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

Percentage Accurate: 60.2% → 91.9%

Time: 5.0s

Alternatives: 14

Speedup: 7.5×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    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: 60.2% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    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: 91.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{\left(x\_m \cdot y\_m\right) \cdot z\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}}\\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\left(x\_m \cdot y\_m\right) \cdot \frac{z\_m}{\mathsf{fma}\left(a, \frac{t}{z\_m \cdot z\_m} \cdot -0.5, 1\right) \cdot z\_m}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+207}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (let* ((t_1 (/ (* (* x_m y_m) z_m) (sqrt (- (* z_m z_m) (* t a))))))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= t_1 0.0)
        (* (* x_m y_m) (/ z_m (* (fma a (* (/ t (* z_m z_m)) -0.5) 1.0) z_m)))
        (if (<= t_1 2e+207) t_1 (* y_m x_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double t_1 = ((x_m * y_m) * z_m) / sqrt(((z_m * z_m) - (t * a)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (x_m * y_m) * (z_m / (fma(a, ((t / (z_m * z_m)) * -0.5), 1.0) * z_m));
	} else if (t_1 <= 2e+207) {
		tmp = t_1;
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = Float64(Float64(Float64(x_m * y_m) * z_m) / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(x_m * y_m) * Float64(z_m / Float64(fma(a, Float64(Float64(t / Float64(z_m * z_m)) * -0.5), 1.0) * z_m)));
	elseif (t_1 <= 2e+207)
		tmp = t_1;
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x$95$m * y$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$1, 0.0], N[(N[(x$95$m * y$95$m), $MachinePrecision] * N[(z$95$m / N[(N[(a * N[(N[(t / N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+207], t$95$1, N[(y$95$m * x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a)))) < 0.0

   1. Initial program 66.6%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 55.7

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

   5. Applied rewrites 55.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 57.4

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

   7. Applied rewrites 57.4%

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

   if 0.0 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a)))) < 2.0000000000000001e207

   1. Initial program 95.7%

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

   if 2.0000000000000001e207 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))))

   1. Initial program 13.9%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 38.7

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

   5. Applied rewrites 38.7%

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

Alternative 2: 90.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.38 \cdot 10^{+45}:\\ \;\;\;\;x\_m \cdot \left(\left(z\_m \cdot y\_m\right) \cdot {\left(\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}\right)}^{-1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot y\_m\right) \cdot \frac{z\_m}{\mathsf{fma}\left(a, \frac{t}{z\_m \cdot z\_m} \cdot -0.5, 1\right) \cdot z\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1.38e+45)
      (* x_m (* (* z_m y_m) (pow (sqrt (fma (- a) t (* z_m z_m))) -1.0)))
      (*
       (* x_m y_m)
       (/ z_m (* (fma a (* (/ t (* z_m z_m)) -0.5) 1.0) z_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.38e+45) {
		tmp = x_m * ((z_m * y_m) * pow(sqrt(fma(-a, t, (z_m * z_m))), -1.0));
	} else {
		tmp = (x_m * y_m) * (z_m / (fma(a, ((t / (z_m * z_m)) * -0.5), 1.0) * z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.38e+45)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) * (sqrt(fma(Float64(-a), t, Float64(z_m * z_m))) ^ -1.0)));
	else
		tmp = Float64(Float64(x_m * y_m) * Float64(z_m / Float64(fma(a, Float64(Float64(t / Float64(z_m * z_m)) * -0.5), 1.0) * z_m)));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.38e+45], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] * N[Power[N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * y$95$m), $MachinePrecision] * N[(z$95$m / N[(N[(a * N[(N[(t / N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.3799999999999999e45

   1. Initial program 65.2%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 65.2%

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

   7. Applied rewrites 64.0%

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

   if 1.3799999999999999e45 < z

   1. Initial program 38.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 79.5

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

   5. Applied rewrites 79.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 95.9

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

   7. Applied rewrites 95.9%

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

Alternative 3: 91.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \left(x\_m \cdot y\_m\right) \cdot z\_m\\ t_2 := \frac{t\_1}{\sqrt{z\_m \cdot z\_m - t \cdot a}}\\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(a \cdot \frac{t}{z\_m}, -0.5, z\_m\right)}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+207}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (let* ((t_1 (* (* x_m y_m) z_m)) (t_2 (/ t_1 (sqrt (- (* z_m z_m) (* t a))))))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= t_2 0.0)
        (/ t_1 (fma (* a (/ t z_m)) -0.5 z_m))
        (if (<= t_2 2e+207) t_2 (* y_m x_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double t_1 = (x_m * y_m) * z_m;
	double t_2 = t_1 / sqrt(((z_m * z_m) - (t * a)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = t_1 / fma((a * (t / z_m)), -0.5, z_m);
	} else if (t_2 <= 2e+207) {
		tmp = t_2;
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = Float64(Float64(x_m * y_m) * z_m)
	t_2 = Float64(t_1 / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(t_1 / fma(Float64(a * Float64(t / z_m)), -0.5, z_m));
	elseif (t_2 <= 2e+207)
		tmp = t_2;
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := Block[{t$95$1 = N[(N[(x$95$m * y$95$m), $MachinePrecision] * z$95$m), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$2, 0.0], N[(t$95$1 / N[(N[(a * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision] * -0.5 + z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+207], t$95$2, N[(y$95$m * x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a)))) < 0.0

   1. Initial program 66.6%

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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 55.8

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

   5. Applied rewrites 55.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 56.4

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

   7. Applied rewrites 56.4%

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

   if 0.0 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a)))) < 2.0000000000000001e207

   1. Initial program 95.7%

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

   if 2.0000000000000001e207 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))))

   1. Initial program 13.9%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 38.7

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

   5. Applied rewrites 38.7%

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

Alternative 4: 91.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \left(x\_m \cdot y\_m\right) \cdot z\_m\\ t_2 := \frac{t\_1}{\sqrt{z\_m \cdot z\_m - t \cdot a}}\\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(a \cdot \frac{t}{z\_m}, -0.5, z\_m\right)}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+207}:\\ \;\;\;\;\left(y\_m \cdot x\_m\right) \cdot \frac{z\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (let* ((t_1 (* (* x_m y_m) z_m)) (t_2 (/ t_1 (sqrt (- (* z_m z_m) (* t a))))))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= t_2 0.0)
        (/ t_1 (fma (* a (/ t z_m)) -0.5 z_m))
        (if (<= t_2 2e+207)
          (* (* y_m x_m) (/ z_m (sqrt (fma (- a) t (* z_m z_m)))))
          (* y_m x_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double t_1 = (x_m * y_m) * z_m;
	double t_2 = t_1 / sqrt(((z_m * z_m) - (t * a)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = t_1 / fma((a * (t / z_m)), -0.5, z_m);
	} else if (t_2 <= 2e+207) {
		tmp = (y_m * x_m) * (z_m / sqrt(fma(-a, t, (z_m * z_m))));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = Float64(Float64(x_m * y_m) * z_m)
	t_2 = Float64(t_1 / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(t_1 / fma(Float64(a * Float64(t / z_m)), -0.5, z_m));
	elseif (t_2 <= 2e+207)
		tmp = Float64(Float64(y_m * x_m) * Float64(z_m / sqrt(fma(Float64(-a), t, Float64(z_m * z_m)))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := Block[{t$95$1 = N[(N[(x$95$m * y$95$m), $MachinePrecision] * z$95$m), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$2, 0.0], N[(t$95$1 / N[(N[(a * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision] * -0.5 + z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+207], N[(N[(y$95$m * x$95$m), $MachinePrecision] * N[(z$95$m / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a)))) < 0.0

   1. Initial program 66.6%

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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 55.8

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

   5. Applied rewrites 55.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 56.4

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

   7. Applied rewrites 56.4%

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

   if 0.0 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a)))) < 2.0000000000000001e207

   1. Initial program 95.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. pow2 N/A

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

      15. *-commutative N/A

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

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

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

      17. mul-1-neg N/A

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

      18. associate-*r* N/A

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

      19. +-commutative N/A

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

      20. associate-*r* N/A

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

      21. mul-1-neg N/A

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

      22. lower-fma.f64 N/A

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

      23. lower-neg.f64 N/A

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

      24. pow2 N/A

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

      25. lift-*.f64 90.7

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

   4. Applied rewrites 90.7%

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

   if 2.0000000000000001e207 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))))

   1. Initial program 13.9%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 38.7

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

   5. Applied rewrites 38.7%

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

Alternative 5: 73.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x\_m \cdot y\_m\right) \cdot z\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}} \leq 10^{-241}:\\ \;\;\;\;\frac{y\_m \cdot \left(x\_m \cdot z\_m\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= (/ (* (* x_m y_m) z_m) (sqrt (- (* z_m z_m) (* t a)))) 1e-241)
      (/ (* y_m (* x_m z_m)) z_m)
      (* y_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if ((((x_m * y_m) * z_m) / sqrt(((z_m * z_m) - (t * a)))) <= 1e-241) {
		tmp = (y_m * (x_m * z_m)) / z_m;
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((((x_m * y_m) * z_m) / sqrt(((z_m * z_m) - (t * a)))) <= 1d-241) then
        tmp = (y_m * (x_m * z_m)) / z_m
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if ((((x_m * y_m) * z_m) / Math.sqrt(((z_m * z_m) - (t * a)))) <= 1e-241) {
		tmp = (y_m * (x_m * z_m)) / z_m;
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if (((x_m * y_m) * z_m) / math.sqrt(((z_m * z_m) - (t * a)))) <= 1e-241:
		tmp = (y_m * (x_m * z_m)) / z_m
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (Float64(Float64(Float64(x_m * y_m) * z_m) / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))) <= 1e-241)
		tmp = Float64(Float64(y_m * Float64(x_m * z_m)) / z_m);
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if ((((x_m * y_m) * z_m) / sqrt(((z_m * z_m) - (t * a)))) <= 1e-241)
		tmp = (y_m * (x_m * z_m)) / z_m;
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(N[(x$95$m * y$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-241], N[(N[(y$95$m * N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a)))) < 9.9999999999999997e-242

   1. Initial program 68.3%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 50.8%

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

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 48.1

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

   7. Applied rewrites 48.1%

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

   if 9.9999999999999997e-242 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))))

   1. Initial program 36.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 38.1

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

   5. Applied rewrites 38.1%

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

Alternative 6: 72.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x\_m \cdot y\_m\right) \cdot z\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}} \leq 10^{-241}:\\ \;\;\;\;y\_m \cdot \frac{z\_m \cdot x\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= (/ (* (* x_m y_m) z_m) (sqrt (- (* z_m z_m) (* t a)))) 1e-241)
      (* y_m (/ (* z_m x_m) z_m))
      (* y_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if ((((x_m * y_m) * z_m) / sqrt(((z_m * z_m) - (t * a)))) <= 1e-241) {
		tmp = y_m * ((z_m * x_m) / z_m);
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((((x_m * y_m) * z_m) / sqrt(((z_m * z_m) - (t * a)))) <= 1d-241) then
        tmp = y_m * ((z_m * x_m) / z_m)
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if ((((x_m * y_m) * z_m) / Math.sqrt(((z_m * z_m) - (t * a)))) <= 1e-241) {
		tmp = y_m * ((z_m * x_m) / z_m);
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if (((x_m * y_m) * z_m) / math.sqrt(((z_m * z_m) - (t * a)))) <= 1e-241:
		tmp = y_m * ((z_m * x_m) / z_m)
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (Float64(Float64(Float64(x_m * y_m) * z_m) / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))) <= 1e-241)
		tmp = Float64(y_m * Float64(Float64(z_m * x_m) / z_m));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if ((((x_m * y_m) * z_m) / sqrt(((z_m * z_m) - (t * a)))) <= 1e-241)
		tmp = y_m * ((z_m * x_m) / z_m);
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(N[(x$95$m * y$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-241], N[(y$95$m * N[(N[(z$95$m * x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a)))) < 9.9999999999999997e-242

   1. Initial program 68.3%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 50.8%

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

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 48.1

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

   7. Applied rewrites 48.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 45.6

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

   9. Applied rewrites 45.6%

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

   if 9.9999999999999997e-242 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))))

   1. Initial program 36.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 38.1

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

   5. Applied rewrites 38.1%

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

Alternative 7: 90.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5.8 \cdot 10^{-105}:\\ \;\;\;\;\left(z\_m \cdot y\_m\right) \cdot \frac{x\_m}{\sqrt{\mathsf{fma}\left(-t, a, z\_m \cdot z\_m\right)}}\\ \mathbf{elif}\;z\_m \leq 3.3 \cdot 10^{+151}:\\ \;\;\;\;\left(y\_m \cdot x\_m\right) \cdot \frac{z\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 5.8e-105)
      (* (* z_m y_m) (/ x_m (sqrt (fma (- t) a (* z_m z_m)))))
      (if (<= z_m 3.3e+151)
        (* (* y_m x_m) (/ z_m (sqrt (fma (- a) t (* z_m z_m)))))
        (* y_m x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 5.8e-105) {
		tmp = (z_m * y_m) * (x_m / sqrt(fma(-t, a, (z_m * z_m))));
	} else if (z_m <= 3.3e+151) {
		tmp = (y_m * x_m) * (z_m / sqrt(fma(-a, t, (z_m * z_m))));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 5.8e-105)
		tmp = Float64(Float64(z_m * y_m) * Float64(x_m / sqrt(fma(Float64(-t), a, Float64(z_m * z_m)))));
	elseif (z_m <= 3.3e+151)
		tmp = Float64(Float64(y_m * x_m) * Float64(z_m / sqrt(fma(Float64(-a), t, Float64(z_m * z_m)))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 5.8e-105], N[(N[(z$95$m * y$95$m), $MachinePrecision] * N[(x$95$m / N[Sqrt[N[((-t) * a + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 3.3e+151], N[(N[(y$95$m * x$95$m), $MachinePrecision] * N[(z$95$m / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 5.80000000000000007e-105

   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. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. pow2 N/A

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

      15. *-commutative N/A

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

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

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

      17. mul-1-neg N/A

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

      18. associate-*r* N/A

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

      19. +-commutative N/A

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

      20. associate-*r* N/A

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

      21. mul-1-neg N/A

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

      22. lower-fma.f64 N/A

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

      23. lower-neg.f64 N/A

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

      24. pow2 N/A

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

      25. lift-*.f64 60.6

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

   4. Applied rewrites 60.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-/.f64 61.0

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

   6. Applied rewrites 61.0%

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

   8. Applied rewrites 60.0%

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

   if 5.80000000000000007e-105 < z < 3.30000000000000025e151

   1. Initial program 93.0%

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

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. pow2 N/A

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

      15. *-commutative N/A

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

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

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

      17. mul-1-neg N/A

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

      18. associate-*r* N/A

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

      19. +-commutative N/A

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

      20. associate-*r* N/A

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

      21. mul-1-neg N/A

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

      22. lower-fma.f64 N/A

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

      23. lower-neg.f64 N/A

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

      24. pow2 N/A

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

      25. lift-*.f64 97.6

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

   4. Applied rewrites 97.6%

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

   if 3.30000000000000025e151 < z

   1. Initial program 20.3%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 98.1

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

   5. Applied rewrites 98.1%

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

Alternative 8: 91.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 7.5 \cdot 10^{-170}:\\ \;\;\;\;\frac{\left(z\_m \cdot y\_m\right) \cdot x\_m}{\sqrt{\left(-a\right) \cdot t}}\\ \mathbf{elif}\;z\_m \leq 3.3 \cdot 10^{+151}:\\ \;\;\;\;y\_m \cdot \left(x\_m \cdot \frac{z\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 7.5e-170)
      (/ (* (* z_m y_m) x_m) (sqrt (* (- a) t)))
      (if (<= z_m 3.3e+151)
        (* y_m (* x_m (/ z_m (sqrt (fma (- a) t (* z_m z_m))))))
        (* y_m x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 7.5e-170) {
		tmp = ((z_m * y_m) * x_m) / sqrt((-a * t));
	} else if (z_m <= 3.3e+151) {
		tmp = y_m * (x_m * (z_m / sqrt(fma(-a, t, (z_m * z_m)))));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 7.5e-170)
		tmp = Float64(Float64(Float64(z_m * y_m) * x_m) / sqrt(Float64(Float64(-a) * t)));
	elseif (z_m <= 3.3e+151)
		tmp = Float64(y_m * Float64(x_m * Float64(z_m / sqrt(fma(Float64(-a), t, Float64(z_m * z_m))))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 7.5e-170], N[(N[(N[(z$95$m * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] / N[Sqrt[N[((-a) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 3.3e+151], N[(y$95$m * N[(x$95$m * N[(z$95$m / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 7.4999999999999998e-170

   1. Initial program 60.3%

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

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. pow2 N/A

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

      15. *-commutative N/A

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

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

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

      17. mul-1-neg N/A

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

      18. associate-*r* N/A

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

      19. +-commutative N/A

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

      20. associate-*r* N/A

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

      21. mul-1-neg N/A

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

      22. lower-fma.f64 N/A

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

      23. lower-neg.f64 N/A

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

      24. pow2 N/A

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

      25. lift-*.f64 59.8

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

   4. Applied rewrites 59.8%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-out N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 36.2

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

   7. Applied rewrites 36.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 36.3

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

   9. Applied rewrites 36.3%

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

   if 7.4999999999999998e-170 < z < 3.30000000000000025e151

   1. Initial program 88.4%

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

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. pow2 N/A

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

      15. *-commutative N/A

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

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

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

      17. mul-1-neg N/A

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

      18. associate-*r* N/A

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

      19. +-commutative N/A

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

      20. associate-*r* N/A

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

      21. mul-1-neg N/A

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

      22. lower-fma.f64 N/A

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

      23. lower-neg.f64 N/A

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

      24. pow2 N/A

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

      25. lift-*.f64 94.2

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

   4. Applied rewrites 94.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-/.f64 92.3

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

   6. Applied rewrites 92.3%

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

   if 3.30000000000000025e151 < z

   1. Initial program 20.3%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 98.1

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

   5. Applied rewrites 98.1%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.5 \cdot 10^{-170}:\\ \;\;\;\;\frac{\left(z \cdot y\right) \cdot x}{\sqrt{\left(-a\right) \cdot t}}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+151}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 600000:\\ \;\;\;\;\frac{\left(z\_m \cdot y\_m\right) \cdot x\_m}{\sqrt{\left(-a\right) \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 600000.0)
      (/ (* (* z_m y_m) x_m) (sqrt (* (- a) t)))
      (* y_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 600000.0) {
		tmp = ((z_m * y_m) * x_m) / sqrt((-a * t));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 600000.0d0) then
        tmp = ((z_m * y_m) * x_m) / sqrt((-a * t))
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 600000.0) {
		tmp = ((z_m * y_m) * x_m) / Math.sqrt((-a * t));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 600000.0:
		tmp = ((z_m * y_m) * x_m) / math.sqrt((-a * t))
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 600000.0)
		tmp = Float64(Float64(Float64(z_m * y_m) * x_m) / sqrt(Float64(Float64(-a) * t)));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 600000.0)
		tmp = ((z_m * y_m) * x_m) / sqrt((-a * t));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 600000.0], N[(N[(N[(z$95$m * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] / N[Sqrt[N[((-a) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 6e5

   1. Initial program 63.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. pow2 N/A

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

      15. *-commutative N/A

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

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

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

      17. mul-1-neg N/A

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

      18. associate-*r* N/A

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

      19. +-commutative N/A

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

      20. associate-*r* N/A

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

      21. mul-1-neg N/A

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

      22. lower-fma.f64 N/A

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

      23. lower-neg.f64 N/A

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

      24. pow2 N/A

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

      25. lift-*.f64 63.9

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

   4. Applied rewrites 63.9%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-out N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 39.2

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

   7. Applied rewrites 39.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 39.3

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

   9. Applied rewrites 39.3%

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

   if 6e5 < z

   1. Initial program 43.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 95.8

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

   5. Applied rewrites 95.8%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 600000:\\ \;\;\;\;\frac{\left(z \cdot y\right) \cdot x}{\sqrt{\left(-a\right) \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 600000:\\ \;\;\;\;\frac{\left(x\_m \cdot y\_m\right) \cdot z\_m}{\sqrt{\left(-a\right) \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 600000.0)
      (/ (* (* x_m y_m) z_m) (sqrt (* (- a) t)))
      (* y_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 600000.0) {
		tmp = ((x_m * y_m) * z_m) / sqrt((-a * t));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 600000.0d0) then
        tmp = ((x_m * y_m) * z_m) / sqrt((-a * t))
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 600000.0) {
		tmp = ((x_m * y_m) * z_m) / Math.sqrt((-a * t));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 600000.0:
		tmp = ((x_m * y_m) * z_m) / math.sqrt((-a * t))
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 600000.0)
		tmp = Float64(Float64(Float64(x_m * y_m) * z_m) / sqrt(Float64(Float64(-a) * t)));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 600000.0)
		tmp = ((x_m * y_m) * z_m) / sqrt((-a * t));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 600000.0], N[(N[(N[(x$95$m * y$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] / N[Sqrt[N[((-a) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 6e5

   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

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 41.0

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

   5. Applied rewrites 41.0%

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

   if 6e5 < z

   1. Initial program 43.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 95.8

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

   5. Applied rewrites 95.8%

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

Alternative 11: 82.7% accurate, 1.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1e-99)
      (/ (* y_m (* x_m z_m)) (sqrt (* (- a) t)))
      (* y_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1e-99) {
		tmp = (y_m * (x_m * z_m)) / sqrt((-a * t));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1d-99) then
        tmp = (y_m * (x_m * z_m)) / sqrt((-a * t))
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1e-99) {
		tmp = (y_m * (x_m * z_m)) / Math.sqrt((-a * t));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1e-99:
		tmp = (y_m * (x_m * z_m)) / math.sqrt((-a * t))
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1e-99)
		tmp = Float64(Float64(y_m * Float64(x_m * z_m)) / sqrt(Float64(Float64(-a) * t)));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1e-99)
		tmp = (y_m * (x_m * z_m)) / sqrt((-a * t));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1e-99], N[(N[(y$95$m * N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[((-a) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1e-99

   1. Initial program 60.8%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 24.2%

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

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 24.7

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

   7. Applied rewrites 24.7%

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

   8. Taylor expanded in z around 0

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

      1. sqrt-unprod N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

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

      4. distribute-lft-neg-out N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-sqrt.f64 38.2

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

   10. Applied rewrites 38.2%

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

   if 1e-99 < z

   1. Initial program 52.5%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

Alternative 12: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 600000:\\ \;\;\;\;\left(y\_m \cdot x\_m\right) \cdot \frac{z\_m}{\sqrt{\left(-a\right) \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 600000.0)
      (* (* y_m x_m) (/ z_m (sqrt (* (- a) t))))
      (* y_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 600000.0) {
		tmp = (y_m * x_m) * (z_m / sqrt((-a * t)));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 600000.0d0) then
        tmp = (y_m * x_m) * (z_m / sqrt((-a * t)))
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 600000.0) {
		tmp = (y_m * x_m) * (z_m / Math.sqrt((-a * t)));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 600000.0:
		tmp = (y_m * x_m) * (z_m / math.sqrt((-a * t)))
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 600000.0)
		tmp = Float64(Float64(y_m * x_m) * Float64(z_m / sqrt(Float64(Float64(-a) * t))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 600000.0)
		tmp = (y_m * x_m) * (z_m / sqrt((-a * t)));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 600000.0], N[(N[(y$95$m * x$95$m), $MachinePrecision] * N[(z$95$m / N[Sqrt[N[((-a) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 6e5

   1. Initial program 63.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. pow2 N/A

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

      15. *-commutative N/A

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

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

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

      17. mul-1-neg N/A

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

      18. associate-*r* N/A

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

      19. +-commutative N/A

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

      20. associate-*r* N/A

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

      21. mul-1-neg N/A

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

      22. lower-fma.f64 N/A

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

      23. lower-neg.f64 N/A

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

      24. pow2 N/A

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

      25. lift-*.f64 63.9

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

   4. Applied rewrites 63.9%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-out N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 39.2

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

   7. Applied rewrites 39.2%

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

   if 6e5 < z

   1. Initial program 43.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 95.8

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

   5. Applied rewrites 95.8%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 600000:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{\sqrt{\left(-a\right) \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 82.6% accurate, 1.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1e-99)
      (* y_m (* x_m (/ z_m (sqrt (* (- a) t)))))
      (* y_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1e-99) {
		tmp = y_m * (x_m * (z_m / sqrt((-a * t))));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1d-99) then
        tmp = y_m * (x_m * (z_m / sqrt((-a * t))))
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1e-99) {
		tmp = y_m * (x_m * (z_m / Math.sqrt((-a * t))));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1e-99:
		tmp = y_m * (x_m * (z_m / math.sqrt((-a * t))))
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1e-99)
		tmp = Float64(y_m * Float64(x_m * Float64(z_m / sqrt(Float64(Float64(-a) * t)))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1e-99)
		tmp = y_m * (x_m * (z_m / sqrt((-a * t))));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1e-99], N[(y$95$m * N[(x$95$m * N[(z$95$m / N[Sqrt[N[((-a) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1e-99

   1. Initial program 60.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. pow2 N/A

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

      15. *-commutative N/A

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

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

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

      17. mul-1-neg N/A

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

      18. associate-*r* N/A

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

      19. +-commutative N/A

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

      20. associate-*r* N/A

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

      21. mul-1-neg N/A

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

      22. lower-fma.f64 N/A

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

      23. lower-neg.f64 N/A

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

      24. pow2 N/A

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

      25. lift-*.f64 60.9

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

   4. Applied rewrites 60.9%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-out N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 37.0

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

   7. Applied rewrites 37.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 35.1

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

   9. Applied rewrites 35.1%

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

   if 1e-99 < z

   1. Initial program 52.5%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{-99}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{\left(-a\right) \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 73.6% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
z\_m =     private
z\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = z_s * (y_s * (x_s * (y_m * x_m)))
end function

Java

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

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = z_s * (y_s * (x_s * (y_m * x_m)));
end

Wolfram

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

Derivation

1. Initial program 57.9%

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

3. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 43.8

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

5. Applied rewrites 43.8%

   \[\leadsto \color{blue}{y \cdot x} \]
6. Add Preprocessing

Developer Target 1: 89.3% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    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 2025064 
(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)))))