqrt((a * -t))));
	} else {
		tmp = x * (y_m * (z_m / (z_m + (-0.5 * (a * (t / z_m))))));
	}
	return z_s * (y_s * tmp);
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 5.2d-70) then
        tmp = x * (z_m * (y_m / sqrt((a * -t))))
    else
        tmp = x * (y_m * (z_m / (z_m + ((-0.5d0) * (a * (t / z_m))))))
    end if
    code = z_s * (y_s * tmp)
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
assert x < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 5.2e-70) {
		tmp = x * (z_m * (y_m / Math.sqrt((a * -t))));
	} else {
		tmp = x * (y_m * (z_m / (z_m + (-0.5 * (a * (t / z_m))))));
	}
	return z_s * (y_s * tmp);
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
[x, y_m, z_m, t, a] = sort([x, y_m, z_m, t, a])
def code(z_s, y_s, x, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 5.2e-70:
		tmp = x * (z_m * (y_m / math.sqrt((a * -t))))
	else:
		tmp = x * (y_m * (z_m / (z_m + (-0.5 * (a * (t / z_m))))))
	return z_s * (y_s * tmp)

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
x, y_m, z_m, t, a = sort([x, y_m, z_m, t, a])
function code(z_s, y_s, x, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 5.2e-70)
		tmp = Float64(x * Float64(z_m * Float64(y_m / sqrt(Float64(a * Float64(-t))))));
	else
		tmp = Float64(x * Float64(y_m * Float64(z_m / Float64(z_m + Float64(-0.5 * Float64(a * Float64(t / z_m)))))));
	end
	return Float64(z_s * Float64(y_s * tmp))
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
x, y_m, z_m, t, a = num2cell(sort([x, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 5.2e-70)
		tmp = x * (z_m * (y_m / sqrt((a * -t))));
	else
		tmp = x * (y_m * (z_m / (z_m + (-0.5 * (a * (t / z_m))))));
	end
	tmp_2 = z_s * (y_s * tmp);
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * If[LessEqual[z$95$m, 5.2e-70], N[(x * N[(z$95$m * N[(y$95$m / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y$95$m * N[(z$95$m / N[(z$95$m + N[(-0.5 * N[(a * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 5.20000000000000004e-70

   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* 65.4%

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

      2. associate-*l* 64.6%

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

   3. Simplified 64.6%

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

      1. clear-num 64.1%

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

      2. un-div-inv 64.1%

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

      3. pow2 64.1%

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

   6. Applied egg-rr 64.1%

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

      1. associate-/r/ 65.7%

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

      2. *-commutative 65.7%

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

   8. Simplified 65.7%

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

   9. Taylor expanded in z around 0 38.7%

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

       1. neg-mul-1 38.7%

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

       2. distribute-rgt-neg-in 38.7%

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

   11. Simplified 38.7%

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

   if 5.20000000000000004e-70 < z

   1. Initial program 60.2%

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

      1. associate-/l* 62.3%

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

      2. associate-*l* 62.4%

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

   3. Simplified 62.4%

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

   5. Taylor expanded in z around inf 83.3%

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

      1. associate-/l* 85.5%

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

   7. Simplified 85.5%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.2 \cdot 10^{-70}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + -0.5 \cdot \left(a \cdot \frac{t}{z}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.0% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x * N[(y$95$m * N[(z$95$m / N[(z$95$m + N[(-0.5 * N[(a * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.9%

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

   1. associate-/l* 64.3%

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

   2. associate-*l* 63.8%

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

3. Simplified 63.8%

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

5. Taylor expanded in z around inf 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-/l* 49.7%

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

7. Simplified 49.7%

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

8. Final simplification 49.7%

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

Alternative 6: 71.0% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.9%

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

   1. associate-/l* 64.3%

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

   2. *-commutative 64.3%

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

   3. associate-*l* 63.8%

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

   4. associate-*r/ 61.4%

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

3. Simplified 61.4%

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

5. Taylor expanded in z around inf 39.9%

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

6. Final simplification 39.9%

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

Alternative 7: 74.2% accurate, 37.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.9%

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

   1. associate-/l* 64.3%

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

   2. associate-*l* 63.8%

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

3. Simplified 63.8%

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

5. Taylor expanded in z around inf 43.6%

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

   1. *-commutative 43.6%

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

7. Simplified 43.6%

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

8. Final simplification 43.6%

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

Developer target: 89.1% 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 2024062 
(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)))))