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

Percentage Accurate: 61.1% → 91.4%

Time: 10.3s

Alternatives: 6

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: 61.1% 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: 91.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ x\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 9.6 \cdot 10^{+111}:\\ \;\;\;\;x\_m \cdot \left(y \cdot \frac{z\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\\ \end{array}\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s z_s x_m y z_m t a)
 :precision binary64
 (*
  x_s
  (*
   z_s
   (if (<= z_m 9.6e+111)
     (* x_m (* y (/ z_m (sqrt (fma (- a) t (* z_m z_m))))))
     (* x_m y)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double x_s, double z_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 9.6e+111) {
		tmp = x_m * (y * (z_m / sqrt(fma(-a, t, (z_m * z_m)))));
	} else {
		tmp = x_m * y;
	}
	return x_s * (z_s * tmp);
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(x_s, z_s, x_m, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 9.6e+111)
		tmp = Float64(x_m * Float64(y * Float64(z_m / sqrt(fma(Float64(-a), t, Float64(z_m * z_m))))));
	else
		tmp = Float64(x_m * y);
	end
	return Float64(x_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]
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, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, z$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(x$95$s * N[(z$95$s * If[LessEqual[z$95$m, 9.6e+111], N[(x$95$m * N[(y * N[(z$95$m / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 9.60000000000000023e111

   1. Initial program 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 74.0%

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

   if 9.60000000000000023e111 < z

   1. Initial program 16.4%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 98.1

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

   5. Applied rewrites 98.1%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9.6 \cdot 10^{+111}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ x\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4 \cdot 10^{+48}:\\ \;\;\;\;\frac{x\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}} \cdot \left(y \cdot z\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\\ \end{array}\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s z_s x_m y z_m t a)
 :precision binary64
 (*
  x_s
  (*
   z_s
   (if (<= z_m 4e+48)
     (* (/ x_m (sqrt (fma (- a) t (* z_m z_m)))) (* y z_m))
     (* x_m y)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double x_s, double z_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 4e+48) {
		tmp = (x_m / sqrt(fma(-a, t, (z_m * z_m)))) * (y * z_m);
	} else {
		tmp = x_m * y;
	}
	return x_s * (z_s * tmp);
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(x_s, z_s, x_m, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 4e+48)
		tmp = Float64(Float64(x_m / sqrt(fma(Float64(-a), t, Float64(z_m * z_m)))) * Float64(y * z_m));
	else
		tmp = Float64(x_m * y);
	end
	return Float64(x_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]
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, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, z$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(x$95$s * N[(z$95$s * If[LessEqual[z$95$m, 4e+48], N[(N[(x$95$m / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(y * z$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.00000000000000018e48

   1. Initial program 70.5%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 69.7

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

      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 69.7

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

   4. Applied rewrites 69.7%

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

   if 4.00000000000000018e48 < z

   1. Initial program 36.9%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 95.2

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

   5. Applied rewrites 95.2%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{+48}:\\ \;\;\;\;\frac{x}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}} \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ x\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 8 \cdot 10^{-114}:\\ \;\;\;\;\frac{y \cdot z\_m}{\sqrt{t \cdot \left(-a\right)}} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\\ \end{array}\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s z_s x_m y z_m t a)
 :precision binary64
 (*
  x_s
  (*
   z_s
   (if (<= z_m 8e-114) (* (/ (* y z_m) (sqrt (* t (- a)))) x_m) (* x_m y)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double x_s, double z_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 8e-114) {
		tmp = ((y * z_m) / sqrt((t * -a))) * x_m;
	} else {
		tmp = x_m * y;
	}
	return x_s * (z_s * tmp);
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, z_s, x_m, y, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 8d-114) then
        tmp = ((y * z_m) / sqrt((t * -a))) * x_m
    else
        tmp = x_m * y
    end if
    code = x_s * (z_s * tmp)
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z_m && z_m < t && t < a;
public static double code(double x_s, double z_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 8e-114) {
		tmp = ((y * z_m) / Math.sqrt((t * -a))) * x_m;
	} else {
		tmp = x_m * y;
	}
	return x_s * (z_s * tmp);
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z_m, t, a] = sort([x_m, y, z_m, t, a])
def code(x_s, z_s, x_m, y, z_m, t, a):
	tmp = 0
	if z_m <= 8e-114:
		tmp = ((y * z_m) / math.sqrt((t * -a))) * x_m
	else:
		tmp = x_m * y
	return x_s * (z_s * tmp)

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(x_s, z_s, x_m, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 8e-114)
		tmp = Float64(Float64(Float64(y * z_m) / sqrt(Float64(t * Float64(-a)))) * x_m);
	else
		tmp = Float64(x_m * y);
	end
	return Float64(x_s * Float64(z_s * tmp))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z_m, t, a = num2cell(sort([x_m, y, z_m, t, a])){:}
function tmp_2 = code(x_s, z_s, x_m, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 8e-114)
		tmp = ((y * z_m) / sqrt((t * -a))) * x_m;
	else
		tmp = x_m * y;
	end
	tmp_2 = x_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]
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, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, z$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(x$95$s * N[(z$95$s * If[LessEqual[z$95$m, 8e-114], N[(N[(N[(y * z$95$m), $MachinePrecision] / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(x$95$m * y), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 8.0000000000000004e-114

   1. Initial program 65.9%

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

   3. Taylor expanded in a around 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 45.1

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

   5. Applied rewrites 45.1%

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

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 40.3

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

   7. Applied rewrites 40.3%

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

   if 8.0000000000000004e-114 < z

   1. Initial program 54.9%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 87.0

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

   5. Applied rewrites 87.0%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8 \cdot 10^{-114}:\\ \;\;\;\;\frac{y \cdot z}{\sqrt{t \cdot \left(-a\right)}} \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ x\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4.2 \cdot 10^{-132}:\\ \;\;\;\;\frac{x\_m}{\sqrt{t \cdot \left(-a\right)}} \cdot \left(y \cdot z\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\\ \end{array}\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s z_s x_m y z_m t a)
 :precision binary64
 (*
  x_s
  (*
   z_s
   (if (<= z_m 4.2e-132) (* (/ x_m (sqrt (* t (- a)))) (* y z_m)) (* x_m y)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double x_s, double z_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 4.2e-132) {
		tmp = (x_m / sqrt((t * -a))) * (y * z_m);
	} else {
		tmp = x_m * y;
	}
	return x_s * (z_s * tmp);
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, z_s, x_m, y, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 4.2d-132) then
        tmp = (x_m / sqrt((t * -a))) * (y * z_m)
    else
        tmp = x_m * y
    end if
    code = x_s * (z_s * tmp)
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z_m && z_m < t && t < a;
public static double code(double x_s, double z_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 4.2e-132) {
		tmp = (x_m / Math.sqrt((t * -a))) * (y * z_m);
	} else {
		tmp = x_m * y;
	}
	return x_s * (z_s * tmp);
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z_m, t, a] = sort([x_m, y, z_m, t, a])
def code(x_s, z_s, x_m, y, z_m, t, a):
	tmp = 0
	if z_m <= 4.2e-132:
		tmp = (x_m / math.sqrt((t * -a))) * (y * z_m)
	else:
		tmp = x_m * y
	return x_s * (z_s * tmp)

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(x_s, z_s, x_m, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 4.2e-132)
		tmp = Float64(Float64(x_m / sqrt(Float64(t * Float64(-a)))) * Float64(y * z_m));
	else
		tmp = Float64(x_m * y);
	end
	return Float64(x_s * Float64(z_s * tmp))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z_m, t, a = num2cell(sort([x_m, y, z_m, t, a])){:}
function tmp_2 = code(x_s, z_s, x_m, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 4.2e-132)
		tmp = (x_m / sqrt((t * -a))) * (y * z_m);
	else
		tmp = x_m * y;
	end
	tmp_2 = x_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]
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, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, z$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(x$95$s * N[(z$95$s * If[LessEqual[z$95$m, 4.2e-132], N[(N[(x$95$m / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(y * z$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.2000000000000002e-132

   1. Initial program 65.0%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 45.0

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

   5. Applied rewrites 45.0%

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

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* 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 41.3

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

   7. Applied rewrites 41.3%

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

   if 4.2000000000000002e-132 < z

   1. Initial program 56.6%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 85.7

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

   5. Applied rewrites 85.7%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.2 \cdot 10^{-132}:\\ \;\;\;\;\frac{x}{\sqrt{t \cdot \left(-a\right)}} \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ x\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;t \cdot a \leq -1.18 \cdot 10^{+294}:\\ \;\;\;\;\frac{\left(x\_m \cdot z\_m\right) \cdot y}{-z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\\ \end{array}\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s z_s x_m y z_m t a)
 :precision binary64
 (*
  x_s
  (*
   z_s
   (if (<= (* t a) -1.18e+294) (/ (* (* x_m z_m) y) (- z_m)) (* x_m y)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z_m && z_m < t && t < a);
double code(double x_s, double z_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if ((t * a) <= -1.18e+294) {
		tmp = ((x_m * z_m) * y) / -z_m;
	} else {
		tmp = x_m * y;
	}
	return x_s * (z_s * tmp);
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, z_s, x_m, y, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((t * a) <= (-1.18d+294)) then
        tmp = ((x_m * z_m) * y) / -z_m
    else
        tmp = x_m * y
    end if
    code = x_s * (z_s * tmp)
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z_m && z_m < t && t < a;
public static double code(double x_s, double z_s, double x_m, double y, double z_m, double t, double a) {
	double tmp;
	if ((t * a) <= -1.18e+294) {
		tmp = ((x_m * z_m) * y) / -z_m;
	} else {
		tmp = x_m * y;
	}
	return x_s * (z_s * tmp);
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z_m, t, a] = sort([x_m, y, z_m, t, a])
def code(x_s, z_s, x_m, y, z_m, t, a):
	tmp = 0
	if (t * a) <= -1.18e+294:
		tmp = ((x_m * z_m) * y) / -z_m
	else:
		tmp = x_m * y
	return x_s * (z_s * tmp)

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z_m, t, a = sort([x_m, y, z_m, t, a])
function code(x_s, z_s, x_m, y, z_m, t, a)
	tmp = 0.0
	if (Float64(t * a) <= -1.18e+294)
		tmp = Float64(Float64(Float64(x_m * z_m) * y) / Float64(-z_m));
	else
		tmp = Float64(x_m * y);
	end
	return Float64(x_s * Float64(z_s * tmp))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z_m, t, a = num2cell(sort([x_m, y, z_m, t, a])){:}
function tmp_2 = code(x_s, z_s, x_m, y, z_m, t, a)
	tmp = 0.0;
	if ((t * a) <= -1.18e+294)
		tmp = ((x_m * z_m) * y) / -z_m;
	else
		tmp = x_m * y;
	end
	tmp_2 = x_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]
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, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, z$95$s_, x$95$m_, y_, z$95$m_, t_, a_] := N[(x$95$s * N[(z$95$s * If[LessEqual[N[(t * a), $MachinePrecision], -1.18e+294], N[(N[(N[(x$95$m * z$95$m), $MachinePrecision] * y), $MachinePrecision] / (-z$95$m)), $MachinePrecision], N[(x$95$m * y), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t a) < -1.18000000000000002e294

   1. Initial program 48.4%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 35.6

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

   5. Applied rewrites 35.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 39.1

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

   7. Applied rewrites 39.1%

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

   if -1.18000000000000002e294 < (*.f64 t a)

   1. Initial program 63.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 50.0

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

   5. Applied rewrites 50.0%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot a \leq -1.18 \cdot 10^{+294}:\\ \;\;\;\;\frac{\left(x \cdot z\right) \cdot y}{-z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.7% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in a around 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 47.0

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

5. Applied rewrites 47.0%

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

6. Final simplification 47.0%

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

Developer Target 1: 88.0% 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 2024277 
(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)))))