Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, I

Percentage Accurate: 91.6% → 96.3%

Time: 7.7s

Alternatives: 11

Speedup: 0.8×

• 28.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

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 * 9.0d0) * t)) / (a * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

Python

def code(x, y, z, t, a):
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0)

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a * 2.0))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 91.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

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 * 9.0d0) * t)) / (a * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

Python

def code(x, y, z, t, a):
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0)

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a * 2.0))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a\_m \cdot 2}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x, t \cdot \left(\frac{z}{y} \cdot -4.5\right)\right)}{a\_m} \cdot y\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t \cdot -9, z, y \cdot x\right)}{a\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, y, \left(\frac{z}{x} \cdot -4.5\right) \cdot t\right) \cdot \frac{x}{a\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (/ (- (* x y) (* (* z 9.0) t)) (* a_m 2.0))))
   (*
    a_s
    (if (<= t_1 (- INFINITY))
      (* (/ (fma 0.5 x (* t (* (/ z y) -4.5))) a_m) y)
      (if (<= t_1 2e+302)
        (/ (/ (fma (* t -9.0) z (* y x)) a_m) 2.0)
        (* (fma 0.5 y (* (* (/ z x) -4.5) t)) (/ x a_m)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((x * y) - ((z * 9.0) * t)) / (a_m * 2.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma(0.5, x, (t * ((z / y) * -4.5))) / a_m) * y;
	} else if (t_1 <= 2e+302) {
		tmp = (fma((t * -9.0), z, (y * x)) / a_m) / 2.0;
	} else {
		tmp = fma(0.5, y, (((z / x) * -4.5) * t)) * (x / a_m);
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a_m * 2.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(0.5, x, Float64(t * Float64(Float64(z / y) * -4.5))) / a_m) * y);
	elseif (t_1 <= 2e+302)
		tmp = Float64(Float64(fma(Float64(t * -9.0), z, Float64(y * x)) / a_m) / 2.0);
	else
		tmp = Float64(fma(0.5, y, Float64(Float64(Float64(z / x) * -4.5) * t)) * Float64(x / a_m));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(0.5 * x + N[(t * N[(N[(z / y), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 2e+302], N[(N[(N[(N[(t * -9.0), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.5 * y + N[(N[(N[(z / x), $MachinePrecision] * -4.5), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) (*.f64 a #s(literal 2 binary64))) < -inf.0

   1. Initial program 73.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a}\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto y \cdot \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot y} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{x}{a}\right)} \]

      2. metadata-eval N/A

         \[\leadsto y \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot y} - \color{blue}{\frac{-1}{2}} \cdot \frac{x}{a}\right) \]

      3. distribute-rgt-out-- N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right) \cdot y - \left(\frac{-1}{2} \cdot \frac{x}{a}\right) \cdot y} \]

      4. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right) \cdot y + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{x}{a}\right)\right) \cdot y} \]

      5. distribute-rgt-in N/A

         \[\leadsto \color{blue}{y \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot y} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{x}{a}\right)\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2}\right)\right)} \cdot \frac{t \cdot z}{a \cdot y} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{x}{a}\right)\right)\right) \]

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

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right)\right)} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{x}{a}\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{9}{2} \cdot \frac{t \cdot z}{a \cdot y} + \frac{-1}{2} \cdot \frac{x}{a}\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{x}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right)}\right)\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(y \cdot \left(\frac{-1}{2} \cdot \frac{x}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right)\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(\frac{-1}{2} \cdot \frac{x}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto -1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{x}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right) \cdot y\right)} \]

   5. Applied rewrites 91.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, x, t \cdot \left(\frac{z}{y} \cdot -4.5\right)\right)}{a} \cdot y} \]

   if -inf.0 < (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) (*.f64 a #s(literal 2 binary64))) < 2.0000000000000002e302

   1. Initial program 98.5%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{\color{blue}{a \cdot 2}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a}}{2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a}}{2}} \]

      5. lower-/.f64 99.0

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a}}}{2} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a}}{2} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a}}{2} \]

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

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a}}{2} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a}}{2} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{t \cdot \left(\mathsf{neg}\left(z \cdot 9\right)\right)} + x \cdot y}{a}}{2} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{t \cdot \left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) + x \cdot y}{a}}{2} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{t \cdot \left(\mathsf{neg}\left(\color{blue}{9 \cdot z}\right)\right) + x \cdot y}{a}}{2} \]

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

          \[\leadsto \frac{\frac{t \cdot \color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} + x \cdot y}{a}}{2} \]

      14. associate-*r* N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right) \cdot z} + x \cdot y}{a}}{2} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(t \cdot \left(\mathsf{neg}\left(9\right)\right), z, x \cdot y\right)}}{a}}{2} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{t \cdot \left(\mathsf{neg}\left(9\right)\right)}, z, x \cdot y\right)}{a}}{2} \]

      17. metadata-eval 99.0

          \[\leadsto \frac{\frac{\mathsf{fma}\left(t \cdot \color{blue}{-9}, z, x \cdot y\right)}{a}}{2} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(t \cdot -9, z, \color{blue}{x \cdot y}\right)}{a}}{2} \]

      19. *-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(t \cdot -9, z, \color{blue}{y \cdot x}\right)}{a}}{2} \]

      20. lower-*.f64 99.0

          \[\leadsto \frac{\frac{\mathsf{fma}\left(t \cdot -9, z, \color{blue}{y \cdot x}\right)}{a}}{2} \]

   4. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(t \cdot -9, z, y \cdot x\right)}{a}}{2}} \]

   if 2.0000000000000002e302 < (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) (*.f64 a #s(literal 2 binary64)))

   1. Initial program 80.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right)} \]

   5. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, y, t \cdot \left(\frac{z}{x} \cdot -4.5\right)\right)}{a} \cdot x} \]
   6. Step-by-step derivation

   7. Applied rewrites 93.8%

      \[\leadsto \mathsf{fma}\left(0.5, y, \left(\frac{z}{x} \cdot -4.5\right) \cdot t\right) \cdot \color{blue}{\frac{x}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 96.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a\_m \cdot 2}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 2 \cdot 10^{+302}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x, t \cdot \left(\frac{z}{y} \cdot -4.5\right)\right)}{a\_m} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t \cdot -9, z, y \cdot x\right)}{a\_m}}{2}\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (/ (- (* x y) (* (* z 9.0) t)) (* a_m 2.0))))
   (*
    a_s
    (if (or (<= t_1 (- INFINITY)) (not (<= t_1 2e+302)))
      (* (/ (fma 0.5 x (* t (* (/ z y) -4.5))) a_m) y)
      (/ (/ (fma (* t -9.0) z (* y x)) a_m) 2.0)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((x * y) - ((z * 9.0) * t)) / (a_m * 2.0);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 2e+302)) {
		tmp = (fma(0.5, x, (t * ((z / y) * -4.5))) / a_m) * y;
	} else {
		tmp = (fma((t * -9.0), z, (y * x)) / a_m) / 2.0;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a_m * 2.0))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 2e+302))
		tmp = Float64(Float64(fma(0.5, x, Float64(t * Float64(Float64(z / y) * -4.5))) / a_m) * y);
	else
		tmp = Float64(Float64(fma(Float64(t * -9.0), z, Float64(y * x)) / a_m) / 2.0);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 2e+302]], $MachinePrecision]], N[(N[(N[(0.5 * x + N[(t * N[(N[(z / y), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(N[(t * -9.0), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) (*.f64 a #s(literal 2 binary64))) < -inf.0 or 2.0000000000000002e302 < (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) (*.f64 a #s(literal 2 binary64)))

   1. Initial program 77.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a}\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto y \cdot \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot y} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{x}{a}\right)} \]

      2. metadata-eval N/A

         \[\leadsto y \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot y} - \color{blue}{\frac{-1}{2}} \cdot \frac{x}{a}\right) \]

      3. distribute-rgt-out-- N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right) \cdot y - \left(\frac{-1}{2} \cdot \frac{x}{a}\right) \cdot y} \]

      4. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right) \cdot y + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{x}{a}\right)\right) \cdot y} \]

      5. distribute-rgt-in N/A

         \[\leadsto \color{blue}{y \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot y} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{x}{a}\right)\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2}\right)\right)} \cdot \frac{t \cdot z}{a \cdot y} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{x}{a}\right)\right)\right) \]

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

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right)\right)} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{x}{a}\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{9}{2} \cdot \frac{t \cdot z}{a \cdot y} + \frac{-1}{2} \cdot \frac{x}{a}\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{x}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right)}\right)\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(y \cdot \left(\frac{-1}{2} \cdot \frac{x}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right)\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(\frac{-1}{2} \cdot \frac{x}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto -1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{x}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right) \cdot y\right)} \]

   5. Applied rewrites 91.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, x, t \cdot \left(\frac{z}{y} \cdot -4.5\right)\right)}{a} \cdot y} \]

   if -inf.0 < (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) (*.f64 a #s(literal 2 binary64))) < 2.0000000000000002e302

   1. Initial program 98.5%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{\color{blue}{a \cdot 2}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a}}{2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a}}{2}} \]

      5. lower-/.f64 99.0

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a}}}{2} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a}}{2} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a}}{2} \]

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

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a}}{2} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a}}{2} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{t \cdot \left(\mathsf{neg}\left(z \cdot 9\right)\right)} + x \cdot y}{a}}{2} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{t \cdot \left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) + x \cdot y}{a}}{2} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{t \cdot \left(\mathsf{neg}\left(\color{blue}{9 \cdot z}\right)\right) + x \cdot y}{a}}{2} \]

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

          \[\leadsto \frac{\frac{t \cdot \color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} + x \cdot y}{a}}{2} \]

      14. associate-*r* N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right) \cdot z} + x \cdot y}{a}}{2} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(t \cdot \left(\mathsf{neg}\left(9\right)\right), z, x \cdot y\right)}}{a}}{2} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{t \cdot \left(\mathsf{neg}\left(9\right)\right)}, z, x \cdot y\right)}{a}}{2} \]

      17. metadata-eval 99.0

          \[\leadsto \frac{\frac{\mathsf{fma}\left(t \cdot \color{blue}{-9}, z, x \cdot y\right)}{a}}{2} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(t \cdot -9, z, \color{blue}{x \cdot y}\right)}{a}}{2} \]

      19. *-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(t \cdot -9, z, \color{blue}{y \cdot x}\right)}{a}}{2} \]

      20. lower-*.f64 99.0

          \[\leadsto \frac{\frac{\mathsf{fma}\left(t \cdot -9, z, \color{blue}{y \cdot x}\right)}{a}}{2} \]

   4. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(t \cdot -9, z, y \cdot x\right)}{a}}{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \leq -\infty \lor \neg \left(\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \leq 2 \cdot 10^{+302}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x, t \cdot \left(\frac{z}{y} \cdot -4.5\right)\right)}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t \cdot -9, z, y \cdot x\right)}{a}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a\_m \cdot 2}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x, t \cdot \left(\frac{z}{y} \cdot -4.5\right)\right)}{a\_m} \cdot y\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t \cdot -9, z, y \cdot x\right)}{a\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, y, t \cdot \left(\frac{z}{x} \cdot -4.5\right)\right)}{a\_m} \cdot x\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (/ (- (* x y) (* (* z 9.0) t)) (* a_m 2.0))))
   (*
    a_s
    (if (<= t_1 (- INFINITY))
      (* (/ (fma 0.5 x (* t (* (/ z y) -4.5))) a_m) y)
      (if (<= t_1 2e+302)
        (/ (/ (fma (* t -9.0) z (* y x)) a_m) 2.0)
        (* (/ (fma 0.5 y (* t (* (/ z x) -4.5))) a_m) x))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((x * y) - ((z * 9.0) * t)) / (a_m * 2.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma(0.5, x, (t * ((z / y) * -4.5))) / a_m) * y;
	} else if (t_1 <= 2e+302) {
		tmp = (fma((t * -9.0), z, (y * x)) / a_m) / 2.0;
	} else {
		tmp = (fma(0.5, y, (t * ((z / x) * -4.5))) / a_m) * x;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a_m * 2.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(0.5, x, Float64(t * Float64(Float64(z / y) * -4.5))) / a_m) * y);
	elseif (t_1 <= 2e+302)
		tmp = Float64(Float64(fma(Float64(t * -9.0), z, Float64(y * x)) / a_m) / 2.0);
	else
		tmp = Float64(Float64(fma(0.5, y, Float64(t * Float64(Float64(z / x) * -4.5))) / a_m) * x);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(0.5 * x + N[(t * N[(N[(z / y), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 2e+302], N[(N[(N[(N[(t * -9.0), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(0.5 * y + N[(t * N[(N[(z / x), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision] * x), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) (*.f64 a #s(literal 2 binary64))) < -inf.0

   1. Initial program 73.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a}\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto y \cdot \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot y} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{x}{a}\right)} \]

      2. metadata-eval N/A

         \[\leadsto y \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot y} - \color{blue}{\frac{-1}{2}} \cdot \frac{x}{a}\right) \]

      3. distribute-rgt-out-- N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right) \cdot y - \left(\frac{-1}{2} \cdot \frac{x}{a}\right) \cdot y} \]

      4. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right) \cdot y + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{x}{a}\right)\right) \cdot y} \]

      5. distribute-rgt-in N/A

         \[\leadsto \color{blue}{y \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot y} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{x}{a}\right)\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2}\right)\right)} \cdot \frac{t \cdot z}{a \cdot y} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{x}{a}\right)\right)\right) \]

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

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right)\right)} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{x}{a}\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{9}{2} \cdot \frac{t \cdot z}{a \cdot y} + \frac{-1}{2} \cdot \frac{x}{a}\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{x}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right)}\right)\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(y \cdot \left(\frac{-1}{2} \cdot \frac{x}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right)\right)} \]

      11. mul-1-neg N/A

          \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(\frac{-1}{2} \cdot \frac{x}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto -1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{x}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right) \cdot y\right)} \]

   5. Applied rewrites 91.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, x, t \cdot \left(\frac{z}{y} \cdot -4.5\right)\right)}{a} \cdot y} \]

   if -inf.0 < (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) (*.f64 a #s(literal 2 binary64))) < 2.0000000000000002e302

   1. Initial program 98.5%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{\color{blue}{a \cdot 2}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a}}{2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a}}{2}} \]

      5. lower-/.f64 99.0

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a}}}{2} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a}}{2} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a}}{2} \]

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

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a}}{2} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a}}{2} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{t \cdot \left(\mathsf{neg}\left(z \cdot 9\right)\right)} + x \cdot y}{a}}{2} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{t \cdot \left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) + x \cdot y}{a}}{2} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{t \cdot \left(\mathsf{neg}\left(\color{blue}{9 \cdot z}\right)\right) + x \cdot y}{a}}{2} \]

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

          \[\leadsto \frac{\frac{t \cdot \color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} + x \cdot y}{a}}{2} \]

      14. associate-*r* N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right) \cdot z} + x \cdot y}{a}}{2} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(t \cdot \left(\mathsf{neg}\left(9\right)\right), z, x \cdot y\right)}}{a}}{2} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{t \cdot \left(\mathsf{neg}\left(9\right)\right)}, z, x \cdot y\right)}{a}}{2} \]

      17. metadata-eval 99.0

          \[\leadsto \frac{\frac{\mathsf{fma}\left(t \cdot \color{blue}{-9}, z, x \cdot y\right)}{a}}{2} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(t \cdot -9, z, \color{blue}{x \cdot y}\right)}{a}}{2} \]

      19. *-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(t \cdot -9, z, \color{blue}{y \cdot x}\right)}{a}}{2} \]

      20. lower-*.f64 99.0

          \[\leadsto \frac{\frac{\mathsf{fma}\left(t \cdot -9, z, \color{blue}{y \cdot x}\right)}{a}}{2} \]

   4. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(t \cdot -9, z, y \cdot x\right)}{a}}{2}} \]

   if 2.0000000000000002e302 < (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) (*.f64 a #s(literal 2 binary64)))

   1. Initial program 80.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right)} \]

   5. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, y, t \cdot \left(\frac{z}{x} \cdot -4.5\right)\right)}{a} \cdot x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 73.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \frac{0.5 \cdot y}{a\_m} \cdot x\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-248}:\\ \;\;\;\;\left(t \cdot -4.5\right) \cdot \frac{z}{a\_m}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+27}:\\ \;\;\;\;z \cdot \left(\frac{t}{a\_m} \cdot -4.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (/ (* 0.5 y) a_m) x)))
   (*
    a_s
    (if (<= (* x y) -4e-15)
      t_1
      (if (<= (* x y) 4e-248)
        (* (* t -4.5) (/ z a_m))
        (if (<= (* x y) 4e+27) (* z (* (/ t a_m) -4.5)) t_1))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((0.5 * y) / a_m) * x;
	double tmp;
	if ((x * y) <= -4e-15) {
		tmp = t_1;
	} else if ((x * y) <= 4e-248) {
		tmp = (t * -4.5) * (z / a_m);
	} else if ((x * y) <= 4e+27) {
		tmp = z * ((t / a_m) * -4.5);
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Fortran

a\_m =     private
a\_s =     private
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
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(a_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    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_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((0.5d0 * y) / a_m) * x
    if ((x * y) <= (-4d-15)) then
        tmp = t_1
    else if ((x * y) <= 4d-248) then
        tmp = (t * (-4.5d0)) * (z / a_m)
    else if ((x * y) <= 4d+27) then
        tmp = z * ((t / a_m) * (-4.5d0))
    else
        tmp = t_1
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((0.5 * y) / a_m) * x;
	double tmp;
	if ((x * y) <= -4e-15) {
		tmp = t_1;
	} else if ((x * y) <= 4e-248) {
		tmp = (t * -4.5) * (z / a_m);
	} else if ((x * y) <= 4e+27) {
		tmp = z * ((t / a_m) * -4.5);
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = ((0.5 * y) / a_m) * x
	tmp = 0
	if (x * y) <= -4e-15:
		tmp = t_1
	elif (x * y) <= 4e-248:
		tmp = (t * -4.5) * (z / a_m)
	elif (x * y) <= 4e+27:
		tmp = z * ((t / a_m) * -4.5)
	else:
		tmp = t_1
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(Float64(0.5 * y) / a_m) * x)
	tmp = 0.0
	if (Float64(x * y) <= -4e-15)
		tmp = t_1;
	elseif (Float64(x * y) <= 4e-248)
		tmp = Float64(Float64(t * -4.5) * Float64(z / a_m));
	elseif (Float64(x * y) <= 4e+27)
		tmp = Float64(z * Float64(Float64(t / a_m) * -4.5));
	else
		tmp = t_1;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = ((0.5 * y) / a_m) * x;
	tmp = 0.0;
	if ((x * y) <= -4e-15)
		tmp = t_1;
	elseif ((x * y) <= 4e-248)
		tmp = (t * -4.5) * (z / a_m);
	elseif ((x * y) <= 4e+27)
		tmp = z * ((t / a_m) * -4.5);
	else
		tmp = t_1;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(N[(0.5 * y), $MachinePrecision] / a$95$m), $MachinePrecision] * x), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -4e-15], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 4e-248], N[(N[(t * -4.5), $MachinePrecision] * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 4e+27], N[(z * N[(N[(t / a$95$m), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4.0000000000000003e-15 or 4.0000000000000001e27 < (*.f64 x y)

   1. Initial program 89.7%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right)} \]

   5. Applied rewrites 90.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, y, t \cdot \left(\frac{z}{x} \cdot -4.5\right)\right)}{a} \cdot x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{2} \cdot y}{a} \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 75.1%

      \[\leadsto \frac{0.5 \cdot y}{a} \cdot x \]

   if -4.0000000000000003e-15 < (*.f64 x y) < 3.99999999999999992e-248

   1. Initial program 92.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot \frac{-9}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot \frac{-9}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot z}{a}} \cdot \frac{-9}{2} \]

      4. lower-*.f64 79.0

         \[\leadsto \frac{\color{blue}{t \cdot z}}{a} \cdot -4.5 \]

   5. Applied rewrites 79.0%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.9%

      \[\leadsto \left(t \cdot -4.5\right) \cdot \color{blue}{\frac{z}{a}} \]

   if 3.99999999999999992e-248 < (*.f64 x y) < 4.0000000000000001e27

   1. Initial program 92.0%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot \frac{-9}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot \frac{-9}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot z}{a}} \cdot \frac{-9}{2} \]

      4. lower-*.f64 76.3

         \[\leadsto \frac{\color{blue}{t \cdot z}}{a} \cdot -4.5 \]

   5. Applied rewrites 76.3%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 76.3%

      \[\leadsto z \cdot \color{blue}{\left(\frac{t}{a} \cdot -4.5\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 73.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \left(0.5 \cdot y\right) \cdot \frac{x}{a\_m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -20000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-248}:\\ \;\;\;\;\left(t \cdot -4.5\right) \cdot \frac{z}{a\_m}\\ \mathbf{elif}\;x \cdot y \leq 10^{+65}:\\ \;\;\;\;z \cdot \left(\frac{t}{a\_m} \cdot -4.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (* 0.5 y) (/ x a_m))))
   (*
    a_s
    (if (<= (* x y) -20000000000.0)
      t_1
      (if (<= (* x y) 4e-248)
        (* (* t -4.5) (/ z a_m))
        (if (<= (* x y) 1e+65) (* z (* (/ t a_m) -4.5)) t_1))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (0.5 * y) * (x / a_m);
	double tmp;
	if ((x * y) <= -20000000000.0) {
		tmp = t_1;
	} else if ((x * y) <= 4e-248) {
		tmp = (t * -4.5) * (z / a_m);
	} else if ((x * y) <= 1e+65) {
		tmp = z * ((t / a_m) * -4.5);
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Fortran

a\_m =     private
a\_s =     private
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
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(a_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    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_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (0.5d0 * y) * (x / a_m)
    if ((x * y) <= (-20000000000.0d0)) then
        tmp = t_1
    else if ((x * y) <= 4d-248) then
        tmp = (t * (-4.5d0)) * (z / a_m)
    else if ((x * y) <= 1d+65) then
        tmp = z * ((t / a_m) * (-4.5d0))
    else
        tmp = t_1
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (0.5 * y) * (x / a_m);
	double tmp;
	if ((x * y) <= -20000000000.0) {
		tmp = t_1;
	} else if ((x * y) <= 4e-248) {
		tmp = (t * -4.5) * (z / a_m);
	} else if ((x * y) <= 1e+65) {
		tmp = z * ((t / a_m) * -4.5);
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = (0.5 * y) * (x / a_m)
	tmp = 0
	if (x * y) <= -20000000000.0:
		tmp = t_1
	elif (x * y) <= 4e-248:
		tmp = (t * -4.5) * (z / a_m)
	elif (x * y) <= 1e+65:
		tmp = z * ((t / a_m) * -4.5)
	else:
		tmp = t_1
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(0.5 * y) * Float64(x / a_m))
	tmp = 0.0
	if (Float64(x * y) <= -20000000000.0)
		tmp = t_1;
	elseif (Float64(x * y) <= 4e-248)
		tmp = Float64(Float64(t * -4.5) * Float64(z / a_m));
	elseif (Float64(x * y) <= 1e+65)
		tmp = Float64(z * Float64(Float64(t / a_m) * -4.5));
	else
		tmp = t_1;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = (0.5 * y) * (x / a_m);
	tmp = 0.0;
	if ((x * y) <= -20000000000.0)
		tmp = t_1;
	elseif ((x * y) <= 4e-248)
		tmp = (t * -4.5) * (z / a_m);
	elseif ((x * y) <= 1e+65)
		tmp = z * ((t / a_m) * -4.5);
	else
		tmp = t_1;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(0.5 * y), $MachinePrecision] * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -20000000000.0], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 4e-248], N[(N[(t * -4.5), $MachinePrecision] * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+65], N[(z * N[(N[(t / a$95$m), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2e10 or 9.9999999999999999e64 < (*.f64 x y)

   1. Initial program 88.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right)} \]

   5. Applied rewrites 92.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, y, t \cdot \left(\frac{z}{x} \cdot -4.5\right)\right)}{a} \cdot x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{2} \cdot y}{a} \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 79.4%

      \[\leadsto \frac{0.5 \cdot y}{a} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 78.4%

       \[\leadsto \color{blue}{\left(0.5 \cdot y\right) \cdot \frac{x}{a}} \]

   if -2e10 < (*.f64 x y) < 3.99999999999999992e-248

   1. Initial program 92.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot \frac{-9}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot \frac{-9}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot z}{a}} \cdot \frac{-9}{2} \]

      4. lower-*.f64 76.2

         \[\leadsto \frac{\color{blue}{t \cdot z}}{a} \cdot -4.5 \]

   5. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 71.4%

      \[\leadsto \left(t \cdot -4.5\right) \cdot \color{blue}{\frac{z}{a}} \]

   if 3.99999999999999992e-248 < (*.f64 x y) < 9.9999999999999999e64

   1. Initial program 93.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot \frac{-9}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot \frac{-9}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot z}{a}} \cdot \frac{-9}{2} \]

      4. lower-*.f64 71.0

         \[\leadsto \frac{\color{blue}{t \cdot z}}{a} \cdot -4.5 \]

   5. Applied rewrites 71.0%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 71.0%

      \[\leadsto z \cdot \color{blue}{\left(\frac{t}{a} \cdot -4.5\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 93.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t \cdot -9, z, y \cdot x\right)}{a\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \frac{x}{a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* x y) 4e+288)
    (/ (/ (fma (* t -9.0) z (* y x)) a_m) 2.0)
    (* (* 0.5 y) (/ x a_m)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= 4e+288) {
		tmp = (fma((t * -9.0), z, (y * x)) / a_m) / 2.0;
	} else {
		tmp = (0.5 * y) * (x / a_m);
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(x * y) <= 4e+288)
		tmp = Float64(Float64(fma(Float64(t * -9.0), z, Float64(y * x)) / a_m) / 2.0);
	else
		tmp = Float64(Float64(0.5 * y) * Float64(x / a_m));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], 4e+288], N[(N[(N[(N[(t * -9.0), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.5 * y), $MachinePrecision] * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 4e288

   1. Initial program 92.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{\color{blue}{a \cdot 2}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a}}{2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a}}{2}} \]

      5. lower-/.f64 93.0

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a}}}{2} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a}}{2} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a}}{2} \]

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

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a}}{2} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a}}{2} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{t \cdot \left(\mathsf{neg}\left(z \cdot 9\right)\right)} + x \cdot y}{a}}{2} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{t \cdot \left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) + x \cdot y}{a}}{2} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{t \cdot \left(\mathsf{neg}\left(\color{blue}{9 \cdot z}\right)\right) + x \cdot y}{a}}{2} \]

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

          \[\leadsto \frac{\frac{t \cdot \color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} + x \cdot y}{a}}{2} \]

      14. associate-*r* N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right) \cdot z} + x \cdot y}{a}}{2} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(t \cdot \left(\mathsf{neg}\left(9\right)\right), z, x \cdot y\right)}}{a}}{2} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{t \cdot \left(\mathsf{neg}\left(9\right)\right)}, z, x \cdot y\right)}{a}}{2} \]

      17. metadata-eval 93.4

          \[\leadsto \frac{\frac{\mathsf{fma}\left(t \cdot \color{blue}{-9}, z, x \cdot y\right)}{a}}{2} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(t \cdot -9, z, \color{blue}{x \cdot y}\right)}{a}}{2} \]

      19. *-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(t \cdot -9, z, \color{blue}{y \cdot x}\right)}{a}}{2} \]

      20. lower-*.f64 93.4

          \[\leadsto \frac{\frac{\mathsf{fma}\left(t \cdot -9, z, \color{blue}{y \cdot x}\right)}{a}}{2} \]

   4. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(t \cdot -9, z, y \cdot x\right)}{a}}{2}} \]

   if 4e288 < (*.f64 x y)

   1. Initial program 68.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, y, t \cdot \left(\frac{z}{x} \cdot -4.5\right)\right)}{a} \cdot x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{2} \cdot y}{a} \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 99.7%

      \[\leadsto \frac{0.5 \cdot y}{a} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 99.9%

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

Alternative 7: 74.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{-30} \lor \neg \left(x \cdot y \leq 4 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{0.5 \cdot y}{a\_m} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot z}{a\_m} \cdot -4.5\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (or (<= (* x y) -4e-30) (not (<= (* x y) 4e+27)))
    (* (/ (* 0.5 y) a_m) x)
    (* (/ (* t z) a_m) -4.5))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (((x * y) <= -4e-30) || !((x * y) <= 4e+27)) {
		tmp = ((0.5 * y) / a_m) * x;
	} else {
		tmp = ((t * z) / a_m) * -4.5;
	}
	return a_s * tmp;
}

Fortran

a\_m =     private
a\_s =     private
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
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(a_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    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_m
    real(8) :: tmp
    if (((x * y) <= (-4d-30)) .or. (.not. ((x * y) <= 4d+27))) then
        tmp = ((0.5d0 * y) / a_m) * x
    else
        tmp = ((t * z) / a_m) * (-4.5d0)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (((x * y) <= -4e-30) || !((x * y) <= 4e+27)) {
		tmp = ((0.5 * y) / a_m) * x;
	} else {
		tmp = ((t * z) / a_m) * -4.5;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if ((x * y) <= -4e-30) or not ((x * y) <= 4e+27):
		tmp = ((0.5 * y) / a_m) * x
	else:
		tmp = ((t * z) / a_m) * -4.5
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if ((Float64(x * y) <= -4e-30) || !(Float64(x * y) <= 4e+27))
		tmp = Float64(Float64(Float64(0.5 * y) / a_m) * x);
	else
		tmp = Float64(Float64(Float64(t * z) / a_m) * -4.5);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if (((x * y) <= -4e-30) || ~(((x * y) <= 4e+27)))
		tmp = ((0.5 * y) / a_m) * x;
	else
		tmp = ((t * z) / a_m) * -4.5;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[Or[LessEqual[N[(x * y), $MachinePrecision], -4e-30], N[Not[LessEqual[N[(x * y), $MachinePrecision], 4e+27]], $MachinePrecision]], N[(N[(N[(0.5 * y), $MachinePrecision] / a$95$m), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(t * z), $MachinePrecision] / a$95$m), $MachinePrecision] * -4.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4e-30 or 4.0000000000000001e27 < (*.f64 x y)

   1. Initial program 89.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right)} \]

   5. Applied rewrites 88.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, y, t \cdot \left(\frac{z}{x} \cdot -4.5\right)\right)}{a} \cdot x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{2} \cdot y}{a} \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 73.0%

      \[\leadsto \frac{0.5 \cdot y}{a} \cdot x \]

   if -4e-30 < (*.f64 x y) < 4.0000000000000001e27

   1. Initial program 92.5%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot \frac{-9}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot \frac{-9}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot z}{a}} \cdot \frac{-9}{2} \]

      4. lower-*.f64 80.4

         \[\leadsto \frac{\color{blue}{t \cdot z}}{a} \cdot -4.5 \]

   5. Applied rewrites 80.4%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{-30} \lor \neg \left(x \cdot y \leq 4 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{0.5 \cdot y}{a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot z}{a} \cdot -4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-56} \lor \neg \left(x \cdot y \leq 10^{+65}\right):\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \frac{x}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{t}{a\_m} \cdot -4.5\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (or (<= (* x y) -2e-56) (not (<= (* x y) 1e+65)))
    (* (* 0.5 y) (/ x a_m))
    (* z (* (/ t a_m) -4.5)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (((x * y) <= -2e-56) || !((x * y) <= 1e+65)) {
		tmp = (0.5 * y) * (x / a_m);
	} else {
		tmp = z * ((t / a_m) * -4.5);
	}
	return a_s * tmp;
}

Fortran

a\_m =     private
a\_s =     private
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
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(a_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    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_m
    real(8) :: tmp
    if (((x * y) <= (-2d-56)) .or. (.not. ((x * y) <= 1d+65))) then
        tmp = (0.5d0 * y) * (x / a_m)
    else
        tmp = z * ((t / a_m) * (-4.5d0))
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (((x * y) <= -2e-56) || !((x * y) <= 1e+65)) {
		tmp = (0.5 * y) * (x / a_m);
	} else {
		tmp = z * ((t / a_m) * -4.5);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if ((x * y) <= -2e-56) or not ((x * y) <= 1e+65):
		tmp = (0.5 * y) * (x / a_m)
	else:
		tmp = z * ((t / a_m) * -4.5)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if ((Float64(x * y) <= -2e-56) || !(Float64(x * y) <= 1e+65))
		tmp = Float64(Float64(0.5 * y) * Float64(x / a_m));
	else
		tmp = Float64(z * Float64(Float64(t / a_m) * -4.5));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if (((x * y) <= -2e-56) || ~(((x * y) <= 1e+65)))
		tmp = (0.5 * y) * (x / a_m);
	else
		tmp = z * ((t / a_m) * -4.5);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[Or[LessEqual[N[(x * y), $MachinePrecision], -2e-56], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e+65]], $MachinePrecision]], N[(N[(0.5 * y), $MachinePrecision] * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(t / a$95$m), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.0000000000000001e-56 or 9.9999999999999999e64 < (*.f64 x y)

   1. Initial program 89.2%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right)} \]

   5. Applied rewrites 87.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, y, t \cdot \left(\frac{z}{x} \cdot -4.5\right)\right)}{a} \cdot x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{2} \cdot y}{a} \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 73.7%

      \[\leadsto \frac{0.5 \cdot y}{a} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 72.3%

       \[\leadsto \color{blue}{\left(0.5 \cdot y\right) \cdot \frac{x}{a}} \]

   if -2.0000000000000001e-56 < (*.f64 x y) < 9.9999999999999999e64

   1. Initial program 92.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot \frac{-9}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot \frac{-9}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot z}{a}} \cdot \frac{-9}{2} \]

      4. lower-*.f64 78.9

         \[\leadsto \frac{\color{blue}{t \cdot z}}{a} \cdot -4.5 \]

   5. Applied rewrites 78.9%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 77.3%

      \[\leadsto z \cdot \color{blue}{\left(\frac{t}{a} \cdot -4.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-56} \lor \neg \left(x \cdot y \leq 10^{+65}\right):\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{t}{a} \cdot -4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot -9, z, y \cdot x\right)}{a\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \frac{x}{a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* x y) 4e+288)
    (/ (fma (* t -9.0) z (* y x)) (* a_m 2.0))
    (* (* 0.5 y) (/ x a_m)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= 4e+288) {
		tmp = fma((t * -9.0), z, (y * x)) / (a_m * 2.0);
	} else {
		tmp = (0.5 * y) * (x / a_m);
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(x * y) <= 4e+288)
		tmp = Float64(fma(Float64(t * -9.0), z, Float64(y * x)) / Float64(a_m * 2.0));
	else
		tmp = Float64(Float64(0.5 * y) * Float64(x / a_m));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], 4e+288], N[(N[(N[(t * -9.0), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(a$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * y), $MachinePrecision] * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 4e288

   1. Initial program 92.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

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

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a \cdot 2} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(\mathsf{neg}\left(z \cdot 9\right)\right)} + x \cdot y}{a \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{t \cdot \left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) + x \cdot y}{a \cdot 2} \]

      7. *-commutative N/A

         \[\leadsto \frac{t \cdot \left(\mathsf{neg}\left(\color{blue}{9 \cdot z}\right)\right) + x \cdot y}{a \cdot 2} \]

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

         \[\leadsto \frac{t \cdot \color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} + x \cdot y}{a \cdot 2} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(t \cdot \left(\mathsf{neg}\left(9\right)\right)\right) \cdot z} + x \cdot y}{a \cdot 2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(t \cdot \left(\mathsf{neg}\left(9\right)\right), z, x \cdot y\right)}}{a \cdot 2} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{t \cdot \left(\mathsf{neg}\left(9\right)\right)}, z, x \cdot y\right)}{a \cdot 2} \]

      12. metadata-eval 93.1

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot \color{blue}{-9}, z, x \cdot y\right)}{a \cdot 2} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot -9, z, \color{blue}{x \cdot y}\right)}{a \cdot 2} \]

      14. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot -9, z, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      15. lower-*.f64 93.1

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot -9, z, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

   4. Applied rewrites 93.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(t \cdot -9, z, y \cdot x\right)}}{a \cdot 2} \]

   if 4e288 < (*.f64 x y)

   1. Initial program 68.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, y, t \cdot \left(\frac{z}{x} \cdot -4.5\right)\right)}{a} \cdot x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{2} \cdot y}{a} \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 99.7%

      \[\leadsto \frac{0.5 \cdot y}{a} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 99.9%

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

Alternative 10: 93.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a\_m + a\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \frac{x}{a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* x y) 4e+288)
    (/ (fma (* -9.0 z) t (* y x)) (+ a_m a_m))
    (* (* 0.5 y) (/ x a_m)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= 4e+288) {
		tmp = fma((-9.0 * z), t, (y * x)) / (a_m + a_m);
	} else {
		tmp = (0.5 * y) * (x / a_m);
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(x * y) <= 4e+288)
		tmp = Float64(fma(Float64(-9.0 * z), t, Float64(y * x)) / Float64(a_m + a_m));
	else
		tmp = Float64(Float64(0.5 * y) * Float64(x / a_m));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], 4e+288], N[(N[(N[(-9.0 * z), $MachinePrecision] * t + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * y), $MachinePrecision] * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 4e288

   1. Initial program 92.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

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

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a \cdot 2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(z \cdot 9\right), t, x \cdot y\right)}}{a \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right), t, x \cdot y\right)}{a \cdot 2} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{9 \cdot z}\right), t, x \cdot y\right)}{a \cdot 2} \]

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

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(9\right)\right) \cdot z}, t, x \cdot y\right)}{a \cdot 2} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(9\right)\right) \cdot z}, t, x \cdot y\right)}{a \cdot 2} \]

      10. metadata-eval 93.0

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-9} \cdot z, t, x \cdot y\right)}{a \cdot 2} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{x \cdot y}\right)}{a \cdot 2} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      13. lower-*.f64 93.0

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

   4. Applied rewrites 93.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}}{a \cdot 2} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{2 \cdot a}} \]

      3. count-2-rev N/A

         \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

      4. lower-+.f64 93.0

         \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

   6. Applied rewrites 93.0%

      \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

   if 4e288 < (*.f64 x y)

   1. Initial program 68.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, y, t \cdot \left(\frac{z}{x} \cdot -4.5\right)\right)}{a} \cdot x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{2} \cdot y}{a} \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 99.7%

      \[\leadsto \frac{0.5 \cdot y}{a} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 99.9%

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

Alternative 11: 51.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \left(z \cdot \left(\frac{t}{a\_m} \cdot -4.5\right)\right) \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m) :precision binary64 (* a_s (* z (* (/ t a_m) -4.5))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	return a_s * (z * ((t / a_m) * -4.5));
}

Fortran

a\_m =     private
a\_s =     private
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
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(a_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    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_m
    code = a_s * (z * ((t / a_m) * (-4.5d0)))
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	return a_s * (z * ((t / a_m) * -4.5));
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	return a_s * (z * ((t / a_m) * -4.5))

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	return Float64(a_s * Float64(z * Float64(Float64(t / a_m) * -4.5)))
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp = code(a_s, x, y, z, t, a_m)
	tmp = a_s * (z * ((t / a_m) * -4.5));
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * N[(z * N[(N[(t / a$95$m), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.9%

   \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot \frac{-9}{2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot \frac{-9}{2}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{t \cdot z}{a}} \cdot \frac{-9}{2} \]

   4. lower-*.f64 51.4

      \[\leadsto \frac{\color{blue}{t \cdot z}}{a} \cdot -4.5 \]

5. Applied rewrites 51.4%

   \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
6. Step-by-step derivation

7. Applied rewrites 51.6%

   \[\leadsto z \cdot \color{blue}{\left(\frac{t}{a} \cdot -4.5\right)} \]
8. Add Preprocessing

Developer Target 1: 94.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (< a -2.090464557976709e+86)
   (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z))))
   (if (< a 2.144030707833976e+99)
     (/ (- (* x y) (* z (* 9.0 t))) (* a 2.0))
     (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a < -2.090464557976709e+86) {
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)));
	} else if (a < 2.144030707833976e+99) {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	} else {
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5));
	}
	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 (a < (-2.090464557976709d+86)) then
        tmp = (0.5d0 * ((y * x) / a)) - (4.5d0 * (t / (a / z)))
    else if (a < 2.144030707833976d+99) then
        tmp = ((x * y) - (z * (9.0d0 * t))) / (a * 2.0d0)
    else
        tmp = ((y / a) * (x * 0.5d0)) - ((t / a) * (z * 4.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a < -2.090464557976709e+86) {
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)));
	} else if (a < 2.144030707833976e+99) {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	} else {
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a < -2.090464557976709e+86:
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)))
	elif a < 2.144030707833976e+99:
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0)
	else:
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a < -2.090464557976709e+86)
		tmp = Float64(Float64(0.5 * Float64(Float64(y * x) / a)) - Float64(4.5 * Float64(t / Float64(a / z))));
	elseif (a < 2.144030707833976e+99)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * Float64(9.0 * t))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(y / a) * Float64(x * 0.5)) - Float64(Float64(t / a) * Float64(z * 4.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a < -2.090464557976709e+86)
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)));
	elseif (a < 2.144030707833976e+99)
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	else
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Less[a, -2.090464557976709e+86], N[(N[(0.5 * N[(N[(y * x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] - N[(4.5 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[a, 2.144030707833976e+99], N[(N[(N[(x * y), $MachinePrecision] - N[(z * N[(9.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / a), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(t / a), $MachinePrecision] * N[(z * 4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024363 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< a -209046455797670900000000000000000000000000000000000000000000000000000000000000000000000) (- (* 1/2 (/ (* y x) a)) (* 9/2 (/ t (/ a z)))) (if (< a 2144030707833976000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (- (* x y) (* z (* 9 t))) (* a 2)) (- (* (/ y a) (* x 1/2)) (* (/ t a) (* z 9/2))))))

  (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))