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

Percentage Accurate: 61.6% → 93.1%

Time: 16.5s

Alternatives: 11

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: 61.6% 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: 93.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.8 \cdot 10^{-179}:\\ \;\;\;\;x\_m \cdot \left(\frac{1}{\sqrt{{z\_m}^{2} - a \cdot t}} \cdot \left(z\_m \cdot y\right)\right)\\ \mathbf{elif}\;z\_m \leq 10^{+145}:\\ \;\;\;\;x\_m \cdot \left(y \cdot \frac{z\_m}{\sqrt{z\_m \cdot z\_m - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot y\right) \cdot \left(e^{\mathsf{log1p}\left(\frac{z\_m}{\mathsf{fma}\left(a \cdot \frac{t}{z\_m}, -0.5, z\_m\right)}\right)} + -1\right)\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x_s x_m y z_m t a)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 2.8e-179)
     (* x_m (* (/ 1.0 (sqrt (- (pow z_m 2.0) (* a t)))) (* z_m y)))
     (if (<= z_m 1e+145)
       (* x_m (* y (/ z_m (sqrt (- (* z_m z_m) (* a t))))))
       (*
        (* x_m y)
        (+ (exp (log1p (/ z_m (fma (* a (/ t z_m)) -0.5 z_m)))) -1.0)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.8e-179) {
		tmp = x_m * ((1.0 / sqrt((pow(z_m, 2.0) - (a * t)))) * (z_m * y));
	} else if (z_m <= 1e+145) {
		tmp = x_m * (y * (z_m / sqrt(((z_m * z_m) - (a * t)))));
	} else {
		tmp = (x_m * y) * (exp(log1p((z_m / fma((a * (t / z_m)), -0.5, z_m)))) + -1.0);
	}
	return z_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.8e-179)
		tmp = Float64(x_m * Float64(Float64(1.0 / sqrt(Float64((z_m ^ 2.0) - Float64(a * t)))) * Float64(z_m * y)));
	elseif (z_m <= 1e+145)
		tmp = Float64(x_m * Float64(y * Float64(z_m / sqrt(Float64(Float64(z_m * z_m) - Float64(a * t))))));
	else
		tmp = Float64(Float64(x_m * y) * Float64(exp(log1p(Float64(z_m / fma(Float64(a * Float64(t / z_m)), -0.5, z_m)))) + -1.0));
	end
	return Float64(z_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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2.8e-179], N[(x$95$m * N[(N[(1.0 / N[Sqrt[N[(N[Power[z$95$m, 2.0], $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 1e+145], N[(x$95$m * N[(y * N[(z$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * y), $MachinePrecision] * N[(N[Exp[N[Log[1 + N[(z$95$m / N[(N[(a * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision] * -0.5 + z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 2.8000000000000001e-179

   1. Initial program 62.8%

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

      1. associate-/l* 64.0%

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

      2. associate-*l* 64.1%

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

   3. Simplified 64.1%

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

      1. associate-*r/ 61.4%

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

      2. clear-num 61.3%

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

      3. pow2 61.3%

         \[\leadsto x \cdot \frac{1}{\frac{\sqrt{\color{blue}{{z}^{2}} - t \cdot a}}{y \cdot z}} \]

      4. *-commutative 61.3%

         \[\leadsto x \cdot \frac{1}{\frac{\sqrt{{z}^{2} - t \cdot a}}{\color{blue}{z \cdot y}}} \]

   6. Applied egg-rr 61.3%

      \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\sqrt{{z}^{2} - t \cdot a}}{z \cdot y}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 61.3%

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{\sqrt{{z}^{2} - t \cdot a}} \cdot \left(z \cdot y\right)\right)} \]

      2. *-commutative 61.3%

         \[\leadsto x \cdot \left(\frac{1}{\sqrt{{z}^{2} - \color{blue}{a \cdot t}}} \cdot \left(z \cdot y\right)\right) \]

   8. Simplified 61.3%

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{\sqrt{{z}^{2} - a \cdot t}} \cdot \left(z \cdot y\right)\right)} \]

   if 2.8000000000000001e-179 < z < 9.9999999999999999e144

   1. Initial program 89.2%

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

      1. associate-/l* 94.0%

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

      2. associate-*l* 93.4%

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

   3. Simplified 93.4%

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

   if 9.9999999999999999e144 < z

   1. Initial program 25.2%

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

      1. associate-/l* 27.8%

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

   3. Simplified 27.8%

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

   5. Taylor expanded in t around 0 93.2%

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

      1. associate-/l* 95.9%

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

   7. Simplified 95.9%

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

      1. expm1-log1p-u 95.9%

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

      2. expm1-undefine 95.9%

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

      3. associate-*r/ 93.2%

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

      4. +-commutative 93.2%

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

      5. associate-*r/ 95.9%

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

      6. *-commutative 95.9%

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

      7. fma-define 95.9%

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

   9. Applied egg-rr 95.9%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.8 \cdot 10^{-179}:\\ \;\;\;\;x \cdot \left(\frac{1}{\sqrt{{z}^{2} - a \cdot t}} \cdot \left(z \cdot y\right)\right)\\ \mathbf{elif}\;z \leq 10^{+145}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \left(e^{\mathsf{log1p}\left(\frac{z}{\mathsf{fma}\left(a \cdot \frac{t}{z}, -0.5, z\right)}\right)} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4.4 \cdot 10^{-182}:\\ \;\;\;\;x\_m \cdot \left(\frac{1}{\sqrt{{z\_m}^{2} - a \cdot t}} \cdot \left(z\_m \cdot y\right)\right)\\ \mathbf{elif}\;z\_m \leq 10^{+144}:\\ \;\;\;\;x\_m \cdot \left(y \cdot \frac{z\_m}{\sqrt{z\_m \cdot z\_m - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot y\right) \cdot \log \left(e^{\frac{z\_m}{\mathsf{fma}\left(a \cdot \frac{t}{z\_m}, -0.5, z\_m\right)}}\right)\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x_s x_m y z_m t a)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 4.4e-182)
     (* x_m (* (/ 1.0 (sqrt (- (pow z_m 2.0) (* a t)))) (* z_m y)))
     (if (<= z_m 1e+144)
       (* x_m (* y (/ z_m (sqrt (- (* z_m z_m) (* a t))))))
       (* (* x_m y) (log (exp (/ z_m (fma (* a (/ t z_m)) -0.5 z_m))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 4.4e-182) {
		tmp = x_m * ((1.0 / sqrt((pow(z_m, 2.0) - (a * t)))) * (z_m * y));
	} else if (z_m <= 1e+144) {
		tmp = x_m * (y * (z_m / sqrt(((z_m * z_m) - (a * t)))));
	} else {
		tmp = (x_m * y) * log(exp((z_m / fma((a * (t / z_m)), -0.5, z_m))));
	}
	return z_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 4.4e-182)
		tmp = Float64(x_m * Float64(Float64(1.0 / sqrt(Float64((z_m ^ 2.0) - Float64(a * t)))) * Float64(z_m * y)));
	elseif (z_m <= 1e+144)
		tmp = Float64(x_m * Float64(y * Float64(z_m / sqrt(Float64(Float64(z_m * z_m) - Float64(a * t))))));
	else
		tmp = Float64(Float64(x_m * y) * log(exp(Float64(z_m / fma(Float64(a * Float64(t / z_m)), -0.5, z_m)))));
	end
	return Float64(z_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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 4.4e-182], N[(x$95$m * N[(N[(1.0 / N[Sqrt[N[(N[Power[z$95$m, 2.0], $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 1e+144], N[(x$95$m * N[(y * N[(z$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * y), $MachinePrecision] * N[Log[N[Exp[N[(z$95$m / N[(N[(a * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision] * -0.5 + z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 4.3999999999999999e-182

   1. Initial program 62.8%

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

      1. associate-/l* 64.0%

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

      2. associate-*l* 64.1%

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

   3. Simplified 64.1%

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

      1. associate-*r/ 61.4%

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

      2. clear-num 61.3%

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

      3. pow2 61.3%

         \[\leadsto x \cdot \frac{1}{\frac{\sqrt{\color{blue}{{z}^{2}} - t \cdot a}}{y \cdot z}} \]

      4. *-commutative 61.3%

         \[\leadsto x \cdot \frac{1}{\frac{\sqrt{{z}^{2} - t \cdot a}}{\color{blue}{z \cdot y}}} \]

   6. Applied egg-rr 61.3%

      \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\sqrt{{z}^{2} - t \cdot a}}{z \cdot y}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 61.3%

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{\sqrt{{z}^{2} - t \cdot a}} \cdot \left(z \cdot y\right)\right)} \]

      2. *-commutative 61.3%

         \[\leadsto x \cdot \left(\frac{1}{\sqrt{{z}^{2} - \color{blue}{a \cdot t}}} \cdot \left(z \cdot y\right)\right) \]

   8. Simplified 61.3%

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{\sqrt{{z}^{2} - a \cdot t}} \cdot \left(z \cdot y\right)\right)} \]

   if 4.3999999999999999e-182 < z < 1.00000000000000002e144

   1. Initial program 89.2%

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

      1. associate-/l* 94.0%

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

      2. associate-*l* 93.4%

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

   3. Simplified 93.4%

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

   if 1.00000000000000002e144 < z

   1. Initial program 25.2%

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

      1. associate-/l* 27.8%

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

   3. Simplified 27.8%

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

   5. Taylor expanded in t around 0 93.2%

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

      1. associate-/l* 95.9%

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

   7. Simplified 95.9%

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

      1. associate-*r/ 93.2%

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

      2. add-log-exp 93.2%

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

      3. +-commutative 93.2%

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

      4. associate-*r/ 95.9%

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

      5. *-commutative 95.9%

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

      6. fma-define 95.9%

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

   9. Applied egg-rr 95.9%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.4 \cdot 10^{-182}:\\ \;\;\;\;x \cdot \left(\frac{1}{\sqrt{{z}^{2} - a \cdot t}} \cdot \left(z \cdot y\right)\right)\\ \mathbf{elif}\;z \leq 10^{+144}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \log \left(e^{\frac{z}{\mathsf{fma}\left(a \cdot \frac{t}{z}, -0.5, z\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 8 \cdot 10^{-181}:\\ \;\;\;\;x\_m \cdot \left(\frac{1}{\sqrt{{z\_m}^{2} - a \cdot t}} \cdot \left(z\_m \cdot y\right)\right)\\ \mathbf{elif}\;z\_m \leq 10^{+144}:\\ \;\;\;\;x\_m \cdot \left(y \cdot \frac{z\_m}{\sqrt{z\_m \cdot z\_m - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(y \cdot \frac{z\_m}{z\_m + \left(a \cdot \frac{t}{z\_m}\right) \cdot -0.5}\right)\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x_s x_m y z_m t a)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 8e-181)
     (* x_m (* (/ 1.0 (sqrt (- (pow z_m 2.0) (* a t)))) (* z_m y)))
     (if (<= z_m 1e+144)
       (* x_m (* y (/ z_m (sqrt (- (* z_m z_m) (* a t))))))
       (* x_m (* y (/ z_m (+ z_m (* (* a (/ t z_m)) -0.5))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 8e-181) {
		tmp = x_m * ((1.0 / sqrt((pow(z_m, 2.0) - (a * t)))) * (z_m * y));
	} else if (z_m <= 1e+144) {
		tmp = x_m * (y * (z_m / sqrt(((z_m * z_m) - (a * t)))));
	} else {
		tmp = x_m * (y * (z_m / (z_m + ((a * (t / z_m)) * -0.5))));
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, x_s, x_m, y, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 8d-181) then
        tmp = x_m * ((1.0d0 / sqrt(((z_m ** 2.0d0) - (a * t)))) * (z_m * y))
    else if (z_m <= 1d+144) then
        tmp = x_m * (y * (z_m / sqrt(((z_m * z_m) - (a * t)))))
    else
        tmp = x_m * (y * (z_m / (z_m + ((a * (t / z_m)) * (-0.5d0)))))
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y && y < z_m && z_m < t && t < a;
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 8e-181) {
		tmp = x_m * ((1.0 / Math.sqrt((Math.pow(z_m, 2.0) - (a * t)))) * (z_m * y));
	} else if (z_m <= 1e+144) {
		tmp = x_m * (y * (z_m / Math.sqrt(((z_m * z_m) - (a * t)))));
	} else {
		tmp = x_m * (y * (z_m / (z_m + ((a * (t / z_m)) * -0.5))));
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y, z_m, t, a] = sort([x_m, y, z_m, t, a])
def code(z_s, x_s, x_m, y, z_m, t, a):
	tmp = 0
	if z_m <= 8e-181:
		tmp = x_m * ((1.0 / math.sqrt((math.pow(z_m, 2.0) - (a * t)))) * (z_m * y))
	elif z_m <= 1e+144:
		tmp = x_m * (y * (z_m / math.sqrt(((z_m * z_m) - (a * t)))))
	else:
		tmp = x_m * (y * (z_m / (z_m + ((a * (t / z_m)) * -0.5))))
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 8e-181)
		tmp = Float64(x_m * Float64(Float64(1.0 / sqrt(Float64((z_m ^ 2.0) - Float64(a * t)))) * Float64(z_m * y)));
	elseif (z_m <= 1e+144)
		tmp = Float64(x_m * Float64(y * Float64(z_m / sqrt(Float64(Float64(z_m * z_m) - Float64(a * t))))));
	else
		tmp = Float64(x_m * Float64(y * Float64(z_m / Float64(z_m + Float64(Float64(a * Float64(t / z_m)) * -0.5)))));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y, z_m, t, a = num2cell(sort([x_m, y, z_m, t, a])){:}
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 8e-181)
		tmp = x_m * ((1.0 / sqrt(((z_m ^ 2.0) - (a * t)))) * (z_m * y));
	elseif (z_m <= 1e+144)
		tmp = x_m * (y * (z_m / sqrt(((z_m * z_m) - (a * t)))));
	else
		tmp = x_m * (y * (z_m / (z_m + ((a * (t / z_m)) * -0.5))));
	end
	tmp_2 = z_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 8e-181], N[(x$95$m * N[(N[(1.0 / N[Sqrt[N[(N[Power[z$95$m, 2.0], $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 1e+144], N[(x$95$m * N[(y * N[(z$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(y * N[(z$95$m / N[(z$95$m + N[(N[(a * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 8.00000000000000038e-181

   1. Initial program 62.8%

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

      1. associate-/l* 64.0%

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

      2. associate-*l* 64.1%

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

   3. Simplified 64.1%

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

      1. associate-*r/ 61.4%

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

      2. clear-num 61.3%

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

      3. pow2 61.3%

         \[\leadsto x \cdot \frac{1}{\frac{\sqrt{\color{blue}{{z}^{2}} - t \cdot a}}{y \cdot z}} \]

      4. *-commutative 61.3%

         \[\leadsto x \cdot \frac{1}{\frac{\sqrt{{z}^{2} - t \cdot a}}{\color{blue}{z \cdot y}}} \]

   6. Applied egg-rr 61.3%

      \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\sqrt{{z}^{2} - t \cdot a}}{z \cdot y}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 61.3%

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{\sqrt{{z}^{2} - t \cdot a}} \cdot \left(z \cdot y\right)\right)} \]

      2. *-commutative 61.3%

         \[\leadsto x \cdot \left(\frac{1}{\sqrt{{z}^{2} - \color{blue}{a \cdot t}}} \cdot \left(z \cdot y\right)\right) \]

   8. Simplified 61.3%

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{\sqrt{{z}^{2} - a \cdot t}} \cdot \left(z \cdot y\right)\right)} \]

   if 8.00000000000000038e-181 < z < 1.00000000000000002e144

   1. Initial program 89.2%

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

      1. associate-/l* 94.0%

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

      2. associate-*l* 93.4%

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

   3. Simplified 93.4%

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

   if 1.00000000000000002e144 < z

   1. Initial program 25.2%

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

      1. associate-/l* 27.8%

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

      2. associate-*l* 27.9%

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

   3. Simplified 27.9%

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

   5. Taylor expanded in t around 0 93.2%

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

      1. associate-/l* 95.9%

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

   7. Simplified 95.9%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8 \cdot 10^{-181}:\\ \;\;\;\;x \cdot \left(\frac{1}{\sqrt{{z}^{2} - a \cdot t}} \cdot \left(z \cdot y\right)\right)\\ \mathbf{elif}\;z \leq 10^{+144}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + \left(a \cdot \frac{t}{z}\right) \cdot -0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4 \cdot 10^{-234}:\\ \;\;\;\;\frac{x\_m \cdot \left(z\_m \cdot y\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{elif}\;z\_m \leq 10^{+144}:\\ \;\;\;\;x\_m \cdot \left(y \cdot \frac{z\_m}{\sqrt{z\_m \cdot z\_m - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(y \cdot \frac{z\_m}{z\_m + \left(a \cdot \frac{t}{z\_m}\right) \cdot -0.5}\right)\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x_s x_m y z_m t a)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 4e-234)
     (/ (* x_m (* z_m y)) (sqrt (* a (- t))))
     (if (<= z_m 1e+144)
       (* x_m (* y (/ z_m (sqrt (- (* z_m z_m) (* a t))))))
       (* x_m (* y (/ z_m (+ z_m (* (* a (/ t z_m)) -0.5))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 4e-234) {
		tmp = (x_m * (z_m * y)) / sqrt((a * -t));
	} else if (z_m <= 1e+144) {
		tmp = x_m * (y * (z_m / sqrt(((z_m * z_m) - (a * t)))));
	} else {
		tmp = x_m * (y * (z_m / (z_m + ((a * (t / z_m)) * -0.5))));
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, x_s, x_m, y, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 4d-234) then
        tmp = (x_m * (z_m * y)) / sqrt((a * -t))
    else if (z_m <= 1d+144) then
        tmp = x_m * (y * (z_m / sqrt(((z_m * z_m) - (a * t)))))
    else
        tmp = x_m * (y * (z_m / (z_m + ((a * (t / z_m)) * (-0.5d0)))))
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y && y < z_m && z_m < t && t < a;
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 4e-234) {
		tmp = (x_m * (z_m * y)) / Math.sqrt((a * -t));
	} else if (z_m <= 1e+144) {
		tmp = x_m * (y * (z_m / Math.sqrt(((z_m * z_m) - (a * t)))));
	} else {
		tmp = x_m * (y * (z_m / (z_m + ((a * (t / z_m)) * -0.5))));
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y, z_m, t, a] = sort([x_m, y, z_m, t, a])
def code(z_s, x_s, x_m, y, z_m, t, a):
	tmp = 0
	if z_m <= 4e-234:
		tmp = (x_m * (z_m * y)) / math.sqrt((a * -t))
	elif z_m <= 1e+144:
		tmp = x_m * (y * (z_m / math.sqrt(((z_m * z_m) - (a * t)))))
	else:
		tmp = x_m * (y * (z_m / (z_m + ((a * (t / z_m)) * -0.5))))
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 4e-234)
		tmp = Float64(Float64(x_m * Float64(z_m * y)) / sqrt(Float64(a * Float64(-t))));
	elseif (z_m <= 1e+144)
		tmp = Float64(x_m * Float64(y * Float64(z_m / sqrt(Float64(Float64(z_m * z_m) - Float64(a * t))))));
	else
		tmp = Float64(x_m * Float64(y * Float64(z_m / Float64(z_m + Float64(Float64(a * Float64(t / z_m)) * -0.5)))));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y, z_m, t, a = num2cell(sort([x_m, y, z_m, t, a])){:}
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 4e-234)
		tmp = (x_m * (z_m * y)) / sqrt((a * -t));
	elseif (z_m <= 1e+144)
		tmp = x_m * (y * (z_m / sqrt(((z_m * z_m) - (a * t)))));
	else
		tmp = x_m * (y * (z_m / (z_m + ((a * (t / z_m)) * -0.5))));
	end
	tmp_2 = z_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 4e-234], N[(N[(x$95$m * N[(z$95$m * y), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 1e+144], N[(x$95$m * N[(y * N[(z$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(y * N[(z$95$m / N[(z$95$m + N[(N[(a * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 3.9999999999999998e-234

   1. Initial program 61.9%

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

   3. Taylor expanded in x around 0 58.3%

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

   4. Taylor expanded in z around 0 31.4%

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

      1. associate-*r* 31.4%

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

      2. neg-mul-1 31.4%

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

      3. *-commutative 31.4%

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

   6. Simplified 31.4%

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

   if 3.9999999999999998e-234 < z < 1.00000000000000002e144

   1. Initial program 86.8%

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

      1. associate-/l* 92.1%

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

      2. associate-*l* 94.4%

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

   3. Simplified 94.4%

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

   if 1.00000000000000002e144 < z

   1. Initial program 25.2%

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

      1. associate-/l* 27.8%

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

      2. associate-*l* 27.9%

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

   3. Simplified 27.9%

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

   5. Taylor expanded in t around 0 93.2%

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

      1. associate-/l* 95.9%

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

   7. Simplified 95.9%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{-234}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{elif}\;z \leq 10^{+144}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + \left(a \cdot \frac{t}{z}\right) \cdot -0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.35 \cdot 10^{-179}:\\ \;\;\;\;\frac{x\_m \cdot \left(z\_m \cdot y\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot y\right) \cdot \frac{z\_m}{z\_m + t \cdot \frac{a \cdot -0.5}{z\_m}}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x_s x_m y z_m t a)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 2.35e-179)
     (/ (* x_m (* z_m y)) (sqrt (* a (- t))))
     (* (* x_m y) (/ z_m (+ z_m (* t (/ (* a -0.5) z_m)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.35e-179) {
		tmp = (x_m * (z_m * y)) / sqrt((a * -t));
	} else {
		tmp = (x_m * y) * (z_m / (z_m + (t * ((a * -0.5) / z_m))));
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, x_s, x_m, y, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 2.35d-179) then
        tmp = (x_m * (z_m * y)) / sqrt((a * -t))
    else
        tmp = (x_m * y) * (z_m / (z_m + (t * ((a * (-0.5d0)) / z_m))))
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y && y < z_m && z_m < t && t < a;
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.35e-179) {
		tmp = (x_m * (z_m * y)) / Math.sqrt((a * -t));
	} else {
		tmp = (x_m * y) * (z_m / (z_m + (t * ((a * -0.5) / z_m))));
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y, z_m, t, a] = sort([x_m, y, z_m, t, a])
def code(z_s, x_s, x_m, y, z_m, t, a):
	tmp = 0
	if z_m <= 2.35e-179:
		tmp = (x_m * (z_m * y)) / math.sqrt((a * -t))
	else:
		tmp = (x_m * y) * (z_m / (z_m + (t * ((a * -0.5) / z_m))))
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.35e-179)
		tmp = Float64(Float64(x_m * Float64(z_m * y)) / sqrt(Float64(a * Float64(-t))));
	else
		tmp = Float64(Float64(x_m * y) * Float64(z_m / Float64(z_m + Float64(t * Float64(Float64(a * -0.5) / z_m)))));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y, z_m, t, a = num2cell(sort([x_m, y, z_m, t, a])){:}
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2.35e-179)
		tmp = (x_m * (z_m * y)) / sqrt((a * -t));
	else
		tmp = (x_m * y) * (z_m / (z_m + (t * ((a * -0.5) / z_m))));
	end
	tmp_2 = z_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2.35e-179], N[(N[(x$95$m * N[(z$95$m * y), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * y), $MachinePrecision] * N[(z$95$m / N[(z$95$m + N[(t * N[(N[(a * -0.5), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.3500000000000001e-179

   1. Initial program 62.8%

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

   3. Taylor expanded in x around 0 60.7%

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

   4. Taylor expanded in z around 0 35.6%

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

      1. associate-*r* 35.6%

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

      2. neg-mul-1 35.6%

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

      3. *-commutative 35.6%

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

   6. Simplified 35.6%

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

   if 2.3500000000000001e-179 < z

   1. Initial program 61.9%

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

      1. associate-/l* 65.8%

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

   3. Simplified 65.8%

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

   5. Taylor expanded in t around 0 83.1%

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

      1. associate-/l* 84.2%

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

   7. Simplified 84.2%

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

      1. associate-*r* 84.2%

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

      2. clear-num 84.2%

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

      3. un-div-inv 84.2%

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

   9. Applied egg-rr 84.2%

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

       1. associate-/r/ 84.2%

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

   11. Applied egg-rr 84.2%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.35 \cdot 10^{-179}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{z + t \cdot \frac{a \cdot -0.5}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.9 \cdot 10^{-179}:\\ \;\;\;\;x\_m \cdot \left(y \cdot \frac{z\_m}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot y\right) \cdot \frac{z\_m}{z\_m + t \cdot \frac{a \cdot -0.5}{z\_m}}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x_s x_m y z_m t a)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 2.9e-179)
     (* x_m (* y (/ z_m (sqrt (* a (- t))))))
     (* (* x_m y) (/ z_m (+ z_m (* t (/ (* a -0.5) z_m)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.9e-179) {
		tmp = x_m * (y * (z_m / sqrt((a * -t))));
	} else {
		tmp = (x_m * y) * (z_m / (z_m + (t * ((a * -0.5) / z_m))));
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, x_s, x_m, y, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 2.9d-179) then
        tmp = x_m * (y * (z_m / sqrt((a * -t))))
    else
        tmp = (x_m * y) * (z_m / (z_m + (t * ((a * (-0.5d0)) / z_m))))
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y && y < z_m && z_m < t && t < a;
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.9e-179) {
		tmp = x_m * (y * (z_m / Math.sqrt((a * -t))));
	} else {
		tmp = (x_m * y) * (z_m / (z_m + (t * ((a * -0.5) / z_m))));
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y, z_m, t, a] = sort([x_m, y, z_m, t, a])
def code(z_s, x_s, x_m, y, z_m, t, a):
	tmp = 0
	if z_m <= 2.9e-179:
		tmp = x_m * (y * (z_m / math.sqrt((a * -t))))
	else:
		tmp = (x_m * y) * (z_m / (z_m + (t * ((a * -0.5) / z_m))))
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.9e-179)
		tmp = Float64(x_m * Float64(y * Float64(z_m / sqrt(Float64(a * Float64(-t))))));
	else
		tmp = Float64(Float64(x_m * y) * Float64(z_m / Float64(z_m + Float64(t * Float64(Float64(a * -0.5) / z_m)))));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y, z_m, t, a = num2cell(sort([x_m, y, z_m, t, a])){:}
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2.9e-179)
		tmp = x_m * (y * (z_m / sqrt((a * -t))));
	else
		tmp = (x_m * y) * (z_m / (z_m + (t * ((a * -0.5) / z_m))));
	end
	tmp_2 = z_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2.9e-179], N[(x$95$m * N[(y * N[(z$95$m / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * y), $MachinePrecision] * N[(z$95$m / N[(z$95$m + N[(t * N[(N[(a * -0.5), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.8999999999999999e-179

   1. Initial program 62.8%

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

      1. associate-/l* 64.0%

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

      2. associate-*l* 64.1%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in z around 0 33.8%

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

      1. associate-*r* 35.6%

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

      2. neg-mul-1 35.6%

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

      3. *-commutative 35.6%

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

   7. Simplified 33.8%

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

   if 2.8999999999999999e-179 < z

   1. Initial program 61.9%

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

      1. associate-/l* 65.8%

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

   3. Simplified 65.8%

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

   5. Taylor expanded in t around 0 83.1%

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

      1. associate-/l* 84.2%

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

   7. Simplified 84.2%

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

      1. associate-*r* 84.2%

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

      2. clear-num 84.2%

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

      3. un-div-inv 84.2%

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

   9. Applied egg-rr 84.2%

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

       1. associate-/r/ 84.2%

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

   11. Applied egg-rr 84.2%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.9 \cdot 10^{-179}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{z + t \cdot \frac{a \cdot -0.5}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.9% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

   1. associate-/l* 64.7%

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

3. Simplified 64.7%

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

5. Taylor expanded in t around 0 47.4%

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

   1. associate-/l* 47.8%

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

7. Simplified 47.8%

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

   1. associate-*r* 47.8%

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

   2. clear-num 47.8%

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

   3. un-div-inv 47.8%

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

9. Applied egg-rr 47.8%

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

    1. associate-/r/ 47.8%

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

11. Applied egg-rr 47.8%

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

12. Final simplification 47.8%

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

Alternative 8: 79.8% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

   1. associate-/l* 64.7%

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

3. Simplified 64.7%

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

5. Taylor expanded in t around 0 47.4%

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

   1. associate-/l* 47.8%

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

7. Simplified 47.8%

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

8. Final simplification 47.8%

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

Alternative 9: 79.5% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

   1. associate-/l* 64.7%

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

   2. associate-*l* 64.7%

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

3. Simplified 64.7%

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

5. Taylor expanded in t around 0 47.4%

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

   1. associate-/l* 47.8%

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

7. Simplified 47.8%

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

8. Final simplification 47.8%

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

Alternative 10: 72.3% accurate, 9.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 3 \cdot 10^{-197}:\\ \;\;\;\;x\_m \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z\_m \cdot x\_m}{z\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s x_s x_m y z_m t a)
 :precision binary64
 (* z_s (* x_s (if (<= a 3e-197) (* x_m y) (* y (/ (* z_m x_m) z_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (a <= 3e-197) {
		tmp = x_m * y;
	} else {
		tmp = y * ((z_m * x_m) / z_m);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, x_s, x_m, y, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (a <= 3d-197) then
        tmp = x_m * y
    else
        tmp = y * ((z_m * x_m) / z_m)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y && y < z_m && z_m < t && t < a;
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (a <= 3e-197) {
		tmp = x_m * y;
	} else {
		tmp = y * ((z_m * x_m) / z_m);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y, z_m, t, a] = sort([x_m, y, z_m, t, a])
def code(z_s, x_s, x_m, y, z_m, t, a):
	tmp = 0
	if a <= 3e-197:
		tmp = x_m * y
	else:
		tmp = y * ((z_m * x_m) / z_m)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0
	if (a <= 3e-197)
		tmp = Float64(x_m * y);
	else
		tmp = Float64(y * Float64(Float64(z_m * x_m) / z_m));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y, z_m, t, a = num2cell(sort([x_m, y, z_m, t, a])){:}
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t, a)
	tmp = 0.0;
	if (a <= 3e-197)
		tmp = x_m * y;
	else
		tmp = y * ((z_m * x_m) / z_m);
	end
	tmp_2 = z_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(x$95$s * If[LessEqual[a, 3e-197], N[(x$95$m * y), $MachinePrecision], N[(y * N[(N[(z$95$m * x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 3.00000000000000026e-197

   1. Initial program 63.5%

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

      1. associate-/l* 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* 66.0%

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

      4. associate-*r/ 64.1%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in z around inf 41.0%

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

   if 3.00000000000000026e-197 < a

   1. Initial program 60.8%

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

      1. associate-/l* 62.8%

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

      2. *-commutative 62.8%

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

      3. associate-*l* 63.9%

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

      4. associate-*r/ 63.5%

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

   3. Simplified 63.5%

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

   5. Taylor expanded in z around inf 37.2%

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

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3 \cdot 10^{-197}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.1% accurate, 37.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

   1. associate-/l* 64.7%

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

   2. *-commutative 64.7%

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

   3. associate-*l* 65.2%

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

   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 40.8%

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

6. Final simplification 40.8%

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

Developer Target 1: 88.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024116 
(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)))))