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

Percentage Accurate: 61.7% → 92.1%

Time: 10.4s

Alternatives: 8

Speedup: 7.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z_m, t, a] = \mathsf{sort}([x, y_m, z_m, t, a])\\ \\ y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 10^{+115}:\\ \;\;\;\;x \cdot \left(y\_m \cdot \frac{z\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\_m\\ \end{array}\right) \end{array} \]

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)
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (y_s z_s x y_m z_m t a)
 :precision binary64
 (*
  y_s
  (*
   z_s
   (if (<= z_m 1e+115)
     (* x (* y_m (/ z_m (sqrt (fma (- a) t (* z_m z_m))))))
     (* x y_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z_m && z_m < t && t < a);
double code(double y_s, double z_s, double x, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1e+115) {
		tmp = x * (y_m * (z_m / sqrt(fma(-a, t, (z_m * z_m)))));
	} else {
		tmp = x * y_m;
	}
	return 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, y_m, z_m, t, a = sort([x, y_m, z_m, t, a])
function code(y_s, z_s, x, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1e+115)
		tmp = Float64(x * Float64(y_m * Float64(z_m / sqrt(fma(Float64(-a), t, Float64(z_m * z_m))))));
	else
		tmp = Float64(x * y_m);
	end
	return 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]
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[y$95$s_, z$95$s_, x_, y$95$m_, z$95$m_, t_, a_] := N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 1e+115], N[(x * N[(y$95$m * N[(z$95$m / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1e115

   1. Initial program 65.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 71.7%

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

   if 1e115 < z

   1. Initial program 35.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{+115}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z_m, t, a] = \mathsf{sort}([x, y_m, z_m, t, a])\\ \\ y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot y\_m\right) \cdot z\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}} \leq 2 \cdot 10^{-322}:\\ \;\;\;\;\frac{y\_m}{z\_m} \cdot \left(x \cdot z\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\_m\\ \end{array}\right) \end{array} \]

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)
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (y_s z_s x y_m z_m t a)
 :precision binary64
 (*
  y_s
  (*
   z_s
   (if (<= (/ (* (* x y_m) z_m) (sqrt (- (* z_m z_m) (* t a)))) 2e-322)
     (* (/ y_m z_m) (* x z_m))
     (* x y_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z_m && z_m < t && t < a);
double code(double y_s, double z_s, double x, double y_m, double z_m, double t, double a) {
	double tmp;
	if ((((x * y_m) * z_m) / sqrt(((z_m * z_m) - (t * a)))) <= 2e-322) {
		tmp = (y_m / z_m) * (x * z_m);
	} else {
		tmp = x * y_m;
	}
	return 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)
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(y_s, z_s, x, y_m, z_m, t, a)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((((x * y_m) * z_m) / sqrt(((z_m * z_m) - (t * a)))) <= 2d-322) then
        tmp = (y_m / z_m) * (x * z_m)
    else
        tmp = x * y_m
    end if
    code = 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);
assert x < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double y_s, double z_s, double x, double y_m, double z_m, double t, double a) {
	double tmp;
	if ((((x * y_m) * z_m) / Math.sqrt(((z_m * z_m) - (t * a)))) <= 2e-322) {
		tmp = (y_m / z_m) * (x * z_m);
	} else {
		tmp = x * y_m;
	}
	return 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, y_m, z_m, t, a] = sort([x, y_m, z_m, t, a])
def code(y_s, z_s, x, y_m, z_m, t, a):
	tmp = 0
	if (((x * y_m) * z_m) / math.sqrt(((z_m * z_m) - (t * a)))) <= 2e-322:
		tmp = (y_m / z_m) * (x * z_m)
	else:
		tmp = x * y_m
	return 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, y_m, z_m, t, a = sort([x, y_m, z_m, t, a])
function code(y_s, z_s, x, y_m, z_m, t, a)
	tmp = 0.0
	if (Float64(Float64(Float64(x * y_m) * z_m) / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))) <= 2e-322)
		tmp = Float64(Float64(y_m / z_m) * Float64(x * z_m));
	else
		tmp = Float64(x * y_m);
	end
	return 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, y_m, z_m, t, a = num2cell(sort([x, y_m, z_m, t, a])){:}
function tmp_2 = code(y_s, z_s, x, y_m, z_m, t, a)
	tmp = 0.0;
	if ((((x * y_m) * z_m) / sqrt(((z_m * z_m) - (t * a)))) <= 2e-322)
		tmp = (y_m / z_m) * (x * z_m);
	else
		tmp = x * y_m;
	end
	tmp_2 = 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]
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[y$95$s_, z$95$s_, x_, y$95$m_, z$95$m_, t_, a_] := N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(N[(x * y$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e-322], N[(N[(y$95$m / z$95$m), $MachinePrecision] * N[(x * z$95$m), $MachinePrecision]), $MachinePrecision], N[(x * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a)))) < 1.97626e-322

   1. Initial program 65.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 65.6

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

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

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 65.6

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

   4. Applied rewrites 65.6%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 38.6

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

   7. Applied rewrites 38.6%

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

   if 1.97626e-322 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))))

   1. Initial program 48.7%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 46.0

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

   5. Applied rewrites 46.0%

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

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \leq 2 \cdot 10^{-322}:\\ \;\;\;\;\frac{y}{z} \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z_m, t, a] = \mathsf{sort}([x, y_m, z_m, t, a])\\ \\ y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.2 \cdot 10^{-166}:\\ \;\;\;\;\frac{x}{\sqrt{t \cdot \left(-a\right)}} \cdot \left(y\_m \cdot z\_m\right)\\ \mathbf{elif}\;z\_m \leq 1.15 \cdot 10^{+115}:\\ \;\;\;\;\frac{y\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}} \cdot \left(x \cdot z\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\_m\\ \end{array}\right) \end{array} \]

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)
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (y_s z_s x y_m z_m t a)
 :precision binary64
 (*
  y_s
  (*
   z_s
   (if (<= z_m 1.2e-166)
     (* (/ x (sqrt (* t (- a)))) (* y_m z_m))
     (if (<= z_m 1.15e+115)
       (* (/ y_m (sqrt (fma (- a) t (* z_m z_m)))) (* x z_m))
       (* x y_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z_m && z_m < t && t < a);
double code(double y_s, double z_s, double x, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.2e-166) {
		tmp = (x / sqrt((t * -a))) * (y_m * z_m);
	} else if (z_m <= 1.15e+115) {
		tmp = (y_m / sqrt(fma(-a, t, (z_m * z_m)))) * (x * z_m);
	} else {
		tmp = x * y_m;
	}
	return 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, y_m, z_m, t, a = sort([x, y_m, z_m, t, a])
function code(y_s, z_s, x, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.2e-166)
		tmp = Float64(Float64(x / sqrt(Float64(t * Float64(-a)))) * Float64(y_m * z_m));
	elseif (z_m <= 1.15e+115)
		tmp = Float64(Float64(y_m / sqrt(fma(Float64(-a), t, Float64(z_m * z_m)))) * Float64(x * z_m));
	else
		tmp = Float64(x * y_m);
	end
	return 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]
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[y$95$s_, z$95$s_, x_, y$95$m_, z$95$m_, t_, a_] := N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 1.2e-166], N[(N[(x / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 1.15e+115], N[(N[(y$95$m / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(x * z$95$m), $MachinePrecision]), $MachinePrecision], N[(x * y$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 1.1999999999999999e-166

   1. Initial program 56.5%

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

   3. Taylor expanded in 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 34.4

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

   5. Applied rewrites 34.4%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{\left(-a\right) \cdot t}}} \]
   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. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 37.9

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

   7. Applied rewrites 37.9%

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

   if 1.1999999999999999e-166 < z < 1.15000000000000002e115

   1. Initial program 84.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 87.4

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

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

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 87.4

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

   4. Applied rewrites 87.4%

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

   if 1.15000000000000002e115 < z

   1. Initial program 35.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{-166}:\\ \;\;\;\;\frac{x}{\sqrt{t \cdot \left(-a\right)}} \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+115}:\\ \;\;\;\;\frac{y}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}} \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z_m, t, a] = \mathsf{sort}([x, y_m, z_m, t, a])\\ \\ y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.9 \cdot 10^{+31}:\\ \;\;\;\;\frac{x}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}} \cdot \left(y\_m \cdot z\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\_m\\ \end{array}\right) \end{array} \]

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)
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (y_s z_s x y_m z_m t a)
 :precision binary64
 (*
  y_s
  (*
   z_s
   (if (<= z_m 2.9e+31)
     (* (/ x (sqrt (fma (- a) t (* z_m z_m)))) (* y_m z_m))
     (* x y_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z_m && z_m < t && t < a);
double code(double y_s, double z_s, double x, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.9e+31) {
		tmp = (x / sqrt(fma(-a, t, (z_m * z_m)))) * (y_m * z_m);
	} else {
		tmp = x * y_m;
	}
	return 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, y_m, z_m, t, a = sort([x, y_m, z_m, t, a])
function code(y_s, z_s, x, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.9e+31)
		tmp = Float64(Float64(x / sqrt(fma(Float64(-a), t, Float64(z_m * z_m)))) * Float64(y_m * z_m));
	else
		tmp = Float64(x * y_m);
	end
	return 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]
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[y$95$s_, z$95$s_, x_, y$95$m_, z$95$m_, t_, a_] := N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 2.9e+31], N[(N[(x / 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[(x * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.9e31

   1. Initial program 62.6%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 63.9

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

      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 64.4

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

   4. Applied rewrites 64.4%

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

   if 2.9e31 < z

   1. Initial program 49.9%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 95.4

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

   5. Applied rewrites 95.4%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.9 \cdot 10^{+31}:\\ \;\;\;\;\frac{x}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}} \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z_m, t, a] = \mathsf{sort}([x, y_m, z_m, t, a])\\ \\ y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.7 \cdot 10^{-55}:\\ \;\;\;\;\frac{x}{\sqrt{t \cdot \left(-a\right)}} \cdot \left(y\_m \cdot z\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\_m\\ \end{array}\right) \end{array} \]

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)
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (y_s z_s x y_m z_m t a)
 :precision binary64
 (*
  y_s
  (*
   z_s
   (if (<= z_m 1.7e-55) (* (/ x (sqrt (* t (- a)))) (* y_m z_m)) (* x y_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z_m && z_m < t && t < a);
double code(double y_s, double z_s, double x, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.7e-55) {
		tmp = (x / sqrt((t * -a))) * (y_m * z_m);
	} else {
		tmp = x * y_m;
	}
	return 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)
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(y_s, z_s, x, y_m, z_m, t, a)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    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 <= 1.7d-55) then
        tmp = (x / sqrt((t * -a))) * (y_m * z_m)
    else
        tmp = x * y_m
    end if
    code = 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);
assert x < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double y_s, double z_s, double x, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.7e-55) {
		tmp = (x / Math.sqrt((t * -a))) * (y_m * z_m);
	} else {
		tmp = x * y_m;
	}
	return 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, y_m, z_m, t, a] = sort([x, y_m, z_m, t, a])
def code(y_s, z_s, x, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.7e-55:
		tmp = (x / math.sqrt((t * -a))) * (y_m * z_m)
	else:
		tmp = x * y_m
	return 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, y_m, z_m, t, a = sort([x, y_m, z_m, t, a])
function code(y_s, z_s, x, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.7e-55)
		tmp = Float64(Float64(x / sqrt(Float64(t * Float64(-a)))) * Float64(y_m * z_m));
	else
		tmp = Float64(x * y_m);
	end
	return 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, y_m, z_m, t, a = num2cell(sort([x, y_m, z_m, t, a])){:}
function tmp_2 = code(y_s, z_s, x, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.7e-55)
		tmp = (x / sqrt((t * -a))) * (y_m * z_m);
	else
		tmp = x * y_m;
	end
	tmp_2 = 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]
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[y$95$s_, z$95$s_, x_, y$95$m_, z$95$m_, t_, a_] := N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 1.7e-55], N[(N[(x / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], N[(x * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.69999999999999986e-55

   1. Initial program 59.7%

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

   3. Taylor expanded in 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 36.9

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

   5. Applied rewrites 36.9%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{\left(-a\right) \cdot t}}} \]
   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. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 41.1

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

   7. Applied rewrites 41.1%

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

   if 1.69999999999999986e-55 < z

   1. Initial program 57.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 89.4

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

   5. Applied rewrites 89.4%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.7 \cdot 10^{-55}:\\ \;\;\;\;\frac{x}{\sqrt{t \cdot \left(-a\right)}} \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z_m, t, a] = \mathsf{sort}([x, y_m, z_m, t, a])\\ \\ y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.7 \cdot 10^{-55}:\\ \;\;\;\;\left(\frac{z\_m}{\sqrt{t \cdot \left(-a\right)}} \cdot x\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\_m\\ \end{array}\right) \end{array} \]

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)
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (y_s z_s x y_m z_m t a)
 :precision binary64
 (*
  y_s
  (*
   z_s
   (if (<= z_m 1.7e-55) (* (* (/ z_m (sqrt (* t (- a)))) x) y_m) (* x y_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z_m && z_m < t && t < a);
double code(double y_s, double z_s, double x, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.7e-55) {
		tmp = ((z_m / sqrt((t * -a))) * x) * y_m;
	} else {
		tmp = x * y_m;
	}
	return 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)
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(y_s, z_s, x, y_m, z_m, t, a)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    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 <= 1.7d-55) then
        tmp = ((z_m / sqrt((t * -a))) * x) * y_m
    else
        tmp = x * y_m
    end if
    code = 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);
assert x < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double y_s, double z_s, double x, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.7e-55) {
		tmp = ((z_m / Math.sqrt((t * -a))) * x) * y_m;
	} else {
		tmp = x * y_m;
	}
	return 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, y_m, z_m, t, a] = sort([x, y_m, z_m, t, a])
def code(y_s, z_s, x, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.7e-55:
		tmp = ((z_m / math.sqrt((t * -a))) * x) * y_m
	else:
		tmp = x * y_m
	return 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, y_m, z_m, t, a = sort([x, y_m, z_m, t, a])
function code(y_s, z_s, x, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.7e-55)
		tmp = Float64(Float64(Float64(z_m / sqrt(Float64(t * Float64(-a)))) * x) * y_m);
	else
		tmp = Float64(x * y_m);
	end
	return 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, y_m, z_m, t, a = num2cell(sort([x, y_m, z_m, t, a])){:}
function tmp_2 = code(y_s, z_s, x, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.7e-55)
		tmp = ((z_m / sqrt((t * -a))) * x) * y_m;
	else
		tmp = x * y_m;
	end
	tmp_2 = 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]
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[y$95$s_, z$95$s_, x_, y$95$m_, z$95$m_, t_, a_] := N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 1.7e-55], N[(N[(N[(z$95$m / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * y$95$m), $MachinePrecision], N[(x * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.69999999999999986e-55

   1. Initial program 59.7%

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

   3. Taylor expanded in 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 36.9

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

   5. Applied rewrites 36.9%

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

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 39.9

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

   7. Applied rewrites 39.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 38.8%

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

   if 1.69999999999999986e-55 < z

   1. Initial program 57.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 89.4

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

   5. Applied rewrites 89.4%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.7 \cdot 10^{-55}:\\ \;\;\;\;\left(\frac{z}{\sqrt{t \cdot \left(-a\right)}} \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z_m, t, a] = \mathsf{sort}([x, y_m, z_m, t, a])\\ \\ y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.3 \cdot 10^{-170}:\\ \;\;\;\;\frac{\left(x \cdot z\_m\right) \cdot y\_m}{-z\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\_m\\ \end{array}\right) \end{array} \]

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)
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (y_s z_s x y_m z_m t a)
 :precision binary64
 (*
  y_s
  (* z_s (if (<= z_m 3.3e-170) (/ (* (* x z_m) y_m) (- z_m)) (* x y_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z_m && z_m < t && t < a);
double code(double y_s, double z_s, double x, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.3e-170) {
		tmp = ((x * z_m) * y_m) / -z_m;
	} else {
		tmp = x * y_m;
	}
	return 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)
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(y_s, z_s, x, y_m, z_m, t, a)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 3.3d-170) then
        tmp = ((x * z_m) * y_m) / -z_m
    else
        tmp = x * y_m
    end if
    code = 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);
assert x < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double y_s, double z_s, double x, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.3e-170) {
		tmp = ((x * z_m) * y_m) / -z_m;
	} else {
		tmp = x * y_m;
	}
	return 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, y_m, z_m, t, a] = sort([x, y_m, z_m, t, a])
def code(y_s, z_s, x, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 3.3e-170:
		tmp = ((x * z_m) * y_m) / -z_m
	else:
		tmp = x * y_m
	return 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, y_m, z_m, t, a = sort([x, y_m, z_m, t, a])
function code(y_s, z_s, x, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 3.3e-170)
		tmp = Float64(Float64(Float64(x * z_m) * y_m) / Float64(-z_m));
	else
		tmp = Float64(x * y_m);
	end
	return 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, y_m, z_m, t, a = num2cell(sort([x, y_m, z_m, t, a])){:}
function tmp_2 = code(y_s, z_s, x, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 3.3e-170)
		tmp = ((x * z_m) * y_m) / -z_m;
	else
		tmp = x * y_m;
	end
	tmp_2 = 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]
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[y$95$s_, z$95$s_, x_, y$95$m_, z$95$m_, t_, a_] := N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 3.3e-170], N[(N[(N[(x * z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / (-z$95$m)), $MachinePrecision], N[(x * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.30000000000000004e-170

   1. Initial program 56.2%

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

   3. Taylor expanded in z around -inf

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{-1 \cdot z}} \]
   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 57.1

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

   5. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 54.5

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

   7. Applied rewrites 54.5%

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

   if 3.30000000000000004e-170 < z

   1. Initial program 61.2%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 81.6

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

   5. Applied rewrites 81.6%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.3 \cdot 10^{-170}:\\ \;\;\;\;\frac{\left(x \cdot z\right) \cdot y}{-z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.3% accurate, 7.5× speedup

Math

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

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)
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (y_s z_s x y_m z_m t a) :precision binary64 (* y_s (* z_s (* x y_m))))

C

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

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(y_s, z_s, x, y_m, z_m, t, a)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = y_s * (z_s * (x * 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);
assert x < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double y_s, double z_s, double x, double y_m, double z_m, double t, double a) {
	return y_s * (z_s * (x * 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, y_m, z_m, t, a] = sort([x, y_m, z_m, t, a])
def code(y_s, z_s, x, y_m, z_m, t, a):
	return y_s * (z_s * (x * y_m))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z_m, t, a = sort([x, y_m, z_m, t, a])
function code(y_s, z_s, x, y_m, z_m, t, a)
	return Float64(y_s * Float64(z_s * Float64(x * 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, y_m, z_m, t, a = num2cell(sort([x, y_m, z_m, t, a])){:}
function tmp = code(y_s, z_s, x, y_m, z_m, t, a)
	tmp = y_s * (z_s * (x * 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]
NOTE: x, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[y$95$s_, z$95$s_, x_, y$95$m_, z$95$m_, t_, a_] := N[(y$95$s * N[(z$95$s * N[(x * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.6%

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

3. Taylor expanded in 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 44.8

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

5. Applied rewrites 44.8%

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

6. Final simplification 44.8%

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

Developer Target 1: 88.1% 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 2024254 
(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)))))