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

Percentage Accurate: 60.2% → 91.5%

Time: 18.2s

Alternatives: 11

Speedup: 18.6×

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: 60.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} t_1 := z \cdot z - a \cdot t\\ \mathbf{if}\;z \leq -1.5 \cdot 10^{+146}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{-216}:\\ \;\;\;\;x \cdot \left(y \cdot \left(z \cdot {t_1}^{-0.5}\right)\right)\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-147}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{-t} \cdot \sqrt{a}}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+133}:\\ \;\;\;\;x \cdot \frac{z}{\frac{\sqrt{t_1}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* z z) (* a t))))
   (if (<= z -1.5e+146)
     (* y (- x))
     (if (<= z -6.5e-216)
       (* x (* y (* z (pow t_1 -0.5))))
       (if (<= z 5.4e-147)
         (* x (/ (* z y) (* (sqrt (- t)) (sqrt a))))
         (if (<= z 6.2e+133) (* x (/ z (/ (sqrt t_1) y))) (* y x)))))))

C

assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * z) - (a * t);
	double tmp;
	if (z <= -1.5e+146) {
		tmp = y * -x;
	} else if (z <= -6.5e-216) {
		tmp = x * (y * (z * pow(t_1, -0.5)));
	} else if (z <= 5.4e-147) {
		tmp = x * ((z * y) / (sqrt(-t) * sqrt(a)));
	} else if (z <= 6.2e+133) {
		tmp = x * (z / (sqrt(t_1) / y));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: t and a should be sorted in increasing order before calling this function.
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) :: t_1
    real(8) :: tmp
    t_1 = (z * z) - (a * t)
    if (z <= (-1.5d+146)) then
        tmp = y * -x
    else if (z <= (-6.5d-216)) then
        tmp = x * (y * (z * (t_1 ** (-0.5d0))))
    else if (z <= 5.4d-147) then
        tmp = x * ((z * y) / (sqrt(-t) * sqrt(a)))
    else if (z <= 6.2d+133) then
        tmp = x * (z / (sqrt(t_1) / y))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * z) - (a * t);
	double tmp;
	if (z <= -1.5e+146) {
		tmp = y * -x;
	} else if (z <= -6.5e-216) {
		tmp = x * (y * (z * Math.pow(t_1, -0.5)));
	} else if (z <= 5.4e-147) {
		tmp = x * ((z * y) / (Math.sqrt(-t) * Math.sqrt(a)));
	} else if (z <= 6.2e+133) {
		tmp = x * (z / (Math.sqrt(t_1) / y));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[t, a] = sort([t, a])
def code(x, y, z, t, a):
	t_1 = (z * z) - (a * t)
	tmp = 0
	if z <= -1.5e+146:
		tmp = y * -x
	elif z <= -6.5e-216:
		tmp = x * (y * (z * math.pow(t_1, -0.5)))
	elif z <= 5.4e-147:
		tmp = x * ((z * y) / (math.sqrt(-t) * math.sqrt(a)))
	elif z <= 6.2e+133:
		tmp = x * (z / (math.sqrt(t_1) / y))
	else:
		tmp = y * x
	return tmp

Julia

t, a = sort([t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * z) - Float64(a * t))
	tmp = 0.0
	if (z <= -1.5e+146)
		tmp = Float64(y * Float64(-x));
	elseif (z <= -6.5e-216)
		tmp = Float64(x * Float64(y * Float64(z * (t_1 ^ -0.5))));
	elseif (z <= 5.4e-147)
		tmp = Float64(x * Float64(Float64(z * y) / Float64(sqrt(Float64(-t)) * sqrt(a))));
	elseif (z <= 6.2e+133)
		tmp = Float64(x * Float64(z / Float64(sqrt(t_1) / y)));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * z) - (a * t);
	tmp = 0.0;
	if (z <= -1.5e+146)
		tmp = y * -x;
	elseif (z <= -6.5e-216)
		tmp = x * (y * (z * (t_1 ^ -0.5)));
	elseif (z <= 5.4e-147)
		tmp = x * ((z * y) / (sqrt(-t) * sqrt(a)));
	elseif (z <= 6.2e+133)
		tmp = x * (z / (sqrt(t_1) / y));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.5e+146], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, -6.5e-216], N[(x * N[(y * N[(z * N[Power[t$95$1, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.4e-147], N[(x * N[(N[(z * y), $MachinePrecision] / N[(N[Sqrt[(-t)], $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.2e+133], N[(x * N[(z / N[(N[Sqrt[t$95$1], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -1.50000000000000001e146

   1. Initial program 12.3%

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

      1. associate-*l* 12.1%

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

      2. associate-*r/ 12.4%

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

      3. *-commutative 12.4%

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

      4. associate-/l* 12.9%

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

   3. Simplified 12.9%

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

   4. Taylor expanded in z around -inf 93.0%

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

      1. associate-*r* 93.0%

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

      2. neg-mul-1 93.0%

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

      3. *-commutative 93.0%

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

   6. Simplified 93.0%

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

   if -1.50000000000000001e146 < z < -6.4999999999999999e-216

   1. Initial program 84.2%

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

      1. associate-*l* 81.1%

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

      2. associate-*r/ 84.2%

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

      3. *-commutative 84.2%

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

      4. associate-/l* 87.2%

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

   3. Simplified 87.2%

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

      1. clear-num 87.1%

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

      2. associate-/r* 84.2%

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

      3. associate-/r/ 84.3%

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

      4. pow1/2 84.3%

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

      5. pow-flip 84.2%

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

      6. metadata-eval 84.2%

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

      7. *-commutative 84.2%

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

   5. Applied egg-rr 84.2%

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

   6. Taylor expanded in y around 0 84.2%

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

      1. associate-*l* 88.5%

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

      2. *-commutative 88.5%

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

      3. unpow2 88.5%

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

      4. unpow-1 88.5%

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

      5. metadata-eval 88.5%

         \[\leadsto x \cdot \left(y \cdot \left(z \cdot \sqrt{{\left(z \cdot z - t \cdot a\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right)\right) \]

      6. pow-sqr 88.5%

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

      7. rem-sqrt-square 88.5%

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

      8. *-commutative 88.5%

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

   8. Simplified 88.5%

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

   9. Taylor expanded in z around 0 88.5%

      \[\leadsto x \cdot \left(y \cdot \color{blue}{\left(z \cdot \left|{\left({z}^{2} - a \cdot t\right)}^{-0.5}\right|\right)}\right) \]
   10. Step-by-step derivation

       1. unpow2 88.5%

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

       2. metadata-eval 88.5%

          \[\leadsto x \cdot \left(y \cdot \left(z \cdot \left|{\left(z \cdot z - a \cdot t\right)}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right|\right)\right) \]

       3. pow-sqr 88.2%

          \[\leadsto x \cdot \left(y \cdot \left(z \cdot \left|\color{blue}{{\left(z \cdot z - a \cdot t\right)}^{-0.25} \cdot {\left(z \cdot z - a \cdot t\right)}^{-0.25}}\right|\right)\right) \]

       4. fabs-sqr 88.2%

          \[\leadsto x \cdot \left(y \cdot \left(z \cdot \color{blue}{\left({\left(z \cdot z - a \cdot t\right)}^{-0.25} \cdot {\left(z \cdot z - a \cdot t\right)}^{-0.25}\right)}\right)\right) \]

       5. pow-sqr 88.5%

          \[\leadsto x \cdot \left(y \cdot \left(z \cdot \color{blue}{{\left(z \cdot z - a \cdot t\right)}^{\left(2 \cdot -0.25\right)}}\right)\right) \]

       6. metadata-eval 88.5%

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

   11. Simplified 88.5%

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

   if -6.4999999999999999e-216 < z < 5.3999999999999999e-147

   1. Initial program 72.4%

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

      1. associate-*l* 70.5%

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

      2. associate-*r/ 70.3%

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

   3. Simplified 70.3%

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

   4. Taylor expanded in z around 0 67.7%

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

      1. mul-1-neg 67.7%

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

      2. distribute-rgt-neg-out 67.7%

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

   6. Simplified 67.7%

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

      1. *-commutative 67.7%

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

      2. sqrt-prod 38.3%

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

   8. Applied egg-rr 38.3%

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

   if 5.3999999999999999e-147 < z < 6.2e133

   1. Initial program 91.5%

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

      1. associate-*l* 82.7%

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

      2. associate-*r/ 87.8%

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

      3. *-commutative 87.8%

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

      4. associate-/l* 96.4%

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

   3. Simplified 96.4%

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

   if 6.2e133 < z

   1. Initial program 22.8%

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

      1. associate-*l* 22.5%

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

      2. associate-*r/ 22.7%

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

      3. *-commutative 22.7%

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

      4. associate-/l* 21.2%

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

   3. Simplified 21.2%

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

   4. Taylor expanded in z around inf 100.0%

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

      1. *-commutative 100.0%

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

   6. Simplified 100.0%

      \[\leadsto \color{blue}{y \cdot x} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+146}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{-216}:\\ \;\;\;\;x \cdot \left(y \cdot \left(z \cdot {\left(z \cdot z - a \cdot t\right)}^{-0.5}\right)\right)\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-147}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{-t} \cdot \sqrt{a}}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+133}:\\ \;\;\;\;x \cdot \frac{z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2: 89.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{+156}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+133}:\\ \;\;\;\;x \cdot \frac{z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.6e+156)
   (* y (- x))
   (if (<= z 6.5e+133) (* x (/ z (/ (sqrt (- (* z z) (* a t))) y))) (* y x))))

C

assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.6e+156) {
		tmp = y * -x;
	} else if (z <= 6.5e+133) {
		tmp = x * (z / (sqrt(((z * z) - (a * t))) / y));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: t and a should be sorted in increasing order before calling this function.
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 <= (-5.6d+156)) then
        tmp = y * -x
    else if (z <= 6.5d+133) then
        tmp = x * (z / (sqrt(((z * z) - (a * t))) / y))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.6e+156) {
		tmp = y * -x;
	} else if (z <= 6.5e+133) {
		tmp = x * (z / (Math.sqrt(((z * z) - (a * t))) / y));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.6e+156:
		tmp = y * -x
	elif z <= 6.5e+133:
		tmp = x * (z / (math.sqrt(((z * z) - (a * t))) / y))
	else:
		tmp = y * x
	return tmp

Julia

t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.6e+156)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 6.5e+133)
		tmp = Float64(x * Float64(z / Float64(sqrt(Float64(Float64(z * z) - Float64(a * t))) / y)));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.6e+156)
		tmp = y * -x;
	elseif (z <= 6.5e+133)
		tmp = x * (z / (sqrt(((z * z) - (a * t))) / y));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.6e+156], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 6.5e+133], N[(x * N[(z / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.59999999999999975e156

   1. Initial program 7.0%

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

      1. associate-*l* 6.8%

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

      2. associate-*r/ 7.2%

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

      3. *-commutative 7.2%

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

      4. associate-/l* 7.6%

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

   3. Simplified 7.6%

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

   4. Taylor expanded in z around -inf 94.4%

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

      1. associate-*r* 94.4%

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

      2. neg-mul-1 94.4%

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

      3. *-commutative 94.4%

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

   6. Simplified 94.4%

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

   if -5.59999999999999975e156 < z < 6.5000000000000004e133

   1. Initial program 83.8%

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

      1. associate-*l* 79.2%

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

      2. associate-*r/ 82.2%

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

      3. *-commutative 82.2%

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

      4. associate-/l* 86.9%

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

   3. Simplified 86.9%

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

   if 6.5000000000000004e133 < z

   1. Initial program 22.8%

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

      1. associate-*l* 22.5%

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

      2. associate-*r/ 22.7%

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

      3. *-commutative 22.7%

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

      4. associate-/l* 21.2%

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

   3. Simplified 21.2%

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

   4. Taylor expanded in z around inf 100.0%

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

      1. *-commutative 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{+156}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+133}:\\ \;\;\;\;x \cdot \frac{z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3: 82.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-94}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 950000000:\\ \;\;\;\;x \cdot \left(y \cdot \left(z \cdot {\left(t \cdot \left(-a\right)\right)}^{-0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.6e-94)
   (* y (- x))
   (if (<= z 950000000.0) (* x (* y (* z (pow (* t (- a)) -0.5)))) (* y x))))

C

assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.6e-94) {
		tmp = y * -x;
	} else if (z <= 950000000.0) {
		tmp = x * (y * (z * pow((t * -a), -0.5)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: t and a should be sorted in increasing order before calling this function.
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 <= (-1.6d-94)) then
        tmp = y * -x
    else if (z <= 950000000.0d0) then
        tmp = x * (y * (z * ((t * -a) ** (-0.5d0))))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.6e-94) {
		tmp = y * -x;
	} else if (z <= 950000000.0) {
		tmp = x * (y * (z * Math.pow((t * -a), -0.5)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.6e-94:
		tmp = y * -x
	elif z <= 950000000.0:
		tmp = x * (y * (z * math.pow((t * -a), -0.5)))
	else:
		tmp = y * x
	return tmp

Julia

t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.6e-94)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 950000000.0)
		tmp = Float64(x * Float64(y * Float64(z * (Float64(t * Float64(-a)) ^ -0.5))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.6e-94)
		tmp = y * -x;
	elseif (z <= 950000000.0)
		tmp = x * (y * (z * ((t * -a) ^ -0.5)));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.6e-94], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 950000000.0], N[(x * N[(y * N[(z * N[Power[N[(t * (-a)), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.59999999999999998e-94

   1. Initial program 48.1%

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

      1. associate-*l* 45.0%

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

      2. associate-*r/ 47.3%

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

      3. *-commutative 47.3%

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

      4. associate-/l* 49.5%

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

   3. Simplified 49.5%

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

   4. Taylor expanded in z around -inf 88.0%

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

      1. associate-*r* 88.0%

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

      2. neg-mul-1 88.0%

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

      3. *-commutative 88.0%

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

   6. Simplified 88.0%

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

   if -1.59999999999999998e-94 < z < 9.5e8

   1. Initial program 78.3%

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

      1. associate-*l* 77.3%

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

      2. associate-*r/ 79.6%

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

   3. Simplified 79.6%

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

   4. Taylor expanded in z around 0 69.5%

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

      1. mul-1-neg 69.5%

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

      2. distribute-rgt-neg-out 69.5%

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

   6. Simplified 69.5%

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

      1. div-inv 69.4%

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

      2. pow1/2 70.8%

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

      3. pow-flip 70.9%

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

      4. metadata-eval 70.9%

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

   8. Applied egg-rr 70.9%

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

      1. associate-*l* 69.2%

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

   10. Simplified 69.2%

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

   if 9.5e8 < z

   1. Initial program 54.1%

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

      1. associate-*l* 48.9%

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

      2. associate-*r/ 50.3%

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

      3. *-commutative 50.3%

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

      4. associate-/l* 54.5%

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

   3. Simplified 54.5%

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

   4. Taylor expanded in z around inf 92.7%

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

      1. *-commutative 92.7%

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

   6. Simplified 92.7%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-94}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 950000000:\\ \;\;\;\;x \cdot \left(y \cdot \left(z \cdot {\left(t \cdot \left(-a\right)\right)}^{-0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 4: 81.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-93}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 950000000:\\ \;\;\;\;x \cdot \frac{z}{\frac{\sqrt{t \cdot \left(-a\right)}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.9e-93)
   (* y (- x))
   (if (<= z 950000000.0) (* x (/ z (/ (sqrt (* t (- a))) y))) (* y x))))

C

assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.9e-93) {
		tmp = y * -x;
	} else if (z <= 950000000.0) {
		tmp = x * (z / (sqrt((t * -a)) / y));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: t and a should be sorted in increasing order before calling this function.
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 <= (-2.9d-93)) then
        tmp = y * -x
    else if (z <= 950000000.0d0) then
        tmp = x * (z / (sqrt((t * -a)) / y))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.9e-93) {
		tmp = y * -x;
	} else if (z <= 950000000.0) {
		tmp = x * (z / (Math.sqrt((t * -a)) / y));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.9e-93:
		tmp = y * -x
	elif z <= 950000000.0:
		tmp = x * (z / (math.sqrt((t * -a)) / y))
	else:
		tmp = y * x
	return tmp

Julia

t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.9e-93)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 950000000.0)
		tmp = Float64(x * Float64(z / Float64(sqrt(Float64(t * Float64(-a))) / y)));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.9e-93)
		tmp = y * -x;
	elseif (z <= 950000000.0)
		tmp = x * (z / (sqrt((t * -a)) / y));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.9e-93], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 950000000.0], N[(x * N[(z / N[(N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.8999999999999998e-93

   1. Initial program 48.1%

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

      1. associate-*l* 45.0%

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

      2. associate-*r/ 47.3%

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

      3. *-commutative 47.3%

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

      4. associate-/l* 49.5%

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

   3. Simplified 49.5%

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

   4. Taylor expanded in z around -inf 88.0%

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

      1. associate-*r* 88.0%

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

      2. neg-mul-1 88.0%

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

      3. *-commutative 88.0%

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

   6. Simplified 88.0%

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

   if -2.8999999999999998e-93 < z < 9.5e8

   1. Initial program 78.3%

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

      1. associate-*l* 77.3%

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

      2. associate-*r/ 79.6%

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

      3. *-commutative 79.6%

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

      4. associate-/l* 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in z around 0 70.7%

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

      1. mul-1-neg 69.5%

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

      2. distribute-rgt-neg-out 69.5%

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

   6. Simplified 70.7%

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

   if 9.5e8 < z

   1. Initial program 54.1%

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

      1. associate-*l* 48.9%

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

      2. associate-*r/ 50.3%

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

      3. *-commutative 50.3%

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

      4. associate-/l* 54.5%

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

   3. Simplified 54.5%

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

   4. Taylor expanded in z around inf 92.7%

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

      1. *-commutative 92.7%

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

   6. Simplified 92.7%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-93}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 950000000:\\ \;\;\;\;x \cdot \frac{z}{\frac{\sqrt{t \cdot \left(-a\right)}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 5: 76.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{+156}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{-179}:\\ \;\;\;\;x \cdot \frac{z}{\frac{0.5 \cdot \frac{a}{\frac{z}{t}} - z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.6e+156)
   (* y (- x))
   (if (<= z 5.7e-179) (* x (/ z (/ (- (* 0.5 (/ a (/ z t))) z) y))) (* y x))))

C

assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.6e+156) {
		tmp = y * -x;
	} else if (z <= 5.7e-179) {
		tmp = x * (z / (((0.5 * (a / (z / t))) - z) / y));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: t and a should be sorted in increasing order before calling this function.
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 <= (-5.6d+156)) then
        tmp = y * -x
    else if (z <= 5.7d-179) then
        tmp = x * (z / (((0.5d0 * (a / (z / t))) - z) / y))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.6e+156) {
		tmp = y * -x;
	} else if (z <= 5.7e-179) {
		tmp = x * (z / (((0.5 * (a / (z / t))) - z) / y));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.6e+156:
		tmp = y * -x
	elif z <= 5.7e-179:
		tmp = x * (z / (((0.5 * (a / (z / t))) - z) / y))
	else:
		tmp = y * x
	return tmp

Julia

t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.6e+156)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 5.7e-179)
		tmp = Float64(x * Float64(z / Float64(Float64(Float64(0.5 * Float64(a / Float64(z / t))) - z) / y)));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.6e+156)
		tmp = y * -x;
	elseif (z <= 5.7e-179)
		tmp = x * (z / (((0.5 * (a / (z / t))) - z) / y));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.6e+156], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 5.7e-179], N[(x * N[(z / N[(N[(N[(0.5 * N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.59999999999999975e156

   1. Initial program 7.0%

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

      1. associate-*l* 6.8%

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

      2. associate-*r/ 7.2%

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

      3. *-commutative 7.2%

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

      4. associate-/l* 7.6%

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

   3. Simplified 7.6%

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

   4. Taylor expanded in z around -inf 94.4%

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

      1. associate-*r* 94.4%

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

      2. neg-mul-1 94.4%

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

      3. *-commutative 94.4%

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

   6. Simplified 94.4%

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

   if -5.59999999999999975e156 < z < 5.6999999999999999e-179

   1. Initial program 80.7%

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

      1. associate-*l* 78.0%

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

      2. associate-*r/ 80.1%

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

      3. *-commutative 80.1%

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

      4. associate-/l* 82.9%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in z around -inf 68.1%

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

      1. neg-mul-1 68.1%

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

      2. +-commutative 68.1%

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

      3. unsub-neg 68.1%

         \[\leadsto x \cdot \frac{z}{\frac{\color{blue}{0.5 \cdot \frac{a \cdot t}{z} - z}}{y}} \]

      4. associate-/l* 68.2%

         \[\leadsto x \cdot \frac{z}{\frac{0.5 \cdot \color{blue}{\frac{a}{\frac{z}{t}}} - z}{y}} \]

   6. Simplified 68.2%

      \[\leadsto x \cdot \frac{z}{\frac{\color{blue}{0.5 \cdot \frac{a}{\frac{z}{t}} - z}}{y}} \]

   if 5.6999999999999999e-179 < z

   1. Initial program 62.1%

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

      1. associate-*l* 57.2%

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

      2. associate-*r/ 60.0%

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

      3. *-commutative 60.0%

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

      4. associate-/l* 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in z around inf 80.9%

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

      1. *-commutative 80.9%

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

   6. Simplified 80.9%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{+156}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{-179}:\\ \;\;\;\;x \cdot \frac{z}{\frac{0.5 \cdot \frac{a}{\frac{z}{t}} - z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 6: 75.5% accurate, 6.6× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-210}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-186}:\\ \;\;\;\;-2 \cdot \left(\frac{x}{a} \cdot \frac{y \cdot \left(z \cdot z\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.6e-210)
   (* y (- x))
   (if (<= z 1.25e-186) (* -2.0 (* (/ x a) (/ (* y (* z z)) t))) (* y x))))

C

assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.6e-210) {
		tmp = y * -x;
	} else if (z <= 1.25e-186) {
		tmp = -2.0 * ((x / a) * ((y * (z * z)) / t));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: t and a should be sorted in increasing order before calling this function.
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 <= (-3.6d-210)) then
        tmp = y * -x
    else if (z <= 1.25d-186) then
        tmp = (-2.0d0) * ((x / a) * ((y * (z * z)) / t))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.6e-210) {
		tmp = y * -x;
	} else if (z <= 1.25e-186) {
		tmp = -2.0 * ((x / a) * ((y * (z * z)) / t));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.6e-210:
		tmp = y * -x
	elif z <= 1.25e-186:
		tmp = -2.0 * ((x / a) * ((y * (z * z)) / t))
	else:
		tmp = y * x
	return tmp

Julia

t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.6e-210)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.25e-186)
		tmp = Float64(-2.0 * Float64(Float64(x / a) * Float64(Float64(y * Float64(z * z)) / t)));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -3.6e-210)
		tmp = y * -x;
	elseif (z <= 1.25e-186)
		tmp = -2.0 * ((x / a) * ((y * (z * z)) / t));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.6e-210], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.25e-186], N[(-2.0 * N[(N[(x / a), $MachinePrecision] * N[(N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.5999999999999999e-210

   1. Initial program 52.3%

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

      1. associate-*l* 50.5%

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

      2. associate-*r/ 52.4%

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

      3. *-commutative 52.4%

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

      4. associate-/l* 54.3%

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

   3. Simplified 54.3%

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

   4. Taylor expanded in z around -inf 81.2%

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

      1. associate-*r* 81.2%

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

      2. neg-mul-1 81.2%

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

      3. *-commutative 81.2%

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

   6. Simplified 81.2%

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

   if -3.5999999999999999e-210 < z < 1.25e-186

   1. Initial program 74.7%

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

      1. associate-*l* 72.5%

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

      2. associate-*r/ 72.4%

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

      3. *-commutative 72.4%

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

      4. associate-/l* 75.3%

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

   3. Simplified 75.3%

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

      1. clear-num 75.3%

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

      2. un-div-inv 75.4%

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

      3. associate-/l/ 72.4%

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

   5. Applied egg-rr 72.4%

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

   6. Taylor expanded in z around inf 48.2%

      \[\leadsto \frac{x}{\frac{\color{blue}{z + -0.5 \cdot \frac{a \cdot t}{z}}}{z \cdot y}} \]
   7. Step-by-step derivation

      1. associate-/l* 48.1%

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

   8. Simplified 48.1%

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

   9. Taylor expanded in z around 0 47.7%

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

       1. times-frac 47.6%

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

       2. *-commutative 47.6%

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

       3. unpow2 47.6%

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

   11. Simplified 47.6%

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

   if 1.25e-186 < z

   1. Initial program 62.1%

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

      1. associate-*l* 57.2%

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

      2. associate-*r/ 60.0%

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

      3. *-commutative 60.0%

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

      4. associate-/l* 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in z around inf 80.9%

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

      1. *-commutative 80.9%

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

   6. Simplified 80.9%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-210}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-186}:\\ \;\;\;\;-2 \cdot \left(\frac{x}{a} \cdot \frac{y \cdot \left(z \cdot z\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 7: 75.5% accurate, 6.6× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-210}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-186}:\\ \;\;\;\;\frac{2}{a} \cdot \frac{x \cdot \left(y \cdot \left(z \cdot z\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.6e-210)
   (* y (- x))
   (if (<= z 1.6e-186) (* (/ 2.0 a) (/ (* x (* y (* z z))) t)) (* y x))))

C

assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.6e-210) {
		tmp = y * -x;
	} else if (z <= 1.6e-186) {
		tmp = (2.0 / a) * ((x * (y * (z * z))) / t);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: t and a should be sorted in increasing order before calling this function.
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 <= (-3.6d-210)) then
        tmp = y * -x
    else if (z <= 1.6d-186) then
        tmp = (2.0d0 / a) * ((x * (y * (z * z))) / t)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.6e-210) {
		tmp = y * -x;
	} else if (z <= 1.6e-186) {
		tmp = (2.0 / a) * ((x * (y * (z * z))) / t);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.6e-210:
		tmp = y * -x
	elif z <= 1.6e-186:
		tmp = (2.0 / a) * ((x * (y * (z * z))) / t)
	else:
		tmp = y * x
	return tmp

Julia

t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.6e-210)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.6e-186)
		tmp = Float64(Float64(2.0 / a) * Float64(Float64(x * Float64(y * Float64(z * z))) / t));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -3.6e-210)
		tmp = y * -x;
	elseif (z <= 1.6e-186)
		tmp = (2.0 / a) * ((x * (y * (z * z))) / t);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.6e-210], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.6e-186], N[(N[(2.0 / a), $MachinePrecision] * N[(N[(x * N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.5999999999999999e-210

   1. Initial program 52.3%

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

      1. associate-*l* 50.5%

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

      2. associate-*r/ 52.4%

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

      3. *-commutative 52.4%

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

      4. associate-/l* 54.3%

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

   3. Simplified 54.3%

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

   4. Taylor expanded in z around -inf 81.2%

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

      1. associate-*r* 81.2%

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

      2. neg-mul-1 81.2%

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

      3. *-commutative 81.2%

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

   6. Simplified 81.2%

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

   if -3.5999999999999999e-210 < z < 1.6e-186

   1. Initial program 74.7%

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

      1. *-commutative 74.7%

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

      2. associate-/l* 73.2%

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

   3. Simplified 73.2%

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

   4. Taylor expanded in z around -inf 47.6%

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

   5. Taylor expanded in z around 0 47.7%

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

      1. associate-*r/ 47.7%

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

      2. *-commutative 47.7%

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

      3. unpow2 47.7%

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

   7. Simplified 47.7%

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

      1. times-frac 48.0%

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

      2. *-commutative 48.0%

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

   9. Applied egg-rr 48.0%

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

   if 1.6e-186 < z

   1. Initial program 62.1%

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

      1. associate-*l* 57.2%

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

      2. associate-*r/ 60.0%

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

      3. *-commutative 60.0%

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

      4. associate-/l* 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in z around inf 80.9%

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

      1. *-commutative 80.9%

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

   6. Simplified 80.9%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-210}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-186}:\\ \;\;\;\;\frac{2}{a} \cdot \frac{x \cdot \left(y \cdot \left(z \cdot z\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 8: 74.8% accurate, 7.5× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-131}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-197}:\\ \;\;\;\;\frac{-1}{z} \cdot \frac{z}{\frac{1}{y \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1e-131)
   (* y (- x))
   (if (<= z 3e-197) (* (/ -1.0 z) (/ z (/ 1.0 (* y x)))) (* y x))))

C

assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1e-131) {
		tmp = y * -x;
	} else if (z <= 3e-197) {
		tmp = (-1.0 / z) * (z / (1.0 / (y * x)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: t and a should be sorted in increasing order before calling this function.
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 <= (-1d-131)) then
        tmp = y * -x
    else if (z <= 3d-197) then
        tmp = ((-1.0d0) / z) * (z / (1.0d0 / (y * x)))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1e-131) {
		tmp = y * -x;
	} else if (z <= 3e-197) {
		tmp = (-1.0 / z) * (z / (1.0 / (y * x)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1e-131:
		tmp = y * -x
	elif z <= 3e-197:
		tmp = (-1.0 / z) * (z / (1.0 / (y * x)))
	else:
		tmp = y * x
	return tmp

Julia

t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1e-131)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 3e-197)
		tmp = Float64(Float64(-1.0 / z) * Float64(z / Float64(1.0 / Float64(y * x))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1e-131)
		tmp = y * -x;
	elseif (z <= 3e-197)
		tmp = (-1.0 / z) * (z / (1.0 / (y * x)));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1e-131], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 3e-197], N[(N[(-1.0 / z), $MachinePrecision] * N[(z / N[(1.0 / N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.9999999999999999e-132

   1. Initial program 50.6%

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

      1. associate-*l* 47.7%

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

      2. associate-*r/ 49.8%

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

      3. *-commutative 49.8%

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

      4. associate-/l* 51.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in z around -inf 84.4%

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

      1. associate-*r* 84.4%

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

      2. neg-mul-1 84.4%

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

      3. *-commutative 84.4%

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

   6. Simplified 84.4%

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

   if -9.9999999999999999e-132 < z < 3.00000000000000026e-197

   1. Initial program 73.9%

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

      1. *-commutative 73.9%

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

      2. associate-/l* 70.6%

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

   3. Simplified 70.6%

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

   4. Taylor expanded in z around -inf 32.5%

      \[\leadsto \frac{z}{\color{blue}{-1 \cdot \frac{z}{x \cdot y}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 32.5%

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

      2. neg-mul-1 32.5%

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

      3. *-commutative 32.5%

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

   6. Simplified 32.5%

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

      1. *-un-lft-identity 32.5%

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

      2. div-inv 32.4%

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

      3. times-frac 48.2%

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

      4. add-sqr-sqrt 29.0%

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

      5. sqrt-unprod 6.3%

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

      6. sqr-neg 6.3%

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

      7. sqrt-prod 19.7%

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

      8. add-sqr-sqrt 43.5%

         \[\leadsto \frac{1}{\color{blue}{z}} \cdot \frac{z}{\frac{1}{y \cdot x}} \]

      9. frac-2neg 43.5%

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

      10. metadata-eval 43.5%

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

      11. add-sqr-sqrt 23.8%

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

      12. sqrt-unprod 5.2%

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

      13. sqr-neg 5.2%

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

      14. sqrt-prod 19.2%

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

      15. add-sqr-sqrt 48.2%

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

   8. Applied egg-rr 48.2%

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

   if 3.00000000000000026e-197 < z

   1. Initial program 61.8%

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

      1. associate-*l* 57.1%

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

      2. associate-*r/ 59.9%

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

      3. *-commutative 59.9%

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

      4. associate-/l* 63.8%

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

   3. Simplified 63.8%

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

   4. Taylor expanded in z around inf 79.4%

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

      1. *-commutative 79.4%

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

   6. Simplified 79.4%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-131}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-197}:\\ \;\;\;\;\frac{-1}{z} \cdot \frac{z}{\frac{1}{y \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 9: 73.4% accurate, 10.2× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 7.2 \cdot 10^{-302}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-132}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z 7.2e-302)
   (* y (- x))
   (if (<= z 2.8e-132) (* x (/ (* z y) z)) (* y x))))

C

assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 7.2e-302) {
		tmp = y * -x;
	} else if (z <= 2.8e-132) {
		tmp = x * ((z * y) / z);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: t and a should be sorted in increasing order before calling this function.
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 <= 7.2d-302) then
        tmp = y * -x
    else if (z <= 2.8d-132) then
        tmp = x * ((z * y) / z)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 7.2e-302) {
		tmp = y * -x;
	} else if (z <= 2.8e-132) {
		tmp = x * ((z * y) / z);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= 7.2e-302:
		tmp = y * -x
	elif z <= 2.8e-132:
		tmp = x * ((z * y) / z)
	else:
		tmp = y * x
	return tmp

Julia

t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= 7.2e-302)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 2.8e-132)
		tmp = Float64(x * Float64(Float64(z * y) / z));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= 7.2e-302)
		tmp = y * -x;
	elseif (z <= 2.8e-132)
		tmp = x * ((z * y) / z);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, 7.2e-302], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 2.8e-132], N[(x * N[(N[(z * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 7.2000000000000001e-302

   1. Initial program 55.2%

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

      1. associate-*l* 52.9%

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

      2. associate-*r/ 54.6%

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

      3. *-commutative 54.6%

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

      4. associate-/l* 56.3%

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

   3. Simplified 56.3%

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

   4. Taylor expanded in z around -inf 76.5%

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

      1. associate-*r* 76.5%

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

      2. neg-mul-1 76.5%

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

      3. *-commutative 76.5%

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

   6. Simplified 76.5%

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

   if 7.2000000000000001e-302 < z < 2.80000000000000002e-132

   1. Initial program 67.6%

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

      1. associate-*l* 68.1%

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

      2. associate-*r/ 71.2%

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

   3. Simplified 71.2%

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

   4. Taylor expanded in z around inf 36.4%

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

   if 2.80000000000000002e-132 < z

   1. Initial program 62.2%

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

      1. associate-*l* 56.9%

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

      2. associate-*r/ 59.1%

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

      3. *-commutative 59.1%

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

      4. associate-/l* 63.4%

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

   3. Simplified 63.4%

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

   4. Taylor expanded in z around inf 84.4%

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

      1. *-commutative 84.4%

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

   6. Simplified 84.4%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.2 \cdot 10^{-302}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-132}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 10: 72.7% accurate, 18.6× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 7.2 \cdot 10^{-302}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z 7.2e-302) (* y (- x)) (* y x)))

C

assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 7.2e-302) {
		tmp = y * -x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

NOTE: t and a should be sorted in increasing order before calling this function.
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 <= 7.2d-302) then
        tmp = y * -x
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 7.2e-302) {
		tmp = y * -x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= 7.2e-302:
		tmp = y * -x
	else:
		tmp = y * x
	return tmp

Julia

t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= 7.2e-302)
		tmp = Float64(y * Float64(-x));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= 7.2e-302)
		tmp = y * -x;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, 7.2e-302], N[(y * (-x)), $MachinePrecision], N[(y * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 7.2000000000000001e-302

   1. Initial program 55.2%

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

      1. associate-*l* 52.9%

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

      2. associate-*r/ 54.6%

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

      3. *-commutative 54.6%

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

      4. associate-/l* 56.3%

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

   3. Simplified 56.3%

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

   4. Taylor expanded in z around -inf 76.5%

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

      1. associate-*r* 76.5%

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

      2. neg-mul-1 76.5%

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

      3. *-commutative 76.5%

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

   6. Simplified 76.5%

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

   if 7.2000000000000001e-302 < z

   1. Initial program 63.4%

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

      1. associate-*l* 59.4%

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

      2. associate-*r/ 61.8%

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

      3. *-commutative 61.8%

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

      4. associate-/l* 65.9%

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

   3. Simplified 65.9%

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

   4. Taylor expanded in z around inf 70.8%

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

      1. *-commutative 70.8%

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

   6. Simplified 70.8%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.2 \cdot 10^{-302}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 11: 42.1% accurate, 37.7× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ y \cdot x \end{array} \]

FPCore

NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a) :precision binary64 (* y x))

C

assert(t < a);
double code(double x, double y, double z, double t, double a) {
	return y * x;
}

Fortran

NOTE: t and a should be sorted in increasing order before calling this function.
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 = y * x
end function

Java

assert t < a;
public static double code(double x, double y, double z, double t, double a) {
	return y * x;
}

Python

[t, a] = sort([t, a])
def code(x, y, z, t, a):
	return y * x

Julia

t, a = sort([t, a])
function code(x, y, z, t, a)
	return Float64(y * x)
end

MATLAB

t, a = num2cell(sort([t, a])){:}
function tmp = code(x, y, z, t, a)
	tmp = y * x;
end

Wolfram

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

Derivation

1. Initial program 59.1%

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

   1. associate-*l* 56.0%

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

   2. associate-*r/ 58.0%

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

   3. *-commutative 58.0%

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

   4. associate-/l* 60.9%

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

3. Simplified 60.9%

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

4. Taylor expanded in z around inf 41.1%

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

   1. *-commutative 41.1%

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

6. Simplified 41.1%

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

7. Final simplification 41.1%

   \[\leadsto y \cdot x \]

Developer target: 89.3% accurate, 1.0× 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 2023271 
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"
  :precision binary64

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

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