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

Percentage Accurate: 61.1% → 89.6%

Time: 10.0s

Alternatives: 11

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: 61.1% 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: 89.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{\left(x \cdot y\right) \cdot z\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m}{\mathsf{fma}\left(\frac{t}{z\_m}, -0.5 \cdot a, z\_m\right)} \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t a)
 :precision binary64
 (*
  z_s
  (if (<= z_m 5.1e+93)
    (/ (* (* x y) z_m) (sqrt (- (* z_m z_m) (* t a))))
    (* (/ z_m (fma (/ t z_m) (* -0.5 a) z_m)) (* y x)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 5.1e+93) {
		tmp = ((x * y) * z_m) / sqrt(((z_m * z_m) - (t * a)));
	} else {
		tmp = (z_m / fma((t / z_m), (-0.5 * a), z_m)) * (y * x);
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 5.1e+93)
		tmp = Float64(Float64(Float64(x * y) * z_m) / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a))));
	else
		tmp = Float64(Float64(z_m / fma(Float64(t / z_m), Float64(-0.5 * a), z_m)) * Float64(y * x));
	end
	return 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]
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * If[LessEqual[z$95$m, 5.1e+93], N[(N[(N[(x * y), $MachinePrecision] * z$95$m), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(z$95$m / N[(N[(t / z$95$m), $MachinePrecision] * N[(-0.5 * a), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision] * N[(y * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 5.0999999999999996e93

   1. Initial program 70.6%

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

   if 5.0999999999999996e93 < z

   1. Initial program 20.5%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 79.9

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

   5. Applied rewrites 79.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 96.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

   7. Applied rewrites 96.4%

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

Alternative 2: 76.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.25 \cdot 10^{-69}:\\ \;\;\;\;{z\_m}^{-1} \cdot \left(\left(z\_m \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t a)
 :precision binary64
 (*
  z_s
  (if (<= z_m 1.25e-69) (* (pow z_m -1.0) (* (* z_m y) x)) (* 1.0 (* y x)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.25e-69) {
		tmp = pow(z_m, -1.0) * ((z_m * y) * x);
	} else {
		tmp = 1.0 * (y * x);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    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 <= 1.25d-69) then
        tmp = (z_m ** (-1.0d0)) * ((z_m * y) * x)
    else
        tmp = 1.0d0 * (y * x)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.25e-69) {
		tmp = Math.pow(z_m, -1.0) * ((z_m * y) * x);
	} else {
		tmp = 1.0 * (y * x);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t, a):
	tmp = 0
	if z_m <= 1.25e-69:
		tmp = math.pow(z_m, -1.0) * ((z_m * y) * x)
	else:
		tmp = 1.0 * (y * x)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.25e-69)
		tmp = Float64((z_m ^ -1.0) * Float64(Float64(z_m * y) * x));
	else
		tmp = Float64(1.0 * Float64(y * x));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.25e-69)
		tmp = (z_m ^ -1.0) * ((z_m * y) * x);
	else
		tmp = 1.0 * (y * x);
	end
	tmp_2 = 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]
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * If[LessEqual[z$95$m, 1.25e-69], N[(N[Power[z$95$m, -1.0], $MachinePrecision] * N[(N[(z$95$m * y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(y * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.25000000000000008e-69

   1. Initial program 64.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 60.6

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

   5. Applied rewrites 60.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 19.6%

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

   if 1.25000000000000008e-69 < z

   1. Initial program 51.7%

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

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 76.5

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

   5. Applied rewrites 76.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 86.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

   7. Applied rewrites 86.8%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 85.4%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.25 \cdot 10^{-69}:\\ \;\;\;\;{z}^{-1} \cdot \left(\left(z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 6 \cdot 10^{-119}:\\ \;\;\;\;\sqrt{\frac{-1}{a \cdot t}} \cdot \left(\left(z\_m \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m}{\mathsf{fma}\left(\frac{t}{z\_m}, -0.5 \cdot a, z\_m\right)} \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t a)
 :precision binary64
 (*
  z_s
  (if (<= z_m 6e-119)
    (* (sqrt (/ -1.0 (* a t))) (* (* z_m y) x))
    (* (/ z_m (fma (/ t z_m) (* -0.5 a) z_m)) (* y x)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 6e-119) {
		tmp = sqrt((-1.0 / (a * t))) * ((z_m * y) * x);
	} else {
		tmp = (z_m / fma((t / z_m), (-0.5 * a), z_m)) * (y * x);
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 6e-119)
		tmp = Float64(sqrt(Float64(-1.0 / Float64(a * t))) * Float64(Float64(z_m * y) * x));
	else
		tmp = Float64(Float64(z_m / fma(Float64(t / z_m), Float64(-0.5 * a), z_m)) * Float64(y * x));
	end
	return 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]
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * If[LessEqual[z$95$m, 6e-119], N[(N[Sqrt[N[(-1.0 / N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(z$95$m * y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(z$95$m / N[(N[(t / z$95$m), $MachinePrecision] * N[(-0.5 * a), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision] * N[(y * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 6.0000000000000004e-119

   1. Initial program 63.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 59.9

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

   5. Applied rewrites 59.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 34.7%

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

   if 6.0000000000000004e-119 < z

   1. Initial program 53.2%

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

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 76.7

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

   5. Applied rewrites 76.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

   7. Applied rewrites 87.4%

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

Alternative 4: 83.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 6 \cdot 10^{-119}:\\ \;\;\;\;\sqrt{\frac{-1}{a \cdot t}} \cdot \left(\left(z\_m \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z\_m}{\mathsf{fma}\left(\frac{t}{z\_m}, -0.5 \cdot a, z\_m\right)}\right)\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t a)
 :precision binary64
 (*
  z_s
  (if (<= z_m 6e-119)
    (* (sqrt (/ -1.0 (* a t))) (* (* z_m y) x))
    (* y (* x (/ z_m (fma (/ t z_m) (* -0.5 a) z_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 6e-119) {
		tmp = sqrt((-1.0 / (a * t))) * ((z_m * y) * x);
	} else {
		tmp = y * (x * (z_m / fma((t / z_m), (-0.5 * a), z_m)));
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 6e-119)
		tmp = Float64(sqrt(Float64(-1.0 / Float64(a * t))) * Float64(Float64(z_m * y) * x));
	else
		tmp = Float64(y * Float64(x * Float64(z_m / fma(Float64(t / z_m), Float64(-0.5 * a), z_m))));
	end
	return 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]
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * If[LessEqual[z$95$m, 6e-119], N[(N[Sqrt[N[(-1.0 / N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(z$95$m * y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(y * N[(x * N[(z$95$m / N[(N[(t / z$95$m), $MachinePrecision] * N[(-0.5 * a), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 6.0000000000000004e-119

   1. Initial program 63.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 59.9

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

   5. Applied rewrites 59.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 34.7%

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

   if 6.0000000000000004e-119 < z

   1. Initial program 53.2%

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

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 76.7

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

   5. Applied rewrites 76.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 87.4

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

   7. Applied rewrites 87.4%

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

Alternative 5: 82.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 6.4 \cdot 10^{-119}:\\ \;\;\;\;\sqrt{\frac{-1}{a \cdot t}} \cdot \left(\left(z\_m \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t a)
 :precision binary64
 (*
  z_s
  (if (<= z_m 6.4e-119)
    (* (sqrt (/ -1.0 (* a t))) (* (* z_m y) x))
    (* 1.0 (* y x)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 6.4e-119) {
		tmp = sqrt((-1.0 / (a * t))) * ((z_m * y) * x);
	} else {
		tmp = 1.0 * (y * x);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    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 <= 6.4d-119) then
        tmp = sqrt(((-1.0d0) / (a * t))) * ((z_m * y) * x)
    else
        tmp = 1.0d0 * (y * x)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 6.4e-119) {
		tmp = Math.sqrt((-1.0 / (a * t))) * ((z_m * y) * x);
	} else {
		tmp = 1.0 * (y * x);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t, a):
	tmp = 0
	if z_m <= 6.4e-119:
		tmp = math.sqrt((-1.0 / (a * t))) * ((z_m * y) * x)
	else:
		tmp = 1.0 * (y * x)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 6.4e-119)
		tmp = Float64(sqrt(Float64(-1.0 / Float64(a * t))) * Float64(Float64(z_m * y) * x));
	else
		tmp = Float64(1.0 * Float64(y * x));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 6.4e-119)
		tmp = sqrt((-1.0 / (a * t))) * ((z_m * y) * x);
	else
		tmp = 1.0 * (y * x);
	end
	tmp_2 = 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]
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * If[LessEqual[z$95$m, 6.4e-119], N[(N[Sqrt[N[(-1.0 / N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(z$95$m * y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(y * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 6.39999999999999986e-119

   1. Initial program 63.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 59.9

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

   5. Applied rewrites 59.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 34.7%

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

   if 6.39999999999999986e-119 < z

   1. Initial program 53.2%

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

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 76.7

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

   5. Applied rewrites 76.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

   7. Applied rewrites 87.4%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 84.4%

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

Alternative 6: 82.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 6.4 \cdot 10^{-119}:\\ \;\;\;\;\frac{\left(z\_m \cdot y\right) \cdot x}{\sqrt{\left(-a\right) \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t a)
 :precision binary64
 (*
  z_s
  (if (<= z_m 6.4e-119)
    (/ (* (* z_m y) x) (sqrt (* (- a) t)))
    (* 1.0 (* y x)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 6.4e-119) {
		tmp = ((z_m * y) * x) / sqrt((-a * t));
	} else {
		tmp = 1.0 * (y * x);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    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 <= 6.4d-119) then
        tmp = ((z_m * y) * x) / sqrt((-a * t))
    else
        tmp = 1.0d0 * (y * x)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 6.4e-119) {
		tmp = ((z_m * y) * x) / Math.sqrt((-a * t));
	} else {
		tmp = 1.0 * (y * x);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t, a):
	tmp = 0
	if z_m <= 6.4e-119:
		tmp = ((z_m * y) * x) / math.sqrt((-a * t))
	else:
		tmp = 1.0 * (y * x)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 6.4e-119)
		tmp = Float64(Float64(Float64(z_m * y) * x) / sqrt(Float64(Float64(-a) * t)));
	else
		tmp = Float64(1.0 * Float64(y * x));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 6.4e-119)
		tmp = ((z_m * y) * x) / sqrt((-a * t));
	else
		tmp = 1.0 * (y * x);
	end
	tmp_2 = 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]
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * If[LessEqual[z$95$m, 6.4e-119], N[(N[(N[(z$95$m * y), $MachinePrecision] * x), $MachinePrecision] / N[Sqrt[N[((-a) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(y * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 6.39999999999999986e-119

   1. Initial program 63.8%

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

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 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 38.7

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

   5. Applied rewrites 38.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 34.0

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

   7. Applied rewrites 34.0%

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

   if 6.39999999999999986e-119 < z

   1. Initial program 53.2%

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

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 76.7

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

   5. Applied rewrites 76.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

   7. Applied rewrites 87.4%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 84.4%

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

Alternative 7: 82.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 6.4 \cdot 10^{-119}:\\ \;\;\;\;\frac{\left(x \cdot y\right) \cdot z\_m}{\sqrt{\left(-a\right) \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t a)
 :precision binary64
 (*
  z_s
  (if (<= z_m 6.4e-119)
    (/ (* (* x y) z_m) (sqrt (* (- a) t)))
    (* 1.0 (* y x)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 6.4e-119) {
		tmp = ((x * y) * z_m) / sqrt((-a * t));
	} else {
		tmp = 1.0 * (y * x);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    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 <= 6.4d-119) then
        tmp = ((x * y) * z_m) / sqrt((-a * t))
    else
        tmp = 1.0d0 * (y * x)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 6.4e-119) {
		tmp = ((x * y) * z_m) / Math.sqrt((-a * t));
	} else {
		tmp = 1.0 * (y * x);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t, a):
	tmp = 0
	if z_m <= 6.4e-119:
		tmp = ((x * y) * z_m) / math.sqrt((-a * t))
	else:
		tmp = 1.0 * (y * x)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 6.4e-119)
		tmp = Float64(Float64(Float64(x * y) * z_m) / sqrt(Float64(Float64(-a) * t)));
	else
		tmp = Float64(1.0 * Float64(y * x));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 6.4e-119)
		tmp = ((x * y) * z_m) / sqrt((-a * t));
	else
		tmp = 1.0 * (y * x);
	end
	tmp_2 = 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]
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * If[LessEqual[z$95$m, 6.4e-119], N[(N[(N[(x * y), $MachinePrecision] * z$95$m), $MachinePrecision] / N[Sqrt[N[((-a) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(y * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 6.39999999999999986e-119

   1. Initial program 63.8%

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

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 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 38.7

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

   5. Applied rewrites 38.7%

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

   if 6.39999999999999986e-119 < z

   1. Initial program 53.2%

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

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 76.7

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

   5. Applied rewrites 76.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

   7. Applied rewrites 87.4%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 84.4%

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

Alternative 8: 75.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4 \cdot 10^{-173}:\\ \;\;\;\;\frac{\left(z\_m \cdot x\right) \cdot y}{-z\_m}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t a)
 :precision binary64
 (* z_s (if (<= z_m 4e-173) (/ (* (* z_m x) y) (- z_m)) (* 1.0 (* y x)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 4e-173) {
		tmp = ((z_m * x) * y) / -z_m;
	} else {
		tmp = 1.0 * (y * x);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    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 <= 4d-173) then
        tmp = ((z_m * x) * y) / -z_m
    else
        tmp = 1.0d0 * (y * x)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 4e-173) {
		tmp = ((z_m * x) * y) / -z_m;
	} else {
		tmp = 1.0 * (y * x);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t, a):
	tmp = 0
	if z_m <= 4e-173:
		tmp = ((z_m * x) * y) / -z_m
	else:
		tmp = 1.0 * (y * x)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 4e-173)
		tmp = Float64(Float64(Float64(z_m * x) * y) / Float64(-z_m));
	else
		tmp = Float64(1.0 * Float64(y * x));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 4e-173)
		tmp = ((z_m * x) * y) / -z_m;
	else
		tmp = 1.0 * (y * x);
	end
	tmp_2 = 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]
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * If[LessEqual[z$95$m, 4e-173], N[(N[(N[(z$95$m * x), $MachinePrecision] * y), $MachinePrecision] / (-z$95$m)), $MachinePrecision], N[(1.0 * N[(y * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.0000000000000002e-173

   1. Initial program 61.4%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 63.1

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

   5. Applied rewrites 63.1%

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

      1. lift-*.f64 N/A

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

      2. 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 62.4

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

   7. Applied rewrites 62.4%

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

   if 4.0000000000000002e-173 < z

   1. Initial program 57.1%

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

   3. Taylor expanded in 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-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 74.3

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

   5. Applied rewrites 74.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 83.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

   7. Applied rewrites 83.9%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 79.6%

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

Alternative 9: 74.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5.8 \cdot 10^{-176}:\\ \;\;\;\;\frac{\left(x \cdot y\right) \cdot z\_m}{-z\_m}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t a)
 :precision binary64
 (* z_s (if (<= z_m 5.8e-176) (/ (* (* x y) z_m) (- z_m)) (* 1.0 (* y x)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 5.8e-176) {
		tmp = ((x * y) * z_m) / -z_m;
	} else {
		tmp = 1.0 * (y * x);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    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 <= 5.8d-176) then
        tmp = ((x * y) * z_m) / -z_m
    else
        tmp = 1.0d0 * (y * x)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 5.8e-176) {
		tmp = ((x * y) * z_m) / -z_m;
	} else {
		tmp = 1.0 * (y * x);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t, a):
	tmp = 0
	if z_m <= 5.8e-176:
		tmp = ((x * y) * z_m) / -z_m
	else:
		tmp = 1.0 * (y * x)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t, a)
	tmp = 0.0
	if (z_m <= 5.8e-176)
		tmp = Float64(Float64(Float64(x * y) * z_m) / Float64(-z_m));
	else
		tmp = Float64(1.0 * Float64(y * x));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 5.8e-176)
		tmp = ((x * y) * z_m) / -z_m;
	else
		tmp = 1.0 * (y * x);
	end
	tmp_2 = 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]
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * If[LessEqual[z$95$m, 5.8e-176], N[(N[(N[(x * y), $MachinePrecision] * z$95$m), $MachinePrecision] / (-z$95$m)), $MachinePrecision], N[(1.0 * N[(y * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 5.80000000000000012e-176

   1. Initial program 61.4%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 63.1

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

   5. Applied rewrites 63.1%

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

   if 5.80000000000000012e-176 < z

   1. Initial program 57.1%

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

   3. Taylor expanded in 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-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 74.3

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

   5. Applied rewrites 74.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 83.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

   7. Applied rewrites 83.9%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 79.6%

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

Alternative 10: 72.9% accurate, 4.1× speedup

Math

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

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t a) :precision binary64 (* z_s (* 1.0 (* y x))))

C

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

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = z_s * (1.0d0 * (y * x))
end function

Java

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

Python

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

Julia

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

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp = code(z_s, x, y, z_m, t, a)
	tmp = z_s * (1.0 * (y * x));
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]
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[(1.0 * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.4%

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

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 43.3

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

5. Applied rewrites 43.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 47.7

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

7. Applied rewrites 47.7%

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

8. Taylor expanded in z around inf

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

10. Applied rewrites 41.1%

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

Alternative 11: 13.8% accurate, 5.6× speedup

Math

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

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t a) :precision binary64 (* z_s (* (- y) x)))

C

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

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = z_s * (-y * x)
end function

Java

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

Python

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

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t, a)
	return Float64(z_s * Float64(Float64(-y) * x))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp = code(z_s, x, y, z_m, t, a)
	tmp = z_s * (-y * x);
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]
code[z$95$s_, x_, y_, z$95$m_, t_, a_] := N[(z$95$s * N[((-y) * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.4%

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

3. Taylor expanded in z around -inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 43.5

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

5. Applied rewrites 43.5%

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

Developer Target 1: 89.0% 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 2024339 
(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)))))