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

Percentage Accurate: 61.5% → 92.1%

Time: 19.1s

Alternatives: 10

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: 61.5% 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: 92.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.7 \cdot 10^{+104}:\\ \;\;\;\;\frac{z\_m \cdot y\_m}{\sqrt{{z\_m}^{2} - t \cdot a}} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\left(y\_m \cdot x\_m\right) \cdot \left(e^{\mathsf{log1p}\left(\frac{z\_m}{\mathsf{fma}\left(a \cdot -0.5, \frac{t}{z\_m}, z\_m\right)}\right)} + -1\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 3.7e+104)
      (* (/ (* z_m y_m) (sqrt (- (pow z_m 2.0) (* t a)))) x_m)
      (*
       (* y_m x_m)
       (+ (exp (log1p (/ z_m (fma (* a -0.5) (/ t z_m) z_m)))) -1.0)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.7e+104) {
		tmp = ((z_m * y_m) / sqrt((pow(z_m, 2.0) - (t * a)))) * x_m;
	} else {
		tmp = (y_m * x_m) * (exp(log1p((z_m / fma((a * -0.5), (t / z_m), z_m)))) + -1.0);
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 3.7e+104)
		tmp = Float64(Float64(Float64(z_m * y_m) / sqrt(Float64((z_m ^ 2.0) - Float64(t * a)))) * x_m);
	else
		tmp = Float64(Float64(y_m * x_m) * Float64(exp(log1p(Float64(z_m / fma(Float64(a * -0.5), Float64(t / z_m), z_m)))) + -1.0));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, 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$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 3.7e+104], N[(N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[Sqrt[N[(N[Power[z$95$m, 2.0], $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(y$95$m * x$95$m), $MachinePrecision] * N[(N[Exp[N[Log[1 + N[(z$95$m / N[(N[(a * -0.5), $MachinePrecision] * N[(t / z$95$m), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.6999999999999998e104

   1. Initial program 64.1%

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

      1. associate-*r/ 65.2%

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

      2. *-commutative 65.2%

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

      3. *-commutative 65.2%

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

      4. associate-*r* 63.3%

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

      5. associate-*l/ 62.9%

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

      6. pow2 62.9%

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

   4. Applied egg-rr 62.9%

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

   if 3.6999999999999998e104 < z

   1. Initial program 31.6%

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

      1. associate-/l* 36.6%

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

   3. Simplified 36.6%

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

   5. Taylor expanded in t around 0 91.8%

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

      1. associate-/l* 100.0%

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

   7. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-undefine 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-*r* 100.0%

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

      5. fma-define 100.0%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{z}{\mathsf{fma}\left(-0.5 \cdot a, \frac{t}{z}, z\right)}\right)} - 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.7 \cdot 10^{+104}:\\ \;\;\;\;\frac{z \cdot y}{\sqrt{{z}^{2} - t \cdot a}} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \left(e^{\mathsf{log1p}\left(\frac{z}{\mathsf{fma}\left(a \cdot -0.5, \frac{t}{z}, z\right)}\right)} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.7 \cdot 10^{+104}:\\ \;\;\;\;\frac{z\_m \cdot y\_m}{\sqrt{{z\_m}^{2} - t \cdot a}} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\left(y\_m \cdot x\_m\right) \cdot \log \left(e^{\frac{z\_m}{\mathsf{fma}\left(a \cdot -0.5, \frac{t}{z\_m}, z\_m\right)}}\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 3.7e+104)
      (* (/ (* z_m y_m) (sqrt (- (pow z_m 2.0) (* t a)))) x_m)
      (* (* y_m x_m) (log (exp (/ z_m (fma (* a -0.5) (/ t z_m) z_m))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.7e+104) {
		tmp = ((z_m * y_m) / sqrt((pow(z_m, 2.0) - (t * a)))) * x_m;
	} else {
		tmp = (y_m * x_m) * log(exp((z_m / fma((a * -0.5), (t / z_m), z_m))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 3.7e+104)
		tmp = Float64(Float64(Float64(z_m * y_m) / sqrt(Float64((z_m ^ 2.0) - Float64(t * a)))) * x_m);
	else
		tmp = Float64(Float64(y_m * x_m) * log(exp(Float64(z_m / fma(Float64(a * -0.5), Float64(t / z_m), z_m)))));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, 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$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 3.7e+104], N[(N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[Sqrt[N[(N[Power[z$95$m, 2.0], $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(y$95$m * x$95$m), $MachinePrecision] * N[Log[N[Exp[N[(z$95$m / N[(N[(a * -0.5), $MachinePrecision] * N[(t / z$95$m), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.6999999999999998e104

   1. Initial program 64.1%

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

      1. associate-*r/ 65.2%

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

      2. *-commutative 65.2%

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

      3. *-commutative 65.2%

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

      4. associate-*r* 63.3%

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

      5. associate-*l/ 62.9%

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

      6. pow2 62.9%

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

   4. Applied egg-rr 62.9%

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

   if 3.6999999999999998e104 < z

   1. Initial program 31.6%

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

      1. associate-/l* 36.6%

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

   3. Simplified 36.6%

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

   5. Taylor expanded in t around 0 91.8%

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

      1. associate-/l* 100.0%

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

   7. Simplified 100.0%

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

      1. add-log-exp 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. fma-define 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.7 \cdot 10^{+104}:\\ \;\;\;\;\frac{z \cdot y}{\sqrt{{z}^{2} - t \cdot a}} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \log \left(e^{\frac{z}{\mathsf{fma}\left(a \cdot -0.5, \frac{t}{z}, z\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.7 \cdot 10^{+104}:\\ \;\;\;\;\frac{z\_m \cdot y\_m}{\sqrt{{z\_m}^{2} - t \cdot a}} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\left(y\_m \cdot x\_m\right) \cdot \frac{z\_m}{z\_m + -0.5 \cdot \left(a \cdot \frac{t}{z\_m}\right)}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 3.7e+104)
      (* (/ (* z_m y_m) (sqrt (- (pow z_m 2.0) (* t a)))) x_m)
      (* (* y_m x_m) (/ z_m (+ z_m (* -0.5 (* a (/ t z_m)))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.7e+104) {
		tmp = ((z_m * y_m) / sqrt((pow(z_m, 2.0) - (t * a)))) * x_m;
	} else {
		tmp = (y_m * x_m) * (z_m / (z_m + (-0.5 * (a * (t / z_m)))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, 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_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 3.7d+104) then
        tmp = ((z_m * y_m) / sqrt(((z_m ** 2.0d0) - (t * a)))) * x_m
    else
        tmp = (y_m * x_m) * (z_m / (z_m + ((-0.5d0) * (a * (t / z_m)))))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.7e+104) {
		tmp = ((z_m * y_m) / Math.sqrt((Math.pow(z_m, 2.0) - (t * a)))) * x_m;
	} else {
		tmp = (y_m * x_m) * (z_m / (z_m + (-0.5 * (a * (t / z_m)))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 3.7e+104:
		tmp = ((z_m * y_m) / math.sqrt((math.pow(z_m, 2.0) - (t * a)))) * x_m
	else:
		tmp = (y_m * x_m) * (z_m / (z_m + (-0.5 * (a * (t / z_m)))))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 3.7e+104)
		tmp = Float64(Float64(Float64(z_m * y_m) / sqrt(Float64((z_m ^ 2.0) - Float64(t * a)))) * x_m);
	else
		tmp = Float64(Float64(y_m * x_m) * Float64(z_m / Float64(z_m + Float64(-0.5 * Float64(a * Float64(t / z_m))))));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 3.7e+104)
		tmp = ((z_m * y_m) / sqrt(((z_m ^ 2.0) - (t * a)))) * x_m;
	else
		tmp = (y_m * x_m) * (z_m / (z_m + (-0.5 * (a * (t / z_m)))));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, 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$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 3.7e+104], N[(N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[Sqrt[N[(N[Power[z$95$m, 2.0], $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(y$95$m * x$95$m), $MachinePrecision] * 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 < 3.6999999999999998e104

   1. Initial program 64.1%

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

      1. associate-*r/ 65.2%

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

      2. *-commutative 65.2%

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

      3. *-commutative 65.2%

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

      4. associate-*r* 63.3%

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

      5. associate-*l/ 62.9%

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

      6. pow2 62.9%

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

   4. Applied egg-rr 62.9%

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

   if 3.6999999999999998e104 < z

   1. Initial program 31.6%

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

      1. associate-/l* 36.6%

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

   3. Simplified 36.6%

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

   5. Taylor expanded in t around 0 91.8%

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

      1. associate-/l* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.7 \cdot 10^{+104}:\\ \;\;\;\;\frac{z \cdot y}{\sqrt{{z}^{2} - t \cdot a}} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{z + -0.5 \cdot \left(a \cdot \frac{t}{z}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ \begin{array}{l} t_1 := \sqrt{t \cdot \left(-a\right)}\\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 7.2 \cdot 10^{-179}:\\ \;\;\;\;\frac{x\_m \cdot \left(z\_m \cdot y\_m\right)}{t\_1}\\ \mathbf{elif}\;z\_m \leq 1.65 \cdot 10^{-140}:\\ \;\;\;\;x\_m \cdot \left(y\_m \cdot \frac{z\_m}{z\_m + -0.5 \cdot \frac{t \cdot a}{z\_m}}\right)\\ \mathbf{elif}\;z\_m \leq 3.7 \cdot 10^{-59}:\\ \;\;\;\;x\_m \cdot \left(y\_m \cdot \frac{z\_m}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y\_m \cdot x\_m\right) \cdot \frac{z\_m}{z\_m - -0.5 \cdot \frac{-1}{\frac{\frac{z\_m}{a}}{t}}}\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (let* ((t_1 (sqrt (* t (- a)))))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= z_m 7.2e-179)
        (/ (* x_m (* z_m y_m)) t_1)
        (if (<= z_m 1.65e-140)
          (* x_m (* y_m (/ z_m (+ z_m (* -0.5 (/ (* t a) z_m))))))
          (if (<= z_m 3.7e-59)
            (* x_m (* y_m (/ z_m t_1)))
            (*
             (* y_m x_m)
             (/ z_m (- z_m (* -0.5 (/ -1.0 (/ (/ z_m a) t))))))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double t_1 = sqrt((t * -a));
	double tmp;
	if (z_m <= 7.2e-179) {
		tmp = (x_m * (z_m * y_m)) / t_1;
	} else if (z_m <= 1.65e-140) {
		tmp = x_m * (y_m * (z_m / (z_m + (-0.5 * ((t * a) / z_m)))));
	} else if (z_m <= 3.7e-59) {
		tmp = x_m * (y_m * (z_m / t_1));
	} else {
		tmp = (y_m * x_m) * (z_m / (z_m - (-0.5 * (-1.0 / ((z_m / a) / t)))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, 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_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((t * -a))
    if (z_m <= 7.2d-179) then
        tmp = (x_m * (z_m * y_m)) / t_1
    else if (z_m <= 1.65d-140) then
        tmp = x_m * (y_m * (z_m / (z_m + ((-0.5d0) * ((t * a) / z_m)))))
    else if (z_m <= 3.7d-59) then
        tmp = x_m * (y_m * (z_m / t_1))
    else
        tmp = (y_m * x_m) * (z_m / (z_m - ((-0.5d0) * ((-1.0d0) / ((z_m / a) / t)))))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double t_1 = Math.sqrt((t * -a));
	double tmp;
	if (z_m <= 7.2e-179) {
		tmp = (x_m * (z_m * y_m)) / t_1;
	} else if (z_m <= 1.65e-140) {
		tmp = x_m * (y_m * (z_m / (z_m + (-0.5 * ((t * a) / z_m)))));
	} else if (z_m <= 3.7e-59) {
		tmp = x_m * (y_m * (z_m / t_1));
	} else {
		tmp = (y_m * x_m) * (z_m / (z_m - (-0.5 * (-1.0 / ((z_m / a) / t)))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	t_1 = math.sqrt((t * -a))
	tmp = 0
	if z_m <= 7.2e-179:
		tmp = (x_m * (z_m * y_m)) / t_1
	elif z_m <= 1.65e-140:
		tmp = x_m * (y_m * (z_m / (z_m + (-0.5 * ((t * a) / z_m)))))
	elif z_m <= 3.7e-59:
		tmp = x_m * (y_m * (z_m / t_1))
	else:
		tmp = (y_m * x_m) * (z_m / (z_m - (-0.5 * (-1.0 / ((z_m / a) / t)))))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = sqrt(Float64(t * Float64(-a)))
	tmp = 0.0
	if (z_m <= 7.2e-179)
		tmp = Float64(Float64(x_m * Float64(z_m * y_m)) / t_1);
	elseif (z_m <= 1.65e-140)
		tmp = Float64(x_m * Float64(y_m * Float64(z_m / Float64(z_m + Float64(-0.5 * Float64(Float64(t * a) / z_m))))));
	elseif (z_m <= 3.7e-59)
		tmp = Float64(x_m * Float64(y_m * Float64(z_m / t_1)));
	else
		tmp = Float64(Float64(y_m * x_m) * Float64(z_m / Float64(z_m - Float64(-0.5 * Float64(-1.0 / Float64(Float64(z_m / a) / t))))));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = sqrt((t * -a));
	tmp = 0.0;
	if (z_m <= 7.2e-179)
		tmp = (x_m * (z_m * y_m)) / t_1;
	elseif (z_m <= 1.65e-140)
		tmp = x_m * (y_m * (z_m / (z_m + (-0.5 * ((t * a) / z_m)))));
	elseif (z_m <= 3.7e-59)
		tmp = x_m * (y_m * (z_m / t_1));
	else
		tmp = (y_m * x_m) * (z_m / (z_m - (-0.5 * (-1.0 / ((z_m / a) / t)))));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, 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$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := Block[{t$95$1 = N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 7.2e-179], N[(N[(x$95$m * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z$95$m, 1.65e-140], N[(x$95$m * N[(y$95$m * N[(z$95$m / N[(z$95$m + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 3.7e-59], N[(x$95$m * N[(y$95$m * N[(z$95$m / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * x$95$m), $MachinePrecision] * N[(z$95$m / N[(z$95$m - N[(-0.5 * N[(-1.0 / N[(N[(z$95$m / a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if z < 7.20000000000000015e-179

   1. Initial program 56.8%

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

   3. Taylor expanded in z around 0 38.4%

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

      1. mul-1-neg 38.4%

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

      2. distribute-rgt-neg-out 38.4%

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

   5. Simplified 38.4%

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

   6. Taylor expanded in x around 0 38.3%

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

   if 7.20000000000000015e-179 < z < 1.64999999999999994e-140

   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* 69.6%

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

      2. associate-*l* 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in t around 0 72.1%

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

   if 1.64999999999999994e-140 < z < 3.6999999999999999e-59

   1. Initial program 82.3%

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

      1. associate-/l* 82.2%

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

      2. associate-*l* 74.1%

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

   3. Simplified 74.1%

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

   5. Taylor expanded in z around 0 56.6%

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

      1. mul-1-neg 64.8%

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

      2. distribute-rgt-neg-out 64.8%

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

   7. Simplified 56.6%

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

   if 3.6999999999999999e-59 < z

   1. Initial program 52.3%

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

      1. associate-/l* 56.0%

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

   3. Simplified 56.0%

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

   5. Taylor expanded in t around 0 85.8%

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

      1. associate-/l* 91.2%

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

   7. Simplified 91.2%

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

      1. associate-*r/ 85.8%

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

      2. clear-num 85.8%

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

      3. frac-2neg 85.8%

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

      4. metadata-eval 85.8%

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

   9. Applied egg-rr 85.8%

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

       1. distribute-frac-neg2 85.8%

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

       2. associate-/r* 91.2%

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

   11. Simplified 91.2%

       \[\leadsto \left(x \cdot y\right) \cdot \frac{z}{z + -0.5 \cdot \color{blue}{\left(-\frac{-1}{\frac{\frac{z}{a}}{t}}\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.2 \cdot 10^{-179}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-140}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + -0.5 \cdot \frac{t \cdot a}{z}}\right)\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-59}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{t \cdot \left(-a\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{z - -0.5 \cdot \frac{-1}{\frac{\frac{z}{a}}{t}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ \begin{array}{l} t_1 := x\_m \cdot \left(y\_m \cdot \frac{z\_m}{\sqrt{t \cdot \left(-a\right)}}\right)\\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 7.2 \cdot 10^{-179}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z\_m \leq 1.65 \cdot 10^{-140}:\\ \;\;\;\;x\_m \cdot \left(y\_m \cdot \frac{z\_m}{z\_m + -0.5 \cdot \frac{t \cdot a}{z\_m}}\right)\\ \mathbf{elif}\;z\_m \leq 1.35 \cdot 10^{-57}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(y\_m \cdot x\_m\right) \cdot \frac{z\_m}{z\_m - -0.5 \cdot \frac{-1}{\frac{\frac{z\_m}{a}}{t}}}\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (let* ((t_1 (* x_m (* y_m (/ z_m (sqrt (* t (- a))))))))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= z_m 7.2e-179)
        t_1
        (if (<= z_m 1.65e-140)
          (* x_m (* y_m (/ z_m (+ z_m (* -0.5 (/ (* t a) z_m))))))
          (if (<= z_m 1.35e-57)
            t_1
            (*
             (* y_m x_m)
             (/ z_m (- z_m (* -0.5 (/ -1.0 (/ (/ z_m a) t))))))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double t_1 = x_m * (y_m * (z_m / sqrt((t * -a))));
	double tmp;
	if (z_m <= 7.2e-179) {
		tmp = t_1;
	} else if (z_m <= 1.65e-140) {
		tmp = x_m * (y_m * (z_m / (z_m + (-0.5 * ((t * a) / z_m)))));
	} else if (z_m <= 1.35e-57) {
		tmp = t_1;
	} else {
		tmp = (y_m * x_m) * (z_m / (z_m - (-0.5 * (-1.0 / ((z_m / a) / t)))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, 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_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x_m * (y_m * (z_m / sqrt((t * -a))))
    if (z_m <= 7.2d-179) then
        tmp = t_1
    else if (z_m <= 1.65d-140) then
        tmp = x_m * (y_m * (z_m / (z_m + ((-0.5d0) * ((t * a) / z_m)))))
    else if (z_m <= 1.35d-57) then
        tmp = t_1
    else
        tmp = (y_m * x_m) * (z_m / (z_m - ((-0.5d0) * ((-1.0d0) / ((z_m / a) / t)))))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double t_1 = x_m * (y_m * (z_m / Math.sqrt((t * -a))));
	double tmp;
	if (z_m <= 7.2e-179) {
		tmp = t_1;
	} else if (z_m <= 1.65e-140) {
		tmp = x_m * (y_m * (z_m / (z_m + (-0.5 * ((t * a) / z_m)))));
	} else if (z_m <= 1.35e-57) {
		tmp = t_1;
	} else {
		tmp = (y_m * x_m) * (z_m / (z_m - (-0.5 * (-1.0 / ((z_m / a) / t)))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	t_1 = x_m * (y_m * (z_m / math.sqrt((t * -a))))
	tmp = 0
	if z_m <= 7.2e-179:
		tmp = t_1
	elif z_m <= 1.65e-140:
		tmp = x_m * (y_m * (z_m / (z_m + (-0.5 * ((t * a) / z_m)))))
	elif z_m <= 1.35e-57:
		tmp = t_1
	else:
		tmp = (y_m * x_m) * (z_m / (z_m - (-0.5 * (-1.0 / ((z_m / a) / t)))))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = Float64(x_m * Float64(y_m * Float64(z_m / sqrt(Float64(t * Float64(-a))))))
	tmp = 0.0
	if (z_m <= 7.2e-179)
		tmp = t_1;
	elseif (z_m <= 1.65e-140)
		tmp = Float64(x_m * Float64(y_m * Float64(z_m / Float64(z_m + Float64(-0.5 * Float64(Float64(t * a) / z_m))))));
	elseif (z_m <= 1.35e-57)
		tmp = t_1;
	else
		tmp = Float64(Float64(y_m * x_m) * Float64(z_m / Float64(z_m - Float64(-0.5 * Float64(-1.0 / Float64(Float64(z_m / a) / t))))));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = x_m * (y_m * (z_m / sqrt((t * -a))));
	tmp = 0.0;
	if (z_m <= 7.2e-179)
		tmp = t_1;
	elseif (z_m <= 1.65e-140)
		tmp = x_m * (y_m * (z_m / (z_m + (-0.5 * ((t * a) / z_m)))));
	elseif (z_m <= 1.35e-57)
		tmp = t_1;
	else
		tmp = (y_m * x_m) * (z_m / (z_m - (-0.5 * (-1.0 / ((z_m / a) / t)))));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, 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$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := Block[{t$95$1 = N[(x$95$m * N[(y$95$m * N[(z$95$m / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 7.2e-179], t$95$1, If[LessEqual[z$95$m, 1.65e-140], N[(x$95$m * N[(y$95$m * N[(z$95$m / N[(z$95$m + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 1.35e-57], t$95$1, N[(N[(y$95$m * x$95$m), $MachinePrecision] * N[(z$95$m / N[(z$95$m - N[(-0.5 * N[(-1.0 / N[(N[(z$95$m / a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < 7.20000000000000015e-179 or 1.64999999999999994e-140 < z < 1.3500000000000001e-57

   1. Initial program 58.6%

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

      1. associate-/l* 59.8%

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

      2. associate-*l* 59.0%

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

   3. Simplified 59.0%

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

   5. Taylor expanded in z around 0 36.4%

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

      1. mul-1-neg 40.2%

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

      2. distribute-rgt-neg-out 40.2%

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

   7. Simplified 36.4%

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

   if 7.20000000000000015e-179 < z < 1.64999999999999994e-140

   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* 69.6%

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

      2. associate-*l* 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in t around 0 72.1%

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

   if 1.3500000000000001e-57 < z

   1. Initial program 52.3%

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

      1. associate-/l* 56.0%

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

   3. Simplified 56.0%

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

   5. Taylor expanded in t around 0 85.8%

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

      1. associate-/l* 91.2%

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

   7. Simplified 91.2%

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

      1. associate-*r/ 85.8%

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

      2. clear-num 85.8%

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

      3. frac-2neg 85.8%

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

      4. metadata-eval 85.8%

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

   9. Applied egg-rr 85.8%

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

       1. distribute-frac-neg2 85.8%

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

       2. associate-/r* 91.2%

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

   11. Simplified 91.2%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.2 \cdot 10^{-179}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{t \cdot \left(-a\right)}}\right)\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-140}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + -0.5 \cdot \frac{t \cdot a}{z}}\right)\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-57}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{t \cdot \left(-a\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{z - -0.5 \cdot \frac{-1}{\frac{\frac{z}{a}}{t}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5 \cdot 10^{+104}:\\ \;\;\;\;x\_m \cdot \left(y\_m \cdot \frac{z\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y\_m \cdot x\_m\right) \cdot \frac{z\_m}{z\_m + -0.5 \cdot \left(a \cdot \frac{t}{z\_m}\right)}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 5e+104)
      (* x_m (* y_m (/ z_m (sqrt (- (* z_m z_m) (* t a))))))
      (* (* y_m x_m) (/ z_m (+ z_m (* -0.5 (* a (/ t z_m)))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 5e+104) {
		tmp = x_m * (y_m * (z_m / sqrt(((z_m * z_m) - (t * a)))));
	} else {
		tmp = (y_m * x_m) * (z_m / (z_m + (-0.5 * (a * (t / z_m)))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, 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_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 5d+104) then
        tmp = x_m * (y_m * (z_m / sqrt(((z_m * z_m) - (t * a)))))
    else
        tmp = (y_m * x_m) * (z_m / (z_m + ((-0.5d0) * (a * (t / z_m)))))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 5e+104) {
		tmp = x_m * (y_m * (z_m / Math.sqrt(((z_m * z_m) - (t * a)))));
	} else {
		tmp = (y_m * x_m) * (z_m / (z_m + (-0.5 * (a * (t / z_m)))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 5e+104:
		tmp = x_m * (y_m * (z_m / math.sqrt(((z_m * z_m) - (t * a)))))
	else:
		tmp = (y_m * x_m) * (z_m / (z_m + (-0.5 * (a * (t / z_m)))))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 5e+104)
		tmp = Float64(x_m * Float64(y_m * Float64(z_m / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a))))));
	else
		tmp = Float64(Float64(y_m * x_m) * Float64(z_m / Float64(z_m + Float64(-0.5 * Float64(a * Float64(t / z_m))))));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 5e+104)
		tmp = x_m * (y_m * (z_m / sqrt(((z_m * z_m) - (t * a)))));
	else
		tmp = (y_m * x_m) * (z_m / (z_m + (-0.5 * (a * (t / z_m)))));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, 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$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 5e+104], N[(x$95$m * N[(y$95$m * 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[(N[(y$95$m * x$95$m), $MachinePrecision] * 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 < 4.9999999999999997e104

   1. Initial program 64.1%

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

      1. associate-/l* 65.2%

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

      2. associate-*l* 63.3%

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

   3. Simplified 63.3%

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

   if 4.9999999999999997e104 < z

   1. Initial program 31.6%

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

      1. associate-/l* 36.6%

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

   3. Simplified 36.6%

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

   5. Taylor expanded in t around 0 91.8%

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

      1. associate-/l* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{+104}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{z + -0.5 \cdot \left(a \cdot \frac{t}{z}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.8 \cdot 10^{-96}:\\ \;\;\;\;x\_m \cdot \left(y\_m \cdot \frac{z\_m}{z\_m + -0.5 \cdot \frac{t \cdot a}{z\_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1.8e-96)
      (* x_m (* y_m (/ z_m (+ z_m (* -0.5 (/ (* t a) z_m))))))
      (* y_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.8e-96) {
		tmp = x_m * (y_m * (z_m / (z_m + (-0.5 * ((t * a) / z_m)))));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, 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_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.8d-96) then
        tmp = x_m * (y_m * (z_m / (z_m + ((-0.5d0) * ((t * a) / z_m)))))
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.8e-96) {
		tmp = x_m * (y_m * (z_m / (z_m + (-0.5 * ((t * a) / z_m)))));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.8e-96:
		tmp = x_m * (y_m * (z_m / (z_m + (-0.5 * ((t * a) / z_m)))))
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.8e-96)
		tmp = Float64(x_m * Float64(y_m * Float64(z_m / Float64(z_m + Float64(-0.5 * Float64(Float64(t * a) / z_m))))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.8e-96)
		tmp = x_m * (y_m * (z_m / (z_m + (-0.5 * ((t * a) / z_m)))));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, 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$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.8e-96], N[(x$95$m * N[(y$95$m * N[(z$95$m / N[(z$95$m + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.80000000000000004e-96

   1. Initial program 58.4%

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

      1. associate-/l* 59.6%

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

      2. associate-*l* 58.3%

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

   3. Simplified 58.3%

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

   5. Taylor expanded in t around 0 27.5%

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

   if 1.80000000000000004e-96 < z

   1. Initial program 53.9%

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

      1. associate-/l* 57.3%

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

      2. *-commutative 57.3%

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

      3. associate-*l* 57.4%

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

      4. associate-*r/ 52.0%

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

   3. Simplified 52.0%

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

   5. Taylor expanded in z around inf 87.9%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.8 \cdot 10^{-96}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + -0.5 \cdot \frac{t \cdot a}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.5% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, 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$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * N[(N[(y$95$m * x$95$m), $MachinePrecision] * N[(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 56.8%

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

   1. associate-/l* 58.8%

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

3. Simplified 58.8%

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

5. Taylor expanded in t around 0 47.8%

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

   1. associate-/l* 49.6%

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

7. Simplified 49.6%

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

8. Final simplification 49.6%

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

Alternative 9: 73.8% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, 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$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * N[(y$95$m * N[(N[(z$95$m * x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.8%

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

   1. associate-/l* 58.8%

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

   2. *-commutative 58.8%

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

   3. associate-*l* 59.9%

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

   4. associate-*r/ 57.6%

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

3. Simplified 57.6%

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

5. Taylor expanded in z around inf 38.8%

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

6. Final simplification 38.8%

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

Alternative 10: 72.1% accurate, 37.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, 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$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.8%

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

   1. associate-/l* 58.8%

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

   2. *-commutative 58.8%

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

   3. associate-*l* 59.9%

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

   4. associate-*r/ 57.6%

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

3. Simplified 57.6%

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

5. Taylor expanded in z around inf 44.3%

   \[\leadsto y \cdot \color{blue}{x} \]
6. Add Preprocessing

Developer target: 87.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024091 
(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)))))