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

Percentage Accurate: 61.3% → 94.6%

Time: 13.7s

Alternatives: 13

Speedup: 7.5×

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: 61.3% 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: 94.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.55 \cdot 10^{-148}:\\ \;\;\;\;\frac{z\_m \cdot y\_m}{\sqrt{a}} \cdot \frac{x\_m}{\sqrt{-t}}\\ \mathbf{elif}\;z\_m \leq 4 \cdot 10^{+91}:\\ \;\;\;\;\frac{y\_m \cdot x\_m}{\frac{\sqrt{z\_m \cdot z\_m - a \cdot t}}{z\_m}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 1.55e-148)
      (* (/ (* z_m y_m) (sqrt a)) (/ x_m (sqrt (- t))))
      (if (<= z_m 4e+91)
        (/ (* y_m x_m) (/ (sqrt (- (* z_m z_m) (* a t))) z_m))
        (* y_m x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.55e-148) {
		tmp = ((z_m * y_m) / sqrt(a)) * (x_m / sqrt(-t));
	} else if (z_m <= 4e+91) {
		tmp = (y_m * x_m) / (sqrt(((z_m * z_m) - (a * t))) / z_m);
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_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 <= 1.55d-148) then
        tmp = ((z_m * y_m) / sqrt(a)) * (x_m / sqrt(-t))
    else if (z_m <= 4d+91) then
        tmp = (y_m * x_m) / (sqrt(((z_m * z_m) - (a * t))) / z_m)
    else
        tmp = y_m * x_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.55e-148) {
		tmp = ((z_m * y_m) / Math.sqrt(a)) * (x_m / Math.sqrt(-t));
	} else if (z_m <= 4e+91) {
		tmp = (y_m * x_m) / (Math.sqrt(((z_m * z_m) - (a * t))) / z_m);
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.55e-148:
		tmp = ((z_m * y_m) / math.sqrt(a)) * (x_m / math.sqrt(-t))
	elif z_m <= 4e+91:
		tmp = (y_m * x_m) / (math.sqrt(((z_m * z_m) - (a * t))) / z_m)
	else:
		tmp = y_m * x_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.55e-148)
		tmp = Float64(Float64(Float64(z_m * y_m) / sqrt(a)) * Float64(x_m / sqrt(Float64(-t))));
	elseif (z_m <= 4e+91)
		tmp = Float64(Float64(y_m * x_m) / Float64(sqrt(Float64(Float64(z_m * z_m) - Float64(a * t))) / z_m));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.55e-148)
		tmp = ((z_m * y_m) / sqrt(a)) * (x_m / sqrt(-t));
	elseif (z_m <= 4e+91)
		tmp = (y_m * x_m) / (sqrt(((z_m * z_m) - (a * t))) / z_m);
	else
		tmp = y_m * x_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 1.55e-148], N[(N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[Sqrt[a], $MachinePrecision]), $MachinePrecision] * N[(x$95$m / N[Sqrt[(-t)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 4e+91], N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 1.5500000000000001e-148

   1. Initial program 65.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 66.2

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

   4. Applied egg-rr 66.2%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. *-lowering-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 44.2

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

   7. Simplified 44.2%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. pow1/2 N/A

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

      6. unpow-prod-down N/A

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

      7. times-frac N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. sqrt-lowering-sqrt.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. pow1/2 N/A

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

      15. sqrt-lowering-sqrt.f64 N/A

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

      16. neg-lowering-neg.f64 21.0

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

   9. Applied egg-rr 21.0%

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

   if 1.5500000000000001e-148 < z < 4.00000000000000032e91

   1. Initial program 93.7%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 93.9

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

   4. Applied egg-rr 93.9%

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

   if 4.00000000000000032e91 < z

   1. Initial program 31.2%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 95.8

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

   5. Simplified 95.8%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.55 \cdot 10^{-148}:\\ \;\;\;\;\frac{z \cdot y}{\sqrt{a}} \cdot \frac{x}{\sqrt{-t}}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+91}:\\ \;\;\;\;\frac{y \cdot x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5.4 \cdot 10^{-113}:\\ \;\;\;\;x\_m \cdot \frac{z\_m \cdot y\_m}{\sqrt{-a \cdot t}}\\ \mathbf{elif}\;z\_m \leq 1.85 \cdot 10^{+94}:\\ \;\;\;\;\left(z\_m \cdot x\_m\right) \cdot \frac{y\_m}{\sqrt{z\_m \cdot z\_m - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 5.4e-113)
      (* x_m (/ (* z_m y_m) (sqrt (- (* a t)))))
      (if (<= z_m 1.85e+94)
        (* (* z_m x_m) (/ y_m (sqrt (- (* z_m z_m) (* a t)))))
        (* y_m x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 5.4e-113) {
		tmp = x_m * ((z_m * y_m) / sqrt(-(a * t)));
	} else if (z_m <= 1.85e+94) {
		tmp = (z_m * x_m) * (y_m / sqrt(((z_m * z_m) - (a * t))));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_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 <= 5.4d-113) then
        tmp = x_m * ((z_m * y_m) / sqrt(-(a * t)))
    else if (z_m <= 1.85d+94) then
        tmp = (z_m * x_m) * (y_m / sqrt(((z_m * z_m) - (a * t))))
    else
        tmp = y_m * x_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 5.4e-113) {
		tmp = x_m * ((z_m * y_m) / Math.sqrt(-(a * t)));
	} else if (z_m <= 1.85e+94) {
		tmp = (z_m * x_m) * (y_m / Math.sqrt(((z_m * z_m) - (a * t))));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 5.4e-113:
		tmp = x_m * ((z_m * y_m) / math.sqrt(-(a * t)))
	elif z_m <= 1.85e+94:
		tmp = (z_m * x_m) * (y_m / math.sqrt(((z_m * z_m) - (a * t))))
	else:
		tmp = y_m * x_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 5.4e-113)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / sqrt(Float64(-Float64(a * t)))));
	elseif (z_m <= 1.85e+94)
		tmp = Float64(Float64(z_m * x_m) * Float64(y_m / sqrt(Float64(Float64(z_m * z_m) - Float64(a * t)))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 5.4e-113)
		tmp = x_m * ((z_m * y_m) / sqrt(-(a * t)));
	elseif (z_m <= 1.85e+94)
		tmp = (z_m * x_m) * (y_m / sqrt(((z_m * z_m) - (a * t))));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 5.4e-113], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[Sqrt[(-N[(a * t), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 1.85e+94], N[(N[(z$95$m * x$95$m), $MachinePrecision] * N[(y$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 5.39999999999999991e-113

   1. Initial program 66.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 66.6

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

   4. Applied egg-rr 66.6%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. *-lowering-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 44.8

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

   7. Simplified 44.8%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-neg-in N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. neg-lowering-neg.f64 44.2

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

   9. Applied egg-rr 44.2%

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. associate-*l/ N/A

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

       4. associate-/l* N/A

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

       5. *-lowering-*.f64 N/A

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

       6. /-lowering-/.f64 N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 N/A

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

       9. sqrt-lowering-sqrt.f64 N/A

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

       10. *-commutative N/A

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

       11. distribute-lft-neg-out N/A

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

       12. neg-lowering-neg.f64 N/A

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

       13. *-lowering-*.f64 43.6

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

   11. Applied egg-rr 43.6%

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

   if 5.39999999999999991e-113 < z < 1.8500000000000001e94

   1. Initial program 93.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 84.8

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

   4. Applied egg-rr 84.8%

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

   if 1.8500000000000001e94 < z

   1. Initial program 31.2%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 95.8

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

   5. Simplified 95.8%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.4 \cdot 10^{-113}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{-a \cdot t}}\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+94}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \frac{y}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 7.5 \cdot 10^{+41}:\\ \;\;\;\;x\_m \cdot \frac{z\_m \cdot y\_m}{\sqrt{z\_m \cdot z\_m - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 7.5e+41)
      (* x_m (/ (* z_m y_m) (sqrt (- (* z_m z_m) (* a t)))))
      (* y_m x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 7.5e+41) {
		tmp = x_m * ((z_m * y_m) / sqrt(((z_m * z_m) - (a * t))));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_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 <= 7.5d+41) then
        tmp = x_m * ((z_m * y_m) / sqrt(((z_m * z_m) - (a * t))))
    else
        tmp = y_m * x_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 7.5e+41) {
		tmp = x_m * ((z_m * y_m) / Math.sqrt(((z_m * z_m) - (a * t))));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 7.5e+41:
		tmp = x_m * ((z_m * y_m) / math.sqrt(((z_m * z_m) - (a * t))))
	else:
		tmp = y_m * x_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 7.5e+41)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / sqrt(Float64(Float64(z_m * z_m) - Float64(a * t)))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 7.5e+41)
		tmp = x_m * ((z_m * y_m) / sqrt(((z_m * z_m) - (a * t))));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 7.5e+41], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(a * t), $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 < 7.50000000000000072e41

   1. Initial program 71.2%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 69.8

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

   4. Applied egg-rr 69.8%

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

   if 7.50000000000000072e41 < z

   1. Initial program 42.1%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 94.8

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

   5. Simplified 94.8%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.5 \cdot 10^{+41}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 6.7 \cdot 10^{+41}:\\ \;\;\;\;\left(z\_m \cdot y\_m\right) \cdot \frac{x\_m}{\sqrt{\mathsf{fma}\left(z\_m, z\_m, -a \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 6.7e+41)
      (* (* z_m y_m) (/ x_m (sqrt (fma z_m z_m (- (* a t))))))
      (* y_m x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 6.7e+41) {
		tmp = (z_m * y_m) * (x_m / sqrt(fma(z_m, z_m, -(a * t))));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 6.7e+41)
		tmp = Float64(Float64(z_m * y_m) * Float64(x_m / sqrt(fma(z_m, z_m, Float64(-Float64(a * t))))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 6.7e+41], N[(N[(z$95$m * y$95$m), $MachinePrecision] * N[(x$95$m / N[Sqrt[N[(z$95$m * z$95$m + (-N[(a * t), $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 < 6.6999999999999996e41

   1. Initial program 71.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 70.9

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

   4. Applied egg-rr 70.9%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. sub-neg N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. *-lowering-*.f64 N/A

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

      13. neg-lowering-neg.f64 N/A

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

      14. *-lowering-*.f64 69.0

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

   6. Applied egg-rr 69.0%

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

   if 6.6999999999999996e41 < z

   1. Initial program 42.1%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 94.8

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

   5. Simplified 94.8%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.7 \cdot 10^{+41}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{x}{\sqrt{\mathsf{fma}\left(z, z, -a \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 8 \cdot 10^{+41}:\\ \;\;\;\;\left(z\_m \cdot y\_m\right) \cdot \frac{x\_m}{\sqrt{z\_m \cdot z\_m - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 8e+41)
      (* (* z_m y_m) (/ x_m (sqrt (- (* z_m z_m) (* a t)))))
      (* y_m x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 8e+41) {
		tmp = (z_m * y_m) * (x_m / sqrt(((z_m * z_m) - (a * t))));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_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 <= 8d+41) then
        tmp = (z_m * y_m) * (x_m / sqrt(((z_m * z_m) - (a * t))))
    else
        tmp = y_m * x_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 8e+41) {
		tmp = (z_m * y_m) * (x_m / Math.sqrt(((z_m * z_m) - (a * t))));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 8e+41:
		tmp = (z_m * y_m) * (x_m / math.sqrt(((z_m * z_m) - (a * t))))
	else:
		tmp = y_m * x_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 8e+41)
		tmp = Float64(Float64(z_m * y_m) * Float64(x_m / sqrt(Float64(Float64(z_m * z_m) - Float64(a * t)))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 8e+41)
		tmp = (z_m * y_m) * (x_m / sqrt(((z_m * z_m) - (a * t))));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 8e+41], N[(N[(z$95$m * y$95$m), $MachinePrecision] * N[(x$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(a * t), $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 < 8.00000000000000005e41

   1. Initial program 71.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 69.0

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

   4. Applied egg-rr 69.0%

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

   if 8.00000000000000005e41 < z

   1. Initial program 42.1%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 94.8

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

   5. Simplified 94.8%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8 \cdot 10^{+41}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.55 \cdot 10^{-43}:\\ \;\;\;\;x\_m \cdot \frac{z\_m \cdot y\_m}{\sqrt{-a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 1.55e-43)
      (* x_m (/ (* z_m y_m) (sqrt (- (* a t)))))
      (* y_m x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.55e-43) {
		tmp = x_m * ((z_m * y_m) / sqrt(-(a * t)));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_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 <= 1.55d-43) then
        tmp = x_m * ((z_m * y_m) / sqrt(-(a * t)))
    else
        tmp = y_m * x_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.55e-43) {
		tmp = x_m * ((z_m * y_m) / Math.sqrt(-(a * t)));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.55e-43:
		tmp = x_m * ((z_m * y_m) / math.sqrt(-(a * t)))
	else:
		tmp = y_m * x_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.55e-43)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / sqrt(Float64(-Float64(a * t)))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.55e-43)
		tmp = x_m * ((z_m * y_m) / sqrt(-(a * t)));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 1.55e-43], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[Sqrt[(-N[(a * t), $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 < 1.55e-43

   1. Initial program 68.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 68.8

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

   4. Applied egg-rr 68.8%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. *-lowering-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 45.4

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

   7. Simplified 45.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-neg-in N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. neg-lowering-neg.f64 45.0

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

   9. Applied egg-rr 45.0%

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. associate-*l/ N/A

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

       4. associate-/l* N/A

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

       5. *-lowering-*.f64 N/A

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

       6. /-lowering-/.f64 N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 N/A

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

       9. sqrt-lowering-sqrt.f64 N/A

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

       10. *-commutative N/A

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

       11. distribute-lft-neg-out N/A

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

       12. neg-lowering-neg.f64 N/A

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

       13. *-lowering-*.f64 43.4

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

   11. Applied egg-rr 43.4%

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

   if 1.55e-43 < z

   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 inf

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

      1. *-lowering-*.f64 81.4

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

   5. Simplified 81.4%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.55 \cdot 10^{-43}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{-a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.62 \cdot 10^{-43}:\\ \;\;\;\;x\_m \cdot \left(z\_m \cdot \frac{y\_m}{\sqrt{-a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 1.62e-43)
      (* x_m (* z_m (/ y_m (sqrt (- (* a t))))))
      (* y_m x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.62e-43) {
		tmp = x_m * (z_m * (y_m / sqrt(-(a * t))));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_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 <= 1.62d-43) then
        tmp = x_m * (z_m * (y_m / sqrt(-(a * t))))
    else
        tmp = y_m * x_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.62e-43) {
		tmp = x_m * (z_m * (y_m / Math.sqrt(-(a * t))));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.62e-43:
		tmp = x_m * (z_m * (y_m / math.sqrt(-(a * t))))
	else:
		tmp = y_m * x_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.62e-43)
		tmp = Float64(x_m * Float64(z_m * Float64(y_m / sqrt(Float64(-Float64(a * t))))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.62e-43)
		tmp = x_m * (z_m * (y_m / sqrt(-(a * t))));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 1.62e-43], N[(x$95$m * N[(z$95$m * N[(y$95$m / N[Sqrt[(-N[(a * t), $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 < 1.6199999999999999e-43

   1. Initial program 68.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 68.8

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

   4. Applied egg-rr 68.8%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. *-lowering-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 45.4

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

   7. Simplified 45.4%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-neg-in N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. neg-lowering-neg.f64 41.7

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

   9. Applied egg-rr 41.7%

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

   if 1.6199999999999999e-43 < z

   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 inf

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

      1. *-lowering-*.f64 81.4

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

   5. Simplified 81.4%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.62 \cdot 10^{-43}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{\sqrt{-a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 7.5 \cdot 10^{-125}:\\ \;\;\;\;x\_m \cdot \frac{y\_m \cdot \left(\left(z\_m \cdot z\_m\right) \cdot -2\right)}{a \cdot t}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 7.5e-125)
      (* x_m (/ (* y_m (* (* z_m z_m) -2.0)) (* a t)))
      (* y_m x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 7.5e-125) {
		tmp = x_m * ((y_m * ((z_m * z_m) * -2.0)) / (a * t));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_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 <= 7.5d-125) then
        tmp = x_m * ((y_m * ((z_m * z_m) * (-2.0d0))) / (a * t))
    else
        tmp = y_m * x_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 7.5e-125) {
		tmp = x_m * ((y_m * ((z_m * z_m) * -2.0)) / (a * t));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 7.5e-125:
		tmp = x_m * ((y_m * ((z_m * z_m) * -2.0)) / (a * t))
	else:
		tmp = y_m * x_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 7.5e-125)
		tmp = Float64(x_m * Float64(Float64(y_m * Float64(Float64(z_m * z_m) * -2.0)) / Float64(a * t)));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 7.5e-125)
		tmp = x_m * ((y_m * ((z_m * z_m) * -2.0)) / (a * t));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 7.5e-125], N[(x$95$m * N[(N[(y$95$m * N[(N[(z$95$m * z$95$m), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] / N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 7.5e-125

   1. Initial program 65.6%

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

   3. Taylor expanded in t around 0

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z + \frac{-1}{2} \cdot \frac{a \cdot t}{z}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 30.8

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

   5. Simplified 30.8%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 30.2

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

   7. Applied egg-rr 30.2%

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

   8. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-2 \cdot \frac{y \cdot {z}^{2}}{a \cdot t}\right)} \cdot x \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 21.9

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

   10. Simplified 21.9%

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

   if 7.5e-125 < z

   1. Initial program 64.1%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 76.0

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

   5. Simplified 76.0%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.5 \cdot 10^{-125}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(\left(z \cdot z\right) \cdot -2\right)}{a \cdot t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.06 \cdot 10^{-129}:\\ \;\;\;\;y\_m \cdot \frac{-2 \cdot \left(x\_m \cdot \left(z\_m \cdot z\_m\right)\right)}{a \cdot t}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 1.06e-129)
      (* y_m (/ (* -2.0 (* x_m (* z_m z_m))) (* a t)))
      (* y_m x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.06e-129) {
		tmp = y_m * ((-2.0 * (x_m * (z_m * z_m))) / (a * t));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_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 <= 1.06d-129) then
        tmp = y_m * (((-2.0d0) * (x_m * (z_m * z_m))) / (a * t))
    else
        tmp = y_m * x_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.06e-129) {
		tmp = y_m * ((-2.0 * (x_m * (z_m * z_m))) / (a * t));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.06e-129:
		tmp = y_m * ((-2.0 * (x_m * (z_m * z_m))) / (a * t))
	else:
		tmp = y_m * x_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.06e-129)
		tmp = Float64(y_m * Float64(Float64(-2.0 * Float64(x_m * Float64(z_m * z_m))) / Float64(a * t)));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.06e-129)
		tmp = y_m * ((-2.0 * (x_m * (z_m * z_m))) / (a * t));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 1.06e-129], N[(y$95$m * N[(N[(-2.0 * N[(x$95$m * N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.0600000000000001e-129

   1. Initial program 65.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 66.4

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

   4. Applied egg-rr 66.4%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 30.2

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

   7. Simplified 30.2%

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

   8. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-2 \cdot \frac{x \cdot {z}^{2}}{a \cdot t}\right)} \cdot y \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 21.9

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

   10. Simplified 21.9%

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

   if 1.0600000000000001e-129 < z

   1. Initial program 64.1%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 76.0

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

   5. Simplified 76.0%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.06 \cdot 10^{-129}:\\ \;\;\;\;y \cdot \frac{-2 \cdot \left(x \cdot \left(z \cdot z\right)\right)}{a \cdot t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.32 \cdot 10^{-132}:\\ \;\;\;\;\frac{y\_m \cdot \left(z\_m \cdot x\_m\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (* z_s (if (<= z_m 1.32e-132) (/ (* y_m (* z_m x_m)) z_m) (* y_m x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.32e-132) {
		tmp = (y_m * (z_m * x_m)) / z_m;
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_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 <= 1.32d-132) then
        tmp = (y_m * (z_m * x_m)) / z_m
    else
        tmp = y_m * x_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.32e-132) {
		tmp = (y_m * (z_m * x_m)) / z_m;
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.32e-132:
		tmp = (y_m * (z_m * x_m)) / z_m
	else:
		tmp = y_m * x_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.32e-132)
		tmp = Float64(Float64(y_m * Float64(z_m * x_m)) / z_m);
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.32e-132)
		tmp = (y_m * (z_m * x_m)) / z_m;
	else
		tmp = y_m * x_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 1.32e-132], N[(N[(y$95$m * N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.32000000000000004e-132

   1. Initial program 65.6%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 25.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 26.1

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

   7. Applied egg-rr 26.1%

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

   if 1.32000000000000004e-132 < z

   1. Initial program 64.1%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 76.0

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

   5. Simplified 76.0%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.32 \cdot 10^{-132}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.5 \cdot 10^{-135}:\\ \;\;\;\;\frac{z\_m \cdot \left(y\_m \cdot x\_m\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (* z_s (if (<= z_m 2.5e-135) (/ (* z_m (* y_m x_m)) z_m) (* y_m x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.5e-135) {
		tmp = (z_m * (y_m * x_m)) / z_m;
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_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 <= 2.5d-135) then
        tmp = (z_m * (y_m * x_m)) / z_m
    else
        tmp = y_m * x_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.5e-135) {
		tmp = (z_m * (y_m * x_m)) / z_m;
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 2.5e-135:
		tmp = (z_m * (y_m * x_m)) / z_m
	else:
		tmp = y_m * x_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.5e-135)
		tmp = Float64(Float64(z_m * Float64(y_m * x_m)) / z_m);
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2.5e-135)
		tmp = (z_m * (y_m * x_m)) / z_m;
	else
		tmp = y_m * x_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 2.5e-135], N[(N[(z$95$m * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.5000000000000001e-135

   1. Initial program 65.6%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 25.5%

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

   if 2.5000000000000001e-135 < z

   1. Initial program 64.1%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 76.0

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

   5. Simplified 76.0%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{-135}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4 \cdot 10^{-198}:\\ \;\;\;\;\left(z\_m \cdot x\_m\right) \cdot \frac{y\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (* z_s (if (<= z_m 4e-198) (* (* z_m x_m) (/ y_m z_m)) (* y_m x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 4e-198) {
		tmp = (z_m * x_m) * (y_m / z_m);
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_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 <= 4d-198) then
        tmp = (z_m * x_m) * (y_m / z_m)
    else
        tmp = y_m * x_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 4e-198) {
		tmp = (z_m * x_m) * (y_m / z_m);
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 4e-198:
		tmp = (z_m * x_m) * (y_m / z_m)
	else:
		tmp = y_m * x_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 4e-198)
		tmp = Float64(Float64(z_m * x_m) * Float64(y_m / z_m));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 4e-198)
		tmp = (z_m * x_m) * (y_m / z_m);
	else
		tmp = y_m * x_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 4e-198], N[(N[(z$95$m * x$95$m), $MachinePrecision] * N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.9999999999999996e-198

   1. Initial program 64.9%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 19.1

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

   5. Simplified 19.1%

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

      1. *-rgt-identity N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot 1} \]

      2. *-inverses N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 20.1

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

   7. Applied egg-rr 20.1%

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

   if 3.9999999999999996e-198 < z

   1. Initial program 65.3%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 73.6

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

   5. Simplified 73.6%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{-198}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.4% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

3. Taylor expanded in z around inf

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

   1. *-lowering-*.f64 40.2

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

5. Simplified 40.2%

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

6. Final simplification 40.2%

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

Developer Target 1: 88.3% accurate, 0.7× 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 2024205 
(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)))))