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

Percentage Accurate: 61.3% → 91.8%

Time: 9.0s

Alternatives: 9

Speedup: 4.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.3% 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: 91.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ x\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.6 \cdot 10^{+142}:\\ \;\;\;\;\left(\frac{z\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}} \cdot y\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot y\right) \cdot x\_m\\ \end{array}\right) \end{array} \]

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 1.6e+142)
     (* (* (/ z_m (sqrt (fma (- a) t (* z_m z_m)))) y) x_m)
     (* (* 1.0 y) x_m)))))

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 <= 1.6e+142) {
		tmp = ((z_m / sqrt(fma(-a, t, (z_m * z_m)))) * y) * x_m;
	} else {
		tmp = (1.0 * y) * x_m;
	}
	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 <= 1.6e+142)
		tmp = Float64(Float64(Float64(z_m / sqrt(fma(Float64(-a), t, Float64(z_m * z_m)))) * y) * x_m);
	else
		tmp = Float64(Float64(1.0 * y) * x_m);
	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, 1.6e+142], N[(N[(N[(z$95$m / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(1.0 * y), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.60000000000000003e142

   1. Initial program 74.4%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 75.4%

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

   if 1.60000000000000003e142 < z

   1. Initial program 11.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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 12.0%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 100.0%

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

Alternative 2: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ x\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.05 \cdot 10^{+43}:\\ \;\;\;\;\left(z\_m \cdot y\right) \cdot \frac{x\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m \cdot z\_m}{\mathsf{fma}\left(a, \frac{t \cdot -0.5}{z\_m}, z\_m\right)}\\ \end{array}\right) \end{array} \]

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 1.05e+43)
     (* (* z_m y) (/ x_m (sqrt (fma (- a) t (* z_m z_m)))))
     (* y (/ (* x_m z_m) (fma a (/ (* t -0.5) z_m) z_m)))))))

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 <= 1.05e+43) {
		tmp = (z_m * y) * (x_m / sqrt(fma(-a, t, (z_m * z_m))));
	} else {
		tmp = y * ((x_m * z_m) / fma(a, ((t * -0.5) / z_m), z_m));
	}
	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 <= 1.05e+43)
		tmp = Float64(Float64(z_m * y) * Float64(x_m / sqrt(fma(Float64(-a), t, Float64(z_m * z_m)))));
	else
		tmp = Float64(y * Float64(Float64(x_m * z_m) / fma(a, Float64(Float64(t * -0.5) / z_m), z_m)));
	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, 1.05e+43], N[(N[(z$95$m * y), $MachinePrecision] * N[(x$95$m / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(x$95$m * z$95$m), $MachinePrecision] / N[(a * N[(N[(t * -0.5), $MachinePrecision] / z$95$m), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.05000000000000001e43

   1. Initial program 71.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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 71.8

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

      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 70.7

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

   4. Applied rewrites 70.7%

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

   if 1.05000000000000001e43 < z

   1. Initial program 49.7%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 86.4

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

   5. Applied rewrites 86.4%

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

   7. Applied rewrites 86.4%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\mathsf{fma}\left(a, \color{blue}{{z}^{-1} \cdot \left(-0.5 \cdot t\right)}, z\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 83.1

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 83.1

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

   9. Applied rewrites 83.1%

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

Alternative 3: 82.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ x\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.6 \cdot 10^{-99}:\\ \;\;\;\;\frac{\left(z\_m \cdot y\right) \cdot x\_m}{\sqrt{\left(-t\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot y\right) \cdot x\_m\\ \end{array}\right) \end{array} \]

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 1.6e-99)
     (/ (* (* z_m y) x_m) (sqrt (* (- t) a)))
     (* (* 1.0 y) x_m)))))

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 <= 1.6e-99) {
		tmp = ((z_m * y) * x_m) / sqrt((-t * a));
	} else {
		tmp = (1.0 * y) * x_m;
	}
	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 <= 1.6d-99) then
        tmp = ((z_m * y) * x_m) / sqrt((-t * a))
    else
        tmp = (1.0d0 * y) * x_m
    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 <= 1.6e-99) {
		tmp = ((z_m * y) * x_m) / Math.sqrt((-t * a));
	} else {
		tmp = (1.0 * y) * x_m;
	}
	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 <= 1.6e-99:
		tmp = ((z_m * y) * x_m) / math.sqrt((-t * a))
	else:
		tmp = (1.0 * y) * x_m
	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 <= 1.6e-99)
		tmp = Float64(Float64(Float64(z_m * y) * x_m) / sqrt(Float64(Float64(-t) * a)));
	else
		tmp = Float64(Float64(1.0 * y) * x_m);
	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 <= 1.6e-99)
		tmp = ((z_m * y) * x_m) / sqrt((-t * a));
	else
		tmp = (1.0 * y) * x_m;
	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, 1.6e-99], N[(N[(N[(z$95$m * y), $MachinePrecision] * x$95$m), $MachinePrecision] / N[Sqrt[N[((-t) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * y), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.6e-99

   1. Initial program 68.8%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 41.0

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

   5. Applied rewrites 41.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 40.4

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

   7. Applied rewrites 40.4%

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

   if 1.6e-99 < z

   1. Initial program 60.5%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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. 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 60.8%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 87.4%

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

Alternative 4: 83.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ x\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.6 \cdot 10^{-99}:\\ \;\;\;\;x\_m \cdot \frac{z\_m \cdot y}{\sqrt{\left(-a\right) \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot y\right) \cdot x\_m\\ \end{array}\right) \end{array} \]

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 1.6e-99)
     (* x_m (/ (* z_m y) (sqrt (* (- a) t))))
     (* (* 1.0 y) x_m)))))

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 <= 1.6e-99) {
		tmp = x_m * ((z_m * y) / sqrt((-a * t)));
	} else {
		tmp = (1.0 * y) * x_m;
	}
	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 <= 1.6d-99) then
        tmp = x_m * ((z_m * y) / sqrt((-a * t)))
    else
        tmp = (1.0d0 * y) * x_m
    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 <= 1.6e-99) {
		tmp = x_m * ((z_m * y) / Math.sqrt((-a * t)));
	} else {
		tmp = (1.0 * y) * x_m;
	}
	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 <= 1.6e-99:
		tmp = x_m * ((z_m * y) / math.sqrt((-a * t)))
	else:
		tmp = (1.0 * y) * x_m
	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 <= 1.6e-99)
		tmp = Float64(x_m * Float64(Float64(z_m * y) / sqrt(Float64(Float64(-a) * t))));
	else
		tmp = Float64(Float64(1.0 * y) * x_m);
	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 <= 1.6e-99)
		tmp = x_m * ((z_m * y) / sqrt((-a * t)));
	else
		tmp = (1.0 * y) * x_m;
	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, 1.6e-99], N[(x$95$m * N[(N[(z$95$m * y), $MachinePrecision] / N[Sqrt[N[((-a) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * y), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.6e-99

   1. Initial program 68.8%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 41.0

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

   5. Applied rewrites 41.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 38.4

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

   7. Applied rewrites 38.4%

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

   if 1.6e-99 < z

   1. Initial program 60.5%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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. 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 60.8%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 87.4%

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

Alternative 5: 82.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ x\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.6 \cdot 10^{-99}:\\ \;\;\;\;x\_m \cdot \left(y \cdot \frac{z\_m}{\sqrt{\left(-a\right) \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot y\right) \cdot x\_m\\ \end{array}\right) \end{array} \]

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 1.6e-99)
     (* x_m (* y (/ z_m (sqrt (* (- a) t)))))
     (* (* 1.0 y) x_m)))))

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 <= 1.6e-99) {
		tmp = x_m * (y * (z_m / sqrt((-a * t))));
	} else {
		tmp = (1.0 * y) * x_m;
	}
	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 <= 1.6d-99) then
        tmp = x_m * (y * (z_m / sqrt((-a * t))))
    else
        tmp = (1.0d0 * y) * x_m
    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 <= 1.6e-99) {
		tmp = x_m * (y * (z_m / Math.sqrt((-a * t))));
	} else {
		tmp = (1.0 * y) * x_m;
	}
	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 <= 1.6e-99:
		tmp = x_m * (y * (z_m / math.sqrt((-a * t))))
	else:
		tmp = (1.0 * y) * x_m
	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 <= 1.6e-99)
		tmp = Float64(x_m * Float64(y * Float64(z_m / sqrt(Float64(Float64(-a) * t)))));
	else
		tmp = Float64(Float64(1.0 * y) * x_m);
	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 <= 1.6e-99)
		tmp = x_m * (y * (z_m / sqrt((-a * t))));
	else
		tmp = (1.0 * y) * x_m;
	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, 1.6e-99], N[(x$95$m * N[(y * N[(z$95$m / N[Sqrt[N[((-a) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * y), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.6e-99

   1. Initial program 68.8%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 41.0

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

   5. Applied rewrites 41.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 39.4

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

   7. Applied rewrites 39.4%

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

   if 1.6e-99 < z

   1. Initial program 60.5%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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. 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 60.8%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 87.4%

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

Alternative 6: 75.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ x\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.15 \cdot 10^{-194}:\\ \;\;\;\;\frac{\left(x\_m \cdot z\_m\right) \cdot y}{-z\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot y\right) \cdot x\_m\\ \end{array}\right) \end{array} \]

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 1.15e-194) (/ (* (* x_m z_m) y) (- z_m)) (* (* 1.0 y) x_m)))))

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 <= 1.15e-194) {
		tmp = ((x_m * z_m) * y) / -z_m;
	} else {
		tmp = (1.0 * y) * x_m;
	}
	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 <= 1.15d-194) then
        tmp = ((x_m * z_m) * y) / -z_m
    else
        tmp = (1.0d0 * y) * x_m
    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 <= 1.15e-194) {
		tmp = ((x_m * z_m) * y) / -z_m;
	} else {
		tmp = (1.0 * y) * x_m;
	}
	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 <= 1.15e-194:
		tmp = ((x_m * z_m) * y) / -z_m
	else:
		tmp = (1.0 * y) * x_m
	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 <= 1.15e-194)
		tmp = Float64(Float64(Float64(x_m * z_m) * y) / Float64(-z_m));
	else
		tmp = Float64(Float64(1.0 * y) * x_m);
	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 <= 1.15e-194)
		tmp = ((x_m * z_m) * y) / -z_m;
	else
		tmp = (1.0 * y) * x_m;
	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, 1.15e-194], N[(N[(N[(x$95$m * z$95$m), $MachinePrecision] * y), $MachinePrecision] / (-z$95$m)), $MachinePrecision], N[(N[(1.0 * y), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.15000000000000001e-194

   1. Initial program 66.1%

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

   3. Taylor expanded in z around -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 67.3

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

   5. Applied rewrites 67.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 60.3

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

   7. Applied rewrites 60.3%

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

   if 1.15000000000000001e-194 < z

   1. Initial program 66.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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 64.3%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 80.1%

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

Alternative 7: 74.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z_m, t, a] = \mathsf{sort}([x_m, y, z_m, t, a])\\ \\ x\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.12 \cdot 10^{-194}:\\ \;\;\;\;\frac{\left(x\_m \cdot y\right) \cdot z\_m}{-z\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot y\right) \cdot x\_m\\ \end{array}\right) \end{array} \]

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 1.12e-194) (/ (* (* x_m y) z_m) (- z_m)) (* (* 1.0 y) x_m)))))

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 <= 1.12e-194) {
		tmp = ((x_m * y) * z_m) / -z_m;
	} else {
		tmp = (1.0 * y) * x_m;
	}
	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 <= 1.12d-194) then
        tmp = ((x_m * y) * z_m) / -z_m
    else
        tmp = (1.0d0 * y) * x_m
    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 <= 1.12e-194) {
		tmp = ((x_m * y) * z_m) / -z_m;
	} else {
		tmp = (1.0 * y) * x_m;
	}
	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 <= 1.12e-194:
		tmp = ((x_m * y) * z_m) / -z_m
	else:
		tmp = (1.0 * y) * x_m
	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 <= 1.12e-194)
		tmp = Float64(Float64(Float64(x_m * y) * z_m) / Float64(-z_m));
	else
		tmp = Float64(Float64(1.0 * y) * x_m);
	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 <= 1.12e-194)
		tmp = ((x_m * y) * z_m) / -z_m;
	else
		tmp = (1.0 * y) * x_m;
	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, 1.12e-194], N[(N[(N[(x$95$m * y), $MachinePrecision] * z$95$m), $MachinePrecision] / (-z$95$m)), $MachinePrecision], N[(N[(1.0 * y), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.12000000000000001e-194

   1. Initial program 66.1%

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

   3. Taylor expanded in z around -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 67.3

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

   5. Applied rewrites 67.3%

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

   if 1.12000000000000001e-194 < z

   1. Initial program 66.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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 64.3%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 80.1%

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

Alternative 8: 72.9% accurate, 4.1× speedup

Math

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

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 (* (* 1.0 y) x_m))))

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 * ((1.0 * y) * x_m));
}

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 * ((1.0d0 * y) * x_m))
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 * ((1.0 * y) * x_m));
}

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 * ((1.0 * y) * x_m))

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(Float64(1.0 * y) * x_m)))
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 * ((1.0 * y) * x_m));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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[(N[(1.0 * y), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.1%

   \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
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. 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 67.0%

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

5. Taylor expanded in z around inf

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

7. Applied rewrites 39.8%

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

Alternative 9: 14.4% accurate, 5.6× speedup

Math

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

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 (* (- y) x_m))))

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 * (-y * x_m));
}

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 * (-y * x_m))
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 * (-y * x_m));
}

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 * (-y * x_m))

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(Float64(-y) * x_m)))
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 * (-y * x_m));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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[((-y) * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.1%

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

3. Taylor expanded in z around -inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. neg-mul-1 N/A

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

   5. lower-neg.f64 47.7

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

5. Applied rewrites 47.7%

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

Developer Target 1: 88.2% 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 2024313 
(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)))))