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

Percentage Accurate: 61.2% → 91.4%

Time: 15.9s

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: 91.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x, y, z_m, t, a] = \mathsf{sort}([x, y, z_m, t, a])\\ \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.55 \cdot 10^{+131}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z\_m}{z\_m + -0.5 \cdot \left(a \cdot \frac{t}{z\_m}\right)}\right)\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x y z_m t a)
 :precision binary64
 (*
  z_s
  (if (<= z_m 1.55e+131)
    (* x (* y (/ z_m (sqrt (- (* z_m z_m) (* t a))))))
    (* x (* y (/ z_m (+ z_m (* -0.5 (* a (/ t z_m))))))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.55e+131) {
		tmp = x * (y * (z_m / sqrt(((z_m * z_m) - (t * a)))));
	} else {
		tmp = x * (y * (z_m / (z_m + (-0.5 * (a * (t / z_m))))));
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, x, y, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.55d+131) then
        tmp = x * (y * (z_m / sqrt(((z_m * z_m) - (t * a)))))
    else
        tmp = x * (y * (z_m / (z_m + ((-0.5d0) * (a * (t / z_m))))))
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x < y && y < z_m && z_m < t && t < a;
public static double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.55e+131) {
		tmp = x * (y * (z_m / Math.sqrt(((z_m * z_m) - (t * a)))));
	} else {
		tmp = x * (y * (z_m / (z_m + (-0.5 * (a * (t / z_m))))));
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x, y, z_m, t, a] = sort([x, y, z_m, t, a])
def code(z_s, x, y, z_m, t, a):
	tmp = 0
	if z_m <= 1.55e+131:
		tmp = x * (y * (z_m / math.sqrt(((z_m * z_m) - (t * a)))))
	else:
		tmp = x * (y * (z_m / (z_m + (-0.5 * (a * (t / z_m))))))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
x, y, z_m, t, a = sort([x, y, z_m, t, a])
function code(z_s, x, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.55e+131)
		tmp = Float64(x * Float64(y * Float64(z_m / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a))))));
	else
		tmp = Float64(x * Float64(y * Float64(z_m / Float64(z_m + Float64(-0.5 * Float64(a * Float64(t / z_m)))))));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x, y, z_m, t, a = num2cell(sort([x, y, z_m, t, a])){:}
function tmp_2 = code(z_s, x, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.55e+131)
		tmp = x * (y * (z_m / sqrt(((z_m * z_m) - (t * a)))));
	else
		tmp = x * (y * (z_m / (z_m + (-0.5 * (a * (t / z_m))))));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * If[LessEqual[z$95$m, 1.55e+131], N[(x * 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 * N[(y * 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]

Derivation

1. Split input into 2 regimes

2. if z < 1.55000000000000008e131

   1. Initial program 72.2%

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

      1. associate-/l* 74.0%

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

      2. associate-*l* 74.4%

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

   3. Simplified 74.4%

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

   if 1.55000000000000008e131 < z

   1. Initial program 32.4%

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

      1. associate-/l* 32.8%

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

      2. associate-*l* 33.1%

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

   3. Simplified 33.1%

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

   5. Taylor expanded in t around 0 91.5%

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

      1. associate-/l* 98.0%

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

   7. Simplified 98.0%

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

Alternative 2: 83.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x, y, z_m, t, a] = \mathsf{sort}([x, y, z_m, t, a])\\ \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.7 \cdot 10^{-95}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z\_m}{\sqrt{t \cdot \left(-a\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z\_m}{z\_m + -0.5 \cdot \left(a \cdot \frac{t}{z\_m}\right)}\right)\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x y z_m t a)
 :precision binary64
 (*
  z_s
  (if (<= z_m 2.7e-95)
    (* x (* y (/ z_m (sqrt (* t (- a))))))
    (* x (* y (/ z_m (+ z_m (* -0.5 (* a (/ t z_m))))))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.7e-95) {
		tmp = x * (y * (z_m / sqrt((t * -a))));
	} else {
		tmp = x * (y * (z_m / (z_m + (-0.5 * (a * (t / z_m))))));
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, x, y, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 2.7d-95) then
        tmp = x * (y * (z_m / sqrt((t * -a))))
    else
        tmp = x * (y * (z_m / (z_m + ((-0.5d0) * (a * (t / z_m))))))
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x < y && y < z_m && z_m < t && t < a;
public static double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.7e-95) {
		tmp = x * (y * (z_m / Math.sqrt((t * -a))));
	} else {
		tmp = x * (y * (z_m / (z_m + (-0.5 * (a * (t / z_m))))));
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x, y, z_m, t, a] = sort([x, y, z_m, t, a])
def code(z_s, x, y, z_m, t, a):
	tmp = 0
	if z_m <= 2.7e-95:
		tmp = x * (y * (z_m / math.sqrt((t * -a))))
	else:
		tmp = x * (y * (z_m / (z_m + (-0.5 * (a * (t / z_m))))))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
x, y, z_m, t, a = sort([x, y, z_m, t, a])
function code(z_s, x, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.7e-95)
		tmp = Float64(x * Float64(y * Float64(z_m / sqrt(Float64(t * Float64(-a))))));
	else
		tmp = Float64(x * Float64(y * Float64(z_m / Float64(z_m + Float64(-0.5 * Float64(a * Float64(t / z_m)))))));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x, y, z_m, t, a = num2cell(sort([x, y, z_m, t, a])){:}
function tmp_2 = code(z_s, x, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2.7e-95)
		tmp = x * (y * (z_m / sqrt((t * -a))));
	else
		tmp = x * (y * (z_m / (z_m + (-0.5 * (a * (t / z_m))))));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * If[LessEqual[z$95$m, 2.7e-95], N[(x * N[(y * N[(z$95$m / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * 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]

Derivation

1. Split input into 2 regimes

2. if z < 2.7e-95

   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* 66.7%

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

      2. associate-*l* 67.3%

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

   3. Simplified 67.3%

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

   5. Taylor expanded in z around 0 37.9%

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

      1. associate-*r* 37.9%

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

      2. neg-mul-1 37.9%

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

      3. *-commutative 37.9%

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

   7. Simplified 37.9%

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

   if 2.7e-95 < z

   1. Initial program 66.1%

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

      1. associate-/l* 66.3%

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

      2. associate-*l* 66.4%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in t around 0 86.6%

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

      1. associate-/l* 89.6%

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

   7. Simplified 89.6%

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

Alternative 3: 78.9% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x * N[(y * 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]

Derivation

1. Initial program 65.0%

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

   1. associate-/l* 66.6%

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

   2. associate-*l* 66.9%

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

3. Simplified 66.9%

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

5. Taylor expanded in t around 0 49.1%

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

   1. associate-/l* 50.2%

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

7. Simplified 50.2%

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

Alternative 4: 76.2% accurate, 9.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x, y, z_m, t, a] = \mathsf{sort}([x, y, z_m, t, a])\\ \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.4 \cdot 10^{-60}:\\ \;\;\;\;\frac{y \cdot \left(z\_m \cdot x\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x y z_m t a)
 :precision binary64
 (* z_s (if (<= z_m 1.4e-60) (/ (* y (* z_m x)) z_m) (* x y))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.4e-60) {
		tmp = (y * (z_m * x)) / z_m;
	} else {
		tmp = x * y;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, x, y, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.4d-60) then
        tmp = (y * (z_m * x)) / z_m
    else
        tmp = x * y
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x < y && y < z_m && z_m < t && t < a;
public static double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.4e-60) {
		tmp = (y * (z_m * x)) / z_m;
	} else {
		tmp = x * y;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x, y, z_m, t, a] = sort([x, y, z_m, t, a])
def code(z_s, x, y, z_m, t, a):
	tmp = 0
	if z_m <= 1.4e-60:
		tmp = (y * (z_m * x)) / z_m
	else:
		tmp = x * y
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
x, y, z_m, t, a = sort([x, y, z_m, t, a])
function code(z_s, x, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.4e-60)
		tmp = Float64(Float64(y * Float64(z_m * x)) / z_m);
	else
		tmp = Float64(x * y);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x, y, z_m, t, a = num2cell(sort([x, y, z_m, t, a])){:}
function tmp_2 = code(z_s, x, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.4e-60)
		tmp = (y * (z_m * x)) / z_m;
	else
		tmp = x * y;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * If[LessEqual[z$95$m, 1.4e-60], N[(N[(y * N[(z$95$m * x), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(x * y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.4000000000000001e-60

   1. Initial program 65.7%

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

      1. associate-/l* 67.9%

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

      2. *-commutative 67.9%

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

      3. associate-*l* 66.9%

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

      4. associate-*r/ 65.5%

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

   3. Simplified 65.5%

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

   5. Taylor expanded in z around inf 20.5%

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

      1. associate-*r/ 24.8%

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

   7. Applied egg-rr 24.8%

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

   if 1.4000000000000001e-60 < z

   1. Initial program 63.9%

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

      1. associate-/l* 64.1%

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

      2. *-commutative 64.1%

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

      3. associate-*l* 64.2%

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

      4. associate-*r/ 62.9%

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

   3. Simplified 62.9%

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

   5. Taylor expanded in z around inf 89.1%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.4 \cdot 10^{-60}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x, y, z_m, t, a] = \mathsf{sort}([x, y, z_m, t, a])\\ \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 7.5 \cdot 10^{-94}:\\ \;\;\;\;\frac{x \cdot \left(z\_m \cdot y\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x y z_m t a)
 :precision binary64
 (* z_s (if (<= z_m 7.5e-94) (/ (* x (* z_m y)) z_m) (* x y))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 7.5e-94) {
		tmp = (x * (z_m * y)) / z_m;
	} else {
		tmp = x * y;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, x, y, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    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 <= 7.5d-94) then
        tmp = (x * (z_m * y)) / z_m
    else
        tmp = x * y
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x < y && y < z_m && z_m < t && t < a;
public static double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 7.5e-94) {
		tmp = (x * (z_m * y)) / z_m;
	} else {
		tmp = x * y;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x, y, z_m, t, a] = sort([x, y, z_m, t, a])
def code(z_s, x, y, z_m, t, a):
	tmp = 0
	if z_m <= 7.5e-94:
		tmp = (x * (z_m * y)) / z_m
	else:
		tmp = x * y
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
x, y, z_m, t, a = sort([x, y, z_m, t, a])
function code(z_s, x, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 7.5e-94)
		tmp = Float64(Float64(x * Float64(z_m * y)) / z_m);
	else
		tmp = Float64(x * y);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x, y, z_m, t, a = num2cell(sort([x, y, z_m, t, a])){:}
function tmp_2 = code(z_s, x, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 7.5e-94)
		tmp = (x * (z_m * y)) / z_m;
	else
		tmp = x * y;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * If[LessEqual[z$95$m, 7.5e-94], N[(N[(x * N[(z$95$m * y), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(x * y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 7.5000000000000003e-94

   1. Initial program 64.6%

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

      1. associate-/l* 66.9%

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

      2. associate-*l* 67.5%

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

   3. Simplified 67.5%

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

   5. Taylor expanded in t around 0 25.9%

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

      1. associate-/l* 25.8%

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

   7. Simplified 25.8%

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

   8. Taylor expanded in x around 0 25.8%

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

   9. Taylor expanded in z around inf 23.6%

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

   if 7.5000000000000003e-94 < z

   1. Initial program 65.7%

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

      1. associate-/l* 66.0%

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

      2. *-commutative 66.0%

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

      3. associate-*l* 66.1%

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

      4. associate-*r/ 64.8%

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

   3. Simplified 64.8%

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

   5. Taylor expanded in z around inf 87.7%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.5 \cdot 10^{-94}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.9% accurate, 9.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x, y, z_m, t, a] = \mathsf{sort}([x, y, z_m, t, a])\\ \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.8 \cdot 10^{-186}:\\ \;\;\;\;y \cdot \frac{z\_m \cdot x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x y z_m t a)
 :precision binary64
 (* z_s (if (<= z_m 1.8e-186) (* y (/ (* z_m x) z_m)) (* x y))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.8e-186) {
		tmp = y * ((z_m * x) / z_m);
	} else {
		tmp = x * y;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x, y, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, x, y, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.8d-186) then
        tmp = y * ((z_m * x) / z_m)
    else
        tmp = x * y
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x < y && y < z_m && z_m < t && t < a;
public static double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.8e-186) {
		tmp = y * ((z_m * x) / z_m);
	} else {
		tmp = x * y;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x, y, z_m, t, a] = sort([x, y, z_m, t, a])
def code(z_s, x, y, z_m, t, a):
	tmp = 0
	if z_m <= 1.8e-186:
		tmp = y * ((z_m * x) / z_m)
	else:
		tmp = x * y
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
x, y, z_m, t, a = sort([x, y, z_m, t, a])
function code(z_s, x, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.8e-186)
		tmp = Float64(y * Float64(Float64(z_m * x) / z_m));
	else
		tmp = Float64(x * y);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x, y, z_m, t, a = num2cell(sort([x, y, z_m, t, a])){:}
function tmp_2 = code(z_s, x, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.8e-186)
		tmp = y * ((z_m * x) / z_m);
	else
		tmp = x * y;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * If[LessEqual[z$95$m, 1.8e-186], N[(y * N[(N[(z$95$m * x), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.7999999999999999e-186

   1. Initial program 63.4%

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

      1. associate-/l* 65.9%

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

      2. *-commutative 65.9%

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

      3. associate-*l* 65.4%

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

      4. associate-*r/ 63.8%

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

   3. Simplified 63.8%

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

   5. Taylor expanded in z around inf 17.1%

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

   if 1.7999999999999999e-186 < z

   1. Initial program 67.2%

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

      1. associate-/l* 67.4%

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

      2. *-commutative 67.4%

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

      3. associate-*l* 66.6%

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

      4. associate-*r/ 65.5%

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

   3. Simplified 65.5%

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

   5. Taylor expanded in z around inf 82.4%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.8 \cdot 10^{-186}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.0% accurate, 37.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.0%

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

   1. associate-/l* 66.6%

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

   2. *-commutative 66.6%

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

   3. associate-*l* 66.0%

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

   4. associate-*r/ 64.6%

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

3. Simplified 64.6%

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

5. Taylor expanded in z around inf 44.3%

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

6. Final simplification 44.3%

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

Developer Target 1: 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 2024146 
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -31921305903852764000000000000000000000000000000) (- (* y x)) (if (< z 5976268120920894000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y)) (* y x))))

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