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

Percentage Accurate: 61.3% → 91.8%

Time: 23.7s

Alternatives: 15

Speedup: 37.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{z\_m \cdot \left(x\_m \cdot y\_m\right)}{\sqrt{z\_m \cdot z\_m - t \cdot a}}\\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{\frac{0.5 \cdot \left(t \cdot \frac{a}{z\_m}\right) - z\_m}{-z\_m}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+173}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (let* ((t_1 (/ (* z_m (* x_m y_m)) (sqrt (- (* z_m z_m) (* t a))))))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= t_1 0.0)
        (* x_m (/ y_m (/ (- (* 0.5 (* t (/ a z_m))) z_m) (- z_m))))
        (if (<= t_1 2e+173) t_1 (* x_m y_m))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double t_1 = (z_m * (x_m * y_m)) / sqrt(((z_m * z_m) - (t * a)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = x_m * (y_m / (((0.5 * (t * (a / z_m))) - z_m) / -z_m));
	} else if (t_1 <= 2e+173) {
		tmp = t_1;
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z_m * (x_m * y_m)) / sqrt(((z_m * z_m) - (t * a)))
    if (t_1 <= 0.0d0) then
        tmp = x_m * (y_m / (((0.5d0 * (t * (a / z_m))) - z_m) / -z_m))
    else if (t_1 <= 2d+173) then
        tmp = t_1
    else
        tmp = x_m * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double t_1 = (z_m * (x_m * y_m)) / Math.sqrt(((z_m * z_m) - (t * a)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = x_m * (y_m / (((0.5 * (t * (a / z_m))) - z_m) / -z_m));
	} else if (t_1 <= 2e+173) {
		tmp = t_1;
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	t_1 = (z_m * (x_m * y_m)) / math.sqrt(((z_m * z_m) - (t * a)))
	tmp = 0
	if t_1 <= 0.0:
		tmp = x_m * (y_m / (((0.5 * (t * (a / z_m))) - z_m) / -z_m))
	elif t_1 <= 2e+173:
		tmp = t_1
	else:
		tmp = x_m * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = Float64(Float64(z_m * Float64(x_m * y_m)) / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(x_m * Float64(y_m / Float64(Float64(Float64(0.5 * Float64(t * Float64(a / z_m))) - z_m) / Float64(-z_m))));
	elseif (t_1 <= 2e+173)
		tmp = t_1;
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = (z_m * (x_m * y_m)) / sqrt(((z_m * z_m) - (t * a)));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = x_m * (y_m / (((0.5 * (t * (a / z_m))) - z_m) / -z_m));
	elseif (t_1 <= 2e+173)
		tmp = t_1;
	else
		tmp = x_m * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := Block[{t$95$1 = N[(N[(z$95$m * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$1, 0.0], N[(x$95$m * N[(y$95$m / N[(N[(N[(0.5 * N[(t * N[(a / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z$95$m), $MachinePrecision] / (-z$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+173], t$95$1, N[(x$95$m * y$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a)))) < 0.0

   1. Initial program 69.5%

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

      1. associate-/l* 72.3%

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

      2. *-commutative 72.3%

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

      3. associate-*l/ 69.7%

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

      4. *-commutative 69.7%

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

      5. associate-/r/ 69.4%

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

   3. Simplified 69.4%

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

   5. Taylor expanded in z around inf 51.7%

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

      1. associate-*l/ 51.8%

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

      2. associate-/l* 57.7%

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

      3. +-commutative 57.7%

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

      4. fma-define 57.7%

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

      5. *-commutative 57.7%

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

      6. associate-/l* 58.0%

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

   7. Applied egg-rr 58.0%

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

      1. expm1-log1p-u 58.0%

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

      2. expm1-undefine 58.0%

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

      3. div-inv 58.0%

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

      4. clear-num 58.0%

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

   9. Applied egg-rr 58.0%

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

       1. expm1-define 58.0%

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

   11. Simplified 58.0%

       \[\leadsto x \cdot \frac{y}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-0.5, t \cdot \frac{a}{z}, z\right)}{z}\right)\right)}} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 58.0%

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

       2. frac-2neg 58.0%

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

   13. Applied egg-rr 58.0%

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

       1. fma-undefine 58.0%

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

       2. associate-*r/ 57.7%

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

       3. *-commutative 57.7%

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

       4. associate-*r/ 57.7%

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

       5. *-commutative 57.7%

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

       6. distribute-neg-in 57.7%

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

       7. distribute-frac-neg 57.7%

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

       8. distribute-lft-neg-in 57.7%

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

       9. metadata-eval 57.7%

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

       10. *-commutative 57.7%

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

       11. associate-*r/ 57.7%

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

       12. unsub-neg 57.7%

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

       13. *-commutative 57.7%

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

       14. associate-*r/ 58.0%

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

   15. Simplified 58.0%

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

   if 0.0 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a)))) < 2e173

   1. Initial program 99.7%

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

   if 2e173 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))))

   1. Initial program 22.0%

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

      1. associate-/l* 22.9%

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

      2. *-commutative 22.9%

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

      3. associate-*l/ 24.9%

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

      4. *-commutative 24.9%

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

      5. associate-/r/ 24.9%

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

   3. Simplified 24.9%

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

   5. Taylor expanded in z around inf 44.0%

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

      1. *-commutative 44.0%

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

   7. Simplified 44.0%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(x \cdot y\right)}{\sqrt{z \cdot z - t \cdot a}} \leq 0:\\ \;\;\;\;x \cdot \frac{y}{\frac{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}{-z}}\\ \mathbf{elif}\;\frac{z \cdot \left(x \cdot y\right)}{\sqrt{z \cdot z - t \cdot a}} \leq 2 \cdot 10^{+173}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4 \cdot 10^{-172} \lor \neg \left(z\_m \leq 2.05 \cdot 10^{-150}\right) \land z\_m \leq 4.8 \cdot 10^{-106}:\\ \;\;\;\;x\_m \cdot \left(z\_m \cdot \frac{y\_m}{\sqrt{t \cdot \left(-a\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{\frac{0.5 \cdot \left(t \cdot \frac{a}{z\_m}\right) - z\_m}{-z\_m}}\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (or (<= z_m 4e-172) (and (not (<= z_m 2.05e-150)) (<= z_m 4.8e-106)))
      (* x_m (* z_m (/ y_m (sqrt (* t (- a))))))
      (* x_m (/ y_m (/ (- (* 0.5 (* t (/ a z_m))) z_m) (- z_m)))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if ((z_m <= 4e-172) || (!(z_m <= 2.05e-150) && (z_m <= 4.8e-106))) {
		tmp = x_m * (z_m * (y_m / sqrt((t * -a))));
	} else {
		tmp = x_m * (y_m / (((0.5 * (t * (a / z_m))) - z_m) / -z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((z_m <= 4d-172) .or. (.not. (z_m <= 2.05d-150)) .and. (z_m <= 4.8d-106)) then
        tmp = x_m * (z_m * (y_m / sqrt((t * -a))))
    else
        tmp = x_m * (y_m / (((0.5d0 * (t * (a / z_m))) - z_m) / -z_m))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if ((z_m <= 4e-172) || (!(z_m <= 2.05e-150) && (z_m <= 4.8e-106))) {
		tmp = x_m * (z_m * (y_m / Math.sqrt((t * -a))));
	} else {
		tmp = x_m * (y_m / (((0.5 * (t * (a / z_m))) - z_m) / -z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if (z_m <= 4e-172) or (not (z_m <= 2.05e-150) and (z_m <= 4.8e-106)):
		tmp = x_m * (z_m * (y_m / math.sqrt((t * -a))))
	else:
		tmp = x_m * (y_m / (((0.5 * (t * (a / z_m))) - z_m) / -z_m))
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if ((z_m <= 4e-172) || (!(z_m <= 2.05e-150) && (z_m <= 4.8e-106)))
		tmp = Float64(x_m * Float64(z_m * Float64(y_m / sqrt(Float64(t * Float64(-a))))));
	else
		tmp = Float64(x_m * Float64(y_m / Float64(Float64(Float64(0.5 * Float64(t * Float64(a / z_m))) - z_m) / Float64(-z_m))));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if ((z_m <= 4e-172) || (~((z_m <= 2.05e-150)) && (z_m <= 4.8e-106)))
		tmp = x_m * (z_m * (y_m / sqrt((t * -a))));
	else
		tmp = x_m * (y_m / (((0.5 * (t * (a / z_m))) - z_m) / -z_m));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[Or[LessEqual[z$95$m, 4e-172], And[N[Not[LessEqual[z$95$m, 2.05e-150]], $MachinePrecision], LessEqual[z$95$m, 4.8e-106]]], N[(x$95$m * N[(z$95$m * N[(y$95$m / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(y$95$m / N[(N[(N[(0.5 * N[(t * N[(a / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z$95$m), $MachinePrecision] / (-z$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.0000000000000002e-172 or 2.0499999999999999e-150 < z < 4.7999999999999995e-106

   1. Initial program 67.5%

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

      1. associate-/l* 68.3%

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

      2. *-commutative 68.3%

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

      3. associate-*l/ 65.8%

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

      4. *-commutative 65.8%

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

      5. associate-/r/ 67.0%

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

   3. Simplified 67.0%

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

   5. Taylor expanded in z around 0 41.4%

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

      1. associate-*r* 41.4%

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

      2. neg-mul-1 41.4%

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

      3. *-commutative 41.4%

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

   7. Simplified 41.4%

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

   if 4.0000000000000002e-172 < z < 2.0499999999999999e-150 or 4.7999999999999995e-106 < z

   1. Initial program 58.4%

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

      1. associate-/l* 59.1%

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

      2. *-commutative 59.1%

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

      3. associate-*l/ 58.3%

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

      4. *-commutative 58.3%

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

      5. associate-/r/ 55.0%

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

   3. Simplified 55.0%

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

   5. Taylor expanded in z around inf 77.6%

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

      1. associate-*l/ 71.6%

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

      2. associate-/l* 89.6%

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

      3. +-commutative 89.6%

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

      4. fma-define 89.6%

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

      5. *-commutative 89.6%

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

      6. associate-/l* 90.6%

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

   7. Applied egg-rr 90.6%

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

      1. expm1-log1p-u 90.6%

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

      2. expm1-undefine 90.6%

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

      3. div-inv 90.6%

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

      4. clear-num 90.6%

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

   9. Applied egg-rr 90.6%

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

       1. expm1-define 90.6%

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

   11. Simplified 90.6%

       \[\leadsto x \cdot \frac{y}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-0.5, t \cdot \frac{a}{z}, z\right)}{z}\right)\right)}} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 90.6%

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

       2. frac-2neg 90.6%

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

   13. Applied egg-rr 90.6%

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

       1. fma-undefine 90.6%

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

       2. associate-*r/ 89.6%

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

       3. *-commutative 89.6%

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

       4. associate-*r/ 89.6%

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

       5. *-commutative 89.6%

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

       6. distribute-neg-in 89.6%

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

       7. distribute-frac-neg 89.6%

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

       8. distribute-lft-neg-in 89.6%

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

       9. metadata-eval 89.6%

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

       10. *-commutative 89.6%

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

       11. associate-*r/ 89.6%

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

       12. unsub-neg 89.6%

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

       13. *-commutative 89.6%

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

       14. associate-*r/ 90.6%

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

   15. Simplified 90.6%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{-172} \lor \neg \left(z \leq 2.05 \cdot 10^{-150}\right) \land z \leq 4.8 \cdot 10^{-106}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{\sqrt{t \cdot \left(-a\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{\frac{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}{-z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \sqrt{t \cdot \left(-a\right)}\\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4 \cdot 10^{-172}:\\ \;\;\;\;x\_m \cdot \left(z\_m \cdot \frac{y\_m}{t\_1}\right)\\ \mathbf{elif}\;z\_m \leq 5.2 \cdot 10^{-151} \lor \neg \left(z\_m \leq 4 \cdot 10^{-106}\right):\\ \;\;\;\;x\_m \cdot \frac{y\_m}{\frac{0.5 \cdot \left(t \cdot \frac{a}{z\_m}\right) - z\_m}{-z\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot y\_m}{\frac{t\_1}{z\_m}}\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (let* ((t_1 (sqrt (* t (- a)))))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= z_m 4e-172)
        (* x_m (* z_m (/ y_m t_1)))
        (if (or (<= z_m 5.2e-151) (not (<= z_m 4e-106)))
          (* x_m (/ y_m (/ (- (* 0.5 (* t (/ a z_m))) z_m) (- z_m))))
          (/ (* x_m y_m) (/ t_1 z_m)))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double t_1 = sqrt((t * -a));
	double tmp;
	if (z_m <= 4e-172) {
		tmp = x_m * (z_m * (y_m / t_1));
	} else if ((z_m <= 5.2e-151) || !(z_m <= 4e-106)) {
		tmp = x_m * (y_m / (((0.5 * (t * (a / z_m))) - z_m) / -z_m));
	} else {
		tmp = (x_m * y_m) / (t_1 / z_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((t * -a))
    if (z_m <= 4d-172) then
        tmp = x_m * (z_m * (y_m / t_1))
    else if ((z_m <= 5.2d-151) .or. (.not. (z_m <= 4d-106))) then
        tmp = x_m * (y_m / (((0.5d0 * (t * (a / z_m))) - z_m) / -z_m))
    else
        tmp = (x_m * y_m) / (t_1 / z_m)
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double t_1 = Math.sqrt((t * -a));
	double tmp;
	if (z_m <= 4e-172) {
		tmp = x_m * (z_m * (y_m / t_1));
	} else if ((z_m <= 5.2e-151) || !(z_m <= 4e-106)) {
		tmp = x_m * (y_m / (((0.5 * (t * (a / z_m))) - z_m) / -z_m));
	} else {
		tmp = (x_m * y_m) / (t_1 / z_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	t_1 = math.sqrt((t * -a))
	tmp = 0
	if z_m <= 4e-172:
		tmp = x_m * (z_m * (y_m / t_1))
	elif (z_m <= 5.2e-151) or not (z_m <= 4e-106):
		tmp = x_m * (y_m / (((0.5 * (t * (a / z_m))) - z_m) / -z_m))
	else:
		tmp = (x_m * y_m) / (t_1 / z_m)
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = sqrt(Float64(t * Float64(-a)))
	tmp = 0.0
	if (z_m <= 4e-172)
		tmp = Float64(x_m * Float64(z_m * Float64(y_m / t_1)));
	elseif ((z_m <= 5.2e-151) || !(z_m <= 4e-106))
		tmp = Float64(x_m * Float64(y_m / Float64(Float64(Float64(0.5 * Float64(t * Float64(a / z_m))) - z_m) / Float64(-z_m))));
	else
		tmp = Float64(Float64(x_m * y_m) / Float64(t_1 / z_m));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = sqrt((t * -a));
	tmp = 0.0;
	if (z_m <= 4e-172)
		tmp = x_m * (z_m * (y_m / t_1));
	elseif ((z_m <= 5.2e-151) || ~((z_m <= 4e-106)))
		tmp = x_m * (y_m / (((0.5 * (t * (a / z_m))) - z_m) / -z_m));
	else
		tmp = (x_m * y_m) / (t_1 / z_m);
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := Block[{t$95$1 = N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 4e-172], N[(x$95$m * N[(z$95$m * N[(y$95$m / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z$95$m, 5.2e-151], N[Not[LessEqual[z$95$m, 4e-106]], $MachinePrecision]], N[(x$95$m * N[(y$95$m / N[(N[(N[(0.5 * N[(t * N[(a / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z$95$m), $MachinePrecision] / (-z$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[(t$95$1 / z$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < 4.0000000000000002e-172

   1. Initial program 66.0%

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

      1. associate-/l* 66.8%

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

      2. *-commutative 66.8%

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

      3. associate-*l/ 64.2%

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

      4. *-commutative 64.2%

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

      5. associate-/r/ 65.5%

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

   3. Simplified 65.5%

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

   5. Taylor expanded in z around 0 39.4%

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

      1. associate-*r* 39.4%

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

      2. neg-mul-1 39.4%

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

      3. *-commutative 39.4%

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

   7. Simplified 39.4%

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

   if 4.0000000000000002e-172 < z < 5.2000000000000001e-151 or 3.99999999999999976e-106 < z

   1. Initial program 58.4%

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

      1. associate-/l* 59.1%

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

      2. *-commutative 59.1%

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

      3. associate-*l/ 58.3%

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

      4. *-commutative 58.3%

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

      5. associate-/r/ 55.0%

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

   3. Simplified 55.0%

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

   5. Taylor expanded in z around inf 77.6%

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

      1. associate-*l/ 71.6%

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

      2. associate-/l* 89.6%

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

      3. +-commutative 89.6%

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

      4. fma-define 89.6%

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

      5. *-commutative 89.6%

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

      6. associate-/l* 90.6%

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

   7. Applied egg-rr 90.6%

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

      1. expm1-log1p-u 90.6%

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

      2. expm1-undefine 90.6%

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

      3. div-inv 90.6%

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

      4. clear-num 90.6%

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

   9. Applied egg-rr 90.6%

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

       1. expm1-define 90.6%

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

   11. Simplified 90.6%

       \[\leadsto x \cdot \frac{y}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-0.5, t \cdot \frac{a}{z}, z\right)}{z}\right)\right)}} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 90.6%

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

       2. frac-2neg 90.6%

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

   13. Applied egg-rr 90.6%

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

       1. fma-undefine 90.6%

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

       2. associate-*r/ 89.6%

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

       3. *-commutative 89.6%

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

       4. associate-*r/ 89.6%

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

       5. *-commutative 89.6%

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

       6. distribute-neg-in 89.6%

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

       7. distribute-frac-neg 89.6%

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

       8. distribute-lft-neg-in 89.6%

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

       9. metadata-eval 89.6%

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

       10. *-commutative 89.6%

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

       11. associate-*r/ 89.6%

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

       12. unsub-neg 89.6%

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

       13. *-commutative 89.6%

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

       14. associate-*r/ 90.6%

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

   15. Simplified 90.6%

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

   if 5.2000000000000001e-151 < z < 3.99999999999999976e-106

   1. Initial program 99.8%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 85.7%

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

      1. associate-*r* 85.3%

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

      2. neg-mul-1 85.3%

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

      3. *-commutative 85.3%

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

   7. Simplified 85.7%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{-172}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{\sqrt{t \cdot \left(-a\right)}}\right)\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-151} \lor \neg \left(z \leq 4 \cdot 10^{-106}\right):\\ \;\;\;\;x \cdot \frac{y}{\frac{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}{-z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\sqrt{t \cdot \left(-a\right)}}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \sqrt{t \cdot \left(-a\right)}\\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.6 \cdot 10^{-171}:\\ \;\;\;\;\frac{x\_m \cdot \left(y\_m \cdot z\_m\right)}{t\_1}\\ \mathbf{elif}\;z\_m \leq 2.05 \cdot 10^{-150} \lor \neg \left(z\_m \leq 3.65 \cdot 10^{-106}\right):\\ \;\;\;\;x\_m \cdot \frac{y\_m}{\frac{0.5 \cdot \left(t \cdot \frac{a}{z\_m}\right) - z\_m}{-z\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot y\_m}{\frac{t\_1}{z\_m}}\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (let* ((t_1 (sqrt (* t (- a)))))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= z_m 1.6e-171)
        (/ (* x_m (* y_m z_m)) t_1)
        (if (or (<= z_m 2.05e-150) (not (<= z_m 3.65e-106)))
          (* x_m (/ y_m (/ (- (* 0.5 (* t (/ a z_m))) z_m) (- z_m))))
          (/ (* x_m y_m) (/ t_1 z_m)))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double t_1 = sqrt((t * -a));
	double tmp;
	if (z_m <= 1.6e-171) {
		tmp = (x_m * (y_m * z_m)) / t_1;
	} else if ((z_m <= 2.05e-150) || !(z_m <= 3.65e-106)) {
		tmp = x_m * (y_m / (((0.5 * (t * (a / z_m))) - z_m) / -z_m));
	} else {
		tmp = (x_m * y_m) / (t_1 / z_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((t * -a))
    if (z_m <= 1.6d-171) then
        tmp = (x_m * (y_m * z_m)) / t_1
    else if ((z_m <= 2.05d-150) .or. (.not. (z_m <= 3.65d-106))) then
        tmp = x_m * (y_m / (((0.5d0 * (t * (a / z_m))) - z_m) / -z_m))
    else
        tmp = (x_m * y_m) / (t_1 / z_m)
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double t_1 = Math.sqrt((t * -a));
	double tmp;
	if (z_m <= 1.6e-171) {
		tmp = (x_m * (y_m * z_m)) / t_1;
	} else if ((z_m <= 2.05e-150) || !(z_m <= 3.65e-106)) {
		tmp = x_m * (y_m / (((0.5 * (t * (a / z_m))) - z_m) / -z_m));
	} else {
		tmp = (x_m * y_m) / (t_1 / z_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	t_1 = math.sqrt((t * -a))
	tmp = 0
	if z_m <= 1.6e-171:
		tmp = (x_m * (y_m * z_m)) / t_1
	elif (z_m <= 2.05e-150) or not (z_m <= 3.65e-106):
		tmp = x_m * (y_m / (((0.5 * (t * (a / z_m))) - z_m) / -z_m))
	else:
		tmp = (x_m * y_m) / (t_1 / z_m)
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = sqrt(Float64(t * Float64(-a)))
	tmp = 0.0
	if (z_m <= 1.6e-171)
		tmp = Float64(Float64(x_m * Float64(y_m * z_m)) / t_1);
	elseif ((z_m <= 2.05e-150) || !(z_m <= 3.65e-106))
		tmp = Float64(x_m * Float64(y_m / Float64(Float64(Float64(0.5 * Float64(t * Float64(a / z_m))) - z_m) / Float64(-z_m))));
	else
		tmp = Float64(Float64(x_m * y_m) / Float64(t_1 / z_m));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = sqrt((t * -a));
	tmp = 0.0;
	if (z_m <= 1.6e-171)
		tmp = (x_m * (y_m * z_m)) / t_1;
	elseif ((z_m <= 2.05e-150) || ~((z_m <= 3.65e-106)))
		tmp = x_m * (y_m / (((0.5 * (t * (a / z_m))) - z_m) / -z_m));
	else
		tmp = (x_m * y_m) / (t_1 / z_m);
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := Block[{t$95$1 = N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.6e-171], N[(N[(x$95$m * N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[Or[LessEqual[z$95$m, 2.05e-150], N[Not[LessEqual[z$95$m, 3.65e-106]], $MachinePrecision]], N[(x$95$m * N[(y$95$m / N[(N[(N[(0.5 * N[(t * N[(a / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z$95$m), $MachinePrecision] / (-z$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[(t$95$1 / z$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < 1.6000000000000001e-171

   1. Initial program 66.0%

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

   3. Taylor expanded in z around 0 43.8%

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

      1. associate-*r* 39.4%

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

      2. neg-mul-1 39.4%

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

      3. *-commutative 39.4%

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

   5. Simplified 43.8%

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

   6. Taylor expanded in x around 0 41.2%

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

   if 1.6000000000000001e-171 < z < 2.0499999999999999e-150 or 3.64999999999999996e-106 < z

   1. Initial program 58.4%

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

      1. associate-/l* 59.1%

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

      2. *-commutative 59.1%

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

      3. associate-*l/ 58.3%

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

      4. *-commutative 58.3%

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

      5. associate-/r/ 55.0%

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

   3. Simplified 55.0%

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

   5. Taylor expanded in z around inf 77.6%

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

      1. associate-*l/ 71.6%

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

      2. associate-/l* 89.6%

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

      3. +-commutative 89.6%

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

      4. fma-define 89.6%

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

      5. *-commutative 89.6%

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

      6. associate-/l* 90.6%

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

   7. Applied egg-rr 90.6%

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

      1. expm1-log1p-u 90.6%

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

      2. expm1-undefine 90.6%

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

      3. div-inv 90.6%

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

      4. clear-num 90.6%

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

   9. Applied egg-rr 90.6%

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

       1. expm1-define 90.6%

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

   11. Simplified 90.6%

       \[\leadsto x \cdot \frac{y}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-0.5, t \cdot \frac{a}{z}, z\right)}{z}\right)\right)}} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 90.6%

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

       2. frac-2neg 90.6%

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

   13. Applied egg-rr 90.6%

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

       1. fma-undefine 90.6%

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

       2. associate-*r/ 89.6%

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

       3. *-commutative 89.6%

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

       4. associate-*r/ 89.6%

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

       5. *-commutative 89.6%

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

       6. distribute-neg-in 89.6%

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

       7. distribute-frac-neg 89.6%

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

       8. distribute-lft-neg-in 89.6%

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

       9. metadata-eval 89.6%

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

       10. *-commutative 89.6%

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

       11. associate-*r/ 89.6%

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

       12. unsub-neg 89.6%

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

       13. *-commutative 89.6%

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

       14. associate-*r/ 90.6%

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

   15. Simplified 90.6%

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

   if 2.0499999999999999e-150 < z < 3.64999999999999996e-106

   1. Initial program 99.8%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 85.7%

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

      1. associate-*r* 85.3%

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

      2. neg-mul-1 85.3%

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

      3. *-commutative 85.3%

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

   7. Simplified 85.7%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.6 \cdot 10^{-171}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{-150} \lor \neg \left(z \leq 3.65 \cdot 10^{-106}\right):\\ \;\;\;\;x \cdot \frac{y}{\frac{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}{-z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\sqrt{t \cdot \left(-a\right)}}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 10^{-175}:\\ \;\;\;\;\frac{x\_m \cdot \left(y\_m \cdot z\_m\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{elif}\;z\_m \leq 5 \cdot 10^{+129}:\\ \;\;\;\;\frac{x\_m \cdot y\_m}{\frac{\sqrt{z\_m \cdot z\_m - t \cdot a}}{z\_m}}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1e-175)
      (/ (* x_m (* y_m z_m)) (sqrt (* t (- a))))
      (if (<= z_m 5e+129)
        (/ (* x_m y_m) (/ (sqrt (- (* z_m z_m) (* t a))) z_m))
        (* x_m y_m)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1e-175) {
		tmp = (x_m * (y_m * z_m)) / sqrt((t * -a));
	} else if (z_m <= 5e+129) {
		tmp = (x_m * y_m) / (sqrt(((z_m * z_m) - (t * a))) / z_m);
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1d-175) then
        tmp = (x_m * (y_m * z_m)) / sqrt((t * -a))
    else if (z_m <= 5d+129) then
        tmp = (x_m * y_m) / (sqrt(((z_m * z_m) - (t * a))) / z_m)
    else
        tmp = x_m * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1e-175) {
		tmp = (x_m * (y_m * z_m)) / Math.sqrt((t * -a));
	} else if (z_m <= 5e+129) {
		tmp = (x_m * y_m) / (Math.sqrt(((z_m * z_m) - (t * a))) / z_m);
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1e-175:
		tmp = (x_m * (y_m * z_m)) / math.sqrt((t * -a))
	elif z_m <= 5e+129:
		tmp = (x_m * y_m) / (math.sqrt(((z_m * z_m) - (t * a))) / z_m)
	else:
		tmp = x_m * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1e-175)
		tmp = Float64(Float64(x_m * Float64(y_m * z_m)) / sqrt(Float64(t * Float64(-a))));
	elseif (z_m <= 5e+129)
		tmp = Float64(Float64(x_m * y_m) / Float64(sqrt(Float64(Float64(z_m * z_m) - Float64(t * a))) / z_m));
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1e-175)
		tmp = (x_m * (y_m * z_m)) / sqrt((t * -a));
	elseif (z_m <= 5e+129)
		tmp = (x_m * y_m) / (sqrt(((z_m * z_m) - (t * a))) / z_m);
	else
		tmp = x_m * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1e-175], N[(N[(x$95$m * N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 5e+129], N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[(N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 1e-175

   1. Initial program 66.0%

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

   3. Taylor expanded in z around 0 43.8%

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

      1. associate-*r* 39.4%

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

      2. neg-mul-1 39.4%

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

      3. *-commutative 39.4%

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

   5. Simplified 43.8%

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

   6. Taylor expanded in x around 0 41.2%

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

   if 1e-175 < z < 5.0000000000000003e129

   1. Initial program 92.6%

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

      1. associate-/l* 93.3%

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

   3. Simplified 93.3%

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

   if 5.0000000000000003e129 < z

   1. Initial program 20.9%

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

      1. associate-/l* 21.5%

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

      2. *-commutative 21.5%

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

      3. associate-*l/ 21.6%

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

      4. *-commutative 21.6%

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

      5. associate-/r/ 21.6%

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

   3. Simplified 21.6%

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

   5. Taylor expanded in z around inf 96.1%

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

      1. *-commutative 96.1%

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

   7. Simplified 96.1%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{-175}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+129}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 6.2 \cdot 10^{+129}:\\ \;\;\;\;x\_m \cdot \left(z\_m \cdot \frac{y\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 6.2e+129)
      (* x_m (* z_m (/ y_m (sqrt (- (* z_m z_m) (* t a))))))
      (* x_m y_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 6.2e+129) {
		tmp = x_m * (z_m * (y_m / sqrt(((z_m * z_m) - (t * a)))));
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 6.2d+129) then
        tmp = x_m * (z_m * (y_m / sqrt(((z_m * z_m) - (t * a)))))
    else
        tmp = x_m * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 6.2e+129) {
		tmp = x_m * (z_m * (y_m / Math.sqrt(((z_m * z_m) - (t * a)))));
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 6.2e+129:
		tmp = x_m * (z_m * (y_m / math.sqrt(((z_m * z_m) - (t * a)))))
	else:
		tmp = x_m * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 6.2e+129)
		tmp = Float64(x_m * Float64(z_m * Float64(y_m / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a))))));
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 6.2e+129)
		tmp = x_m * (z_m * (y_m / sqrt(((z_m * z_m) - (t * a)))));
	else
		tmp = x_m * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 6.2e+129], N[(x$95$m * N[(z$95$m * N[(y$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 6.1999999999999999e129

   1. Initial program 73.6%

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

      1. associate-/l* 74.4%

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

      2. *-commutative 74.4%

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

      3. associate-*l/ 72.1%

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

      4. *-commutative 72.1%

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

      5. associate-/r/ 71.5%

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

   3. Simplified 71.5%

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

   if 6.1999999999999999e129 < z

   1. Initial program 20.9%

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

      1. associate-/l* 21.5%

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

      2. *-commutative 21.5%

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

      3. associate-*l/ 21.6%

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

      4. *-commutative 21.6%

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

      5. associate-/r/ 21.6%

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

   3. Simplified 21.6%

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

   5. Taylor expanded in z around inf 96.1%

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

      1. *-commutative 96.1%

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

   7. Simplified 96.1%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.2 \cdot 10^{+129}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.15 \cdot 10^{+31}:\\ \;\;\;\;\frac{x\_m \cdot \left(y\_m \cdot z\_m\right)}{\sqrt{z\_m \cdot z\_m - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{\frac{0.5 \cdot \left(t \cdot \frac{a}{z\_m}\right) - z\_m}{-z\_m}}\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1.15e+31)
      (/ (* x_m (* y_m z_m)) (sqrt (- (* z_m z_m) (* t a))))
      (* x_m (/ y_m (/ (- (* 0.5 (* t (/ a z_m))) z_m) (- z_m)))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.15e+31) {
		tmp = (x_m * (y_m * z_m)) / sqrt(((z_m * z_m) - (t * a)));
	} else {
		tmp = x_m * (y_m / (((0.5 * (t * (a / z_m))) - z_m) / -z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.15d+31) then
        tmp = (x_m * (y_m * z_m)) / sqrt(((z_m * z_m) - (t * a)))
    else
        tmp = x_m * (y_m / (((0.5d0 * (t * (a / z_m))) - z_m) / -z_m))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.15e+31) {
		tmp = (x_m * (y_m * z_m)) / Math.sqrt(((z_m * z_m) - (t * a)));
	} else {
		tmp = x_m * (y_m / (((0.5 * (t * (a / z_m))) - z_m) / -z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.15e+31:
		tmp = (x_m * (y_m * z_m)) / math.sqrt(((z_m * z_m) - (t * a)))
	else:
		tmp = x_m * (y_m / (((0.5 * (t * (a / z_m))) - z_m) / -z_m))
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.15e+31)
		tmp = Float64(Float64(x_m * Float64(y_m * z_m)) / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a))));
	else
		tmp = Float64(x_m * Float64(y_m / Float64(Float64(Float64(0.5 * Float64(t * Float64(a / z_m))) - z_m) / Float64(-z_m))));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.15e+31)
		tmp = (x_m * (y_m * z_m)) / sqrt(((z_m * z_m) - (t * a)));
	else
		tmp = x_m * (y_m / (((0.5 * (t * (a / z_m))) - z_m) / -z_m));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.15e+31], N[(N[(x$95$m * N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(y$95$m / N[(N[(N[(0.5 * N[(t * N[(a / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z$95$m), $MachinePrecision] / (-z$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.15e31

   1. Initial program 70.7%

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

   3. Taylor expanded in x around 0 67.4%

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

   if 1.15e31 < z

   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* 48.0%

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

      2. *-commutative 48.0%

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

      3. associate-*l/ 48.1%

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

      4. *-commutative 48.1%

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

      5. associate-/r/ 43.8%

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

   3. Simplified 43.8%

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

   5. Taylor expanded in z around inf 76.3%

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

      1. associate-*l/ 72.8%

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

      2. associate-/l* 92.2%

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

      3. +-commutative 92.2%

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

      4. fma-define 92.2%

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

      5. *-commutative 92.2%

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

      6. associate-/l* 93.6%

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

   7. Applied egg-rr 93.6%

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

      1. expm1-log1p-u 93.6%

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

      2. expm1-undefine 93.6%

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

      3. div-inv 93.6%

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

      4. clear-num 93.6%

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

   9. Applied egg-rr 93.6%

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

       1. expm1-define 93.6%

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

   11. Simplified 93.6%

       \[\leadsto x \cdot \frac{y}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-0.5, t \cdot \frac{a}{z}, z\right)}{z}\right)\right)}} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 93.6%

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

       2. frac-2neg 93.6%

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

   13. Applied egg-rr 93.6%

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

       1. fma-undefine 93.6%

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

       2. associate-*r/ 92.2%

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

       3. *-commutative 92.2%

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

       4. associate-*r/ 92.2%

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

       5. *-commutative 92.2%

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

       6. distribute-neg-in 92.2%

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

       7. distribute-frac-neg 92.2%

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

       8. distribute-lft-neg-in 92.2%

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

       9. metadata-eval 92.2%

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

       10. *-commutative 92.2%

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

       11. associate-*r/ 92.2%

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

       12. unsub-neg 92.2%

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

       13. *-commutative 92.2%

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

       14. associate-*r/ 93.6%

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

   15. Simplified 93.6%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.15 \cdot 10^{+31}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{\frac{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}{-z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.8% accurate, 5.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot y\_m \leq 5 \cdot 10^{+53}:\\ \;\;\;\;\frac{z\_m \cdot \left(x\_m \cdot y\_m\right)}{z\_m + -0.5 \cdot \frac{t \cdot a}{z\_m}}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= (* x_m y_m) 5e+53)
      (/ (* z_m (* x_m y_m)) (+ z_m (* -0.5 (/ (* t a) z_m))))
      (* x_m y_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if ((x_m * y_m) <= 5e+53) {
		tmp = (z_m * (x_m * y_m)) / (z_m + (-0.5 * ((t * a) / z_m)));
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((x_m * y_m) <= 5d+53) then
        tmp = (z_m * (x_m * y_m)) / (z_m + ((-0.5d0) * ((t * a) / z_m)))
    else
        tmp = x_m * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if ((x_m * y_m) <= 5e+53) {
		tmp = (z_m * (x_m * y_m)) / (z_m + (-0.5 * ((t * a) / z_m)));
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if (x_m * y_m) <= 5e+53:
		tmp = (z_m * (x_m * y_m)) / (z_m + (-0.5 * ((t * a) / z_m)))
	else:
		tmp = x_m * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (Float64(x_m * y_m) <= 5e+53)
		tmp = Float64(Float64(z_m * Float64(x_m * y_m)) / Float64(z_m + Float64(-0.5 * Float64(Float64(t * a) / z_m))));
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if ((x_m * y_m) <= 5e+53)
		tmp = (z_m * (x_m * y_m)) / (z_m + (-0.5 * ((t * a) / z_m)));
	else
		tmp = x_m * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[N[(x$95$m * y$95$m), $MachinePrecision], 5e+53], N[(N[(z$95$m * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[(z$95$m + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 5.0000000000000004e53

   1. Initial program 66.3%

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

   3. Taylor expanded in z around inf 52.5%

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

   if 5.0000000000000004e53 < (*.f64 x y)

   1. Initial program 55.8%

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

      1. associate-/l* 56.0%

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

      2. *-commutative 56.0%

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

      3. associate-*l/ 54.6%

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

      4. *-commutative 54.6%

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

      5. associate-/r/ 59.6%

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

   3. Simplified 59.6%

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

   5. Taylor expanded in z around inf 31.8%

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

      1. *-commutative 31.8%

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

   7. Simplified 31.8%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq 5 \cdot 10^{+53}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{z + -0.5 \cdot \frac{t \cdot a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.8% accurate, 5.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -9.6 \cdot 10^{-108}:\\ \;\;\;\;x\_m \cdot \left(z\_m \cdot \frac{y\_m}{z\_m + -0.5 \cdot \left(a \cdot \frac{t}{z\_m}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= t -9.6e-108)
      (* x_m (* z_m (/ y_m (+ z_m (* -0.5 (* a (/ t z_m)))))))
      (* x_m y_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (t <= -9.6e-108) {
		tmp = x_m * (z_m * (y_m / (z_m + (-0.5 * (a * (t / z_m))))));
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-9.6d-108)) then
        tmp = x_m * (z_m * (y_m / (z_m + ((-0.5d0) * (a * (t / z_m))))))
    else
        tmp = x_m * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (t <= -9.6e-108) {
		tmp = x_m * (z_m * (y_m / (z_m + (-0.5 * (a * (t / z_m))))));
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if t <= -9.6e-108:
		tmp = x_m * (z_m * (y_m / (z_m + (-0.5 * (a * (t / z_m))))))
	else:
		tmp = x_m * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (t <= -9.6e-108)
		tmp = Float64(x_m * Float64(z_m * Float64(y_m / Float64(z_m + Float64(-0.5 * Float64(a * Float64(t / z_m)))))));
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (t <= -9.6e-108)
		tmp = x_m * (z_m * (y_m / (z_m + (-0.5 * (a * (t / z_m))))));
	else
		tmp = x_m * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[t, -9.6e-108], N[(x$95$m * N[(z$95$m * N[(y$95$m / N[(z$95$m + N[(-0.5 * N[(a * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -9.60000000000000068e-108

   1. Initial program 65.2%

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

      1. associate-/l* 65.6%

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

      2. *-commutative 65.6%

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

      3. associate-*l/ 64.4%

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

      4. *-commutative 64.4%

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

      5. associate-/r/ 64.5%

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

   3. Simplified 64.5%

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

   5. Taylor expanded in z around inf 49.2%

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

      1. add0 49.2%

         \[\leadsto x \cdot \left(\frac{y}{z + -0.5 \cdot \color{blue}{\left(\frac{a \cdot t}{z} + 0\right)}} \cdot z\right) \]

      2. *-commutative 49.2%

         \[\leadsto x \cdot \left(\frac{y}{z + -0.5 \cdot \left(\frac{\color{blue}{t \cdot a}}{z} + 0\right)} \cdot z\right) \]

      3. associate-/l* 49.1%

         \[\leadsto x \cdot \left(\frac{y}{z + -0.5 \cdot \left(\color{blue}{\frac{t}{\frac{z}{a}}} + 0\right)} \cdot z\right) \]

   7. Applied egg-rr 49.1%

      \[\leadsto x \cdot \left(\frac{y}{z + -0.5 \cdot \color{blue}{\left(\frac{t}{\frac{z}{a}} + 0\right)}} \cdot z\right) \]
   8. Step-by-step derivation

      1. associate-/r/ 49.1%

         \[\leadsto x \cdot \left(\frac{y}{z + -0.5 \cdot \left(\color{blue}{\frac{t}{z} \cdot a} + 0\right)} \cdot z\right) \]

      2. *-commutative 49.1%

         \[\leadsto x \cdot \left(\frac{y}{z + -0.5 \cdot \left(\color{blue}{a \cdot \frac{t}{z}} + 0\right)} \cdot z\right) \]

      3. add0 49.1%

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

   9. Simplified 49.1%

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

   if -9.60000000000000068e-108 < t

   1. Initial program 63.3%

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

      1. associate-/l* 64.3%

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

      2. *-commutative 64.3%

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

      3. associate-*l/ 62.2%

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

      4. *-commutative 62.2%

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

      5. associate-/r/ 61.4%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in z around inf 43.9%

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

      1. *-commutative 43.9%

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

   7. Simplified 43.9%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.6 \cdot 10^{-108}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{z + -0.5 \cdot \left(a \cdot \frac{t}{z}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.9 \cdot 10^{-93}:\\ \;\;\;\;\frac{x\_m \cdot \left(y\_m \cdot z\_m\right)}{z\_m + \left(t \cdot \frac{a}{z\_m}\right) \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(z\_m \cdot \frac{y\_m}{z\_m + -0.5 \cdot \left(a \cdot \frac{t}{z\_m}\right)}\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1.9e-93)
      (/ (* x_m (* y_m z_m)) (+ z_m (* (* t (/ a z_m)) -0.5)))
      (* x_m (* z_m (/ y_m (+ z_m (* -0.5 (* a (/ t z_m))))))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.9e-93) {
		tmp = (x_m * (y_m * z_m)) / (z_m + ((t * (a / z_m)) * -0.5));
	} else {
		tmp = x_m * (z_m * (y_m / (z_m + (-0.5 * (a * (t / z_m))))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.9d-93) then
        tmp = (x_m * (y_m * z_m)) / (z_m + ((t * (a / z_m)) * (-0.5d0)))
    else
        tmp = x_m * (z_m * (y_m / (z_m + ((-0.5d0) * (a * (t / z_m))))))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.9e-93) {
		tmp = (x_m * (y_m * z_m)) / (z_m + ((t * (a / z_m)) * -0.5));
	} else {
		tmp = x_m * (z_m * (y_m / (z_m + (-0.5 * (a * (t / z_m))))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.9e-93:
		tmp = (x_m * (y_m * z_m)) / (z_m + ((t * (a / z_m)) * -0.5))
	else:
		tmp = x_m * (z_m * (y_m / (z_m + (-0.5 * (a * (t / z_m))))))
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.9e-93)
		tmp = Float64(Float64(x_m * Float64(y_m * z_m)) / Float64(z_m + Float64(Float64(t * Float64(a / z_m)) * -0.5)));
	else
		tmp = Float64(x_m * Float64(z_m * Float64(y_m / Float64(z_m + Float64(-0.5 * Float64(a * Float64(t / z_m)))))));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.9e-93)
		tmp = (x_m * (y_m * z_m)) / (z_m + ((t * (a / z_m)) * -0.5));
	else
		tmp = x_m * (z_m * (y_m / (z_m + (-0.5 * (a * (t / z_m))))));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.9e-93], N[(N[(x$95$m * N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(z$95$m + N[(N[(t * N[(a / z$95$m), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(z$95$m * N[(y$95$m / N[(z$95$m + N[(-0.5 * N[(a * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.8999999999999999e-93

   1. Initial program 66.6%

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

      1. associate-/l* 67.7%

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

      2. *-commutative 67.7%

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

      3. associate-*l/ 65.2%

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

      4. *-commutative 65.2%

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

      5. associate-/r/ 66.4%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in z around inf 24.8%

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

   6. Taylor expanded in x around 0 23.5%

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

      1. add0 23.5%

         \[\leadsto \frac{x \cdot \left(y \cdot z\right)}{z + \color{blue}{\left(-0.5 \cdot \frac{a \cdot t}{z} + 0\right)}} \]

      2. associate-*r/ 23.5%

         \[\leadsto \frac{x \cdot \left(y \cdot z\right)}{z + \left(\color{blue}{\frac{-0.5 \cdot \left(a \cdot t\right)}{z}} + 0\right)} \]

      3. *-commutative 23.5%

         \[\leadsto \frac{x \cdot \left(y \cdot z\right)}{z + \left(\frac{-0.5 \cdot \color{blue}{\left(t \cdot a\right)}}{z} + 0\right)} \]

   8. Applied egg-rr 23.5%

      \[\leadsto \frac{x \cdot \left(y \cdot z\right)}{z + \color{blue}{\left(\frac{-0.5 \cdot \left(t \cdot a\right)}{z} + 0\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 23.5%

         \[\leadsto \frac{x \cdot \left(y \cdot z\right)}{z + \left(\frac{-0.5 \cdot \color{blue}{\left(a \cdot t\right)}}{z} + 0\right)} \]

      2. associate-*r/ 23.5%

         \[\leadsto \frac{x \cdot \left(y \cdot z\right)}{z + \left(\color{blue}{-0.5 \cdot \frac{a \cdot t}{z}} + 0\right)} \]

      3. add0 23.5%

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

      4. *-commutative 23.5%

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

      5. associate-*r/ 23.8%

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

   10. Simplified 23.8%

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

   if 1.8999999999999999e-93 < z

   1. Initial program 59.5%

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

      1. associate-/l* 59.8%

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

      2. *-commutative 59.8%

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

      3. associate-*l/ 59.0%

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

      4. *-commutative 59.0%

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

      5. associate-/r/ 55.6%

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

   3. Simplified 55.6%

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

   5. Taylor expanded in z around inf 77.8%

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

      1. add0 77.8%

         \[\leadsto x \cdot \left(\frac{y}{z + -0.5 \cdot \color{blue}{\left(\frac{a \cdot t}{z} + 0\right)}} \cdot z\right) \]

      2. *-commutative 77.8%

         \[\leadsto x \cdot \left(\frac{y}{z + -0.5 \cdot \left(\frac{\color{blue}{t \cdot a}}{z} + 0\right)} \cdot z\right) \]

      3. associate-/l* 78.9%

         \[\leadsto x \cdot \left(\frac{y}{z + -0.5 \cdot \left(\color{blue}{\frac{t}{\frac{z}{a}}} + 0\right)} \cdot z\right) \]

   7. Applied egg-rr 78.9%

      \[\leadsto x \cdot \left(\frac{y}{z + -0.5 \cdot \color{blue}{\left(\frac{t}{\frac{z}{a}} + 0\right)}} \cdot z\right) \]
   8. Step-by-step derivation

      1. associate-/r/ 78.9%

         \[\leadsto x \cdot \left(\frac{y}{z + -0.5 \cdot \left(\color{blue}{\frac{t}{z} \cdot a} + 0\right)} \cdot z\right) \]

      2. *-commutative 78.9%

         \[\leadsto x \cdot \left(\frac{y}{z + -0.5 \cdot \left(\color{blue}{a \cdot \frac{t}{z}} + 0\right)} \cdot z\right) \]

      3. add0 78.9%

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

   9. Simplified 78.9%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.9 \cdot 10^{-93}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{z + \left(t \cdot \frac{a}{z}\right) \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{z + -0.5 \cdot \left(a \cdot \frac{t}{z}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.7% accurate, 7.1× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (* y_s (* x_s (* x_m (/ y_m (/ (- (* 0.5 (* t (/ a z_m))) z_m) (- z_m))))))))

C

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

Fortran

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

Java

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

Python

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

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	return Float64(z_s * Float64(y_s * Float64(x_s * Float64(x_m * Float64(y_m / Float64(Float64(Float64(0.5 * Float64(t * Float64(a / z_m))) - z_m) / Float64(-z_m)))))))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = z_s * (y_s * (x_s * (x_m * (y_m / (((0.5 * (t * (a / z_m))) - z_m) / -z_m)))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * N[(x$95$m * N[(y$95$m / N[(N[(N[(0.5 * N[(t * N[(a / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z$95$m), $MachinePrecision] / (-z$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.9%

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

   1. associate-/l* 64.7%

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

   2. *-commutative 64.7%

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

   3. associate-*l/ 62.9%

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

   4. *-commutative 62.9%

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

   5. associate-/r/ 62.3%

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

3. Simplified 62.3%

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

5. Taylor expanded in z around inf 44.9%

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

   1. associate-*l/ 42.9%

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

   2. associate-/l* 49.9%

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

   3. +-commutative 49.9%

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

   4. fma-define 49.9%

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

   5. *-commutative 49.9%

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

   6. associate-/l* 50.5%

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

7. Applied egg-rr 50.5%

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

   1. expm1-log1p-u 50.5%

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

   2. expm1-undefine 50.5%

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

   3. div-inv 50.5%

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

   4. clear-num 50.5%

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

9. Applied egg-rr 50.5%

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

    1. expm1-define 50.5%

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

11. Simplified 50.5%

    \[\leadsto x \cdot \frac{y}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-0.5, t \cdot \frac{a}{z}, z\right)}{z}\right)\right)}} \]
12. Step-by-step derivation

    1. expm1-log1p-u 50.5%

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

    2. frac-2neg 50.5%

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

13. Applied egg-rr 50.5%

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

    1. fma-undefine 50.5%

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

    2. associate-*r/ 49.9%

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

    3. *-commutative 49.9%

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

    4. associate-*r/ 49.9%

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

    5. *-commutative 49.9%

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

    6. distribute-neg-in 49.9%

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

    7. distribute-frac-neg 49.9%

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

    8. distribute-lft-neg-in 49.9%

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

    9. metadata-eval 49.9%

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

    10. *-commutative 49.9%

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

    11. associate-*r/ 49.9%

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

    12. unsub-neg 49.9%

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

    13. *-commutative 49.9%

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

    14. associate-*r/ 50.5%

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

15. Simplified 50.5%

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

16. Final simplification 50.5%

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

Alternative 12: 76.7% accurate, 9.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.85 \cdot 10^{-142}:\\ \;\;\;\;\frac{x\_m \cdot \left(y\_m \cdot z\_m\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (* x_s (if (<= z_m 2.85e-142) (/ (* x_m (* y_m z_m)) z_m) (* x_m y_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.85e-142) {
		tmp = (x_m * (y_m * z_m)) / z_m;
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 2.85d-142) then
        tmp = (x_m * (y_m * z_m)) / z_m
    else
        tmp = x_m * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.85e-142) {
		tmp = (x_m * (y_m * z_m)) / z_m;
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 2.85e-142:
		tmp = (x_m * (y_m * z_m)) / z_m
	else:
		tmp = x_m * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.85e-142)
		tmp = Float64(Float64(x_m * Float64(y_m * z_m)) / z_m);
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2.85e-142)
		tmp = (x_m * (y_m * z_m)) / z_m;
	else
		tmp = x_m * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2.85e-142], N[(N[(x$95$m * N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.84999999999999997e-142

   1. Initial program 65.3%

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

   3. Taylor expanded in z around inf 17.8%

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

   4. Taylor expanded in x around 0 18.8%

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

   if 2.84999999999999997e-142 < z

   1. Initial program 61.8%

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

      1. associate-/l* 62.1%

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

      2. *-commutative 62.1%

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

      3. associate-*l/ 61.4%

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

      4. *-commutative 61.4%

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

      5. associate-/r/ 58.2%

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

   3. Simplified 58.2%

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

   5. Taylor expanded in z around inf 87.3%

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

      1. *-commutative 87.3%

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

   7. Simplified 87.3%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.85 \cdot 10^{-142}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 76.8% accurate, 9.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 7.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{y\_m \cdot \left(x\_m \cdot z\_m\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (* x_s (if (<= z_m 7.6e-96) (/ (* y_m (* x_m z_m)) z_m) (* x_m y_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 7.6e-96) {
		tmp = (y_m * (x_m * z_m)) / z_m;
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 7.6d-96) then
        tmp = (y_m * (x_m * z_m)) / z_m
    else
        tmp = x_m * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 7.6e-96) {
		tmp = (y_m * (x_m * z_m)) / z_m;
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 7.6e-96:
		tmp = (y_m * (x_m * z_m)) / z_m
	else:
		tmp = x_m * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 7.6e-96)
		tmp = Float64(Float64(y_m * Float64(x_m * z_m)) / z_m);
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 7.6e-96)
		tmp = (y_m * (x_m * z_m)) / z_m;
	else
		tmp = x_m * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 7.6e-96], N[(N[(y$95$m * N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 7.6000000000000001e-96

   1. Initial program 66.6%

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

   3. Taylor expanded in z around inf 19.7%

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

   4. Taylor expanded in x around 0 20.6%

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

      1. associate-*r* 19.7%

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

      2. *-commutative 19.7%

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

      3. associate-*r* 21.0%

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

      4. *-commutative 21.0%

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

   6. Simplified 21.0%

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

   if 7.6000000000000001e-96 < z

   1. Initial program 59.5%

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

      1. associate-/l* 59.8%

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

      2. *-commutative 59.8%

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

      3. associate-*l/ 59.0%

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

      4. *-commutative 59.0%

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

      5. associate-/r/ 55.6%

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

   3. Simplified 55.6%

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

   5. Taylor expanded in z around inf 88.5%

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

      1. *-commutative 88.5%

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

   7. Simplified 88.5%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 71.0% accurate, 9.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{-162}:\\ \;\;\;\;y\_m \cdot \frac{x\_m \cdot z\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (* x_s (if (<= t -2.6e-162) (* y_m (/ (* x_m z_m) z_m)) (* x_m y_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (t <= -2.6e-162) {
		tmp = y_m * ((x_m * z_m) / z_m);
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-2.6d-162)) then
        tmp = y_m * ((x_m * z_m) / z_m)
    else
        tmp = x_m * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (t <= -2.6e-162) {
		tmp = y_m * ((x_m * z_m) / z_m);
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if t <= -2.6e-162:
		tmp = y_m * ((x_m * z_m) / z_m)
	else:
		tmp = x_m * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (t <= -2.6e-162)
		tmp = Float64(y_m * Float64(Float64(x_m * z_m) / z_m));
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (t <= -2.6e-162)
		tmp = y_m * ((x_m * z_m) / z_m);
	else
		tmp = x_m * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[t, -2.6e-162], N[(y$95$m * N[(N[(x$95$m * z$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -2.6e-162

   1. Initial program 66.2%

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

   3. Taylor expanded in z around inf 43.5%

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

      1. expm1-log1p-u 35.1%

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

      2. expm1-undefine 32.6%

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

      3. *-commutative 32.6%

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

      4. associate-*l* 31.7%

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

   5. Applied egg-rr 31.7%

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

      1. expm1-define 36.2%

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

      2. *-commutative 36.2%

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

   7. Simplified 36.2%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(z \cdot x\right)\right)\right)}}{z} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 42.3%

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

      2. *-un-lft-identity 42.3%

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

      3. times-frac 41.5%

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

   9. Applied egg-rr 41.5%

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

   if -2.6e-162 < t

   1. Initial program 62.7%

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

      1. associate-/l* 63.6%

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

      2. *-commutative 63.6%

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

      3. associate-*l/ 61.4%

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

      4. *-commutative 61.4%

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

      5. associate-/r/ 60.0%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in z around inf 43.8%

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

      1. *-commutative 43.8%

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

   7. Simplified 43.8%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{-162}:\\ \;\;\;\;y \cdot \frac{x \cdot z}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 73.8% accurate, 37.7× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (* z_s (* y_s (* x_s (* x_m y_m)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = z_s * (y_s * (x_s * (x_m * y_m)));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.9%

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

   1. associate-/l* 64.7%

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

   2. *-commutative 64.7%

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

   3. associate-*l/ 62.9%

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

   4. *-commutative 62.9%

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

   5. associate-/r/ 62.3%

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

3. Simplified 62.3%

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

5. Taylor expanded in z around inf 43.0%

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

   1. *-commutative 43.0%

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

7. Simplified 43.0%

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

8. Final simplification 43.0%

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

Developer target: 89.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024034 
(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)))))