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

Percentage Accurate: 61.5% → 92.8%

Time: 10.2s

Alternatives: 9

Speedup: 7.5×

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: 61.5% 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: 92.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5.2 \cdot 10^{+41}:\\ \;\;\;\;x\_m \cdot \frac{z\_m \cdot y\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m}{\mathsf{fma}\left(\frac{a}{z\_m}, -0.5 \cdot t, z\_m\right)} \cdot \left(y\_m \cdot x\_m\right)\\ \end{array}\right)\right) \end{array} \]

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 5.2e+41)
      (* x_m (/ (* z_m y_m) (sqrt (fma (- a) t (* z_m z_m)))))
      (* (/ z_m (fma (/ a z_m) (* -0.5 t) 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 <= 5.2e+41) {
		tmp = x_m * ((z_m * y_m) / sqrt(fma(-a, t, (z_m * z_m))));
	} else {
		tmp = (z_m / fma((a / z_m), (-0.5 * t), 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 <= 5.2e+41)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / sqrt(fma(Float64(-a), t, Float64(z_m * z_m)))));
	else
		tmp = Float64(Float64(z_m / fma(Float64(a / z_m), Float64(-0.5 * t), 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, 5.2e+41], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z$95$m / N[(N[(a / z$95$m), $MachinePrecision] * N[(-0.5 * t), $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 < 5.2000000000000001e41

   1. Initial program 70.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

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

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. frac-2neg N/A

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

      11. remove-double-neg N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 68.6

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

      17. lift--.f64 N/A

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

      18. sub-neg N/A

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

      19. +-commutative N/A

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

   4. Applied rewrites 68.0%

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

   if 5.2000000000000001e41 < z

   1. Initial program 39.5%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 76.9

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

   5. Applied rewrites 76.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 97.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 97.1%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.2 \cdot 10^{+41}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(\frac{a}{z}, -0.5 \cdot t, z\right)} \cdot \left(y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.55 \cdot 10^{-135}:\\ \;\;\;\;x\_m \cdot \frac{z\_m \cdot y\_m}{\sqrt{\left(-a\right) \cdot t}}\\ \mathbf{elif}\;z\_m \leq 3 \cdot 10^{+52}:\\ \;\;\;\;\left(\frac{z\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}} \cdot x\_m\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m}{\mathsf{fma}\left(\frac{a}{z\_m}, -0.5 \cdot t, z\_m\right)} \cdot \left(y\_m \cdot x\_m\right)\\ \end{array}\right)\right) \end{array} \]

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 1.55e-135)
      (* x_m (/ (* z_m y_m) (sqrt (* (- a) t))))
      (if (<= z_m 3e+52)
        (* (* (/ z_m (sqrt (fma (- a) t (* z_m z_m)))) x_m) y_m)
        (* (/ z_m (fma (/ a z_m) (* -0.5 t) 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 <= 1.55e-135) {
		tmp = x_m * ((z_m * y_m) / sqrt((-a * t)));
	} else if (z_m <= 3e+52) {
		tmp = ((z_m / sqrt(fma(-a, t, (z_m * z_m)))) * x_m) * y_m;
	} else {
		tmp = (z_m / fma((a / z_m), (-0.5 * t), 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 <= 1.55e-135)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / sqrt(Float64(Float64(-a) * t))));
	elseif (z_m <= 3e+52)
		tmp = Float64(Float64(Float64(z_m / sqrt(fma(Float64(-a), t, Float64(z_m * z_m)))) * x_m) * y_m);
	else
		tmp = Float64(Float64(z_m / fma(Float64(a / z_m), Float64(-0.5 * t), 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, 1.55e-135], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[Sqrt[N[((-a) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 3e+52], N[(N[(N[(z$95$m / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(z$95$m / N[(N[(a / z$95$m), $MachinePrecision] * N[(-0.5 * t), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision] * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 1.55e-135

   1. Initial program 66.1%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 45.5

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

   5. Applied rewrites 45.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 42.5

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

   7. Applied rewrites 42.5%

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

   if 1.55e-135 < z < 3e52

   1. Initial program 93.3%

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

      1. lift-/.f64 N/A

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

      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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 90.1%

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

   if 3e52 < z

   1. Initial program 39.5%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 76.9

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

   5. Applied rewrites 76.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 97.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 97.1%

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

Alternative 3: 90.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3 \cdot 10^{-135}:\\ \;\;\;\;x\_m \cdot \frac{z\_m \cdot y\_m}{\sqrt{\left(-a\right) \cdot t}}\\ \mathbf{elif}\;z\_m \leq 3 \cdot 10^{+52}:\\ \;\;\;\;\left(z\_m \cdot x\_m\right) \cdot \frac{y\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m}{\mathsf{fma}\left(\frac{a}{z\_m}, -0.5 \cdot t, z\_m\right)} \cdot \left(y\_m \cdot x\_m\right)\\ \end{array}\right)\right) \end{array} \]

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 3e-135)
      (* x_m (/ (* z_m y_m) (sqrt (* (- a) t))))
      (if (<= z_m 3e+52)
        (* (* z_m x_m) (/ y_m (sqrt (fma (- a) t (* z_m z_m)))))
        (* (/ z_m (fma (/ a z_m) (* -0.5 t) 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 <= 3e-135) {
		tmp = x_m * ((z_m * y_m) / sqrt((-a * t)));
	} else if (z_m <= 3e+52) {
		tmp = (z_m * x_m) * (y_m / sqrt(fma(-a, t, (z_m * z_m))));
	} else {
		tmp = (z_m / fma((a / z_m), (-0.5 * t), 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 <= 3e-135)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / sqrt(Float64(Float64(-a) * t))));
	elseif (z_m <= 3e+52)
		tmp = Float64(Float64(z_m * x_m) * Float64(y_m / sqrt(fma(Float64(-a), t, Float64(z_m * z_m)))));
	else
		tmp = Float64(Float64(z_m / fma(Float64(a / z_m), Float64(-0.5 * t), 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, 3e-135], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[Sqrt[N[((-a) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 3e+52], N[(N[(z$95$m * x$95$m), $MachinePrecision] * N[(y$95$m / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z$95$m / N[(N[(a / z$95$m), $MachinePrecision] * N[(-0.5 * t), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision] * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 3.00000000000000012e-135

   1. Initial program 66.1%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 45.5

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

   5. Applied rewrites 45.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 42.5

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

   7. Applied rewrites 42.5%

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

   if 3.00000000000000012e-135 < z < 3e52

   1. Initial program 93.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 86.7

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

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

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 86.7

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

   4. Applied rewrites 86.7%

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

   if 3e52 < z

   1. Initial program 39.5%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 76.9

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

   5. Applied rewrites 76.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 97.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 97.1%

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

Alternative 4: 92.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3 \cdot 10^{+52}:\\ \;\;\;\;\left(\frac{z\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}} \cdot y\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m}{\mathsf{fma}\left(\frac{a}{z\_m}, -0.5 \cdot t, z\_m\right)} \cdot \left(y\_m \cdot x\_m\right)\\ \end{array}\right)\right) \end{array} \]

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 3e+52)
      (* (* (/ z_m (sqrt (fma (- a) t (* z_m z_m)))) y_m) x_m)
      (* (/ z_m (fma (/ a z_m) (* -0.5 t) 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 <= 3e+52) {
		tmp = ((z_m / sqrt(fma(-a, t, (z_m * z_m)))) * y_m) * x_m;
	} else {
		tmp = (z_m / fma((a / z_m), (-0.5 * t), 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 <= 3e+52)
		tmp = Float64(Float64(Float64(z_m / sqrt(fma(Float64(-a), t, Float64(z_m * z_m)))) * y_m) * x_m);
	else
		tmp = Float64(Float64(z_m / fma(Float64(a / z_m), Float64(-0.5 * t), 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, 3e+52], N[(N[(N[(z$95$m / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(z$95$m / N[(N[(a / z$95$m), $MachinePrecision] * N[(-0.5 * t), $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 < 3e52

   1. Initial program 70.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 70.0%

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

   if 3e52 < z

   1. Initial program 39.5%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 76.9

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

   5. Applied rewrites 76.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 97.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 97.1%

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

Alternative 5: 86.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 6 \cdot 10^{-98}:\\ \;\;\;\;x\_m \cdot \frac{z\_m \cdot y\_m}{\sqrt{\left(-a\right) \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m}{\mathsf{fma}\left(\frac{a}{z\_m}, -0.5 \cdot t, z\_m\right)} \cdot \left(y\_m \cdot x\_m\right)\\ \end{array}\right)\right) \end{array} \]

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 6e-98)
      (* x_m (/ (* z_m y_m) (sqrt (* (- a) t))))
      (* (/ z_m (fma (/ a z_m) (* -0.5 t) 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 <= 6e-98) {
		tmp = x_m * ((z_m * y_m) / sqrt((-a * t)));
	} else {
		tmp = (z_m / fma((a / z_m), (-0.5 * t), 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 <= 6e-98)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / sqrt(Float64(Float64(-a) * t))));
	else
		tmp = Float64(Float64(z_m / fma(Float64(a / z_m), Float64(-0.5 * t), 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, 6e-98], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[Sqrt[N[((-a) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z$95$m / N[(N[(a / z$95$m), $MachinePrecision] * N[(-0.5 * t), $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 < 6e-98

   1. Initial program 66.7%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 46.5

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

   5. Applied rewrites 46.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 43.6

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

   7. Applied rewrites 43.6%

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

   if 6e-98 < z

   1. Initial program 54.5%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 75.7

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

   5. Applied rewrites 75.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 90.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 90.2%

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

Alternative 6: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 6 \cdot 10^{-98}:\\ \;\;\;\;x\_m \cdot \frac{z\_m \cdot y\_m}{\sqrt{\left(-a\right) \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z\_m}{\mathsf{fma}\left(\frac{a}{z\_m}, -0.5 \cdot t, z\_m\right)} \cdot x\_m\right) \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

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-98)
      (* x_m (/ (* z_m y_m) (sqrt (* (- a) t))))
      (* (* (/ z_m (fma (/ a z_m) (* -0.5 t) z_m)) x_m) y_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-98) {
		tmp = x_m * ((z_m * y_m) / sqrt((-a * t)));
	} else {
		tmp = ((z_m / fma((a / z_m), (-0.5 * t), z_m)) * x_m) * y_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-98)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / sqrt(Float64(Float64(-a) * t))));
	else
		tmp = Float64(Float64(Float64(z_m / fma(Float64(a / z_m), Float64(-0.5 * t), z_m)) * x_m) * y_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, 6e-98], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[Sqrt[N[((-a) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z$95$m / N[(N[(a / z$95$m), $MachinePrecision] * N[(-0.5 * t), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 6e-98

   1. Initial program 66.7%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 46.5

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

   5. Applied rewrites 46.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 43.6

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

   7. Applied rewrites 43.6%

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

   if 6e-98 < z

   1. Initial program 54.5%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 75.7

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

   5. Applied rewrites 75.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   7. Applied rewrites 88.2%

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

Alternative 7: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.5 \cdot 10^{-92}:\\ \;\;\;\;x\_m \cdot \frac{z\_m \cdot y\_m}{\sqrt{\left(-a\right) \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

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-92)
      (* x_m (/ (* z_m y_m) (sqrt (* (- a) t))))
      (* x_m y_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-92) {
		tmp = x_m * ((z_m * y_m) / sqrt((-a * t)));
	} else {
		tmp = x_m * y_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 <= 1.5d-92) then
        tmp = x_m * ((z_m * y_m) / sqrt((-a * t)))
    else
        tmp = x_m * y_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 <= 1.5e-92) {
		tmp = x_m * ((z_m * y_m) / Math.sqrt((-a * t)));
	} else {
		tmp = x_m * y_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 <= 1.5e-92:
		tmp = x_m * ((z_m * y_m) / math.sqrt((-a * t)))
	else:
		tmp = x_m * y_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-92)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / sqrt(Float64(Float64(-a) * t))));
	else
		tmp = Float64(x_m * y_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 <= 1.5e-92)
		tmp = x_m * ((z_m * y_m) / sqrt((-a * t)));
	else
		tmp = x_m * y_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, 1.5e-92], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[Sqrt[N[((-a) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.50000000000000007e-92

   1. Initial program 66.9%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 46.8

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

   5. Applied rewrites 46.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 43.9

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

   7. Applied rewrites 43.9%

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

   if 1.50000000000000007e-92 < z

   1. Initial program 54.0%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 75.5

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

   5. Applied rewrites 75.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 90.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 90.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-/.f64 90.1

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

   9. Applied rewrites 90.1%

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

   10. Taylor expanded in z around inf

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

       1. lower-*.f64 87.3

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

   12. Applied rewrites 87.3%

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

Alternative 8: 74.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.5 \cdot 10^{-219}:\\ \;\;\;\;\frac{\left(x\_m \cdot y\_m\right) \cdot z\_m}{-z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

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-219) (/ (* (* x_m y_m) z_m) (- z_m)) (* x_m y_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-219) {
		tmp = ((x_m * y_m) * z_m) / -z_m;
	} else {
		tmp = x_m * y_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 <= 1.5d-219) then
        tmp = ((x_m * y_m) * z_m) / -z_m
    else
        tmp = x_m * y_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 <= 1.5e-219) {
		tmp = ((x_m * y_m) * z_m) / -z_m;
	} else {
		tmp = x_m * y_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 <= 1.5e-219:
		tmp = ((x_m * y_m) * z_m) / -z_m
	else:
		tmp = x_m * y_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-219)
		tmp = Float64(Float64(Float64(x_m * y_m) * z_m) / Float64(-z_m));
	else
		tmp = Float64(x_m * y_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 <= 1.5e-219)
		tmp = ((x_m * y_m) * z_m) / -z_m;
	else
		tmp = x_m * y_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, 1.5e-219], N[(N[(N[(x$95$m * y$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] / (-z$95$m)), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.5e-219

   1. Initial program 65.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 62.9

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

   5. Applied rewrites 62.9%

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

   if 1.5e-219 < z

   1. Initial program 58.6%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 70.6

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

   5. Applied rewrites 70.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 82.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 82.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-/.f64 82.9

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

   9. Applied rewrites 82.9%

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

   10. Taylor expanded in z around inf

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

       1. lower-*.f64 77.9

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

   12. Applied rewrites 77.9%

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

Alternative 9: 73.0% accurate, 7.5× speedup

Math

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

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 (* x_m y_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 * (x_m * y_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 * (x_m * y_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 * (x_m * y_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 * (x_m * y_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(x_m * y_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 * (x_m * y_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[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.3%

   \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
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-*r/ N/A

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

   3. associate-*r* N/A

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

   4. associate-*l/ N/A

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

   5. associate-*r/ N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 46.9

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

5. Applied rewrites 46.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 52.1

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

7. Applied rewrites 52.1%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. pow-to-exp N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-log.f64 N/A

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

   8. lower-/.f64 52.1

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

9. Applied rewrites 52.1%

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

10. Taylor expanded in z around inf

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

    1. lower-*.f64 43.3

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

12. Applied rewrites 43.3%

    \[\leadsto \color{blue}{x \cdot y} \]
13. Add Preprocessing

Developer Target 1: 87.5% 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 2024318 
(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)))))