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

Percentage Accurate: 62.5% → 91.5%

Time: 12.0s

Alternatives: 11

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: 62.5% 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: 91.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 8 \cdot 10^{+42}:\\ \;\;\;\;\frac{x\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}} \cdot \left(y\_m \cdot z\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot y\_m}{e^{-\log \left(\frac{z\_m}{\mathsf{fma}\left(-0.5 \cdot t, \frac{a}{z\_m}, z\_m\right)}\right)}}\\ \end{array}\right)\right) \end{array} \]

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 8e+42)
      (* (/ x_m (sqrt (fma (- a) t (* z_m z_m)))) (* y_m z_m))
      (/
       (* x_m y_m)
       (exp (- (log (/ z_m (fma (* -0.5 t) (/ a z_m) z_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 <= 8e+42) {
		tmp = (x_m / sqrt(fma(-a, t, (z_m * z_m)))) * (y_m * z_m);
	} else {
		tmp = (x_m * y_m) / exp(-log((z_m / fma((-0.5 * t), (a / z_m), z_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 <= 8e+42)
		tmp = Float64(Float64(x_m / sqrt(fma(Float64(-a), t, Float64(z_m * z_m)))) * Float64(y_m * z_m));
	else
		tmp = Float64(Float64(x_m * y_m) / exp(Float64(-log(Float64(z_m / fma(Float64(-0.5 * t), Float64(a / z_m), z_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, 8e+42], N[(N[(x$95$m / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[Exp[(-N[Log[N[(z$95$m / N[(N[(-0.5 * t), $MachinePrecision] * N[(a / z$95$m), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 8.00000000000000036e42

   1. Initial program 71.5%

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

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 70.0

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 69.9

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

   4. Applied rewrites 69.9%

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

   if 8.00000000000000036e42 < z

   1. Initial program 42.4%

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

   3. Taylor expanded in a 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 73.5

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

   5. Applied rewrites 73.5%

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

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 96.1

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

   7. Applied rewrites 96.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-/.f64 96.1

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

   9. Applied rewrites 96.1%

      \[\leadsto \frac{y \cdot x}{\color{blue}{e^{\log \left(\frac{z}{\mathsf{fma}\left(-0.5 \cdot t, \frac{a}{z}, z\right)}\right) \cdot -1}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8 \cdot 10^{+42}:\\ \;\;\;\;\frac{x}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}} \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{e^{-\log \left(\frac{z}{\mathsf{fma}\left(-0.5 \cdot t, \frac{a}{z}, z\right)}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4.6 \cdot 10^{-161}:\\ \;\;\;\;\frac{x\_m}{\sqrt{t \cdot \left(-a\right)}} \cdot \left(y\_m \cdot z\_m\right)\\ \mathbf{elif}\;z\_m \leq 6 \cdot 10^{+52}:\\ \;\;\;\;\left(\frac{y\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}} \cdot x\_m\right) \cdot z\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z\_m}{\mathsf{fma}\left(\frac{a}{z\_m}, -0.5 \cdot t, z\_m\right)} \cdot x\_m\right) \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

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 4.6e-161)
      (* (/ x_m (sqrt (* t (- a)))) (* y_m z_m))
      (if (<= z_m 6e+52)
        (* (* (/ y_m (sqrt (fma (- a) t (* z_m z_m)))) x_m) z_m)
        (* (* (/ 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 <= 4.6e-161) {
		tmp = (x_m / sqrt((t * -a))) * (y_m * z_m);
	} else if (z_m <= 6e+52) {
		tmp = ((y_m / sqrt(fma(-a, t, (z_m * z_m)))) * x_m) * z_m;
	} 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 <= 4.6e-161)
		tmp = Float64(Float64(x_m / sqrt(Float64(t * Float64(-a)))) * Float64(y_m * z_m));
	elseif (z_m <= 6e+52)
		tmp = Float64(Float64(Float64(y_m / sqrt(fma(Float64(-a), t, Float64(z_m * z_m)))) * x_m) * z_m);
	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, 4.6e-161], N[(N[(x$95$m / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 6e+52], N[(N[(N[(y$95$m / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] * z$95$m), $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 3 regimes

2. if z < 4.6e-161

   1. Initial program 68.3%

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 67.1

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 67.1

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 67.0

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

   4. Applied rewrites 67.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 67.0

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

   6. Applied rewrites 67.0%

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

   7. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 38.7

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

   9. Applied rewrites 38.7%

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

   if 4.6e-161 < z < 6e52

   1. Initial program 87.9%

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 88.3

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 88.3

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 88.3

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

   4. Applied rewrites 88.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 83.1

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

   6. Applied rewrites 83.1%

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

   if 6e52 < z

   1. Initial program 40.8%

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

   3. Taylor expanded in a 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 74.1

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

   5. Applied rewrites 74.1%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{\mathsf{fma}\left(-0.5 \cdot t, \frac{a}{z}, z\right)}} \]
   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 97.4%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.6 \cdot 10^{-161}:\\ \;\;\;\;\frac{x}{\sqrt{t \cdot \left(-a\right)}} \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+52}:\\ \;\;\;\;\left(\frac{y}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}} \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{\mathsf{fma}\left(\frac{a}{z}, -0.5 \cdot t, z\right)} \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.05 \cdot 10^{+41}:\\ \;\;\;\;\frac{x\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}} \cdot \left(y\_m \cdot z\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot y\_m}{\frac{\mathsf{fma}\left(\frac{a}{z\_m}, -0.5 \cdot t, z\_m\right)}{z\_m}}\\ \end{array}\right)\right) \end{array} \]

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 3.05e+41)
      (* (/ x_m (sqrt (fma (- a) t (* z_m z_m)))) (* y_m z_m))
      (/ (* x_m y_m) (/ (fma (/ a z_m) (* -0.5 t) z_m) z_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 <= 3.05e+41) {
		tmp = (x_m / sqrt(fma(-a, t, (z_m * z_m)))) * (y_m * z_m);
	} else {
		tmp = (x_m * y_m) / (fma((a / z_m), (-0.5 * t), z_m) / z_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 <= 3.05e+41)
		tmp = Float64(Float64(x_m / sqrt(fma(Float64(-a), t, Float64(z_m * z_m)))) * Float64(y_m * z_m));
	else
		tmp = Float64(Float64(x_m * y_m) / Float64(fma(Float64(a / z_m), Float64(-0.5 * t), z_m) / z_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, 3.05e+41], N[(N[(x$95$m / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[(N[(N[(a / z$95$m), $MachinePrecision] * N[(-0.5 * t), $MachinePrecision] + z$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.04999999999999999e41

   1. Initial program 71.5%

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

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 70.0

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 69.9

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

   4. Applied rewrites 69.9%

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

   if 3.04999999999999999e41 < z

   1. Initial program 42.4%

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

   3. Taylor expanded in a 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 73.5

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

   5. Applied rewrites 73.5%

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

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 96.1

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

   7. Applied rewrites 96.1%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.05 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}} \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\mathsf{fma}\left(\frac{a}{z}, -0.5 \cdot t, z\right)}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.05 \cdot 10^{+41}:\\ \;\;\;\;\frac{x\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}} \cdot \left(y\_m \cdot z\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z\_m}{\mathsf{fma}\left(\frac{a}{z\_m}, -0.5 \cdot t, z\_m\right)} \cdot x\_m\right) \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

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 3.05e+41)
      (* (/ x_m (sqrt (fma (- a) t (* z_m z_m)))) (* y_m z_m))
      (* (* (/ 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 <= 3.05e+41) {
		tmp = (x_m / sqrt(fma(-a, t, (z_m * z_m)))) * (y_m * z_m);
	} 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 <= 3.05e+41)
		tmp = Float64(Float64(x_m / sqrt(fma(Float64(-a), t, Float64(z_m * z_m)))) * Float64(y_m * z_m));
	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, 3.05e+41], N[(N[(x$95$m / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(y$95$m * z$95$m), $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 < 3.04999999999999999e41

   1. Initial program 71.5%

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

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 70.0

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 69.9

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

   4. Applied rewrites 69.9%

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

   if 3.04999999999999999e41 < z

   1. Initial program 42.4%

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

   3. Taylor expanded in a 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 73.5

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

   5. Applied rewrites 73.5%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{\mathsf{fma}\left(-0.5 \cdot t, \frac{a}{z}, z\right)}} \]
   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. 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 96.1%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.05 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}} \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{\mathsf{fma}\left(\frac{a}{z}, -0.5 \cdot t, z\right)} \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 10^{-69}:\\ \;\;\;\;\frac{x\_m}{\sqrt{t \cdot \left(-a\right)}} \cdot \left(y\_m \cdot z\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z\_m}{\mathsf{fma}\left(\frac{a}{z\_m}, -0.5 \cdot t, z\_m\right)} \cdot x\_m\right) \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

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 1e-69)
      (* (/ x_m (sqrt (* t (- a)))) (* y_m z_m))
      (* (* (/ 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 <= 1e-69) {
		tmp = (x_m / sqrt((t * -a))) * (y_m * z_m);
	} 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 <= 1e-69)
		tmp = Float64(Float64(x_m / sqrt(Float64(t * Float64(-a)))) * Float64(y_m * z_m));
	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, 1e-69], N[(N[(x$95$m / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(y$95$m * z$95$m), $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 < 9.9999999999999996e-70

   1. Initial program 69.8%

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 69.2

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 69.2

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 69.1

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

   4. Applied rewrites 69.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 68.5

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

   6. Applied rewrites 68.5%

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

   7. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 41.5

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

   9. Applied rewrites 41.5%

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

   if 9.9999999999999996e-70 < z

   1. Initial program 51.3%

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

   3. Taylor expanded in a 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 70.0

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

   5. Applied rewrites 70.0%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{\mathsf{fma}\left(-0.5 \cdot t, \frac{a}{z}, z\right)}} \]
   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. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{-69}:\\ \;\;\;\;\frac{x}{\sqrt{t \cdot \left(-a\right)}} \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{\mathsf{fma}\left(\frac{a}{z}, -0.5 \cdot t, z\right)} \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 8 \cdot 10^{-50}:\\ \;\;\;\;\frac{x\_m}{\sqrt{t \cdot \left(-a\right)}} \cdot \left(y\_m \cdot z\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

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 8e-50)
      (* (/ x_m (sqrt (* t (- a)))) (* y_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 <= 8e-50) {
		tmp = (x_m / sqrt((t * -a))) * (y_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 <= 8d-50) then
        tmp = (x_m / sqrt((t * -a))) * (y_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 <= 8e-50) {
		tmp = (x_m / Math.sqrt((t * -a))) * (y_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 <= 8e-50:
		tmp = (x_m / math.sqrt((t * -a))) * (y_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 <= 8e-50)
		tmp = Float64(Float64(x_m / sqrt(Float64(t * Float64(-a)))) * Float64(y_m * 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 <= 8e-50)
		tmp = (x_m / sqrt((t * -a))) * (y_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, 8e-50], N[(N[(x$95$m / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 8.00000000000000006e-50

   1. Initial program 70.0%

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 69.4

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 69.4

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 69.3

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

   4. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 68.7

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

   6. Applied rewrites 68.7%

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

   7. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 41.8

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

   9. Applied rewrites 41.8%

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

   if 8.00000000000000006e-50 < z

   1. Initial program 50.7%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 86.8

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

   5. Applied rewrites 86.8%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{\sqrt{t \cdot \left(-a\right)}} \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 8 \cdot 10^{-50}:\\ \;\;\;\;\left(\frac{z\_m}{\sqrt{t \cdot \left(-a\right)}} \cdot y\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

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 8e-50)
      (* (* (/ z_m (sqrt (* t (- a)))) y_m) x_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 <= 8e-50) {
		tmp = ((z_m / sqrt((t * -a))) * y_m) * x_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 <= 8d-50) then
        tmp = ((z_m / sqrt((t * -a))) * y_m) * x_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 <= 8e-50) {
		tmp = ((z_m / Math.sqrt((t * -a))) * y_m) * x_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 <= 8e-50:
		tmp = ((z_m / math.sqrt((t * -a))) * y_m) * x_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 <= 8e-50)
		tmp = Float64(Float64(Float64(z_m / sqrt(Float64(t * Float64(-a)))) * y_m) * x_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 <= 8e-50)
		tmp = ((z_m / sqrt((t * -a))) * y_m) * x_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, 8e-50], N[(N[(N[(z$95$m / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 8.00000000000000006e-50

   1. Initial program 70.0%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 42.2

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

   5. Applied rewrites 42.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 40.0

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

   7. Applied rewrites 40.0%

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

   if 8.00000000000000006e-50 < z

   1. Initial program 50.7%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 86.8

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

   5. Applied rewrites 86.8%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8 \cdot 10^{-50}:\\ \;\;\;\;\left(\frac{z}{\sqrt{t \cdot \left(-a\right)}} \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2 \cdot 10^{-127}:\\ \;\;\;\;\frac{-1}{z\_m} \cdot \left(\left(\left(-x\_m\right) \cdot y\_m\right) \cdot z\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

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 2e-127)
      (* (/ -1.0 z_m) (* (* (- x_m) y_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 <= 2e-127) {
		tmp = (-1.0 / z_m) * ((-x_m * y_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 <= 2d-127) then
        tmp = ((-1.0d0) / z_m) * ((-x_m * y_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 <= 2e-127) {
		tmp = (-1.0 / z_m) * ((-x_m * y_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 <= 2e-127:
		tmp = (-1.0 / z_m) * ((-x_m * y_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 <= 2e-127)
		tmp = Float64(Float64(-1.0 / z_m) * Float64(Float64(Float64(-x_m) * y_m) * 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 <= 2e-127)
		tmp = (-1.0 / z_m) * ((-x_m * y_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, 2e-127], N[(N[(-1.0 / z$95$m), $MachinePrecision] * N[(N[((-x$95$m) * y$95$m), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.0000000000000001e-127

   1. Initial program 68.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 62.4

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

   5. Applied rewrites 62.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. remove-double-neg N/A

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

      18. lower-/.f64 62.3

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

   7. Applied rewrites 62.3%

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

   8. Taylor expanded in a around 0

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

      1. lower-/.f64 18.7

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

   10. Applied rewrites 18.7%

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

   if 2.0000000000000001e-127 < z

   1. Initial program 55.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-127}:\\ \;\;\;\;\frac{-1}{z} \cdot \left(\left(\left(-x\right) \cdot y\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.5 \cdot 10^{-207}:\\ \;\;\;\;\frac{\left(x\_m \cdot z\_m\right) \cdot y\_m}{-z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

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-207) (/ (* (* x_m z_m) y_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-207) {
		tmp = ((x_m * z_m) * y_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-207) then
        tmp = ((x_m * z_m) * y_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-207) {
		tmp = ((x_m * z_m) * y_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-207:
		tmp = ((x_m * z_m) * y_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-207)
		tmp = Float64(Float64(Float64(x_m * z_m) * y_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-207)
		tmp = ((x_m * z_m) * y_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-207], N[(N[(N[(x$95$m * z$95$m), $MachinePrecision] * y$95$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-207

   1. Initial program 68.7%

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

   3. Taylor expanded in z around -inf

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{-1 \cdot z}} \]
   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 65.4

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

   5. Applied rewrites 65.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 63.0

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

   7. Applied rewrites 63.0%

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

   if 1.5e-207 < z

   1. Initial program 56.0%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 79.5

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

   5. Applied rewrites 79.5%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.5 \cdot 10^{-207}:\\ \;\;\;\;\frac{\left(x \cdot z\right) \cdot y}{-z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.1 \cdot 10^{-207}:\\ \;\;\;\;\frac{\left(x\_m \cdot y\_m\right) \cdot z\_m}{-z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

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 1.1e-207) (/ (* (* 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 <= 1.1e-207) {
		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 <= 1.1d-207) 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 <= 1.1e-207) {
		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 <= 1.1e-207:
		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 <= 1.1e-207)
		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 <= 1.1e-207)
		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, 1.1e-207], 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 < 1.0999999999999999e-207

   1. Initial program 68.7%

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

   3. Taylor expanded in z around -inf

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{-1 \cdot z}} \]
   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 65.4

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

   5. Applied rewrites 65.4%

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

   if 1.0999999999999999e-207 < z

   1. Initial program 56.0%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 79.5

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

   5. Applied rewrites 79.5%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.1 \cdot 10^{-207}:\\ \;\;\;\;\frac{\left(x \cdot y\right) \cdot z}{-z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.9% 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 63.4%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 39.8

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

5. Applied rewrites 39.8%

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

6. Final simplification 39.8%

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

Developer Target 1: 86.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024270 
(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)))))