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

Percentage Accurate: 61.2% → 91.3%

Time: 8.2s

Alternatives: 10

Speedup: 4.1×

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: 61.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.5 \cdot 10^{+24}:\\ \;\;\;\;\frac{\left(z\_m \cdot y\_m\right) \cdot x\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m}{\mathsf{fma}\left(\frac{t}{z\_m}, -0.5 \cdot a, z\_m\right)} \cdot \left(y\_m \cdot x\_m\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 1.5e+24)
      (/ (* (* z_m y_m) x_m) (sqrt (- (* z_m z_m) (* t a))))
      (* (/ z_m (fma (/ t z_m) (* -0.5 a) z_m)) (* y_m x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.5e+24) {
		tmp = ((z_m * y_m) * x_m) / sqrt(((z_m * z_m) - (t * a)));
	} else {
		tmp = (z_m / fma((t / z_m), (-0.5 * a), z_m)) * (y_m * x_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.5e+24)
		tmp = Float64(Float64(Float64(z_m * y_m) * x_m) / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a))));
	else
		tmp = Float64(Float64(z_m / fma(Float64(t / z_m), Float64(-0.5 * a), z_m)) * Float64(y_m * x_m));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 1.5e+24], N[(N[(N[(z$95$m * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(z$95$m / N[(N[(t / z$95$m), $MachinePrecision] * N[(-0.5 * a), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision] * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.49999999999999997e24

   1. Initial program 62.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 61.6

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

   4. Applied rewrites 61.6%

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

   if 1.49999999999999997e24 < z

   1. Initial program 41.6%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 69.2

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

   5. Applied rewrites 69.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 89.6

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

      7. lift-*.f64 N/A

         \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{t}{z}, \frac{-1}{2} \cdot a, z\right)\right)\right)} \cdot \color{blue}{\left(x \cdot y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{t}{z}, \frac{-1}{2} \cdot a, z\right)\right)\right)} \cdot \color{blue}{\left(y \cdot x\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{t}{z}, \frac{-1}{2} \cdot a, z\right)\right)\right)} \cdot \color{blue}{\left(y \cdot x\right)} \]

   7. Applied rewrites 89.6%

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

Alternative 2: 76.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 6 \cdot 10^{-134}:\\ \;\;\;\;{z\_m}^{-1} \cdot \left(\left(z\_m \cdot y\_m\right) \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(y\_m \cdot x\_m\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 6e-134)
      (* (pow z_m -1.0) (* (* z_m y_m) x_m))
      (* 1.0 (* y_m x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 6e-134) {
		tmp = pow(z_m, -1.0) * ((z_m * y_m) * x_m);
	} else {
		tmp = 1.0 * (y_m * x_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 6d-134) then
        tmp = (z_m ** (-1.0d0)) * ((z_m * y_m) * x_m)
    else
        tmp = 1.0d0 * (y_m * x_m)
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 6e-134) {
		tmp = Math.pow(z_m, -1.0) * ((z_m * y_m) * x_m);
	} else {
		tmp = 1.0 * (y_m * x_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 6e-134:
		tmp = math.pow(z_m, -1.0) * ((z_m * y_m) * x_m)
	else:
		tmp = 1.0 * (y_m * x_m)
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 6e-134)
		tmp = Float64((z_m ^ -1.0) * Float64(Float64(z_m * y_m) * x_m));
	else
		tmp = Float64(1.0 * Float64(y_m * x_m));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 6e-134)
		tmp = (z_m ^ -1.0) * ((z_m * y_m) * x_m);
	else
		tmp = 1.0 * (y_m * x_m);
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 6e-134], N[(N[Power[z$95$m, -1.0], $MachinePrecision] * N[(N[(z$95$m * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 6e-134

   1. Initial program 59.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 58.2

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

   5. Applied rewrites 58.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 18.8%

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

   if 6e-134 < z

   1. Initial program 51.3%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 66.1

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

   5. Applied rewrites 66.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 81.5

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

      7. lift-*.f64 N/A

         \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{t}{z}, \frac{-1}{2} \cdot a, z\right)\right)\right)} \cdot \color{blue}{\left(x \cdot y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{t}{z}, \frac{-1}{2} \cdot a, z\right)\right)\right)} \cdot \color{blue}{\left(y \cdot x\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{t}{z}, \frac{-1}{2} \cdot a, z\right)\right)\right)} \cdot \color{blue}{\left(y \cdot x\right)} \]

   7. Applied rewrites 81.5%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 80.2%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6 \cdot 10^{-134}:\\ \;\;\;\;{z}^{-1} \cdot \left(\left(z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.4 \cdot 10^{+22}:\\ \;\;\;\;\left(\frac{x\_m}{\sqrt{\mathsf{fma}\left(-t, a, z\_m \cdot z\_m\right)}} \cdot y\_m\right) \cdot z\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m}{\mathsf{fma}\left(\frac{t}{z\_m}, -0.5 \cdot a, z\_m\right)} \cdot \left(y\_m \cdot x\_m\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 2.4e+22)
      (* (* (/ x_m (sqrt (fma (- t) a (* z_m z_m)))) y_m) z_m)
      (* (/ z_m (fma (/ t z_m) (* -0.5 a) z_m)) (* y_m x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.4e+22) {
		tmp = ((x_m / sqrt(fma(-t, a, (z_m * z_m)))) * y_m) * z_m;
	} else {
		tmp = (z_m / fma((t / z_m), (-0.5 * a), z_m)) * (y_m * x_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.4e+22)
		tmp = Float64(Float64(Float64(x_m / sqrt(fma(Float64(-t), a, Float64(z_m * z_m)))) * y_m) * z_m);
	else
		tmp = Float64(Float64(z_m / fma(Float64(t / z_m), Float64(-0.5 * a), z_m)) * Float64(y_m * x_m));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 2.4e+22], N[(N[(N[(x$95$m / N[Sqrt[N[((-t) * a + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] * z$95$m), $MachinePrecision], N[(N[(z$95$m / N[(N[(t / z$95$m), $MachinePrecision] * N[(-0.5 * a), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision] * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.4e22

   1. Initial program 62.2%

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

   3. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 35.1

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

   5. Applied rewrites 35.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 33.7

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

   7. Applied rewrites 33.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 33.0

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

   9. Applied rewrites 33.0%

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

   10. Taylor expanded in z around 0

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

       1. fp-cancel-sub-sign-inv N/A

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

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

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

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

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

       6. lower-fma.f64 N/A

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

       7. lower-neg.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 62.1

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

   12. Applied rewrites 62.1%

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

   if 2.4e22 < z

   1. Initial program 41.6%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 69.2

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

   5. Applied rewrites 69.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 89.6

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

      7. lift-*.f64 N/A

         \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{t}{z}, \frac{-1}{2} \cdot a, z\right)\right)\right)} \cdot \color{blue}{\left(x \cdot y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{t}{z}, \frac{-1}{2} \cdot a, z\right)\right)\right)} \cdot \color{blue}{\left(y \cdot x\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{t}{z}, \frac{-1}{2} \cdot a, z\right)\right)\right)} \cdot \color{blue}{\left(y \cdot x\right)} \]

   7. Applied rewrites 89.6%

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

Alternative 4: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.38 \cdot 10^{-121}:\\ \;\;\;\;\frac{\left(z\_m \cdot y\_m\right) \cdot x\_m}{\sqrt{\left(-a\right) \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m}{\mathsf{fma}\left(\frac{t}{z\_m}, -0.5 \cdot a, z\_m\right)} \cdot \left(y\_m \cdot x\_m\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 1.38e-121)
      (/ (* (* z_m y_m) x_m) (sqrt (* (- a) t)))
      (* (/ z_m (fma (/ t z_m) (* -0.5 a) z_m)) (* y_m x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.38e-121) {
		tmp = ((z_m * y_m) * x_m) / sqrt((-a * t));
	} else {
		tmp = (z_m / fma((t / z_m), (-0.5 * a), z_m)) * (y_m * x_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.38e-121)
		tmp = Float64(Float64(Float64(z_m * y_m) * x_m) / sqrt(Float64(Float64(-a) * t)));
	else
		tmp = Float64(Float64(z_m / fma(Float64(t / z_m), Float64(-0.5 * a), z_m)) * Float64(y_m * x_m));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 1.38e-121], N[(N[(N[(z$95$m * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] / N[Sqrt[N[((-a) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(z$95$m / N[(N[(t / z$95$m), $MachinePrecision] * N[(-0.5 * a), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision] * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.38e-121

   1. Initial program 59.7%

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

   3. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 32.8

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

   5. Applied rewrites 32.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 31.3

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

   7. Applied rewrites 31.3%

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

   8. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 40.2

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

   10. Applied rewrites 40.2%

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

   if 1.38e-121 < z

   1. Initial program 50.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 66.7

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

   5. Applied rewrites 66.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 82.4

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

      7. lift-*.f64 N/A

         \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{t}{z}, \frac{-1}{2} \cdot a, z\right)\right)\right)} \cdot \color{blue}{\left(x \cdot y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{t}{z}, \frac{-1}{2} \cdot a, z\right)\right)\right)} \cdot \color{blue}{\left(y \cdot x\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{t}{z}, \frac{-1}{2} \cdot a, z\right)\right)\right)} \cdot \color{blue}{\left(y \cdot x\right)} \]

   7. Applied rewrites 82.4%

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

Alternative 5: 83.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.38 \cdot 10^{-121}:\\ \;\;\;\;\frac{\left(z\_m \cdot y\_m\right) \cdot x\_m}{\sqrt{\left(-a\right) \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(y\_m \cdot x\_m\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 1.38e-121)
      (/ (* (* z_m y_m) x_m) (sqrt (* (- a) t)))
      (* 1.0 (* y_m x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.38e-121) {
		tmp = ((z_m * y_m) * x_m) / sqrt((-a * t));
	} else {
		tmp = 1.0 * (y_m * x_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.38d-121) then
        tmp = ((z_m * y_m) * x_m) / sqrt((-a * t))
    else
        tmp = 1.0d0 * (y_m * x_m)
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.38e-121) {
		tmp = ((z_m * y_m) * x_m) / Math.sqrt((-a * t));
	} else {
		tmp = 1.0 * (y_m * x_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.38e-121:
		tmp = ((z_m * y_m) * x_m) / math.sqrt((-a * t))
	else:
		tmp = 1.0 * (y_m * x_m)
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.38e-121)
		tmp = Float64(Float64(Float64(z_m * y_m) * x_m) / sqrt(Float64(Float64(-a) * t)));
	else
		tmp = Float64(1.0 * Float64(y_m * x_m));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.38e-121)
		tmp = ((z_m * y_m) * x_m) / sqrt((-a * t));
	else
		tmp = 1.0 * (y_m * x_m);
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 1.38e-121], N[(N[(N[(z$95$m * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] / N[Sqrt[N[((-a) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.38e-121

   1. Initial program 59.7%

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

   3. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 32.8

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

   5. Applied rewrites 32.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 31.3

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

   7. Applied rewrites 31.3%

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

   8. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 40.2

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

   10. Applied rewrites 40.2%

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

   if 1.38e-121 < z

   1. Initial program 50.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 66.7

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

   5. Applied rewrites 66.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 82.4

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

      7. lift-*.f64 N/A

         \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{t}{z}, \frac{-1}{2} \cdot a, z\right)\right)\right)} \cdot \color{blue}{\left(x \cdot y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{t}{z}, \frac{-1}{2} \cdot a, z\right)\right)\right)} \cdot \color{blue}{\left(y \cdot x\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{t}{z}, \frac{-1}{2} \cdot a, z\right)\right)\right)} \cdot \color{blue}{\left(y \cdot x\right)} \]

   7. Applied rewrites 82.4%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 81.0%

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

Alternative 6: 82.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.38 \cdot 10^{-121}:\\ \;\;\;\;\left(\frac{x\_m}{\sqrt{\left(-t\right) \cdot a}} \cdot y\_m\right) \cdot z\_m\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(y\_m \cdot x\_m\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 1.38e-121)
      (* (* (/ x_m (sqrt (* (- t) a))) y_m) z_m)
      (* 1.0 (* y_m x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.38e-121) {
		tmp = ((x_m / sqrt((-t * a))) * y_m) * z_m;
	} else {
		tmp = 1.0 * (y_m * x_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.38d-121) then
        tmp = ((x_m / sqrt((-t * a))) * y_m) * z_m
    else
        tmp = 1.0d0 * (y_m * x_m)
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.38e-121) {
		tmp = ((x_m / Math.sqrt((-t * a))) * y_m) * z_m;
	} else {
		tmp = 1.0 * (y_m * x_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.38e-121:
		tmp = ((x_m / math.sqrt((-t * a))) * y_m) * z_m
	else:
		tmp = 1.0 * (y_m * x_m)
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.38e-121)
		tmp = Float64(Float64(Float64(x_m / sqrt(Float64(Float64(-t) * a))) * y_m) * z_m);
	else
		tmp = Float64(1.0 * Float64(y_m * x_m));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.38e-121)
		tmp = ((x_m / sqrt((-t * a))) * y_m) * z_m;
	else
		tmp = 1.0 * (y_m * x_m);
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 1.38e-121], N[(N[(N[(x$95$m / N[Sqrt[N[((-t) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] * z$95$m), $MachinePrecision], N[(1.0 * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.38e-121

   1. Initial program 59.7%

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

   3. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 32.8

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

   5. Applied rewrites 32.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 31.3

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

   7. Applied rewrites 31.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 30.5

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

   9. Applied rewrites 30.5%

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

   10. Taylor expanded in z around 0

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

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

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

       4. lower-*.f64 N/A

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

       5. lower-neg.f64 41.7

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

   12. Applied rewrites 41.7%

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

   if 1.38e-121 < z

   1. Initial program 50.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 66.7

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

   5. Applied rewrites 66.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 82.4

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

      7. lift-*.f64 N/A

         \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{t}{z}, \frac{-1}{2} \cdot a, z\right)\right)\right)} \cdot \color{blue}{\left(x \cdot y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{t}{z}, \frac{-1}{2} \cdot a, z\right)\right)\right)} \cdot \color{blue}{\left(y \cdot x\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{t}{z}, \frac{-1}{2} \cdot a, z\right)\right)\right)} \cdot \color{blue}{\left(y \cdot x\right)} \]

   7. Applied rewrites 82.4%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 81.0%

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

Alternative 7: 74.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;t \cdot a \leq -3.6 \cdot 10^{-130}:\\ \;\;\;\;\left(z\_m \cdot x\_m\right) \cdot \frac{y\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(y\_m \cdot x\_m\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (* t a) -3.6e-130)
      (* (* z_m x_m) (/ y_m z_m))
      (* 1.0 (* y_m x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if ((t * a) <= -3.6e-130) {
		tmp = (z_m * x_m) * (y_m / z_m);
	} else {
		tmp = 1.0 * (y_m * x_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((t * a) <= (-3.6d-130)) then
        tmp = (z_m * x_m) * (y_m / z_m)
    else
        tmp = 1.0d0 * (y_m * x_m)
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if ((t * a) <= -3.6e-130) {
		tmp = (z_m * x_m) * (y_m / z_m);
	} else {
		tmp = 1.0 * (y_m * x_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if (t * a) <= -3.6e-130:
		tmp = (z_m * x_m) * (y_m / z_m)
	else:
		tmp = 1.0 * (y_m * x_m)
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (Float64(t * a) <= -3.6e-130)
		tmp = Float64(Float64(z_m * x_m) * Float64(y_m / z_m));
	else
		tmp = Float64(1.0 * Float64(y_m * x_m));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if ((t * a) <= -3.6e-130)
		tmp = (z_m * x_m) * (y_m / z_m);
	else
		tmp = 1.0 * (y_m * x_m);
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(t * a), $MachinePrecision], -3.6e-130], N[(N[(z$95$m * x$95$m), $MachinePrecision] * N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t a) < -3.6000000000000001e-130

   1. Initial program 60.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 37.1

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

   5. Applied rewrites 37.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 32.5

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

   7. Applied rewrites 32.5%

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

   8. Taylor expanded in z around inf

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

      1. lower-/.f64 23.5

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

   10. Applied rewrites 23.5%

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

   if -3.6000000000000001e-130 < (*.f64 t a)

   1. Initial program 52.4%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 37.9

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

   5. Applied rewrites 37.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 48.0

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

      7. lift-*.f64 N/A

         \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{t}{z}, \frac{-1}{2} \cdot a, z\right)\right)\right)} \cdot \color{blue}{\left(x \cdot y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{t}{z}, \frac{-1}{2} \cdot a, z\right)\right)\right)} \cdot \color{blue}{\left(y \cdot x\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{t}{z}, \frac{-1}{2} \cdot a, z\right)\right)\right)} \cdot \color{blue}{\left(y \cdot x\right)} \]

   7. Applied rewrites 48.0%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 47.2%

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

Alternative 8: 75.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.8 \cdot 10^{-209}:\\ \;\;\;\;\frac{\left(x\_m \cdot z\_m\right) \cdot y\_m}{-z\_m}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(y\_m \cdot x\_m\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 1.8e-209)
      (/ (* (* x_m z_m) y_m) (- z_m))
      (* 1.0 (* y_m x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.8e-209) {
		tmp = ((x_m * z_m) * y_m) / -z_m;
	} else {
		tmp = 1.0 * (y_m * x_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.8d-209) then
        tmp = ((x_m * z_m) * y_m) / -z_m
    else
        tmp = 1.0d0 * (y_m * x_m)
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.8e-209) {
		tmp = ((x_m * z_m) * y_m) / -z_m;
	} else {
		tmp = 1.0 * (y_m * x_m);
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.8e-209:
		tmp = ((x_m * z_m) * y_m) / -z_m
	else:
		tmp = 1.0 * (y_m * x_m)
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.8e-209)
		tmp = Float64(Float64(Float64(x_m * z_m) * y_m) / Float64(-z_m));
	else
		tmp = Float64(1.0 * Float64(y_m * x_m));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.8e-209)
		tmp = ((x_m * z_m) * y_m) / -z_m;
	else
		tmp = 1.0 * (y_m * x_m);
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 1.8e-209], N[(N[(N[(x$95$m * z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / (-z$95$m)), $MachinePrecision], N[(1.0 * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.80000000000000008e-209

   1. Initial program 59.1%

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

   3. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 60.7

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

   5. Applied rewrites 60.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 52.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 52.8

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

   7. Applied rewrites 52.8%

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

   if 1.80000000000000008e-209 < z

   1. Initial program 52.7%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 66.0

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

   5. Applied rewrites 66.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 80.7

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

      7. lift-*.f64 N/A

         \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{t}{z}, \frac{-1}{2} \cdot a, z\right)\right)\right)} \cdot \color{blue}{\left(x \cdot y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{t}{z}, \frac{-1}{2} \cdot a, z\right)\right)\right)} \cdot \color{blue}{\left(y \cdot x\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{t}{z}, \frac{-1}{2} \cdot a, z\right)\right)\right)} \cdot \color{blue}{\left(y \cdot x\right)} \]

   7. Applied rewrites 80.7%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 76.8%

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

Alternative 9: 73.2% accurate, 4.1× speedup

Math

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

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (* x_s (* y_s (* z_s (* 1.0 (* y_m x_m))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 37.5

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

5. Applied rewrites 37.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 43.5

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

   7. lift-*.f64 N/A

      \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{t}{z}, \frac{-1}{2} \cdot a, z\right)\right)\right)} \cdot \color{blue}{\left(x \cdot y\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{t}{z}, \frac{-1}{2} \cdot a, z\right)\right)\right)} \cdot \color{blue}{\left(y \cdot x\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{z}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{t}{z}, \frac{-1}{2} \cdot a, z\right)\right)\right)} \cdot \color{blue}{\left(y \cdot x\right)} \]

7. Applied rewrites 43.5%

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

8. Taylor expanded in z around inf

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

10. Applied rewrites 37.0%

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

Alternative 10: 13.5% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in z around -inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 42.7

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

5. Applied rewrites 42.7%

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

Developer Target 1: 87.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024320 
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -31921305903852764000000000000000000000000000000) (- (* y x)) (if (< z 5976268120920894000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y)) (* y x))))

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