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

Percentage Accurate: 62.4% → 93.1%

Time: 13.8s

Alternatives: 11

Speedup: 37.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{\left(x_m \cdot y_m\right) \cdot z_m}{\sqrt{z_m \cdot z_m - t \cdot a}}\\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{y_m}{\frac{1}{x_m} \cdot \frac{z_m + -0.5 \cdot \left(t \cdot \frac{a}{z_m}\right)}{z_m}}\\ \mathbf{elif}\;t_1 \leq 10^{+265}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot y_m\\ \end{array}\right)\right) \end{array} \end{array} \]

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 (/ (* (* x_m y_m) z_m) (sqrt (- (* z_m z_m) (* t a))))))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= t_1 0.0)
        (/ y_m (* (/ 1.0 x_m) (/ (+ z_m (* -0.5 (* t (/ a z_m)))) z_m)))
        (if (<= t_1 1e+265) 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 = ((x_m * y_m) * z_m) / sqrt(((z_m * z_m) - (t * a)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = y_m / ((1.0 / x_m) * ((z_m + (-0.5 * (t * (a / z_m)))) / z_m));
	} else if (t_1 <= 1e+265) {
		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 = ((x_m * y_m) * z_m) / sqrt(((z_m * z_m) - (t * a)))
    if (t_1 <= 0.0d0) then
        tmp = y_m / ((1.0d0 / x_m) * ((z_m + ((-0.5d0) * (t * (a / z_m)))) / z_m))
    else if (t_1 <= 1d+265) 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 = ((x_m * y_m) * z_m) / Math.sqrt(((z_m * z_m) - (t * a)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = y_m / ((1.0 / x_m) * ((z_m + (-0.5 * (t * (a / z_m)))) / z_m));
	} else if (t_1 <= 1e+265) {
		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 = ((x_m * y_m) * z_m) / math.sqrt(((z_m * z_m) - (t * a)))
	tmp = 0
	if t_1 <= 0.0:
		tmp = y_m / ((1.0 / x_m) * ((z_m + (-0.5 * (t * (a / z_m)))) / z_m))
	elif t_1 <= 1e+265:
		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(Float64(x_m * y_m) * z_m) / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(y_m / Float64(Float64(1.0 / x_m) * Float64(Float64(z_m + Float64(-0.5 * Float64(t * Float64(a / z_m)))) / z_m)));
	elseif (t_1 <= 1e+265)
		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 = ((x_m * y_m) * z_m) / sqrt(((z_m * z_m) - (t * a)));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = y_m / ((1.0 / x_m) * ((z_m + (-0.5 * (t * (a / z_m)))) / z_m));
	elseif (t_1 <= 1e+265)
		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[(N[(x$95$m * y$95$m), $MachinePrecision] * z$95$m), $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[(y$95$m / N[(N[(1.0 / x$95$m), $MachinePrecision] * N[(N[(z$95$m + N[(-0.5 * N[(t * N[(a / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+265], 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 65.1%

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

      1. associate-*l/ 69.1%

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

      2. *-commutative 69.1%

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

      3. associate-/l* 67.5%

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

      4. associate-/r/ 68.2%

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

      5. associate-/r* 66.7%

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

   3. Simplified 66.7%

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

   4. Taylor expanded in z around inf 52.8%

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

      1. associate-*r/ 52.8%

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

   6. Simplified 52.8%

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

      1. *-un-lft-identity 52.8%

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

      2. times-frac 56.8%

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

      3. *-commutative 56.8%

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

      4. *-un-lft-identity 56.8%

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

      5. times-frac 56.8%

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

      6. metadata-eval 56.8%

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

      7. *-commutative 56.8%

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

   8. Applied egg-rr 56.8%

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

      1. associate-/l* 56.7%

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

      2. associate-/r/ 56.7%

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

   10. Applied egg-rr 56.7%

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

   if 0.0 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a)))) < 1.00000000000000007e265

   1. Initial program 99.8%

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

   if 1.00000000000000007e265 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))))

   1. Initial program 27.7%

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

      1. associate-*l/ 33.0%

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

      2. *-commutative 33.0%

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

      3. associate-/l* 33.4%

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

      4. associate-/r/ 33.4%

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

      5. associate-/r* 32.8%

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

   3. Simplified 32.8%

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

   4. Taylor expanded in z around inf 48.7%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \leq 0:\\ \;\;\;\;\frac{y}{\frac{1}{x} \cdot \frac{z + -0.5 \cdot \left(t \cdot \frac{a}{z}\right)}{z}}\\ \mathbf{elif}\;\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \leq 10^{+265}:\\ \;\;\;\;\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 2: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 4.8 \cdot 10^{-164}:\\ \;\;\;\;\frac{x_m \cdot \left(y_m \cdot z_m\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{elif}\;z_m \leq 2.5 \cdot 10^{+126}:\\ \;\;\;\;\frac{y_m}{\frac{\sqrt{z_m \cdot z_m - t \cdot a}}{x_m \cdot z_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m}{\frac{1}{x_m} \cdot \frac{z_m + -0.5 \cdot \left(t \cdot \frac{a}{z_m}\right)}{z_m}}\\ \end{array}\right)\right) \end{array} \]

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 4.8e-164)
      (/ (* x_m (* y_m z_m)) (sqrt (* a (- t))))
      (if (<= z_m 2.5e+126)
        (/ y_m (/ (sqrt (- (* z_m z_m) (* t a))) (* x_m z_m)))
        (/ y_m (* (/ 1.0 x_m) (/ (+ z_m (* -0.5 (* t (/ a 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 <= 4.8e-164) {
		tmp = (x_m * (y_m * z_m)) / sqrt((a * -t));
	} else if (z_m <= 2.5e+126) {
		tmp = y_m / (sqrt(((z_m * z_m) - (t * a))) / (x_m * z_m));
	} else {
		tmp = y_m / ((1.0 / x_m) * ((z_m + (-0.5 * (t * (a / 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 <= 4.8d-164) then
        tmp = (x_m * (y_m * z_m)) / sqrt((a * -t))
    else if (z_m <= 2.5d+126) then
        tmp = y_m / (sqrt(((z_m * z_m) - (t * a))) / (x_m * z_m))
    else
        tmp = y_m / ((1.0d0 / x_m) * ((z_m + ((-0.5d0) * (t * (a / 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 <= 4.8e-164) {
		tmp = (x_m * (y_m * z_m)) / Math.sqrt((a * -t));
	} else if (z_m <= 2.5e+126) {
		tmp = y_m / (Math.sqrt(((z_m * z_m) - (t * a))) / (x_m * z_m));
	} else {
		tmp = y_m / ((1.0 / x_m) * ((z_m + (-0.5 * (t * (a / 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 <= 4.8e-164:
		tmp = (x_m * (y_m * z_m)) / math.sqrt((a * -t))
	elif z_m <= 2.5e+126:
		tmp = y_m / (math.sqrt(((z_m * z_m) - (t * a))) / (x_m * z_m))
	else:
		tmp = y_m / ((1.0 / x_m) * ((z_m + (-0.5 * (t * (a / 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 <= 4.8e-164)
		tmp = Float64(Float64(x_m * Float64(y_m * z_m)) / sqrt(Float64(a * Float64(-t))));
	elseif (z_m <= 2.5e+126)
		tmp = Float64(y_m / Float64(sqrt(Float64(Float64(z_m * z_m) - Float64(t * a))) / Float64(x_m * z_m)));
	else
		tmp = Float64(y_m / Float64(Float64(1.0 / x_m) * Float64(Float64(z_m + Float64(-0.5 * Float64(t * Float64(a / z_m)))) / 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 <= 4.8e-164)
		tmp = (x_m * (y_m * z_m)) / sqrt((a * -t));
	elseif (z_m <= 2.5e+126)
		tmp = y_m / (sqrt(((z_m * z_m) - (t * a))) / (x_m * z_m));
	else
		tmp = y_m / ((1.0 / x_m) * ((z_m + (-0.5 * (t * (a / 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, 4.8e-164], N[(N[(x$95$m * N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 2.5e+126], N[(y$95$m / N[(N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m / N[(N[(1.0 / x$95$m), $MachinePrecision] * N[(N[(z$95$m + N[(-0.5 * N[(t * N[(a / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 4.79999999999999966e-164

   1. Initial program 64.9%

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

      1. associate-*l* 62.2%

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

   3. Simplified 62.2%

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

   4. Taylor expanded in z around 0 39.6%

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

      1. mul-1-neg 39.6%

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

      2. distribute-rgt-neg-out 39.6%

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

   6. Simplified 39.6%

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

   if 4.79999999999999966e-164 < z < 2.49999999999999989e126

   1. Initial program 91.7%

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

      1. associate-*l/ 94.4%

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

      2. *-commutative 94.4%

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

      3. associate-/l* 91.8%

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

      4. associate-/r/ 91.9%

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

      5. associate-/r* 90.6%

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

   3. Simplified 90.6%

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

   if 2.49999999999999989e126 < z

   1. Initial program 32.0%

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

      1. associate-*l/ 35.7%

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

      2. *-commutative 35.7%

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

      3. associate-/l* 34.1%

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

      4. associate-/r/ 37.6%

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

      5. associate-/r* 35.3%

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

   3. Simplified 35.3%

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

   4. Taylor expanded in z around inf 78.7%

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

      1. associate-*r/ 78.7%

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

   6. Simplified 78.7%

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

      1. *-un-lft-identity 78.7%

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

      2. times-frac 92.8%

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

      3. *-commutative 92.8%

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

      4. *-un-lft-identity 92.8%

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

      5. times-frac 92.8%

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

      6. metadata-eval 92.8%

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

      7. *-commutative 92.8%

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

   8. Applied egg-rr 92.8%

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

      1. associate-/l* 99.0%

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

      2. associate-/r/ 99.0%

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

   10. Applied egg-rr 99.0%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.8 \cdot 10^{-164}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+126}:\\ \;\;\;\;\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{1}{x} \cdot \frac{z + -0.5 \cdot \left(t \cdot \frac{a}{z}\right)}{z}}\\ \end{array} \]

Alternative 3: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 3.5 \cdot 10^{+35}:\\ \;\;\;\;\frac{x_m \cdot \left(y_m \cdot z_m\right)}{\sqrt{z_m \cdot z_m - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot y_m\\ \end{array}\right)\right) \end{array} \]

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 3.5e+35)
      (/ (* x_m (* y_m z_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 <= 3.5e+35) {
		tmp = (x_m * (y_m * z_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 <= 3.5d+35) then
        tmp = (x_m * (y_m * z_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 <= 3.5e+35) {
		tmp = (x_m * (y_m * z_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 <= 3.5e+35:
		tmp = (x_m * (y_m * z_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 <= 3.5e+35)
		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 * 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 <= 3.5e+35)
		tmp = (x_m * (y_m * z_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, 3.5e+35], 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 * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.5000000000000001e35

   1. Initial program 69.4%

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

      1. associate-*l* 66.6%

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

   3. Simplified 66.6%

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

   if 3.5000000000000001e35 < z

   1. Initial program 50.5%

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

      1. associate-*l/ 54.1%

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

      2. *-commutative 54.1%

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

      3. associate-/l* 51.7%

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

      4. associate-/r/ 54.1%

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

      5. associate-/r* 51.4%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in z around inf 94.6%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{+35}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 4: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 6.5 \cdot 10^{-113}:\\ \;\;\;\;\frac{x_m \cdot \left(y_m \cdot z_m\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot y_m\\ \end{array}\right)\right) \end{array} \]

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 6.5e-113)
      (/ (* x_m (* y_m z_m)) (sqrt (* a (- t))))
      (* 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 <= 6.5e-113) {
		tmp = (x_m * (y_m * z_m)) / sqrt((a * -t));
	} 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 <= 6.5d-113) then
        tmp = (x_m * (y_m * z_m)) / sqrt((a * -t))
    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 <= 6.5e-113) {
		tmp = (x_m * (y_m * z_m)) / Math.sqrt((a * -t));
	} 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 <= 6.5e-113:
		tmp = (x_m * (y_m * z_m)) / math.sqrt((a * -t))
	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 <= 6.5e-113)
		tmp = Float64(Float64(x_m * Float64(y_m * z_m)) / sqrt(Float64(a * Float64(-t))));
	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 <= 6.5e-113)
		tmp = (x_m * (y_m * z_m)) / sqrt((a * -t));
	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, 6.5e-113], N[(N[(x$95$m * N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 6.49999999999999979e-113

   1. Initial program 66.1%

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

      1. associate-*l* 63.6%

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

   3. Simplified 63.6%

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

   4. Taylor expanded in z around 0 41.1%

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

      1. mul-1-neg 41.1%

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

      2. distribute-rgt-neg-out 41.1%

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

   6. Simplified 41.1%

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

   if 6.49999999999999979e-113 < z

   1. Initial program 59.4%

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

      1. associate-*l/ 62.3%

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

      2. *-commutative 62.3%

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

      3. associate-/l* 60.1%

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

      4. associate-/r/ 62.0%

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

      5. associate-/r* 60.1%

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

   3. Simplified 60.1%

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

   4. Taylor expanded in z around inf 88.1%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.5 \cdot 10^{-113}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 5: 72.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 5.7 \cdot 10^{-129}:\\ \;\;\;\;x_m \cdot y_m\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m}{\frac{z_m + -0.5 \cdot \frac{a}{\frac{z_m}{t}}}{x_m \cdot z_m}}\\ \end{array}\right)\right) \end{array} \]

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 (<= a 5.7e-129)
      (* x_m y_m)
      (/ y_m (/ (+ z_m (* -0.5 (/ a (/ z_m t)))) (* x_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 (a <= 5.7e-129) {
		tmp = x_m * y_m;
	} else {
		tmp = y_m / ((z_m + (-0.5 * (a / (z_m / t)))) / (x_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 (a <= 5.7d-129) then
        tmp = x_m * y_m
    else
        tmp = y_m / ((z_m + ((-0.5d0) * (a / (z_m / t)))) / (x_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 (a <= 5.7e-129) {
		tmp = x_m * y_m;
	} else {
		tmp = y_m / ((z_m + (-0.5 * (a / (z_m / t)))) / (x_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 a <= 5.7e-129:
		tmp = x_m * y_m
	else:
		tmp = y_m / ((z_m + (-0.5 * (a / (z_m / t)))) / (x_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 (a <= 5.7e-129)
		tmp = Float64(x_m * y_m);
	else
		tmp = Float64(y_m / Float64(Float64(z_m + Float64(-0.5 * Float64(a / Float64(z_m / t)))) / Float64(x_m * 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 (a <= 5.7e-129)
		tmp = x_m * y_m;
	else
		tmp = y_m / ((z_m + (-0.5 * (a / (z_m / t)))) / (x_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[a, 5.7e-129], N[(x$95$m * y$95$m), $MachinePrecision], N[(y$95$m / N[(N[(z$95$m + N[(-0.5 * N[(a / N[(z$95$m / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 5.7000000000000001e-129

   1. Initial program 60.0%

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

      1. associate-*l/ 61.9%

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

      2. *-commutative 61.9%

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

      3. associate-/l* 60.1%

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

      4. associate-/r/ 61.2%

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

      5. associate-/r* 58.7%

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

   3. Simplified 58.7%

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

   4. Taylor expanded in z around inf 43.9%

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

   if 5.7000000000000001e-129 < a

   1. Initial program 70.3%

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

      1. associate-*l/ 71.5%

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

      2. *-commutative 71.5%

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

      3. associate-/l* 71.4%

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

      4. associate-/r/ 72.5%

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

      5. associate-/r* 70.0%

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

   3. Simplified 70.0%

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

   4. Taylor expanded in z around inf 45.0%

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

      1. associate-*r/ 45.0%

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

   6. Simplified 45.0%

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

   7. Taylor expanded in a around 0 45.0%

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

      1. associate-/l* 46.7%

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

   9. Simplified 46.7%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.7 \cdot 10^{-129}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{x \cdot z}}\\ \end{array} \]

Alternative 6: 79.4% accurate, 6.6× speedup

Math

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

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 (/ y_m (* (/ 1.0 x_m) (/ (+ z_m (* -0.5 (* t (/ a 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 * (y_m / ((1.0 / x_m) * ((z_m + (-0.5 * (t * (a / 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 * (y_m / ((1.0d0 / x_m) * ((z_m + ((-0.5d0) * (t * (a / 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 * (y_m / ((1.0 / x_m) * ((z_m + (-0.5 * (t * (a / 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 * (y_m / ((1.0 / x_m) * ((z_m + (-0.5 * (t * (a / 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(y_m / Float64(Float64(1.0 / x_m) * Float64(Float64(z_m + Float64(-0.5 * Float64(t * Float64(a / z_m)))) / 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 * (y_m / ((1.0 / x_m) * ((z_m + (-0.5 * (t * (a / 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[(y$95$m / N[(N[(1.0 / x$95$m), $MachinePrecision] * N[(N[(z$95$m + N[(-0.5 * N[(t * N[(a / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.5%

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

   1. associate-*l/ 65.2%

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

   2. *-commutative 65.2%

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

   3. associate-/l* 63.9%

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

   4. associate-/r/ 65.0%

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

   5. associate-/r* 62.6%

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

3. Simplified 62.6%

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

4. Taylor expanded in z around inf 44.8%

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

   1. associate-*r/ 44.8%

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

6. Simplified 44.8%

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

   1. *-un-lft-identity 44.8%

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

   2. times-frac 48.9%

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

   3. *-commutative 48.9%

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

   4. *-un-lft-identity 48.9%

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

   5. times-frac 48.9%

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

   6. metadata-eval 48.9%

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

   7. *-commutative 48.9%

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

8. Applied egg-rr 48.9%

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

   1. associate-/l* 50.2%

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

   2. associate-/r/ 50.2%

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

10. Applied egg-rr 50.2%

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

11. Final simplification 50.2%

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

Alternative 7: 70.7% accurate, 8.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 2.6 \cdot 10^{-141}:\\ \;\;\;\;x_m \cdot y_m\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot \left(\frac{1}{z_m} \cdot \frac{z_m}{\frac{1}{x_m}}\right)\\ \end{array}\right)\right) \end{array} \]

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 (<= a 2.6e-141)
      (* x_m y_m)
      (* y_m (* (/ 1.0 z_m) (/ z_m (/ 1.0 x_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 (a <= 2.6e-141) {
		tmp = x_m * y_m;
	} else {
		tmp = y_m * ((1.0 / z_m) * (z_m / (1.0 / x_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 (a <= 2.6d-141) then
        tmp = x_m * y_m
    else
        tmp = y_m * ((1.0d0 / z_m) * (z_m / (1.0d0 / x_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 (a <= 2.6e-141) {
		tmp = x_m * y_m;
	} else {
		tmp = y_m * ((1.0 / z_m) * (z_m / (1.0 / x_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 a <= 2.6e-141:
		tmp = x_m * y_m
	else:
		tmp = y_m * ((1.0 / z_m) * (z_m / (1.0 / x_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 (a <= 2.6e-141)
		tmp = Float64(x_m * y_m);
	else
		tmp = Float64(y_m * Float64(Float64(1.0 / z_m) * Float64(z_m / Float64(1.0 / x_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 (a <= 2.6e-141)
		tmp = x_m * y_m;
	else
		tmp = y_m * ((1.0 / z_m) * (z_m / (1.0 / x_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[a, 2.6e-141], N[(x$95$m * y$95$m), $MachinePrecision], N[(y$95$m * N[(N[(1.0 / z$95$m), $MachinePrecision] * N[(z$95$m / N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 2.60000000000000011e-141

   1. Initial program 60.4%

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

      1. associate-*l/ 62.3%

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

      2. *-commutative 62.3%

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

      3. associate-/l* 60.4%

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

      4. associate-/r/ 61.5%

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

      5. associate-/r* 59.1%

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

   3. Simplified 59.1%

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

   4. Taylor expanded in z around inf 43.6%

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

   if 2.60000000000000011e-141 < a

   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/ 70.7%

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

      2. *-commutative 70.7%

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

      3. associate-/l* 70.7%

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

      4. associate-/r/ 71.7%

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

      5. associate-/r* 69.2%

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

   3. Simplified 69.2%

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

   4. Taylor expanded in z around inf 43.0%

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

      1. expm1-log1p-u 37.5%

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

      2. expm1-udef 33.5%

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

      3. associate-/r/ 32.3%

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

      4. *-commutative 32.3%

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

   6. Applied egg-rr 32.3%

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

      1. expm1-def 33.9%

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

      2. expm1-log1p 39.4%

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

      3. associate-*l/ 41.6%

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

      4. associate-*r/ 43.0%

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

      5. associate-/l* 38.9%

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

   8. Simplified 38.9%

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

      1. *-un-lft-identity 38.9%

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

      2. div-inv 38.8%

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

      3. times-frac 43.0%

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

   10. Applied egg-rr 43.0%

       \[\leadsto y \cdot \color{blue}{\left(\frac{1}{z} \cdot \frac{z}{\frac{1}{x}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.6 \cdot 10^{-141}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{1}{z} \cdot \frac{z}{\frac{1}{x}}\right)\\ \end{array} \]

Alternative 8: 70.8% accurate, 10.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 9.5 \cdot 10^{-126}:\\ \;\;\;\;x_m \cdot y_m\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m}{z_m \cdot \frac{1}{x_m \cdot z_m}}\\ \end{array}\right)\right) \end{array} \]

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 (<= a 9.5e-126) (* x_m y_m) (/ y_m (* z_m (/ 1.0 (* x_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 (a <= 9.5e-126) {
		tmp = x_m * y_m;
	} else {
		tmp = y_m / (z_m * (1.0 / (x_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 (a <= 9.5d-126) then
        tmp = x_m * y_m
    else
        tmp = y_m / (z_m * (1.0d0 / (x_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 (a <= 9.5e-126) {
		tmp = x_m * y_m;
	} else {
		tmp = y_m / (z_m * (1.0 / (x_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 a <= 9.5e-126:
		tmp = x_m * y_m
	else:
		tmp = y_m / (z_m * (1.0 / (x_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 (a <= 9.5e-126)
		tmp = Float64(x_m * y_m);
	else
		tmp = Float64(y_m / Float64(z_m * Float64(1.0 / Float64(x_m * 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 (a <= 9.5e-126)
		tmp = x_m * y_m;
	else
		tmp = y_m / (z_m * (1.0 / (x_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[a, 9.5e-126], N[(x$95$m * y$95$m), $MachinePrecision], N[(y$95$m / N[(z$95$m * N[(1.0 / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 9.5000000000000003e-126

   1. Initial program 60.2%

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

      1. associate-*l/ 62.1%

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

      2. *-commutative 62.1%

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

      3. associate-/l* 60.3%

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

      4. associate-/r/ 61.4%

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

      5. associate-/r* 59.0%

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

   3. Simplified 59.0%

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

   4. Taylor expanded in z around inf 44.1%

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

   if 9.5000000000000003e-126 < a

   1. Initial program 70.0%

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

      1. associate-*l/ 71.2%

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

      2. *-commutative 71.2%

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

      3. associate-/l* 71.1%

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

      4. associate-/r/ 72.2%

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

      5. associate-/r* 69.6%

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

   3. Simplified 69.6%

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

   4. Taylor expanded in z around inf 42.0%

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

      1. div-inv 42.0%

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

      2. *-commutative 42.0%

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

   6. Applied egg-rr 42.0%

      \[\leadsto \frac{y}{\color{blue}{z \cdot \frac{1}{z \cdot x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 9.5 \cdot 10^{-126}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{1}{x \cdot z}}\\ \end{array} \]

Alternative 9: 69.2% accurate, 12.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 5.5 \cdot 10^{-90}:\\ \;\;\;\;x_m \cdot y_m\\ \mathbf{else}:\\ \;\;\;\;\left(x_m \cdot z_m\right) \cdot \frac{y_m}{z_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= a 5.5e-90) (* x_m y_m) (* (* x_m z_m) (/ y_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 (a <= 5.5e-90) {
		tmp = x_m * y_m;
	} else {
		tmp = (x_m * z_m) * (y_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 (a <= 5.5d-90) then
        tmp = x_m * y_m
    else
        tmp = (x_m * z_m) * (y_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 (a <= 5.5e-90) {
		tmp = x_m * y_m;
	} else {
		tmp = (x_m * z_m) * (y_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 a <= 5.5e-90:
		tmp = x_m * y_m
	else:
		tmp = (x_m * z_m) * (y_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 (a <= 5.5e-90)
		tmp = Float64(x_m * y_m);
	else
		tmp = Float64(Float64(x_m * z_m) * Float64(y_m / 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 (a <= 5.5e-90)
		tmp = x_m * y_m;
	else
		tmp = (x_m * z_m) * (y_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[a, 5.5e-90], N[(x$95$m * y$95$m), $MachinePrecision], N[(N[(x$95$m * z$95$m), $MachinePrecision] * N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 5.5000000000000003e-90

   1. Initial program 61.1%

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

      1. associate-*l/ 63.0%

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

      2. *-commutative 63.0%

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

      3. associate-/l* 61.2%

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

      4. associate-/r/ 62.2%

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

      5. associate-/r* 59.9%

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

   3. Simplified 59.9%

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

   4. Taylor expanded in z around inf 43.6%

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

   if 5.5000000000000003e-90 < a

   1. Initial program 69.1%

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

      1. associate-*l/ 70.4%

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

      2. *-commutative 70.4%

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

      3. associate-/l* 70.3%

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

      4. associate-/r/ 71.6%

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

      5. associate-/r* 68.6%

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

   3. Simplified 68.6%

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

   4. Taylor expanded in z around inf 42.9%

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

      1. associate-/r/ 38.8%

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

      2. *-commutative 38.8%

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

   6. Applied egg-rr 38.8%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.5 \cdot 10^{-90}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot \frac{y}{z}\\ \end{array} \]

Alternative 10: 70.7% accurate, 12.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 7.2 \cdot 10^{-128}:\\ \;\;\;\;x_m \cdot y_m\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m}{\frac{z_m}{x_m \cdot z_m}}\\ \end{array}\right)\right) \end{array} \]

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 (<= a 7.2e-128) (* x_m y_m) (/ y_m (/ z_m (* x_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 (a <= 7.2e-128) {
		tmp = x_m * y_m;
	} else {
		tmp = y_m / (z_m / (x_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 (a <= 7.2d-128) then
        tmp = x_m * y_m
    else
        tmp = y_m / (z_m / (x_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 (a <= 7.2e-128) {
		tmp = x_m * y_m;
	} else {
		tmp = y_m / (z_m / (x_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 a <= 7.2e-128:
		tmp = x_m * y_m
	else:
		tmp = y_m / (z_m / (x_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 (a <= 7.2e-128)
		tmp = Float64(x_m * y_m);
	else
		tmp = Float64(y_m / Float64(z_m / Float64(x_m * 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 (a <= 7.2e-128)
		tmp = x_m * y_m;
	else
		tmp = y_m / (z_m / (x_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[a, 7.2e-128], N[(x$95$m * y$95$m), $MachinePrecision], N[(y$95$m / N[(z$95$m / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 7.20000000000000049e-128

   1. Initial program 60.2%

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

      1. associate-*l/ 62.1%

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

      2. *-commutative 62.1%

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

      3. associate-/l* 60.3%

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

      4. associate-/r/ 61.4%

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

      5. associate-/r* 59.0%

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

   3. Simplified 59.0%

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

   4. Taylor expanded in z around inf 44.1%

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

   if 7.20000000000000049e-128 < a

   1. Initial program 70.0%

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

      1. associate-*l/ 71.2%

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

      2. *-commutative 71.2%

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

      3. associate-/l* 71.1%

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

      4. associate-/r/ 72.2%

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

      5. associate-/r* 69.6%

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

   3. Simplified 69.6%

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

   4. Taylor expanded in z around inf 42.0%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7.2 \cdot 10^{-128}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x \cdot z}}\\ \end{array} \]

Alternative 11: 73.3% 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 63.5%

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

   1. associate-*l/ 65.2%

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

   2. *-commutative 65.2%

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

   3. associate-/l* 63.9%

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

   4. associate-/r/ 65.0%

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

   5. associate-/r* 62.6%

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

3. Simplified 62.6%

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

4. Taylor expanded in z around inf 43.5%

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

5. Final simplification 43.5%

   \[\leadsto x \cdot y \]

Developer target: 89.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023321 
(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)))))