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

Percentage Accurate: 61.6% → 91.0%

Time: 35.5s

Alternatives: 23

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: 61.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} t_1 := \sqrt{z \cdot z - a \cdot t}\\ \mathbf{if}\;z \leq -5.1 \cdot 10^{+99}:\\ \;\;\;\;\frac{y}{\frac{\mathsf{fma}\left(0.5, \frac{a}{\log \left(e^{\frac{z \cdot z}{t}}\right)}, -1\right)}{x}}\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{-206}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{t_1}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-156}:\\ \;\;\;\;\left(x \cdot e^{-0.5 \cdot \left(\log \left(-t\right) - \log \left(\frac{1}{a}\right)\right)}\right) \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+137}:\\ \;\;\;\;\frac{y}{\frac{\frac{t_1}{z}}{x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: t and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (sqrt (- (* z z) (* a t)))))
   (if (<= z -5.1e+99)
     (/ y (/ (fma 0.5 (/ a (log (exp (/ (* z z) t)))) -1.0) x))
     (if (<= z -1.9e-206)
       (/ (* z (* y x)) t_1)
       (if (<= z 6e-156)
         (* (* x (exp (* -0.5 (- (log (- t)) (log (/ 1.0 a)))))) (* z y))
         (if (<= z 1.7e+137) (/ y (/ (/ t_1 z) x)) (* y x)))))))

C

assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = sqrt(((z * z) - (a * t)));
	double tmp;
	if (z <= -5.1e+99) {
		tmp = y / (fma(0.5, (a / log(exp(((z * z) / t)))), -1.0) / x);
	} else if (z <= -1.9e-206) {
		tmp = (z * (y * x)) / t_1;
	} else if (z <= 6e-156) {
		tmp = (x * exp((-0.5 * (log(-t) - log((1.0 / a)))))) * (z * y);
	} else if (z <= 1.7e+137) {
		tmp = y / ((t_1 / z) / x);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Julia

t, a = sort([t, a])
function code(x, y, z, t, a)
	t_1 = sqrt(Float64(Float64(z * z) - Float64(a * t)))
	tmp = 0.0
	if (z <= -5.1e+99)
		tmp = Float64(y / Float64(fma(0.5, Float64(a / log(exp(Float64(Float64(z * z) / t)))), -1.0) / x));
	elseif (z <= -1.9e-206)
		tmp = Float64(Float64(z * Float64(y * x)) / t_1);
	elseif (z <= 6e-156)
		tmp = Float64(Float64(x * exp(Float64(-0.5 * Float64(log(Float64(-t)) - log(Float64(1.0 / a)))))) * Float64(z * y));
	elseif (z <= 1.7e+137)
		tmp = Float64(y / Float64(Float64(t_1 / z) / x));
	else
		tmp = Float64(y * x);
	end
	return 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[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -5.1e+99], N[(y / N[(N[(0.5 * N[(a / N[Log[N[Exp[N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.9e-206], N[(N[(z * N[(y * x), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z, 6e-156], N[(N[(x * N[Exp[N[(-0.5 * N[(N[Log[(-t)], $MachinePrecision] - N[Log[N[(1.0 / a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.7e+137], N[(y / N[(N[(t$95$1 / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -5.09999999999999952e99

   1. Initial program 38.4%

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

      1. associate-*l* 36.0%

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

      2. associate-*r/ 38.2%

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

      3. *-commutative 38.2%

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

      4. associate-/l* 36.7%

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

   3. Simplified 36.7%

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

      1. associate-*r/ 33.9%

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

      2. associate-*l/ 34.7%

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

      3. associate-/l* 40.8%

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

      4. associate-/r/ 42.8%

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

      5. *-commutative 42.8%

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

      6. associate-/l* 42.9%

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

   5. Applied egg-rr 42.9%

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

   6. Taylor expanded in z around -inf 91.0%

      \[\leadsto \frac{y}{\frac{\color{blue}{0.5 \cdot \frac{a \cdot t}{{z}^{2}} - 1}}{x}} \]
   7. Step-by-step derivation

      1. fma-neg 91.0%

         \[\leadsto \frac{y}{\frac{\color{blue}{\mathsf{fma}\left(0.5, \frac{a \cdot t}{{z}^{2}}, -1\right)}}{x}} \]

      2. associate-/l* 97.6%

         \[\leadsto \frac{y}{\frac{\mathsf{fma}\left(0.5, \color{blue}{\frac{a}{\frac{{z}^{2}}{t}}}, -1\right)}{x}} \]

      3. unpow2 97.6%

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

      4. metadata-eval 97.6%

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

   8. Simplified 97.6%

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

      1. add-log-exp 99.6%

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

   10. Applied egg-rr 99.6%

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

   if -5.09999999999999952e99 < z < -1.90000000000000001e-206

   1. Initial program 92.1%

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

   if -1.90000000000000001e-206 < z < 6e-156

   1. Initial program 67.3%

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

      1. associate-*l* 72.0%

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

      2. associate-*r/ 74.1%

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

      3. *-commutative 74.1%

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

      4. div-inv 74.1%

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

      5. associate-*l* 71.9%

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

      6. *-commutative 71.9%

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

      7. pow1/2 71.9%

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

      8. pow-flip 72.0%

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

      9. metadata-eval 72.0%

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

   3. Applied egg-rr 72.0%

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

   4. Taylor expanded in a around inf 39.4%

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

   if 6e-156 < z < 1.69999999999999993e137

   1. Initial program 87.8%

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

      1. associate-*l* 79.0%

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

      2. associate-*r/ 83.8%

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

      3. *-commutative 83.8%

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

      4. associate-/l* 89.5%

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

   3. Simplified 89.5%

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

      1. associate-*r/ 89.0%

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

      2. associate-*l/ 87.5%

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

      3. associate-/l* 91.3%

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

      4. associate-/r/ 91.2%

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

      5. *-commutative 91.2%

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

      6. associate-/l* 92.8%

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

   5. Applied egg-rr 92.8%

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

   if 1.69999999999999993e137 < z

   1. Initial program 21.5%

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

      1. associate-*l* 18.9%

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

      2. associate-*r/ 21.0%

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

      3. *-commutative 21.0%

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

      4. associate-/l* 20.1%

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

   3. Simplified 20.1%

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

   4. Taylor expanded in z around inf 98.1%

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

      1. *-commutative 98.1%

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

   6. Simplified 98.1%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{+99}:\\ \;\;\;\;\frac{y}{\frac{\mathsf{fma}\left(0.5, \frac{a}{\log \left(e^{\frac{z \cdot z}{t}}\right)}, -1\right)}{x}}\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{-206}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-156}:\\ \;\;\;\;\left(x \cdot e^{-0.5 \cdot \left(\log \left(-t\right) - \log \left(\frac{1}{a}\right)\right)}\right) \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+137}:\\ \;\;\;\;\frac{y}{\frac{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}{x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2: 89.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+101}:\\ \;\;\;\;\frac{y}{\frac{\mathsf{fma}\left(0.5, \frac{a}{\log \left(e^{\frac{z \cdot z}{t}}\right)}, -1\right)}{x}}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

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 -1.75e+101)
   (/ y (/ (fma 0.5 (/ a (log (exp (/ (* z z) t)))) -1.0) x))
   (if (<= z 1.6e+36) (* x (/ (* z y) (sqrt (- (* z z) (* a t))))) (* y x))))

C

assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.75e+101) {
		tmp = y / (fma(0.5, (a / log(exp(((z * z) / t)))), -1.0) / x);
	} else if (z <= 1.6e+36) {
		tmp = x * ((z * y) / sqrt(((z * z) - (a * 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 <= -1.75e+101)
		tmp = Float64(y / Float64(fma(0.5, Float64(a / log(exp(Float64(Float64(z * z) / t)))), -1.0) / x));
	elseif (z <= 1.6e+36)
		tmp = Float64(x * Float64(Float64(z * y) / sqrt(Float64(Float64(z * z) - Float64(a * t)))));
	else
		tmp = Float64(y * x);
	end
	return 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, -1.75e+101], N[(y / N[(N[(0.5 * N[(a / N[Log[N[Exp[N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.6e+36], N[(x * N[(N[(z * y), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.75000000000000012e101

   1. Initial program 38.4%

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

      1. associate-*l* 36.0%

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

      2. associate-*r/ 38.2%

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

      3. *-commutative 38.2%

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

      4. associate-/l* 36.7%

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

   3. Simplified 36.7%

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

      1. associate-*r/ 33.9%

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

      2. associate-*l/ 34.7%

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

      3. associate-/l* 40.8%

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

      4. associate-/r/ 42.8%

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

      5. *-commutative 42.8%

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

      6. associate-/l* 42.9%

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

   5. Applied egg-rr 42.9%

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

   6. Taylor expanded in z around -inf 91.0%

      \[\leadsto \frac{y}{\frac{\color{blue}{0.5 \cdot \frac{a \cdot t}{{z}^{2}} - 1}}{x}} \]
   7. Step-by-step derivation

      1. fma-neg 91.0%

         \[\leadsto \frac{y}{\frac{\color{blue}{\mathsf{fma}\left(0.5, \frac{a \cdot t}{{z}^{2}}, -1\right)}}{x}} \]

      2. associate-/l* 97.6%

         \[\leadsto \frac{y}{\frac{\mathsf{fma}\left(0.5, \color{blue}{\frac{a}{\frac{{z}^{2}}{t}}}, -1\right)}{x}} \]

      3. unpow2 97.6%

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

      4. metadata-eval 97.6%

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

   8. Simplified 97.6%

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

      1. add-log-exp 99.6%

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

   10. Applied egg-rr 99.6%

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

   if -1.75000000000000012e101 < z < 1.5999999999999999e36

   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* 83.1%

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

      2. associate-*r/ 84.4%

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

   3. Simplified 84.4%

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

   if 1.5999999999999999e36 < z

   1. Initial program 40.0%

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

      1. associate-*l* 33.9%

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

      2. associate-*r/ 36.9%

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

      3. *-commutative 36.9%

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

      4. associate-/l* 37.6%

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

   3. Simplified 37.6%

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

   4. Taylor expanded in z around inf 94.6%

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

      1. *-commutative 94.6%

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

   6. Simplified 94.6%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+101}:\\ \;\;\;\;\frac{y}{\frac{\mathsf{fma}\left(0.5, \frac{a}{\log \left(e^{\frac{z \cdot z}{t}}\right)}, -1\right)}{x}}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+93}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

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 -3.3e+93)
   (* y (- x))
   (if (<= z 1.6e+36) (* x (* z (/ y (sqrt (- (* z z) (* a t)))))) (* y x))))

C

assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.3e+93) {
		tmp = y * -x;
	} else if (z <= 1.6e+36) {
		tmp = x * (z * (y / sqrt(((z * z) - (a * 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 <= (-3.3d+93)) then
        tmp = y * -x
    else if (z <= 1.6d+36) then
        tmp = x * (z * (y / sqrt(((z * z) - (a * 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 <= -3.3e+93) {
		tmp = y * -x;
	} else if (z <= 1.6e+36) {
		tmp = x * (z * (y / Math.sqrt(((z * z) - (a * t)))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.3e+93:
		tmp = y * -x
	elif z <= 1.6e+36:
		tmp = x * (z * (y / math.sqrt(((z * z) - (a * 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 <= -3.3e+93)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.6e+36)
		tmp = Float64(x * Float64(z * Float64(y / sqrt(Float64(Float64(z * z) - Float64(a * 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 <= -3.3e+93)
		tmp = y * -x;
	elseif (z <= 1.6e+36)
		tmp = x * (z * (y / sqrt(((z * z) - (a * 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, -3.3e+93], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.6e+36], N[(x * N[(z * N[(y / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.30000000000000009e93

   1. Initial program 42.2%

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

      1. associate-*l* 39.8%

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

      2. associate-*r/ 41.9%

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

      3. *-commutative 41.9%

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

      4. associate-/l* 38.7%

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

   3. Simplified 38.7%

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

   4. Taylor expanded in z around -inf 96.2%

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

      1. neg-mul-1 96.2%

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

   6. Simplified 96.2%

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

   if -3.30000000000000009e93 < z < 1.5999999999999999e36

   1. Initial program 83.5%

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

      1. associate-*l* 82.8%

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

      2. associate-*r/ 84.0%

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

      3. *-commutative 84.0%

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

      4. associate-/l* 84.7%

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

   3. Simplified 84.7%

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

      1. clear-num 84.6%

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

      2. associate-/r/ 84.2%

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

      3. clear-num 84.2%

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

   5. Applied egg-rr 84.2%

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

   if 1.5999999999999999e36 < z

   1. Initial program 40.0%

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

      1. associate-*l* 33.9%

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

      2. associate-*r/ 36.9%

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

      3. *-commutative 36.9%

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

      4. associate-/l* 37.6%

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

   3. Simplified 37.6%

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

   4. Taylor expanded in z around inf 94.6%

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

      1. *-commutative 94.6%

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

   6. Simplified 94.6%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+93}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 4: 90.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+92}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \frac{z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

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 -2.9e+92)
   (* y (- x))
   (if (<= z 1.6e+36) (* 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 <= -2.9e+92) {
		tmp = y * -x;
	} else if (z <= 1.6e+36) {
		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 <= (-2.9d+92)) then
        tmp = y * -x
    else if (z <= 1.6d+36) 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 <= -2.9e+92) {
		tmp = y * -x;
	} else if (z <= 1.6e+36) {
		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 <= -2.9e+92:
		tmp = y * -x
	elif z <= 1.6e+36:
		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 <= -2.9e+92)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.6e+36)
		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 <= -2.9e+92)
		tmp = y * -x;
	elseif (z <= 1.6e+36)
		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, -2.9e+92], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.6e+36], 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 < -2.9000000000000001e92

   1. Initial program 42.2%

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

      1. associate-*l* 39.8%

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

      2. associate-*r/ 41.9%

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

      3. *-commutative 41.9%

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

      4. associate-/l* 38.7%

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

   3. Simplified 38.7%

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

   4. Taylor expanded in z around -inf 96.2%

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

      1. neg-mul-1 96.2%

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

   6. Simplified 96.2%

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

   if -2.9000000000000001e92 < z < 1.5999999999999999e36

   1. Initial program 83.5%

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

      1. associate-*l* 82.8%

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

      2. associate-*r/ 84.0%

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

      3. *-commutative 84.0%

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

      4. associate-/l* 84.7%

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

   3. Simplified 84.7%

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

   if 1.5999999999999999e36 < z

   1. Initial program 40.0%

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

      1. associate-*l* 33.9%

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

      2. associate-*r/ 36.9%

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

      3. *-commutative 36.9%

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

      4. associate-/l* 37.6%

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

   3. Simplified 37.6%

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

   4. Taylor expanded in z around inf 94.6%

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

      1. *-commutative 94.6%

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

   6. Simplified 94.6%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+92}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \frac{z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 5: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.72 \cdot 10^{+100}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

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 -1.72e+100)
   (* y (- x))
   (if (<= z 1.6e+36) (* x (/ (* z y) (sqrt (- (* z z) (* a t))))) (* y x))))

C

assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.72e+100) {
		tmp = y * -x;
	} else if (z <= 1.6e+36) {
		tmp = x * ((z * y) / sqrt(((z * z) - (a * 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 <= (-1.72d+100)) then
        tmp = y * -x
    else if (z <= 1.6d+36) then
        tmp = x * ((z * y) / sqrt(((z * z) - (a * 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 <= -1.72e+100) {
		tmp = y * -x;
	} else if (z <= 1.6e+36) {
		tmp = x * ((z * y) / Math.sqrt(((z * z) - (a * t))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.72e+100:
		tmp = y * -x
	elif z <= 1.6e+36:
		tmp = x * ((z * y) / math.sqrt(((z * z) - (a * 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 <= -1.72e+100)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.6e+36)
		tmp = Float64(x * Float64(Float64(z * y) / sqrt(Float64(Float64(z * z) - Float64(a * 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 <= -1.72e+100)
		tmp = y * -x;
	elseif (z <= 1.6e+36)
		tmp = x * ((z * y) / sqrt(((z * z) - (a * 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, -1.72e+100], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.6e+36], N[(x * N[(N[(z * y), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.7200000000000001e100

   1. Initial program 38.4%

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

      1. associate-*l* 36.0%

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

      2. associate-*r/ 38.2%

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

      3. *-commutative 38.2%

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

      4. associate-/l* 36.7%

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

   3. Simplified 36.7%

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

   4. Taylor expanded in z around -inf 98.0%

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

      1. neg-mul-1 98.0%

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

   6. Simplified 98.0%

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

   if -1.7200000000000001e100 < z < 1.5999999999999999e36

   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* 83.1%

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

      2. associate-*r/ 84.4%

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

   3. Simplified 84.4%

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

   if 1.5999999999999999e36 < z

   1. Initial program 40.0%

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

      1. associate-*l* 33.9%

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

      2. associate-*r/ 36.9%

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

      3. *-commutative 36.9%

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

      4. associate-/l* 37.6%

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

   3. Simplified 37.6%

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

   4. Taylor expanded in z around inf 94.6%

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

      1. *-commutative 94.6%

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

   6. Simplified 94.6%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.72 \cdot 10^{+100}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 6: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+99}:\\ \;\;\;\;x \cdot \frac{y}{\mathsf{fma}\left(0.5, t \cdot \frac{a}{z \cdot z}, -1\right)}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

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 -4e+99)
   (* x (/ y (fma 0.5 (* t (/ a (* z z))) -1.0)))
   (if (<= z 1.6e+36) (* x (/ (* z y) (sqrt (- (* z z) (* a t))))) (* y x))))

C

assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e+99) {
		tmp = x * (y / fma(0.5, (t * (a / (z * z))), -1.0));
	} else if (z <= 1.6e+36) {
		tmp = x * ((z * y) / sqrt(((z * z) - (a * 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 <= -4e+99)
		tmp = Float64(x * Float64(y / fma(0.5, Float64(t * Float64(a / Float64(z * z))), -1.0)));
	elseif (z <= 1.6e+36)
		tmp = Float64(x * Float64(Float64(z * y) / sqrt(Float64(Float64(z * z) - Float64(a * t)))));
	else
		tmp = Float64(y * x);
	end
	return 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, -4e+99], N[(x * N[(y / N[(0.5 * N[(t * N[(a / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.6e+36], N[(x * N[(N[(z * y), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.9999999999999999e99

   1. Initial program 38.4%

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

      1. associate-*l* 36.0%

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

      2. associate-*r/ 38.2%

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

      3. *-commutative 38.2%

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

      4. associate-/l* 36.7%

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

   3. Simplified 36.7%

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

      1. associate-*r/ 33.9%

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

      2. associate-*l/ 34.7%

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

      3. associate-/l* 40.8%

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

      4. associate-/r/ 42.8%

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

      5. *-commutative 42.8%

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

      6. associate-/l* 42.9%

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

   5. Applied egg-rr 42.9%

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

   6. Taylor expanded in z around -inf 91.0%

      \[\leadsto \frac{y}{\frac{\color{blue}{0.5 \cdot \frac{a \cdot t}{{z}^{2}} - 1}}{x}} \]
   7. Step-by-step derivation

      1. fma-neg 91.0%

         \[\leadsto \frac{y}{\frac{\color{blue}{\mathsf{fma}\left(0.5, \frac{a \cdot t}{{z}^{2}}, -1\right)}}{x}} \]

      2. associate-/l* 97.6%

         \[\leadsto \frac{y}{\frac{\mathsf{fma}\left(0.5, \color{blue}{\frac{a}{\frac{{z}^{2}}{t}}}, -1\right)}{x}} \]

      3. unpow2 97.6%

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

      4. metadata-eval 97.6%

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

   8. Simplified 97.6%

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

      1. associate-/r/ 98.0%

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

      2. associate-/r/ 98.0%

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

   10. Applied egg-rr 98.0%

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

   if -3.9999999999999999e99 < z < 1.5999999999999999e36

   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* 83.1%

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

      2. associate-*r/ 84.4%

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

   3. Simplified 84.4%

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

   if 1.5999999999999999e36 < z

   1. Initial program 40.0%

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

      1. associate-*l* 33.9%

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

      2. associate-*r/ 36.9%

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

      3. *-commutative 36.9%

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

      4. associate-/l* 37.6%

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

   3. Simplified 37.6%

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

   4. Taylor expanded in z around inf 94.6%

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

      1. *-commutative 94.6%

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

   6. Simplified 94.6%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+99}:\\ \;\;\;\;x \cdot \frac{y}{\mathsf{fma}\left(0.5, t \cdot \frac{a}{z \cdot z}, -1\right)}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 7: 81.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} t_1 := \frac{a}{\frac{z}{t}}\\ \mathbf{if}\;z \leq -1.55 \cdot 10^{+140}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -1.16 \cdot 10^{-169}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{0.5 \cdot t_1 - z}\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-36}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{\sqrt{t \cdot \left(-a\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{\frac{z + -0.5 \cdot t_1}{z}}{x}}\\ \end{array} \end{array} \]

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 (/ a (/ z t))))
   (if (<= z -1.55e+140)
     (* y (- x))
     (if (<= z -1.16e-169)
       (/ (* z (* y x)) (- (* 0.5 t_1) z))
       (if (<= z 9.2e-36)
         (* x (* z (/ y (sqrt (* t (- a))))))
         (/ y (/ (/ (+ z (* -0.5 t_1)) z) x)))))))

C

assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = a / (z / t);
	double tmp;
	if (z <= -1.55e+140) {
		tmp = y * -x;
	} else if (z <= -1.16e-169) {
		tmp = (z * (y * x)) / ((0.5 * t_1) - z);
	} else if (z <= 9.2e-36) {
		tmp = x * (z * (y / sqrt((t * -a))));
	} else {
		tmp = y / (((z + (-0.5 * t_1)) / z) / 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 = a / (z / t)
    if (z <= (-1.55d+140)) then
        tmp = y * -x
    else if (z <= (-1.16d-169)) then
        tmp = (z * (y * x)) / ((0.5d0 * t_1) - z)
    else if (z <= 9.2d-36) then
        tmp = x * (z * (y / sqrt((t * -a))))
    else
        tmp = y / (((z + ((-0.5d0) * t_1)) / z) / 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 = a / (z / t);
	double tmp;
	if (z <= -1.55e+140) {
		tmp = y * -x;
	} else if (z <= -1.16e-169) {
		tmp = (z * (y * x)) / ((0.5 * t_1) - z);
	} else if (z <= 9.2e-36) {
		tmp = x * (z * (y / Math.sqrt((t * -a))));
	} else {
		tmp = y / (((z + (-0.5 * t_1)) / z) / x);
	}
	return tmp;
}

Python

[t, a] = sort([t, a])
def code(x, y, z, t, a):
	t_1 = a / (z / t)
	tmp = 0
	if z <= -1.55e+140:
		tmp = y * -x
	elif z <= -1.16e-169:
		tmp = (z * (y * x)) / ((0.5 * t_1) - z)
	elif z <= 9.2e-36:
		tmp = x * (z * (y / math.sqrt((t * -a))))
	else:
		tmp = y / (((z + (-0.5 * t_1)) / z) / x)
	return tmp

Julia

t, a = sort([t, a])
function code(x, y, z, t, a)
	t_1 = Float64(a / Float64(z / t))
	tmp = 0.0
	if (z <= -1.55e+140)
		tmp = Float64(y * Float64(-x));
	elseif (z <= -1.16e-169)
		tmp = Float64(Float64(z * Float64(y * x)) / Float64(Float64(0.5 * t_1) - z));
	elseif (z <= 9.2e-36)
		tmp = Float64(x * Float64(z * Float64(y / sqrt(Float64(t * Float64(-a))))));
	else
		tmp = Float64(y / Float64(Float64(Float64(z + Float64(-0.5 * t_1)) / z) / x));
	end
	return tmp
end

MATLAB

t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = a / (z / t);
	tmp = 0.0;
	if (z <= -1.55e+140)
		tmp = y * -x;
	elseif (z <= -1.16e-169)
		tmp = (z * (y * x)) / ((0.5 * t_1) - z);
	elseif (z <= 9.2e-36)
		tmp = x * (z * (y / sqrt((t * -a))));
	else
		tmp = y / (((z + (-0.5 * t_1)) / z) / 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[(a / N[(z / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.55e+140], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, -1.16e-169], N[(N[(z * N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * t$95$1), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.2e-36], N[(x * N[(z * N[(y / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(N[(N[(z + N[(-0.5 * t$95$1), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.55e140

   1. Initial program 28.1%

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

      1. associate-*l* 27.7%

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

      2. associate-*r/ 30.3%

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

      3. *-commutative 30.3%

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

      4. associate-/l* 28.5%

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

   3. Simplified 28.5%

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

   4. Taylor expanded in z around -inf 100.0%

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

      1. neg-mul-1 100.0%

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

   6. Simplified 100.0%

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

   if -1.55e140 < z < -1.16e-169

   1. Initial program 91.6%

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

   2. Taylor expanded in z around -inf 83.4%

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

      1. neg-mul-1 80.1%

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

      2. +-commutative 80.1%

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

      3. unsub-neg 80.1%

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

      4. associate-/l* 80.2%

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

   4. Simplified 85.2%

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

   if -1.16e-169 < z < 9.19999999999999986e-36

   1. Initial program 75.1%

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

      1. associate-*l* 74.0%

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

      2. associate-*r/ 77.5%

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

      3. *-commutative 77.5%

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

      4. associate-/l* 78.7%

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

   3. Simplified 78.7%

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

      1. clear-num 78.7%

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

      2. associate-/r/ 77.9%

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

      3. clear-num 77.9%

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

   5. Applied egg-rr 77.9%

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

   6. Taylor expanded in z around 0 69.9%

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

      1. mul-1-neg 69.9%

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

      2. distribute-rgt-neg-out 69.9%

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

   8. Simplified 69.9%

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

   if 9.19999999999999986e-36 < z

   1. Initial program 51.8%

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

      1. associate-*l* 47.0%

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

      2. associate-*r/ 49.4%

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

      3. *-commutative 49.4%

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

      4. associate-/l* 50.0%

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

   3. Simplified 50.0%

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

      1. associate-*r/ 49.2%

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

      2. associate-*l/ 49.8%

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

      3. associate-/l* 53.1%

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

      4. associate-/r/ 54.2%

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

      5. *-commutative 54.2%

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

      6. associate-/l* 54.2%

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

   5. Applied egg-rr 54.2%

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

   6. Taylor expanded in z around inf 92.7%

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

      1. associate-/l* 84.2%

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

   8. Simplified 93.8%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+140}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -1.16 \cdot 10^{-169}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{0.5 \cdot \frac{a}{\frac{z}{t}} - z}\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-36}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{\sqrt{t \cdot \left(-a\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}{x}}\\ \end{array} \]

Alternative 8: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} t_1 := \frac{a}{\frac{z}{t}}\\ \mathbf{if}\;z \leq -1.55 \cdot 10^{+140}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-171}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{0.5 \cdot t_1 - z}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-35}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{\frac{z + -0.5 \cdot t_1}{z}}{x}}\\ \end{array} \end{array} \]

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 (/ a (/ z t))))
   (if (<= z -1.55e+140)
     (* y (- x))
     (if (<= z -8.5e-171)
       (/ (* z (* y x)) (- (* 0.5 t_1) z))
       (if (<= z 1.05e-35)
         (* x (/ (* z y) (sqrt (* t (- a)))))
         (/ y (/ (/ (+ z (* -0.5 t_1)) z) x)))))))

C

assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = a / (z / t);
	double tmp;
	if (z <= -1.55e+140) {
		tmp = y * -x;
	} else if (z <= -8.5e-171) {
		tmp = (z * (y * x)) / ((0.5 * t_1) - z);
	} else if (z <= 1.05e-35) {
		tmp = x * ((z * y) / sqrt((t * -a)));
	} else {
		tmp = y / (((z + (-0.5 * t_1)) / z) / 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 = a / (z / t)
    if (z <= (-1.55d+140)) then
        tmp = y * -x
    else if (z <= (-8.5d-171)) then
        tmp = (z * (y * x)) / ((0.5d0 * t_1) - z)
    else if (z <= 1.05d-35) then
        tmp = x * ((z * y) / sqrt((t * -a)))
    else
        tmp = y / (((z + ((-0.5d0) * t_1)) / z) / 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 = a / (z / t);
	double tmp;
	if (z <= -1.55e+140) {
		tmp = y * -x;
	} else if (z <= -8.5e-171) {
		tmp = (z * (y * x)) / ((0.5 * t_1) - z);
	} else if (z <= 1.05e-35) {
		tmp = x * ((z * y) / Math.sqrt((t * -a)));
	} else {
		tmp = y / (((z + (-0.5 * t_1)) / z) / x);
	}
	return tmp;
}

Python

[t, a] = sort([t, a])
def code(x, y, z, t, a):
	t_1 = a / (z / t)
	tmp = 0
	if z <= -1.55e+140:
		tmp = y * -x
	elif z <= -8.5e-171:
		tmp = (z * (y * x)) / ((0.5 * t_1) - z)
	elif z <= 1.05e-35:
		tmp = x * ((z * y) / math.sqrt((t * -a)))
	else:
		tmp = y / (((z + (-0.5 * t_1)) / z) / x)
	return tmp

Julia

t, a = sort([t, a])
function code(x, y, z, t, a)
	t_1 = Float64(a / Float64(z / t))
	tmp = 0.0
	if (z <= -1.55e+140)
		tmp = Float64(y * Float64(-x));
	elseif (z <= -8.5e-171)
		tmp = Float64(Float64(z * Float64(y * x)) / Float64(Float64(0.5 * t_1) - z));
	elseif (z <= 1.05e-35)
		tmp = Float64(x * Float64(Float64(z * y) / sqrt(Float64(t * Float64(-a)))));
	else
		tmp = Float64(y / Float64(Float64(Float64(z + Float64(-0.5 * t_1)) / z) / x));
	end
	return tmp
end

MATLAB

t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = a / (z / t);
	tmp = 0.0;
	if (z <= -1.55e+140)
		tmp = y * -x;
	elseif (z <= -8.5e-171)
		tmp = (z * (y * x)) / ((0.5 * t_1) - z);
	elseif (z <= 1.05e-35)
		tmp = x * ((z * y) / sqrt((t * -a)));
	else
		tmp = y / (((z + (-0.5 * t_1)) / z) / 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[(a / N[(z / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.55e+140], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, -8.5e-171], N[(N[(z * N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * t$95$1), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.05e-35], N[(x * N[(N[(z * y), $MachinePrecision] / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(N[(N[(z + N[(-0.5 * t$95$1), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.55e140

   1. Initial program 28.1%

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

      1. associate-*l* 27.7%

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

      2. associate-*r/ 30.3%

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

      3. *-commutative 30.3%

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

      4. associate-/l* 28.5%

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

   3. Simplified 28.5%

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

   4. Taylor expanded in z around -inf 100.0%

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

      1. neg-mul-1 100.0%

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

   6. Simplified 100.0%

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

   if -1.55e140 < z < -8.50000000000000032e-171

   1. Initial program 91.6%

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

   2. Taylor expanded in z around -inf 83.4%

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

      1. neg-mul-1 80.1%

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

      2. +-commutative 80.1%

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

      3. unsub-neg 80.1%

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

      4. associate-/l* 80.2%

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

   4. Simplified 85.2%

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

   if -8.50000000000000032e-171 < z < 1.05e-35

   1. Initial program 75.1%

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

      1. associate-*l* 74.0%

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

      2. associate-*r/ 77.5%

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

   3. Simplified 77.5%

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

   4. Taylor expanded in z around 0 72.0%

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

      1. mul-1-neg 69.9%

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

      2. distribute-rgt-neg-out 69.9%

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

   6. Simplified 72.0%

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

   if 1.05e-35 < z

   1. Initial program 51.8%

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

      1. associate-*l* 47.0%

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

      2. associate-*r/ 49.4%

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

      3. *-commutative 49.4%

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

      4. associate-/l* 50.0%

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

   3. Simplified 50.0%

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

      1. associate-*r/ 49.2%

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

      2. associate-*l/ 49.8%

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

      3. associate-/l* 53.1%

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

      4. associate-/r/ 54.2%

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

      5. *-commutative 54.2%

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

      6. associate-/l* 54.2%

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

   5. Applied egg-rr 54.2%

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

   6. Taylor expanded in z around inf 92.7%

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

      1. associate-/l* 84.2%

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

   8. Simplified 93.8%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+140}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-171}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{0.5 \cdot \frac{a}{\frac{z}{t}} - z}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-35}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}{x}}\\ \end{array} \]

Alternative 9: 77.4% accurate, 5.3× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+141}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-214}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{0.5 \cdot \frac{a}{\frac{z}{t}} - z}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z + -0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

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 -1.7e+141)
   (* y (- x))
   (if (<= z -8e-214)
     (* x (/ (* z y) (- (* 0.5 (/ a (/ z t))) z)))
     (if (<= z 1.05e+36)
       (* x (/ (* z y) (+ z (* -0.5 (/ (* a t) z)))))
       (* y x)))))

C

assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.7e+141) {
		tmp = y * -x;
	} else if (z <= -8e-214) {
		tmp = x * ((z * y) / ((0.5 * (a / (z / t))) - z));
	} else if (z <= 1.05e+36) {
		tmp = x * ((z * y) / (z + (-0.5 * ((a * t) / 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 <= (-1.7d+141)) then
        tmp = y * -x
    else if (z <= (-8d-214)) then
        tmp = x * ((z * y) / ((0.5d0 * (a / (z / t))) - z))
    else if (z <= 1.05d+36) then
        tmp = x * ((z * y) / (z + ((-0.5d0) * ((a * t) / 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 <= -1.7e+141) {
		tmp = y * -x;
	} else if (z <= -8e-214) {
		tmp = x * ((z * y) / ((0.5 * (a / (z / t))) - z));
	} else if (z <= 1.05e+36) {
		tmp = x * ((z * y) / (z + (-0.5 * ((a * t) / z))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.7e+141:
		tmp = y * -x
	elif z <= -8e-214:
		tmp = x * ((z * y) / ((0.5 * (a / (z / t))) - z))
	elif z <= 1.05e+36:
		tmp = x * ((z * y) / (z + (-0.5 * ((a * t) / 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 <= -1.7e+141)
		tmp = Float64(y * Float64(-x));
	elseif (z <= -8e-214)
		tmp = Float64(x * Float64(Float64(z * y) / Float64(Float64(0.5 * Float64(a / Float64(z / t))) - z)));
	elseif (z <= 1.05e+36)
		tmp = Float64(x * Float64(Float64(z * y) / Float64(z + Float64(-0.5 * Float64(Float64(a * t) / 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 <= -1.7e+141)
		tmp = y * -x;
	elseif (z <= -8e-214)
		tmp = x * ((z * y) / ((0.5 * (a / (z / t))) - z));
	elseif (z <= 1.05e+36)
		tmp = x * ((z * y) / (z + (-0.5 * ((a * t) / 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, -1.7e+141], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, -8e-214], N[(x * N[(N[(z * y), $MachinePrecision] / N[(N[(0.5 * N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.05e+36], N[(x * N[(N[(z * y), $MachinePrecision] / N[(z + N[(-0.5 * N[(N[(a * t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.6999999999999999e141

   1. Initial program 28.7%

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

      1. associate-*l* 28.3%

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

      2. associate-*r/ 28.4%

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

      3. *-commutative 28.4%

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

      4. associate-/l* 26.6%

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

   3. Simplified 26.6%

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

   4. Taylor expanded in z around -inf 100.0%

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

      1. neg-mul-1 100.0%

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

   6. Simplified 100.0%

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

   if -1.6999999999999999e141 < z < -7.9999999999999993e-214

   1. Initial program 89.2%

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

      1. associate-*l* 87.8%

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

      2. associate-*r/ 88.0%

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

   3. Simplified 88.0%

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

   4. Taylor expanded in z around -inf 74.8%

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

      1. neg-mul-1 74.8%

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

      2. +-commutative 74.8%

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

      3. unsub-neg 74.8%

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

      4. associate-/l* 74.8%

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

   6. Simplified 74.8%

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

   if -7.9999999999999993e-214 < z < 1.05000000000000002e36

   1. Initial program 78.2%

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

      1. associate-*l* 77.0%

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

      2. associate-*r/ 80.3%

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

   3. Simplified 80.3%

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

   4. Taylor expanded in z around inf 58.6%

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

   if 1.05000000000000002e36 < z

   1. Initial program 40.0%

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

      1. associate-*l* 33.9%

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

      2. associate-*r/ 36.9%

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

      3. *-commutative 36.9%

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

      4. associate-/l* 37.6%

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

   3. Simplified 37.6%

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

   4. Taylor expanded in z around inf 94.6%

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

      1. *-commutative 94.6%

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

   6. Simplified 94.6%

      \[\leadsto \color{blue}{y \cdot x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+141}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-214}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{0.5 \cdot \frac{a}{\frac{z}{t}} - z}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z + -0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 10: 77.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{-173}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z + -0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

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 -4.3e-173)
   (* y (- x))
   (if (<= z 1.55e+36)
     (* x (/ (* z y) (+ z (* -0.5 (/ (* a t) z)))))
     (* y x))))

C

assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.3e-173) {
		tmp = y * -x;
	} else if (z <= 1.55e+36) {
		tmp = x * ((z * y) / (z + (-0.5 * ((a * t) / 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 <= (-4.3d-173)) then
        tmp = y * -x
    else if (z <= 1.55d+36) then
        tmp = x * ((z * y) / (z + ((-0.5d0) * ((a * t) / 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 <= -4.3e-173) {
		tmp = y * -x;
	} else if (z <= 1.55e+36) {
		tmp = x * ((z * y) / (z + (-0.5 * ((a * t) / z))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -4.3e-173:
		tmp = y * -x
	elif z <= 1.55e+36:
		tmp = x * ((z * y) / (z + (-0.5 * ((a * t) / 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 <= -4.3e-173)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.55e+36)
		tmp = Float64(x * Float64(Float64(z * y) / Float64(z + Float64(-0.5 * Float64(Float64(a * t) / 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 <= -4.3e-173)
		tmp = y * -x;
	elseif (z <= 1.55e+36)
		tmp = x * ((z * y) / (z + (-0.5 * ((a * t) / 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, -4.3e-173], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.55e+36], N[(x * N[(N[(z * y), $MachinePrecision] / N[(z + N[(-0.5 * N[(N[(a * t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.3000000000000003e-173

   1. Initial program 66.5%

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

      1. associate-*l* 65.2%

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

      2. associate-*r/ 65.3%

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

      3. *-commutative 65.3%

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

      4. associate-/l* 63.7%

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

   3. Simplified 63.7%

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

   4. Taylor expanded in z around -inf 84.5%

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

      1. neg-mul-1 84.5%

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

   6. Simplified 84.5%

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

   if -4.3000000000000003e-173 < z < 1.55e36

   1. Initial program 79.1%

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

      1. associate-*l* 78.2%

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

      2. associate-*r/ 81.1%

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

   3. Simplified 81.1%

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

   4. Taylor expanded in z around inf 57.1%

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

   if 1.55e36 < z

   1. Initial program 40.0%

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

      1. associate-*l* 33.9%

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

      2. associate-*r/ 36.9%

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

      3. *-commutative 36.9%

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

      4. associate-/l* 37.6%

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

   3. Simplified 37.6%

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

   4. Taylor expanded in z around inf 94.6%

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

      1. *-commutative 94.6%

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

   6. Simplified 94.6%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{-173}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z + -0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 11: 78.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} t_1 := \frac{a}{\frac{z}{t}}\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{+141}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-291}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{0.5 \cdot t_1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{\frac{z + -0.5 \cdot t_1}{z}}{x}}\\ \end{array} \end{array} \]

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 (/ a (/ z t))))
   (if (<= z -1.7e+141)
     (* y (- x))
     (if (<= z -2.2e-291)
       (* x (/ (* z y) (- (* 0.5 t_1) z)))
       (/ y (/ (/ (+ z (* -0.5 t_1)) z) x))))))

C

assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = a / (z / t);
	double tmp;
	if (z <= -1.7e+141) {
		tmp = y * -x;
	} else if (z <= -2.2e-291) {
		tmp = x * ((z * y) / ((0.5 * t_1) - z));
	} else {
		tmp = y / (((z + (-0.5 * t_1)) / z) / 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 = a / (z / t)
    if (z <= (-1.7d+141)) then
        tmp = y * -x
    else if (z <= (-2.2d-291)) then
        tmp = x * ((z * y) / ((0.5d0 * t_1) - z))
    else
        tmp = y / (((z + ((-0.5d0) * t_1)) / z) / 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 = a / (z / t);
	double tmp;
	if (z <= -1.7e+141) {
		tmp = y * -x;
	} else if (z <= -2.2e-291) {
		tmp = x * ((z * y) / ((0.5 * t_1) - z));
	} else {
		tmp = y / (((z + (-0.5 * t_1)) / z) / x);
	}
	return tmp;
}

Python

[t, a] = sort([t, a])
def code(x, y, z, t, a):
	t_1 = a / (z / t)
	tmp = 0
	if z <= -1.7e+141:
		tmp = y * -x
	elif z <= -2.2e-291:
		tmp = x * ((z * y) / ((0.5 * t_1) - z))
	else:
		tmp = y / (((z + (-0.5 * t_1)) / z) / x)
	return tmp

Julia

t, a = sort([t, a])
function code(x, y, z, t, a)
	t_1 = Float64(a / Float64(z / t))
	tmp = 0.0
	if (z <= -1.7e+141)
		tmp = Float64(y * Float64(-x));
	elseif (z <= -2.2e-291)
		tmp = Float64(x * Float64(Float64(z * y) / Float64(Float64(0.5 * t_1) - z)));
	else
		tmp = Float64(y / Float64(Float64(Float64(z + Float64(-0.5 * t_1)) / z) / x));
	end
	return tmp
end

MATLAB

t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = a / (z / t);
	tmp = 0.0;
	if (z <= -1.7e+141)
		tmp = y * -x;
	elseif (z <= -2.2e-291)
		tmp = x * ((z * y) / ((0.5 * t_1) - z));
	else
		tmp = y / (((z + (-0.5 * t_1)) / z) / 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[(a / N[(z / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.7e+141], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, -2.2e-291], N[(x * N[(N[(z * y), $MachinePrecision] / N[(N[(0.5 * t$95$1), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(N[(N[(z + N[(-0.5 * t$95$1), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.6999999999999999e141

   1. Initial program 28.7%

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

      1. associate-*l* 28.3%

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

      2. associate-*r/ 28.4%

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

      3. *-commutative 28.4%

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

      4. associate-/l* 26.6%

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

   3. Simplified 26.6%

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

   4. Taylor expanded in z around -inf 100.0%

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

      1. neg-mul-1 100.0%

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

   6. Simplified 100.0%

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

   if -1.6999999999999999e141 < z < -2.20000000000000002e-291

   1. Initial program 87.6%

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

      1. associate-*l* 86.3%

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

      2. associate-*r/ 86.4%

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

   3. Simplified 86.4%

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

   4. Taylor expanded in z around -inf 73.2%

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

      1. neg-mul-1 73.2%

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

      2. +-commutative 73.2%

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

      3. unsub-neg 73.2%

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

      4. associate-/l* 73.2%

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

   6. Simplified 73.2%

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

   if -2.20000000000000002e-291 < z

   1. Initial program 59.9%

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

      1. associate-*l* 56.3%

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

      2. associate-*r/ 59.6%

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

      3. *-commutative 59.6%

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

      4. associate-/l* 60.6%

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

   3. Simplified 60.6%

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

      1. associate-*r/ 60.0%

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

      2. associate-*l/ 58.1%

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

      3. associate-/l* 60.6%

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

      4. associate-/r/ 62.1%

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

      5. *-commutative 62.1%

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

      6. associate-/l* 61.5%

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

   5. Applied egg-rr 61.5%

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

   6. Taylor expanded in z around inf 78.4%

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

      1. associate-/l* 71.8%

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

   8. Simplified 79.0%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+141}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-291}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{0.5 \cdot \frac{a}{\frac{z}{t}} - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}{x}}\\ \end{array} \]

Alternative 12: 78.4% accurate, 5.9× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} t_1 := \frac{a}{\frac{z}{t}}\\ \mathbf{if}\;z \leq -1.55 \cdot 10^{+140}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-289}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{0.5 \cdot t_1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{\frac{z + -0.5 \cdot t_1}{z}}{x}}\\ \end{array} \end{array} \]

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 (/ a (/ z t))))
   (if (<= z -1.55e+140)
     (* y (- x))
     (if (<= z -8.2e-289)
       (/ (* z (* y x)) (- (* 0.5 t_1) z))
       (/ y (/ (/ (+ z (* -0.5 t_1)) z) x))))))

C

assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = a / (z / t);
	double tmp;
	if (z <= -1.55e+140) {
		tmp = y * -x;
	} else if (z <= -8.2e-289) {
		tmp = (z * (y * x)) / ((0.5 * t_1) - z);
	} else {
		tmp = y / (((z + (-0.5 * t_1)) / z) / 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 = a / (z / t)
    if (z <= (-1.55d+140)) then
        tmp = y * -x
    else if (z <= (-8.2d-289)) then
        tmp = (z * (y * x)) / ((0.5d0 * t_1) - z)
    else
        tmp = y / (((z + ((-0.5d0) * t_1)) / z) / 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 = a / (z / t);
	double tmp;
	if (z <= -1.55e+140) {
		tmp = y * -x;
	} else if (z <= -8.2e-289) {
		tmp = (z * (y * x)) / ((0.5 * t_1) - z);
	} else {
		tmp = y / (((z + (-0.5 * t_1)) / z) / x);
	}
	return tmp;
}

Python

[t, a] = sort([t, a])
def code(x, y, z, t, a):
	t_1 = a / (z / t)
	tmp = 0
	if z <= -1.55e+140:
		tmp = y * -x
	elif z <= -8.2e-289:
		tmp = (z * (y * x)) / ((0.5 * t_1) - z)
	else:
		tmp = y / (((z + (-0.5 * t_1)) / z) / x)
	return tmp

Julia

t, a = sort([t, a])
function code(x, y, z, t, a)
	t_1 = Float64(a / Float64(z / t))
	tmp = 0.0
	if (z <= -1.55e+140)
		tmp = Float64(y * Float64(-x));
	elseif (z <= -8.2e-289)
		tmp = Float64(Float64(z * Float64(y * x)) / Float64(Float64(0.5 * t_1) - z));
	else
		tmp = Float64(y / Float64(Float64(Float64(z + Float64(-0.5 * t_1)) / z) / x));
	end
	return tmp
end

MATLAB

t, a = num2cell(sort([t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = a / (z / t);
	tmp = 0.0;
	if (z <= -1.55e+140)
		tmp = y * -x;
	elseif (z <= -8.2e-289)
		tmp = (z * (y * x)) / ((0.5 * t_1) - z);
	else
		tmp = y / (((z + (-0.5 * t_1)) / z) / 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[(a / N[(z / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.55e+140], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, -8.2e-289], N[(N[(z * N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * t$95$1), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], N[(y / N[(N[(N[(z + N[(-0.5 * t$95$1), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.55e140

   1. Initial program 28.1%

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

      1. associate-*l* 27.7%

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

      2. associate-*r/ 30.3%

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

      3. *-commutative 30.3%

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

      4. associate-/l* 28.5%

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

   3. Simplified 28.5%

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

   4. Taylor expanded in z around -inf 100.0%

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

      1. neg-mul-1 100.0%

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

   6. Simplified 100.0%

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

   if -1.55e140 < z < -8.1999999999999996e-289

   1. Initial program 88.7%

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

   2. Taylor expanded in z around -inf 75.3%

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

      1. neg-mul-1 72.8%

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

      2. +-commutative 72.8%

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

      3. unsub-neg 72.8%

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

      4. associate-/l* 72.9%

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

   4. Simplified 76.5%

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

   if -8.1999999999999996e-289 < z

   1. Initial program 59.9%

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

      1. associate-*l* 56.3%

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

      2. associate-*r/ 59.6%

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

      3. *-commutative 59.6%

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

      4. associate-/l* 60.6%

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

   3. Simplified 60.6%

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

      1. associate-*r/ 60.0%

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

      2. associate-*l/ 58.1%

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

      3. associate-/l* 60.6%

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

      4. associate-/r/ 62.1%

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

      5. *-commutative 62.1%

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

      6. associate-/l* 61.5%

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

   5. Applied egg-rr 61.5%

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

   6. Taylor expanded in z around inf 78.4%

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

      1. associate-/l* 71.8%

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

   8. Simplified 79.0%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+140}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-289}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{0.5 \cdot \frac{a}{\frac{z}{t}} - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}{x}}\\ \end{array} \]

Alternative 13: 74.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-57}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-183}:\\ \;\;\;\;-2 \cdot \left(\frac{x}{a} \cdot \frac{y \cdot \left(z \cdot z\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

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 -1.9e-57)
   (* y (- x))
   (if (<= z 2.9e-183) (* -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 <= -1.9e-57) {
		tmp = y * -x;
	} else if (z <= 2.9e-183) {
		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 <= (-1.9d-57)) then
        tmp = y * -x
    else if (z <= 2.9d-183) 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 <= -1.9e-57) {
		tmp = y * -x;
	} else if (z <= 2.9e-183) {
		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 <= -1.9e-57:
		tmp = y * -x
	elif z <= 2.9e-183:
		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 <= -1.9e-57)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 2.9e-183)
		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 <= -1.9e-57)
		tmp = y * -x;
	elseif (z <= 2.9e-183)
		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, -1.9e-57], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 2.9e-183], 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 < -1.8999999999999999e-57

   1. Initial program 61.0%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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/ 60.7%

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

      3. *-commutative 60.7%

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

      4. associate-/l* 59.9%

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

   3. Simplified 59.9%

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

   4. Taylor expanded in z around -inf 90.1%

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

      1. neg-mul-1 90.1%

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

   6. Simplified 90.1%

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

   if -1.8999999999999999e-57 < z < 2.9e-183

   1. Initial program 77.5%

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

   2. Taylor expanded in z around inf 50.8%

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

      1. associate-/l* 50.8%

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

   4. Simplified 50.8%

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

   5. Taylor expanded in z around 0 50.5%

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

      1. times-frac 50.8%

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

      2. *-commutative 50.8%

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

      3. unpow2 50.8%

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

   7. Simplified 50.8%

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

   if 2.9e-183 < z

   1. Initial program 57.6%

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

      1. associate-*l* 52.8%

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

      2. associate-*r/ 56.2%

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

      3. *-commutative 56.2%

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

      4. associate-/l* 58.7%

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

   3. Simplified 58.7%

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

   4. Taylor expanded in z around inf 82.4%

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

      1. *-commutative 82.4%

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

   6. Simplified 82.4%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-57}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-183}:\\ \;\;\;\;-2 \cdot \left(\frac{x}{a} \cdot \frac{y \cdot \left(z \cdot z\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 14: 74.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-57}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-176}:\\ \;\;\;\;\frac{-2}{t} \cdot \frac{\left(z \cdot z\right) \cdot \left(y \cdot x\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

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 -1.9e-57)
   (* y (- x))
   (if (<= z 3.2e-176) (* (/ -2.0 t) (/ (* (* z z) (* y x)) a)) (* y x))))

C

assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.9e-57) {
		tmp = y * -x;
	} else if (z <= 3.2e-176) {
		tmp = (-2.0 / t) * (((z * z) * (y * x)) / a);
	} 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 <= (-1.9d-57)) then
        tmp = y * -x
    else if (z <= 3.2d-176) then
        tmp = ((-2.0d0) / t) * (((z * z) * (y * x)) / a)
    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 <= -1.9e-57) {
		tmp = y * -x;
	} else if (z <= 3.2e-176) {
		tmp = (-2.0 / t) * (((z * z) * (y * x)) / a);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.9e-57:
		tmp = y * -x
	elif z <= 3.2e-176:
		tmp = (-2.0 / t) * (((z * z) * (y * x)) / a)
	else:
		tmp = y * x
	return tmp

Julia

t, a = sort([t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.9e-57)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 3.2e-176)
		tmp = Float64(Float64(-2.0 / t) * Float64(Float64(Float64(z * z) * Float64(y * x)) / a));
	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 <= -1.9e-57)
		tmp = y * -x;
	elseif (z <= 3.2e-176)
		tmp = (-2.0 / t) * (((z * z) * (y * x)) / a);
	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, -1.9e-57], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 3.2e-176], N[(N[(-2.0 / t), $MachinePrecision] * N[(N[(N[(z * z), $MachinePrecision] * N[(y * x), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.8999999999999999e-57

   1. Initial program 61.0%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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/ 60.7%

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

      3. *-commutative 60.7%

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

      4. associate-/l* 59.9%

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

   3. Simplified 59.9%

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

   4. Taylor expanded in z around -inf 90.1%

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

      1. neg-mul-1 90.1%

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

   6. Simplified 90.1%

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

   if -1.8999999999999999e-57 < z < 3.19999999999999985e-176

   1. Initial program 77.5%

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

      1. associate-*l* 77.8%

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

      2. associate-*r/ 77.8%

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

      3. *-commutative 77.8%

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

      4. associate-/l* 74.5%

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

   3. Simplified 74.5%

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

      1. associate-*r/ 72.8%

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

      2. associate-*l/ 70.9%

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

      3. associate-/l* 74.3%

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

      4. associate-/r/ 77.7%

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

      5. *-commutative 77.7%

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

      6. associate-/l* 73.6%

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

   5. Applied egg-rr 73.6%

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

   6. Taylor expanded in z around inf 53.6%

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

   7. Taylor expanded in z around 0 50.5%

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

      1. associate-*r/ 50.5%

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

      2. *-commutative 50.5%

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

      3. times-frac 51.0%

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

      4. *-commutative 51.0%

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

      5. *-commutative 51.0%

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

      6. associate-*l* 50.9%

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

      7. unpow2 50.9%

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

   9. Simplified 50.9%

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

   if 3.19999999999999985e-176 < z

   1. Initial program 57.6%

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

      1. associate-*l* 52.8%

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

      2. associate-*r/ 56.2%

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

      3. *-commutative 56.2%

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

      4. associate-/l* 58.7%

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

   3. Simplified 58.7%

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

   4. Taylor expanded in z around inf 82.4%

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

      1. *-commutative 82.4%

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

   6. Simplified 82.4%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-57}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-176}:\\ \;\;\;\;\frac{-2}{t} \cdot \frac{\left(z \cdot z\right) \cdot \left(y \cdot x\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 15: 74.9% accurate, 6.6× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-57}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-146}:\\ \;\;\;\;\frac{y}{-0.5} \cdot \frac{x}{\frac{a}{z} \cdot \frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

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 -1.9e-57)
   (* y (- x))
   (if (<= z 1.95e-146) (* (/ y -0.5) (/ x (* (/ a z) (/ t z)))) (* y x))))

C

assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.9e-57) {
		tmp = y * -x;
	} else if (z <= 1.95e-146) {
		tmp = (y / -0.5) * (x / ((a / z) * (t / 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 <= (-1.9d-57)) then
        tmp = y * -x
    else if (z <= 1.95d-146) then
        tmp = (y / (-0.5d0)) * (x / ((a / z) * (t / 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 <= -1.9e-57) {
		tmp = y * -x;
	} else if (z <= 1.95e-146) {
		tmp = (y / -0.5) * (x / ((a / z) * (t / z)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.9e-57:
		tmp = y * -x
	elif z <= 1.95e-146:
		tmp = (y / -0.5) * (x / ((a / z) * (t / 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 <= -1.9e-57)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.95e-146)
		tmp = Float64(Float64(y / -0.5) * Float64(x / Float64(Float64(a / z) * Float64(t / 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 <= -1.9e-57)
		tmp = y * -x;
	elseif (z <= 1.95e-146)
		tmp = (y / -0.5) * (x / ((a / z) * (t / 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, -1.9e-57], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.95e-146], N[(N[(y / -0.5), $MachinePrecision] * N[(x / N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.8999999999999999e-57

   1. Initial program 61.0%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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/ 60.7%

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

      3. *-commutative 60.7%

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

      4. associate-/l* 59.9%

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

   3. Simplified 59.9%

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

   4. Taylor expanded in z around -inf 90.1%

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

      1. neg-mul-1 90.1%

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

   6. Simplified 90.1%

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

   if -1.8999999999999999e-57 < z < 1.95000000000000001e-146

   1. Initial program 77.4%

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

      1. associate-*l* 79.1%

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

      2. associate-*r/ 79.0%

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

      3. *-commutative 79.0%

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

      4. associate-/l* 75.9%

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

   3. Simplified 75.9%

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

      1. associate-*r/ 74.3%

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

      2. associate-*l/ 71.3%

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

      3. associate-/l* 74.4%

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

      4. associate-/r/ 77.6%

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

      5. *-commutative 77.6%

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

      6. associate-/l* 73.8%

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

   5. Applied egg-rr 73.8%

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

   6. Taylor expanded in z around inf 53.6%

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

   7. Taylor expanded in z around 0 49.2%

      \[\leadsto \frac{y}{\frac{\color{blue}{-0.5 \cdot \frac{a \cdot t}{{z}^{2}}}}{x}} \]
   8. Step-by-step derivation

      1. associate-/l* 49.7%

         \[\leadsto \frac{y}{\frac{-0.5 \cdot \color{blue}{\frac{a}{\frac{{z}^{2}}{t}}}}{x}} \]

      2. unpow2 49.7%

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

      3. associate-/l* 51.2%

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

   9. Simplified 51.2%

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

       1. expm1-log1p-u 49.4%

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

       2. expm1-udef 49.4%

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

       3. associate-/l* 49.4%

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

       4. associate-/r/ 49.5%

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

   11. Applied egg-rr 49.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{y}{\frac{-0.5}{\frac{x}{\frac{a}{z} \cdot \frac{t}{z}}}}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 49.5%

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

       2. expm1-log1p 51.4%

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

       3. associate-/r/ 51.4%

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

   13. Simplified 51.4%

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

   if 1.95000000000000001e-146 < z

   1. Initial program 56.9%

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

      1. associate-*l* 51.0%

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

      2. associate-*r/ 54.5%

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

      3. *-commutative 54.5%

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

      4. associate-/l* 57.1%

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

   3. Simplified 57.1%

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

   4. Taylor expanded in z around inf 83.5%

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

      1. *-commutative 83.5%

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

   6. Simplified 83.5%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-57}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-146}:\\ \;\;\;\;\frac{y}{-0.5} \cdot \frac{x}{\frac{a}{z} \cdot \frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 16: 74.9% accurate, 6.6× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-57}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{-145}:\\ \;\;\;\;x \cdot \frac{y}{-0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

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 -1.9e-57)
   (* y (- x))
   (if (<= z 5.1e-145) (* x (/ y (* -0.5 (* (/ a z) (/ t z))))) (* y x))))

C

assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.9e-57) {
		tmp = y * -x;
	} else if (z <= 5.1e-145) {
		tmp = x * (y / (-0.5 * ((a / z) * (t / 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 <= (-1.9d-57)) then
        tmp = y * -x
    else if (z <= 5.1d-145) then
        tmp = x * (y / ((-0.5d0) * ((a / z) * (t / 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 <= -1.9e-57) {
		tmp = y * -x;
	} else if (z <= 5.1e-145) {
		tmp = x * (y / (-0.5 * ((a / z) * (t / z))));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.9e-57:
		tmp = y * -x
	elif z <= 5.1e-145:
		tmp = x * (y / (-0.5 * ((a / z) * (t / 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 <= -1.9e-57)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 5.1e-145)
		tmp = Float64(x * Float64(y / Float64(-0.5 * Float64(Float64(a / z) * Float64(t / 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 <= -1.9e-57)
		tmp = y * -x;
	elseif (z <= 5.1e-145)
		tmp = x * (y / (-0.5 * ((a / z) * (t / 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, -1.9e-57], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 5.1e-145], N[(x * N[(y / N[(-0.5 * N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.8999999999999999e-57

   1. Initial program 61.0%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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/ 60.7%

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

      3. *-commutative 60.7%

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

      4. associate-/l* 59.9%

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

   3. Simplified 59.9%

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

   4. Taylor expanded in z around -inf 90.1%

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

      1. neg-mul-1 90.1%

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

   6. Simplified 90.1%

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

   if -1.8999999999999999e-57 < z < 5.09999999999999983e-145

   1. Initial program 77.4%

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

      1. associate-*l* 79.1%

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

      2. associate-*r/ 79.0%

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

      3. *-commutative 79.0%

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

      4. associate-/l* 75.9%

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

   3. Simplified 75.9%

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

      1. associate-*r/ 74.3%

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

      2. associate-*l/ 71.3%

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

      3. associate-/l* 74.4%

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

      4. associate-/r/ 77.6%

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

      5. *-commutative 77.6%

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

      6. associate-/l* 73.8%

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

   5. Applied egg-rr 73.8%

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

   6. Taylor expanded in z around inf 53.6%

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

   7. Taylor expanded in z around 0 49.2%

      \[\leadsto \frac{y}{\frac{\color{blue}{-0.5 \cdot \frac{a \cdot t}{{z}^{2}}}}{x}} \]
   8. Step-by-step derivation

      1. associate-/l* 49.7%

         \[\leadsto \frac{y}{\frac{-0.5 \cdot \color{blue}{\frac{a}{\frac{{z}^{2}}{t}}}}{x}} \]

      2. unpow2 49.7%

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

      3. associate-/l* 51.2%

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

   9. Simplified 51.2%

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

       1. associate-/r/ 51.2%

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

       2. associate-/r/ 51.4%

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

   11. Applied egg-rr 51.4%

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

   if 5.09999999999999983e-145 < z

   1. Initial program 56.9%

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

      1. associate-*l* 51.0%

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

      2. associate-*r/ 54.5%

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

      3. *-commutative 54.5%

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

      4. associate-/l* 57.1%

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

   3. Simplified 57.1%

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

   4. Taylor expanded in z around inf 83.5%

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

      1. *-commutative 83.5%

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

   6. Simplified 83.5%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-57}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{-145}:\\ \;\;\;\;x \cdot \frac{y}{-0.5 \cdot \left(\frac{a}{z} \cdot \frac{t}{z}\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 17: 76.2% accurate, 8.0× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{-55}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-178}:\\ \;\;\;\;\frac{1}{\frac{-z}{y \cdot \left(z \cdot x\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

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 -3.5e-55)
   (* y (- x))
   (if (<= z 2.9e-178) (/ 1.0 (/ (- z) (* y (* z x)))) (* y x))))

C

assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.5e-55) {
		tmp = y * -x;
	} else if (z <= 2.9e-178) {
		tmp = 1.0 / (-z / (y * (z * 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 <= (-3.5d-55)) then
        tmp = y * -x
    else if (z <= 2.9d-178) then
        tmp = 1.0d0 / (-z / (y * (z * 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 <= -3.5e-55) {
		tmp = y * -x;
	} else if (z <= 2.9e-178) {
		tmp = 1.0 / (-z / (y * (z * x)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.5e-55:
		tmp = y * -x
	elif z <= 2.9e-178:
		tmp = 1.0 / (-z / (y * (z * 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 <= -3.5e-55)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 2.9e-178)
		tmp = Float64(1.0 / Float64(Float64(-z) / Float64(y * Float64(z * 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 <= -3.5e-55)
		tmp = y * -x;
	elseif (z <= 2.9e-178)
		tmp = 1.0 / (-z / (y * (z * 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, -3.5e-55], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 2.9e-178], N[(1.0 / N[((-z) / N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.50000000000000025e-55

   1. Initial program 61.0%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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/ 60.7%

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

      3. *-commutative 60.7%

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

      4. associate-/l* 59.9%

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

   3. Simplified 59.9%

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

   4. Taylor expanded in z around -inf 90.1%

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

      1. neg-mul-1 90.1%

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

   6. Simplified 90.1%

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

   if -3.50000000000000025e-55 < z < 2.8999999999999998e-178

   1. Initial program 77.5%

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

   2. Taylor expanded in z around -inf 44.0%

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

      1. neg-mul-1 44.0%

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

   4. Simplified 44.0%

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

      1. clear-num 44.0%

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

      2. inv-pow 44.0%

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

      3. *-commutative 44.0%

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

      4. *-commutative 44.0%

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

   6. Applied egg-rr 44.0%

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

      1. unpow-1 44.0%

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

      2. associate-*r* 49.2%

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

      3. *-commutative 49.2%

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

      4. associate-*r* 46.9%

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

   8. Simplified 46.9%

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

   if 2.8999999999999998e-178 < z

   1. Initial program 57.6%

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

      1. associate-*l* 52.8%

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

      2. associate-*r/ 56.2%

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

      3. *-commutative 56.2%

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

      4. associate-/l* 58.7%

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

   3. Simplified 58.7%

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

   4. Taylor expanded in z around inf 82.4%

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

      1. *-commutative 82.4%

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

   6. Simplified 82.4%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{-55}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-178}:\\ \;\;\;\;\frac{1}{\frac{-z}{y \cdot \left(z \cdot x\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 18: 75.7% accurate, 9.3× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+61}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-191}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{-z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

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 -4e+61)
   (* y (- x))
   (if (<= z 6e-191) (/ (* z (* y x)) (- z)) (* y x))))

C

assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e+61) {
		tmp = y * -x;
	} else if (z <= 6e-191) {
		tmp = (z * (y * x)) / -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 <= (-4d+61)) then
        tmp = y * -x
    else if (z <= 6d-191) then
        tmp = (z * (y * x)) / -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 <= -4e+61) {
		tmp = y * -x;
	} else if (z <= 6e-191) {
		tmp = (z * (y * x)) / -z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -4e+61:
		tmp = y * -x
	elif z <= 6e-191:
		tmp = (z * (y * x)) / -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 <= -4e+61)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 6e-191)
		tmp = Float64(Float64(z * Float64(y * x)) / Float64(-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 <= -4e+61)
		tmp = y * -x;
	elseif (z <= 6e-191)
		tmp = (z * (y * x)) / -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, -4e+61], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 6e-191], N[(N[(z * N[(y * x), $MachinePrecision]), $MachinePrecision] / (-z)), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.9999999999999998e61

   1. Initial program 47.6%

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

      1. associate-*l* 45.5%

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

      2. associate-*r/ 47.4%

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

      3. *-commutative 47.4%

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

      4. associate-/l* 44.6%

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

   3. Simplified 44.6%

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

   4. Taylor expanded in z around -inf 93.3%

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

      1. neg-mul-1 93.3%

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

   6. Simplified 93.3%

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

   if -3.9999999999999998e61 < z < 6.0000000000000001e-191

   1. Initial program 81.4%

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

   2. Taylor expanded in z around -inf 54.2%

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

      1. neg-mul-1 54.2%

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

   4. Simplified 54.2%

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

   if 6.0000000000000001e-191 < z

   1. Initial program 57.6%

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

      1. associate-*l* 52.8%

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

      2. associate-*r/ 56.2%

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

      3. *-commutative 56.2%

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

      4. associate-/l* 58.7%

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

   3. Simplified 58.7%

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

   4. Taylor expanded in z around inf 82.4%

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

      1. *-commutative 82.4%

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

   6. Simplified 82.4%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+61}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-191}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{-z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 19: 76.1% accurate, 9.3× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-57}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-184}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{-z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

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 -7e-57)
   (* y (- x))
   (if (<= z 1.4e-184) (/ (* 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 <= -7e-57) {
		tmp = y * -x;
	} else if (z <= 1.4e-184) {
		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 <= (-7d-57)) then
        tmp = y * -x
    else if (z <= 1.4d-184) 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 <= -7e-57) {
		tmp = y * -x;
	} else if (z <= 1.4e-184) {
		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 <= -7e-57:
		tmp = y * -x
	elif z <= 1.4e-184:
		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 <= -7e-57)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.4e-184)
		tmp = Float64(Float64(x * Float64(z * y)) / Float64(-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 <= -7e-57)
		tmp = y * -x;
	elseif (z <= 1.4e-184)
		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, -7e-57], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.4e-184], N[(N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision] / (-z)), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.99999999999999983e-57

   1. Initial program 61.0%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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/ 60.7%

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

      3. *-commutative 60.7%

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

      4. associate-/l* 59.9%

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

   3. Simplified 59.9%

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

   4. Taylor expanded in z around -inf 90.1%

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

      1. neg-mul-1 90.1%

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

   6. Simplified 90.1%

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

   if -6.99999999999999983e-57 < z < 1.3999999999999999e-184

   1. Initial program 77.5%

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

   2. Taylor expanded in z around -inf 44.0%

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

      1. neg-mul-1 44.0%

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

   4. Simplified 44.0%

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

      1. expm1-log1p-u 40.4%

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

      2. expm1-udef 46.2%

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

      3. *-commutative 46.2%

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

      4. *-commutative 46.2%

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

   6. Applied egg-rr 46.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{z \cdot \left(y \cdot x\right)}{-z}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 40.4%

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

      2. expm1-log1p 44.0%

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

      3. associate-*r* 49.2%

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

      4. *-commutative 49.2%

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

   8. Simplified 49.2%

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

   if 1.3999999999999999e-184 < z

   1. Initial program 57.6%

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

      1. associate-*l* 52.8%

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

      2. associate-*r/ 56.2%

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

      3. *-commutative 56.2%

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

      4. associate-/l* 58.7%

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

   3. Simplified 58.7%

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

   4. Taylor expanded in z around inf 82.4%

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

      1. *-commutative 82.4%

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

   6. Simplified 82.4%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-57}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-184}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{-z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 20: 74.5% accurate, 10.2× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-178}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-208}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

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 -1.6e-178)
   (* y (- x))
   (if (<= z 1.1e-208) (* 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 <= -1.6e-178) {
		tmp = y * -x;
	} else if (z <= 1.1e-208) {
		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 <= (-1.6d-178)) then
        tmp = y * -x
    else if (z <= 1.1d-208) 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 <= -1.6e-178) {
		tmp = y * -x;
	} else if (z <= 1.1e-208) {
		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 <= -1.6e-178:
		tmp = y * -x
	elif z <= 1.1e-208:
		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 <= -1.6e-178)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.1e-208)
		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 <= -1.6e-178)
		tmp = y * -x;
	elseif (z <= 1.1e-208)
		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, -1.6e-178], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.1e-208], N[(x * N[(N[(z * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.6e-178

   1. Initial program 66.5%

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

      1. associate-*l* 65.2%

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

      2. associate-*r/ 65.3%

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

      3. *-commutative 65.3%

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

      4. associate-/l* 63.7%

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

   3. Simplified 63.7%

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

   4. Taylor expanded in z around -inf 84.5%

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

      1. neg-mul-1 84.5%

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

   6. Simplified 84.5%

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

   if -1.6e-178 < z < 1.1e-208

   1. Initial program 72.0%

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

      1. associate-*l* 72.4%

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

      2. associate-*r/ 74.4%

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

   3. Simplified 74.4%

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

   4. Taylor expanded in z around inf 35.7%

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

   if 1.1e-208 < z

   1. Initial program 58.2%

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

      1. associate-*l* 53.6%

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

      2. associate-*r/ 56.9%

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

      3. *-commutative 56.9%

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

      4. associate-/l* 59.3%

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

   3. Simplified 59.3%

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

   4. Taylor expanded in z around inf 80.4%

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

      1. *-commutative 80.4%

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

   6. Simplified 80.4%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-178}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-208}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 21: 75.4% accurate, 10.2× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-172}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-41}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

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 -2.3e-172)
   (* y (- x))
   (if (<= z 5e-41) (/ (* z (* y x)) z) (* y x))))

C

assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.3e-172) {
		tmp = y * -x;
	} else if (z <= 5e-41) {
		tmp = (z * (y * x)) / 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 <= (-2.3d-172)) then
        tmp = y * -x
    else if (z <= 5d-41) then
        tmp = (z * (y * x)) / 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 <= -2.3e-172) {
		tmp = y * -x;
	} else if (z <= 5e-41) {
		tmp = (z * (y * x)) / z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

[t, a] = sort([t, a])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.3e-172:
		tmp = y * -x
	elif z <= 5e-41:
		tmp = (z * (y * x)) / 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 <= -2.3e-172)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 5e-41)
		tmp = Float64(Float64(z * Float64(y * x)) / 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 <= -2.3e-172)
		tmp = y * -x;
	elseif (z <= 5e-41)
		tmp = (z * (y * x)) / 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, -2.3e-172], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 5e-41], N[(N[(z * N[(y * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.29999999999999995e-172

   1. Initial program 66.5%

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

      1. associate-*l* 65.2%

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

      2. associate-*r/ 65.3%

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

      3. *-commutative 65.3%

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

      4. associate-/l* 63.7%

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

   3. Simplified 63.7%

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

   4. Taylor expanded in z around -inf 84.5%

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

      1. neg-mul-1 84.5%

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

   6. Simplified 84.5%

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

   if -2.29999999999999995e-172 < z < 4.9999999999999996e-41

   1. Initial program 74.7%

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

   2. Taylor expanded in z around inf 41.9%

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

   if 4.9999999999999996e-41 < z

   1. Initial program 52.3%

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

      1. associate-*l* 47.6%

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

      2. associate-*r/ 49.9%

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

      3. *-commutative 49.9%

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

      4. associate-/l* 50.5%

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

   3. Simplified 50.5%

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

   4. Taylor expanded in z around inf 90.1%

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

      1. *-commutative 90.1%

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

   6. Simplified 90.1%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-172}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-41}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 22: 73.6% accurate, 18.6× speedup

Math

\[\begin{array}{l} [t, a] = \mathsf{sort}([t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-293}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

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 -5e-293) (* y (- x)) (* y x)))

C

assert(t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5e-293) {
		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 <= (-5d-293)) 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 <= -5e-293) {
		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 <= -5e-293:
		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 <= -5e-293)
		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 <= -5e-293)
		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, -5e-293], N[(y * (-x)), $MachinePrecision], N[(y * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.0000000000000003e-293

   1. Initial program 68.1%

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

      1. associate-*l* 67.1%

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

      2. associate-*r/ 67.2%

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

      3. *-commutative 67.2%

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

      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 72.3%

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

      1. neg-mul-1 72.3%

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

   6. Simplified 72.3%

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

   if -5.0000000000000003e-293 < z

   1. Initial program 60.3%

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

      1. associate-*l* 56.6%

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

      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* 61.0%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in z around inf 69.9%

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

      1. *-commutative 69.9%

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

   6. Simplified 69.9%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-293}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 23: 43.8% 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 63.8%

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

   1. associate-*l* 61.4%

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

   2. associate-*r/ 63.3%

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

   3. *-commutative 63.3%

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

   4. associate-/l* 63.2%

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

3. Simplified 63.2%

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

4. Taylor expanded in z around inf 46.5%

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

   1. *-commutative 46.5%

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

6. Simplified 46.5%

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

7. Final simplification 46.5%

   \[\leadsto y \cdot x \]

Developer target: 89.5% 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 2023279 
(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)))))