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

Percentage Accurate: 85.3% → 89.8%

Time: 6.7s

Alternatives: 19

Speedup: 0.5×

• 50.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \end{array} \]

FPCore

(FPCore (x y z t a b c i j k)
 :precision binary64
 (-
  (-
   (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c))
   (* (* x 4.0) i))
  (* (* j 27.0) k)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	return (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k);
}

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, b, c, i, j, k)
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), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    code = (((((((x * 18.0d0) * y) * z) * t) - ((a * 4.0d0) * t)) + (b * c)) - ((x * 4.0d0) * i)) - ((j * 27.0d0) * k)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	return (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k);
}

Python

def code(x, y, z, t, a, b, c, i, j, k):
	return (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k)

Julia

function code(x, y, z, t, a, b, c, i, j, k)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * 18.0) * y) * z) * t) - Float64(Float64(a * 4.0) * t)) + Float64(b * c)) - Float64(Float64(x * 4.0) * i)) - Float64(Float64(j * 27.0) * k))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j, k)
	tmp = (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := N[(N[(N[(N[(N[(N[(N[(N[(x * 18.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] - N[(N[(x * 4.0), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] - N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]

Initial Program: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \end{array} \]

FPCore

(FPCore (x y z t a b c i j k)
 :precision binary64
 (-
  (-
   (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c))
   (* (* x 4.0) i))
  (* (* j 27.0) k)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	return (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k);
}

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, b, c, i, j, k)
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), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    code = (((((((x * 18.0d0) * y) * z) * t) - ((a * 4.0d0) * t)) + (b * c)) - ((x * 4.0d0) * i)) - ((j * 27.0d0) * k)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	return (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k);
}

Python

def code(x, y, z, t, a, b, c, i, j, k):
	return (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k)

Julia

function code(x, y, z, t, a, b, c, i, j, k)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * 18.0) * y) * z) * t) - Float64(Float64(a * 4.0) * t)) + Float64(b * c)) - Float64(Float64(x * 4.0) * i)) - Float64(Float64(j * 27.0) * k))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j, k)
	tmp = (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := N[(N[(N[(N[(N[(N[(N[(N[(x * 18.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] - N[(N[(x * 4.0), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] - N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]

Alternative 1: 89.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(j \cdot 27\right) \cdot k\\ \mathbf{if}\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(\left(x \cdot y\right) \cdot z, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y \cdot x\right) \cdot z\right) \cdot t\right) \cdot 18 - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* (* j 27.0) k)))
   (if (<=
        (-
         (-
          (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c))
          (* (* x 4.0) i))
         t_1)
        INFINITY)
     (fma
      c
      b
      (-
       (* (fma (* (* x y) z) 18.0 (* a -4.0)) t)
       (fma (* i x) 4.0 (* (* j k) 27.0))))
     (- (* (* (* (* y x) z) t) 18.0) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = (j * 27.0) * k;
	double tmp;
	if (((((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - t_1) <= ((double) INFINITY)) {
		tmp = fma(c, b, ((fma(((x * y) * z), 18.0, (a * -4.0)) * t) - fma((i * x), 4.0, ((j * k) * 27.0))));
	} else {
		tmp = ((((y * x) * z) * t) * 18.0) - t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(Float64(j * 27.0) * k)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * 18.0) * y) * z) * t) - Float64(Float64(a * 4.0) * t)) + Float64(b * c)) - Float64(Float64(x * 4.0) * i)) - t_1) <= Inf)
		tmp = fma(c, b, Float64(Float64(fma(Float64(Float64(x * y) * z), 18.0, Float64(a * -4.0)) * t) - fma(Float64(i * x), 4.0, Float64(Float64(j * k) * 27.0))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(y * x) * z) * t) * 18.0) - t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(N[(N[(x * 18.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] - N[(N[(x * 4.0), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], Infinity], N[(c * b + N[(N[(N[(N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision] * 18.0 + N[(a * -4.0), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] - N[(N[(i * x), $MachinePrecision] * 4.0 + N[(N[(j * k), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] * 18.0), $MachinePrecision] - t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x #s(literal 18 binary64)) y) z) t) (*.f64 (*.f64 a #s(literal 4 binary64)) t)) (*.f64 b c)) (*.f64 (*.f64 x #s(literal 4 binary64)) i)) (*.f64 (*.f64 j #s(literal 27 binary64)) k)) < +inf.0

   1. Initial program 95.4%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   4. Applied rewrites 94.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(z \cdot y\right) \cdot x, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right)} \]

   6. Applied rewrites 94.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(\left(y \cdot z\right) \cdot x, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(y \cdot z\right)} \cdot x, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(y \cdot z\right) \cdot x}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{x \cdot \left(y \cdot z\right)}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(x \cdot y\right) \cdot z}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(x \cdot y\right) \cdot z}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      6. lower-*.f64 95.4

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(x \cdot y\right)} \cdot z, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

   8. Applied rewrites 95.4%

      \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(x \cdot y\right) \cdot z}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

   if +inf.0 < (-.f64 (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x #s(literal 18 binary64)) y) z) t) (*.f64 (*.f64 a #s(literal 4 binary64)) t)) (*.f64 b c)) (*.f64 (*.f64 x #s(literal 4 binary64)) i)) (*.f64 (*.f64 j #s(literal 27 binary64)) k))

   1. Initial program 0.0%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot c} - \left(j \cdot 27\right) \cdot k \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto c \cdot \color{blue}{b} - \left(j \cdot 27\right) \cdot k \]

      2. lower-*.f64 27.4

         \[\leadsto c \cdot \color{blue}{b} - \left(j \cdot 27\right) \cdot k \]

   5. Applied rewrites 27.4%

      \[\leadsto \color{blue}{c \cdot b} - \left(j \cdot 27\right) \cdot k \]

   6. Taylor expanded in i around inf

      \[\leadsto \color{blue}{-4 \cdot \left(i \cdot x\right)} - \left(j \cdot 27\right) \cdot k \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-4 \cdot i\right) \cdot \color{blue}{x} - \left(j \cdot 27\right) \cdot k \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-4 \cdot i\right) \cdot \color{blue}{x} - \left(j \cdot 27\right) \cdot k \]

      3. *-commutative N/A

         \[\leadsto \left(i \cdot -4\right) \cdot x - \left(j \cdot 27\right) \cdot k \]

      4. lower-*.f64 33.8

         \[\leadsto \left(i \cdot -4\right) \cdot x - \left(j \cdot 27\right) \cdot k \]

   8. Applied rewrites 33.8%

      \[\leadsto \color{blue}{\left(i \cdot -4\right) \cdot x} - \left(j \cdot 27\right) \cdot k \]

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)} - \left(j \cdot 27\right) \cdot k \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot \color{blue}{18} - \left(j \cdot 27\right) \cdot k \]

       2. lower-*.f64 N/A

          \[\leadsto \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot \color{blue}{18} - \left(j \cdot 27\right) \cdot k \]

       3. *-commutative N/A

          \[\leadsto \left(\left(x \cdot \left(y \cdot z\right)\right) \cdot t\right) \cdot 18 - \left(j \cdot 27\right) \cdot k \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\left(x \cdot \left(y \cdot z\right)\right) \cdot t\right) \cdot 18 - \left(j \cdot 27\right) \cdot k \]

       5. associate-*r* N/A

          \[\leadsto \left(\left(\left(x \cdot y\right) \cdot z\right) \cdot t\right) \cdot 18 - \left(j \cdot 27\right) \cdot k \]

       6. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(x \cdot y\right) \cdot z\right) \cdot t\right) \cdot 18 - \left(j \cdot 27\right) \cdot k \]

       7. *-commutative N/A

          \[\leadsto \left(\left(\left(y \cdot x\right) \cdot z\right) \cdot t\right) \cdot 18 - \left(j \cdot 27\right) \cdot k \]

       8. lower-*.f64 42.2

          \[\leadsto \left(\left(\left(y \cdot x\right) \cdot z\right) \cdot t\right) \cdot 18 - \left(j \cdot 27\right) \cdot k \]

   11. Applied rewrites 42.2%

       \[\leadsto \color{blue}{\left(\left(\left(y \cdot x\right) \cdot z\right) \cdot t\right) \cdot 18} - \left(j \cdot 27\right) \cdot k \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 89.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(j \cdot 27\right) \cdot k\\ \mathbf{if}\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot y\right) \cdot z, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y \cdot x\right) \cdot z\right) \cdot t\right) \cdot 18 - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* (* j 27.0) k)))
   (if (<=
        (-
         (-
          (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c))
          (* (* x 4.0) i))
         t_1)
        INFINITY)
     (-
      (fma (fma (* (* x y) z) 18.0 (* -4.0 a)) t (* c b))
      (fma (* 4.0 x) i (* (* k j) 27.0)))
     (- (* (* (* (* y x) z) t) 18.0) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = (j * 27.0) * k;
	double tmp;
	if (((((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - t_1) <= ((double) INFINITY)) {
		tmp = fma(fma(((x * y) * z), 18.0, (-4.0 * a)), t, (c * b)) - fma((4.0 * x), i, ((k * j) * 27.0));
	} else {
		tmp = ((((y * x) * z) * t) * 18.0) - t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(Float64(j * 27.0) * k)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * 18.0) * y) * z) * t) - Float64(Float64(a * 4.0) * t)) + Float64(b * c)) - Float64(Float64(x * 4.0) * i)) - t_1) <= Inf)
		tmp = Float64(fma(fma(Float64(Float64(x * y) * z), 18.0, Float64(-4.0 * a)), t, Float64(c * b)) - fma(Float64(4.0 * x), i, Float64(Float64(k * j) * 27.0)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(y * x) * z) * t) * 18.0) - t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(N[(N[(x * 18.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] - N[(N[(x * 4.0), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], Infinity], N[(N[(N[(N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision] * 18.0 + N[(-4.0 * a), $MachinePrecision]), $MachinePrecision] * t + N[(c * b), $MachinePrecision]), $MachinePrecision] - N[(N[(4.0 * x), $MachinePrecision] * i + N[(N[(k * j), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] * 18.0), $MachinePrecision] - t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x #s(literal 18 binary64)) y) z) t) (*.f64 (*.f64 a #s(literal 4 binary64)) t)) (*.f64 b c)) (*.f64 (*.f64 x #s(literal 4 binary64)) i)) (*.f64 (*.f64 j #s(literal 27 binary64)) k)) < +inf.0

   1. Initial program 95.4%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   4. Applied rewrites 94.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(z \cdot y\right) \cdot x, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(z \cdot y\right) \cdot x}, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot \left(z \cdot y\right)}, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot \color{blue}{\left(z \cdot y\right)}, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot \color{blue}{\left(y \cdot z\right)}, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(x \cdot y\right) \cdot z}, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(x \cdot y\right) \cdot z}, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right) \]

      7. lower-*.f64 95.4

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(x \cdot y\right)} \cdot z, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right) \]

   6. Applied rewrites 95.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(x \cdot y\right) \cdot z}, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right) \]

   if +inf.0 < (-.f64 (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x #s(literal 18 binary64)) y) z) t) (*.f64 (*.f64 a #s(literal 4 binary64)) t)) (*.f64 b c)) (*.f64 (*.f64 x #s(literal 4 binary64)) i)) (*.f64 (*.f64 j #s(literal 27 binary64)) k))

   1. Initial program 0.0%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot c} - \left(j \cdot 27\right) \cdot k \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto c \cdot \color{blue}{b} - \left(j \cdot 27\right) \cdot k \]

      2. lower-*.f64 27.4

         \[\leadsto c \cdot \color{blue}{b} - \left(j \cdot 27\right) \cdot k \]

   5. Applied rewrites 27.4%

      \[\leadsto \color{blue}{c \cdot b} - \left(j \cdot 27\right) \cdot k \]

   6. Taylor expanded in i around inf

      \[\leadsto \color{blue}{-4 \cdot \left(i \cdot x\right)} - \left(j \cdot 27\right) \cdot k \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-4 \cdot i\right) \cdot \color{blue}{x} - \left(j \cdot 27\right) \cdot k \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-4 \cdot i\right) \cdot \color{blue}{x} - \left(j \cdot 27\right) \cdot k \]

      3. *-commutative N/A

         \[\leadsto \left(i \cdot -4\right) \cdot x - \left(j \cdot 27\right) \cdot k \]

      4. lower-*.f64 33.8

         \[\leadsto \left(i \cdot -4\right) \cdot x - \left(j \cdot 27\right) \cdot k \]

   8. Applied rewrites 33.8%

      \[\leadsto \color{blue}{\left(i \cdot -4\right) \cdot x} - \left(j \cdot 27\right) \cdot k \]

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)} - \left(j \cdot 27\right) \cdot k \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot \color{blue}{18} - \left(j \cdot 27\right) \cdot k \]

       2. lower-*.f64 N/A

          \[\leadsto \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot \color{blue}{18} - \left(j \cdot 27\right) \cdot k \]

       3. *-commutative N/A

          \[\leadsto \left(\left(x \cdot \left(y \cdot z\right)\right) \cdot t\right) \cdot 18 - \left(j \cdot 27\right) \cdot k \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\left(x \cdot \left(y \cdot z\right)\right) \cdot t\right) \cdot 18 - \left(j \cdot 27\right) \cdot k \]

       5. associate-*r* N/A

          \[\leadsto \left(\left(\left(x \cdot y\right) \cdot z\right) \cdot t\right) \cdot 18 - \left(j \cdot 27\right) \cdot k \]

       6. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(x \cdot y\right) \cdot z\right) \cdot t\right) \cdot 18 - \left(j \cdot 27\right) \cdot k \]

       7. *-commutative N/A

          \[\leadsto \left(\left(\left(y \cdot x\right) \cdot z\right) \cdot t\right) \cdot 18 - \left(j \cdot 27\right) \cdot k \]

       8. lower-*.f64 42.2

          \[\leadsto \left(\left(\left(y \cdot x\right) \cdot z\right) \cdot t\right) \cdot 18 - \left(j \cdot 27\right) \cdot k \]

   11. Applied rewrites 42.2%

       \[\leadsto \color{blue}{\left(\left(\left(y \cdot x\right) \cdot z\right) \cdot t\right) \cdot 18} - \left(j \cdot 27\right) \cdot k \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 66.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, c, \mathsf{fma}\left(i, x, t \cdot a\right) \cdot -4\right)\\ t_2 := \left(j \cdot 27\right) \cdot k\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-227}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-99}:\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot z\right) \cdot 18, y, i \cdot -4\right) \cdot x\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(a \cdot t\right) - t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (fma b c (* (fma i x (* t a)) -4.0))) (t_2 (* (* j 27.0) k)))
   (if (<= t_2 -2e+147)
     (fma c b (* (* k j) -27.0))
     (if (<= t_2 2e-227)
       t_1
       (if (<= t_2 1e-99)
         (* (fma (* (* t z) 18.0) y (* i -4.0)) x)
         (if (<= t_2 2e+51) t_1 (- (* -4.0 (* a t)) t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = fma(b, c, (fma(i, x, (t * a)) * -4.0));
	double t_2 = (j * 27.0) * k;
	double tmp;
	if (t_2 <= -2e+147) {
		tmp = fma(c, b, ((k * j) * -27.0));
	} else if (t_2 <= 2e-227) {
		tmp = t_1;
	} else if (t_2 <= 1e-99) {
		tmp = fma(((t * z) * 18.0), y, (i * -4.0)) * x;
	} else if (t_2 <= 2e+51) {
		tmp = t_1;
	} else {
		tmp = (-4.0 * (a * t)) - t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = fma(b, c, Float64(fma(i, x, Float64(t * a)) * -4.0))
	t_2 = Float64(Float64(j * 27.0) * k)
	tmp = 0.0
	if (t_2 <= -2e+147)
		tmp = fma(c, b, Float64(Float64(k * j) * -27.0));
	elseif (t_2 <= 2e-227)
		tmp = t_1;
	elseif (t_2 <= 1e-99)
		tmp = Float64(fma(Float64(Float64(t * z) * 18.0), y, Float64(i * -4.0)) * x);
	elseif (t_2 <= 2e+51)
		tmp = t_1;
	else
		tmp = Float64(Float64(-4.0 * Float64(a * t)) - t_2);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(b * c + N[(N[(i * x + N[(t * a), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+147], N[(c * b + N[(N[(k * j), $MachinePrecision] * -27.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e-227], t$95$1, If[LessEqual[t$95$2, 1e-99], N[(N[(N[(N[(t * z), $MachinePrecision] * 18.0), $MachinePrecision] * y + N[(i * -4.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$2, 2e+51], t$95$1, N[(N[(-4.0 * N[(a * t), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -2e147

   1. Initial program 79.0%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   4. Applied rewrites 81.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(z \cdot y\right) \cdot x, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right)} \]

   6. Applied rewrites 82.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(\left(y \cdot z\right) \cdot x, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right)} \]

   7. Taylor expanded in j around inf

      \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{-27 \cdot \left(j \cdot k\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(j \cdot k\right) \cdot \color{blue}{-27}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(j \cdot k\right) \cdot \color{blue}{-27}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right) \]

      4. lift-*.f64 69.8

         \[\leadsto \mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right) \]

   9. Applied rewrites 69.8%

      \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(k \cdot j\right) \cdot -27}\right) \]

   if -2e147 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 1.99999999999999989e-227 or 1e-99 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 2e51

   1. Initial program 87.6%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in j around 0

      \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + \color{blue}{\left(b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(18 \cdot t\right) \cdot \left(x \cdot \left(y \cdot z\right)\right) + \left(\color{blue}{b \cdot c} - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \color{blue}{x \cdot \left(y \cdot z\right)}, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \color{blue}{x} \cdot \left(y \cdot z\right), b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(y \cdot z\right) \cdot \color{blue}{x}, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(y \cdot z\right) \cdot \color{blue}{x}, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      12. distribute-lft-out N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \left(a \cdot t + i \cdot x\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \left(a \cdot t + i \cdot x\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \mathsf{fma}\left(a, t, i \cdot x\right)\right) \]

      15. lower-*.f64 83.0

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \mathsf{fma}\left(a, t, i \cdot x\right)\right) \]

   5. Applied rewrites 83.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \mathsf{fma}\left(a, t, i \cdot x\right)\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto b \cdot c - \color{blue}{4 \cdot \left(a \cdot t + i \cdot x\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto b \cdot c + \left(\mathsf{neg}\left(4\right)\right) \cdot \color{blue}{\left(a \cdot t + i \cdot x\right)} \]

      2. metadata-eval N/A

         \[\leadsto b \cdot c + -4 \cdot \left(a \cdot t + \color{blue}{i} \cdot x\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, c, -4 \cdot \left(a \cdot t + i \cdot x\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, c, \left(a \cdot t + i \cdot x\right) \cdot -4\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, c, \left(a \cdot t + i \cdot x\right) \cdot -4\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, c, \left(i \cdot x + a \cdot t\right) \cdot -4\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(i, x, a \cdot t\right) \cdot -4\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(i, x, t \cdot a\right) \cdot -4\right) \]

      9. lower-*.f64 70.7

         \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(i, x, t \cdot a\right) \cdot -4\right) \]

   8. Applied rewrites 70.7%

      \[\leadsto \mathsf{fma}\left(b, \color{blue}{c}, \mathsf{fma}\left(i, x, t \cdot a\right) \cdot -4\right) \]

   if 1.99999999999999989e-227 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 1e-99

   1. Initial program 85.9%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right) \cdot \color{blue}{x} \]

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

         \[\leadsto \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot i\right) \cdot x \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(18 \cdot t\right) \cdot \left(y \cdot z\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot i\right) \cdot x \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(18 \cdot t\right) \cdot \left(y \cdot z\right) + -4 \cdot i\right) \cdot x \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, y \cdot z, -4 \cdot i\right) \cdot x \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, y \cdot z, -4 \cdot i\right) \cdot x \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, z \cdot y, -4 \cdot i\right) \cdot x \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, z \cdot y, -4 \cdot i\right) \cdot x \]

      10. lower-*.f64 51.1

          \[\leadsto \mathsf{fma}\left(18 \cdot t, z \cdot y, -4 \cdot i\right) \cdot x \]

   5. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(18 \cdot t, z \cdot y, -4 \cdot i\right) \cdot x} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, z \cdot y, -4 \cdot i\right) \cdot x \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, z \cdot y, -4 \cdot i\right) \cdot x \]

      3. lift-fma.f64 N/A

         \[\leadsto \left(\left(18 \cdot t\right) \cdot \left(z \cdot y\right) + -4 \cdot i\right) \cdot x \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(18 \cdot t\right) \cdot \left(z \cdot y\right) + -4 \cdot i\right) \cdot x \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\left(18 \cdot t\right) \cdot z\right) \cdot y + -4 \cdot i\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(18 \cdot \left(t \cdot z\right)\right) \cdot y + -4 \cdot i\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot \left(t \cdot z\right), y, -4 \cdot i\right) \cdot x \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(t \cdot z\right) \cdot 18, y, -4 \cdot i\right) \cdot x \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(t \cdot z\right) \cdot 18, y, -4 \cdot i\right) \cdot x \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(t \cdot z\right) \cdot 18, y, -4 \cdot i\right) \cdot x \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(t \cdot z\right) \cdot 18, y, i \cdot -4\right) \cdot x \]

      12. lower-*.f64 52.3

          \[\leadsto \mathsf{fma}\left(\left(t \cdot z\right) \cdot 18, y, i \cdot -4\right) \cdot x \]

   7. Applied rewrites 52.3%

      \[\leadsto \mathsf{fma}\left(\left(t \cdot z\right) \cdot 18, y, i \cdot -4\right) \cdot x \]

   if 2e51 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 83.1%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-4 \cdot \left(a \cdot t\right)} - \left(j \cdot 27\right) \cdot k \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\left(a \cdot t\right)} - \left(j \cdot 27\right) \cdot k \]

      2. lower-*.f64 59.7

         \[\leadsto -4 \cdot \left(a \cdot \color{blue}{t}\right) - \left(j \cdot 27\right) \cdot k \]

   5. Applied rewrites 59.7%

      \[\leadsto \color{blue}{-4 \cdot \left(a \cdot t\right)} - \left(j \cdot 27\right) \cdot k \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 67.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, c, \mathsf{fma}\left(i, x, t \cdot a\right) \cdot -4\right)\\ t_2 := \left(j \cdot 27\right) \cdot k\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right)\\ \mathbf{elif}\;t\_2 \leq 10^{-251}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-137}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot x\right) \cdot z, 18, -4 \cdot a\right) \cdot t\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(a \cdot t\right) - t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (fma b c (* (fma i x (* t a)) -4.0))) (t_2 (* (* j 27.0) k)))
   (if (<= t_2 -2e+147)
     (fma c b (* (* k j) -27.0))
     (if (<= t_2 1e-251)
       t_1
       (if (<= t_2 4e-137)
         (* (fma (* (* y x) z) 18.0 (* -4.0 a)) t)
         (if (<= t_2 2e+51) t_1 (- (* -4.0 (* a t)) t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = fma(b, c, (fma(i, x, (t * a)) * -4.0));
	double t_2 = (j * 27.0) * k;
	double tmp;
	if (t_2 <= -2e+147) {
		tmp = fma(c, b, ((k * j) * -27.0));
	} else if (t_2 <= 1e-251) {
		tmp = t_1;
	} else if (t_2 <= 4e-137) {
		tmp = fma(((y * x) * z), 18.0, (-4.0 * a)) * t;
	} else if (t_2 <= 2e+51) {
		tmp = t_1;
	} else {
		tmp = (-4.0 * (a * t)) - t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = fma(b, c, Float64(fma(i, x, Float64(t * a)) * -4.0))
	t_2 = Float64(Float64(j * 27.0) * k)
	tmp = 0.0
	if (t_2 <= -2e+147)
		tmp = fma(c, b, Float64(Float64(k * j) * -27.0));
	elseif (t_2 <= 1e-251)
		tmp = t_1;
	elseif (t_2 <= 4e-137)
		tmp = Float64(fma(Float64(Float64(y * x) * z), 18.0, Float64(-4.0 * a)) * t);
	elseif (t_2 <= 2e+51)
		tmp = t_1;
	else
		tmp = Float64(Float64(-4.0 * Float64(a * t)) - t_2);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(b * c + N[(N[(i * x + N[(t * a), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+147], N[(c * b + N[(N[(k * j), $MachinePrecision] * -27.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-251], t$95$1, If[LessEqual[t$95$2, 4e-137], N[(N[(N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision] * 18.0 + N[(-4.0 * a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$2, 2e+51], t$95$1, N[(N[(-4.0 * N[(a * t), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -2e147

   1. Initial program 79.0%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   4. Applied rewrites 81.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(z \cdot y\right) \cdot x, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right)} \]

   6. Applied rewrites 82.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(\left(y \cdot z\right) \cdot x, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right)} \]

   7. Taylor expanded in j around inf

      \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{-27 \cdot \left(j \cdot k\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(j \cdot k\right) \cdot \color{blue}{-27}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(j \cdot k\right) \cdot \color{blue}{-27}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right) \]

      4. lift-*.f64 69.8

         \[\leadsto \mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right) \]

   9. Applied rewrites 69.8%

      \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(k \cdot j\right) \cdot -27}\right) \]

   if -2e147 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 1.00000000000000002e-251 or 3.99999999999999991e-137 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 2e51

   1. Initial program 87.5%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in j around 0

      \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + \color{blue}{\left(b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(18 \cdot t\right) \cdot \left(x \cdot \left(y \cdot z\right)\right) + \left(\color{blue}{b \cdot c} - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \color{blue}{x \cdot \left(y \cdot z\right)}, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \color{blue}{x} \cdot \left(y \cdot z\right), b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(y \cdot z\right) \cdot \color{blue}{x}, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(y \cdot z\right) \cdot \color{blue}{x}, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      12. distribute-lft-out N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \left(a \cdot t + i \cdot x\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \left(a \cdot t + i \cdot x\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \mathsf{fma}\left(a, t, i \cdot x\right)\right) \]

      15. lower-*.f64 82.9

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \mathsf{fma}\left(a, t, i \cdot x\right)\right) \]

   5. Applied rewrites 82.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \mathsf{fma}\left(a, t, i \cdot x\right)\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto b \cdot c - \color{blue}{4 \cdot \left(a \cdot t + i \cdot x\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto b \cdot c + \left(\mathsf{neg}\left(4\right)\right) \cdot \color{blue}{\left(a \cdot t + i \cdot x\right)} \]

      2. metadata-eval N/A

         \[\leadsto b \cdot c + -4 \cdot \left(a \cdot t + \color{blue}{i} \cdot x\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, c, -4 \cdot \left(a \cdot t + i \cdot x\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, c, \left(a \cdot t + i \cdot x\right) \cdot -4\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, c, \left(a \cdot t + i \cdot x\right) \cdot -4\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, c, \left(i \cdot x + a \cdot t\right) \cdot -4\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(i, x, a \cdot t\right) \cdot -4\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(i, x, t \cdot a\right) \cdot -4\right) \]

      9. lower-*.f64 70.6

         \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(i, x, t \cdot a\right) \cdot -4\right) \]

   8. Applied rewrites 70.6%

      \[\leadsto \mathsf{fma}\left(b, \color{blue}{c}, \mathsf{fma}\left(i, x, t \cdot a\right) \cdot -4\right) \]

   if 1.00000000000000002e-251 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 3.99999999999999991e-137

   1. Initial program 86.5%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   4. Applied rewrites 86.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(z \cdot y\right) \cdot x, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right)} \]

   6. Applied rewrites 88.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(\left(y \cdot z\right) \cdot x, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(y \cdot z\right)} \cdot x, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(y \cdot z\right) \cdot x}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{x \cdot \left(y \cdot z\right)}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(x \cdot y\right) \cdot z}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(x \cdot y\right) \cdot z}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      6. lower-*.f64 91.0

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(x \cdot y\right)} \cdot z, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

   8. Applied rewrites 91.0%

      \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(x \cdot y\right) \cdot z}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-4 \cdot a + 18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto t \cdot \left(-4 \cdot a + 18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto t \cdot \left(-4 \cdot a + 18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \left(-4 \cdot a + 18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot \color{blue}{t} \]

   11. Applied rewrites 51.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y \cdot x\right) \cdot z, 18, -4 \cdot a\right) \cdot t} \]

   if 2e51 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 83.1%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-4 \cdot \left(a \cdot t\right)} - \left(j \cdot 27\right) \cdot k \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\left(a \cdot t\right)} - \left(j \cdot 27\right) \cdot k \]

      2. lower-*.f64 59.7

         \[\leadsto -4 \cdot \left(a \cdot \color{blue}{t}\right) - \left(j \cdot 27\right) \cdot k \]

   5. Applied rewrites 59.7%

      \[\leadsto \color{blue}{-4 \cdot \left(a \cdot t\right)} - \left(j \cdot 27\right) \cdot k \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 66.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, c, \mathsf{fma}\left(i, x, t \cdot a\right) \cdot -4\right)\\ t_2 := \left(j \cdot 27\right) \cdot k\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-207}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-137}:\\ \;\;\;\;\left(\left(\left(y \cdot x\right) \cdot z\right) \cdot t\right) \cdot 18\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(a \cdot t\right) - t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (fma b c (* (fma i x (* t a)) -4.0))) (t_2 (* (* j 27.0) k)))
   (if (<= t_2 -2e+147)
     (fma c b (* (* k j) -27.0))
     (if (<= t_2 5e-207)
       t_1
       (if (<= t_2 4e-137)
         (* (* (* (* y x) z) t) 18.0)
         (if (<= t_2 2e+51) t_1 (- (* -4.0 (* a t)) t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = fma(b, c, (fma(i, x, (t * a)) * -4.0));
	double t_2 = (j * 27.0) * k;
	double tmp;
	if (t_2 <= -2e+147) {
		tmp = fma(c, b, ((k * j) * -27.0));
	} else if (t_2 <= 5e-207) {
		tmp = t_1;
	} else if (t_2 <= 4e-137) {
		tmp = (((y * x) * z) * t) * 18.0;
	} else if (t_2 <= 2e+51) {
		tmp = t_1;
	} else {
		tmp = (-4.0 * (a * t)) - t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = fma(b, c, Float64(fma(i, x, Float64(t * a)) * -4.0))
	t_2 = Float64(Float64(j * 27.0) * k)
	tmp = 0.0
	if (t_2 <= -2e+147)
		tmp = fma(c, b, Float64(Float64(k * j) * -27.0));
	elseif (t_2 <= 5e-207)
		tmp = t_1;
	elseif (t_2 <= 4e-137)
		tmp = Float64(Float64(Float64(Float64(y * x) * z) * t) * 18.0);
	elseif (t_2 <= 2e+51)
		tmp = t_1;
	else
		tmp = Float64(Float64(-4.0 * Float64(a * t)) - t_2);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(b * c + N[(N[(i * x + N[(t * a), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+147], N[(c * b + N[(N[(k * j), $MachinePrecision] * -27.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-207], t$95$1, If[LessEqual[t$95$2, 4e-137], N[(N[(N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] * 18.0), $MachinePrecision], If[LessEqual[t$95$2, 2e+51], t$95$1, N[(N[(-4.0 * N[(a * t), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -2e147

   1. Initial program 79.0%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   4. Applied rewrites 81.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(z \cdot y\right) \cdot x, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right)} \]

   6. Applied rewrites 82.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(\left(y \cdot z\right) \cdot x, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right)} \]

   7. Taylor expanded in j around inf

      \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{-27 \cdot \left(j \cdot k\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(j \cdot k\right) \cdot \color{blue}{-27}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(j \cdot k\right) \cdot \color{blue}{-27}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right) \]

      4. lift-*.f64 69.8

         \[\leadsto \mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right) \]

   9. Applied rewrites 69.8%

      \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(k \cdot j\right) \cdot -27}\right) \]

   if -2e147 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 5.00000000000000014e-207 or 3.99999999999999991e-137 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 2e51

   1. Initial program 87.5%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in j around 0

      \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + \color{blue}{\left(b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(18 \cdot t\right) \cdot \left(x \cdot \left(y \cdot z\right)\right) + \left(\color{blue}{b \cdot c} - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \color{blue}{x \cdot \left(y \cdot z\right)}, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \color{blue}{x} \cdot \left(y \cdot z\right), b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(y \cdot z\right) \cdot \color{blue}{x}, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(y \cdot z\right) \cdot \color{blue}{x}, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      12. distribute-lft-out N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \left(a \cdot t + i \cdot x\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \left(a \cdot t + i \cdot x\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \mathsf{fma}\left(a, t, i \cdot x\right)\right) \]

      15. lower-*.f64 83.1

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \mathsf{fma}\left(a, t, i \cdot x\right)\right) \]

   5. Applied rewrites 83.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \mathsf{fma}\left(a, t, i \cdot x\right)\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto b \cdot c - \color{blue}{4 \cdot \left(a \cdot t + i \cdot x\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto b \cdot c + \left(\mathsf{neg}\left(4\right)\right) \cdot \color{blue}{\left(a \cdot t + i \cdot x\right)} \]

      2. metadata-eval N/A

         \[\leadsto b \cdot c + -4 \cdot \left(a \cdot t + \color{blue}{i} \cdot x\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, c, -4 \cdot \left(a \cdot t + i \cdot x\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, c, \left(a \cdot t + i \cdot x\right) \cdot -4\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, c, \left(a \cdot t + i \cdot x\right) \cdot -4\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, c, \left(i \cdot x + a \cdot t\right) \cdot -4\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(i, x, a \cdot t\right) \cdot -4\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(i, x, t \cdot a\right) \cdot -4\right) \]

      9. lower-*.f64 70.8

         \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(i, x, t \cdot a\right) \cdot -4\right) \]

   8. Applied rewrites 70.8%

      \[\leadsto \mathsf{fma}\left(b, \color{blue}{c}, \mathsf{fma}\left(i, x, t \cdot a\right) \cdot -4\right) \]

   if 5.00000000000000014e-207 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 3.99999999999999991e-137

   1. Initial program 85.0%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   4. Applied rewrites 85.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(z \cdot y\right) \cdot x, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right)} \]

   6. Applied rewrites 87.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(\left(y \cdot z\right) \cdot x, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(y \cdot z\right)} \cdot x, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(y \cdot z\right) \cdot x}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{x \cdot \left(y \cdot z\right)}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(x \cdot y\right) \cdot z}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(x \cdot y\right) \cdot z}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      6. lower-*.f64 90.5

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(x \cdot y\right)} \cdot z, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

   8. Applied rewrites 90.5%

      \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(x \cdot y\right) \cdot z}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot \color{blue}{18} \]

       4. lower-*.f64 N/A

          \[\leadsto \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot \color{blue}{18} \]

       5. *-commutative N/A

          \[\leadsto \left(\left(x \cdot \left(y \cdot z\right)\right) \cdot t\right) \cdot 18 \]

       6. lower-*.f64 N/A

          \[\leadsto \left(\left(x \cdot \left(y \cdot z\right)\right) \cdot t\right) \cdot 18 \]

       7. associate-*r* N/A

          \[\leadsto \left(\left(\left(x \cdot y\right) \cdot z\right) \cdot t\right) \cdot 18 \]

       8. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(x \cdot y\right) \cdot z\right) \cdot t\right) \cdot 18 \]

       9. *-commutative N/A

          \[\leadsto \left(\left(\left(y \cdot x\right) \cdot z\right) \cdot t\right) \cdot 18 \]

       10. lower-*.f64 29.8

           \[\leadsto \left(\left(\left(y \cdot x\right) \cdot z\right) \cdot t\right) \cdot 18 \]

   11. Applied rewrites 29.8%

       \[\leadsto \color{blue}{\left(\left(\left(y \cdot x\right) \cdot z\right) \cdot t\right) \cdot 18} \]

   if 2e51 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 83.1%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-4 \cdot \left(a \cdot t\right)} - \left(j \cdot 27\right) \cdot k \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\left(a \cdot t\right)} - \left(j \cdot 27\right) \cdot k \]

      2. lower-*.f64 59.7

         \[\leadsto -4 \cdot \left(a \cdot \color{blue}{t}\right) - \left(j \cdot 27\right) \cdot k \]

   5. Applied rewrites 59.7%

      \[\leadsto \color{blue}{-4 \cdot \left(a \cdot t\right)} - \left(j \cdot 27\right) \cdot k \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 67.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right)\\ t_2 := \mathsf{fma}\left(b, c, \mathsf{fma}\left(i, x, t \cdot a\right) \cdot -4\right)\\ t_3 := \left(j \cdot 27\right) \cdot k\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+147}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-207}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{-137}:\\ \;\;\;\;\left(\left(\left(y \cdot x\right) \cdot z\right) \cdot t\right) \cdot 18\\ \mathbf{elif}\;t\_3 \leq 10^{+98}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (fma c b (* (* k j) -27.0)))
        (t_2 (fma b c (* (fma i x (* t a)) -4.0)))
        (t_3 (* (* j 27.0) k)))
   (if (<= t_3 -2e+147)
     t_1
     (if (<= t_3 5e-207)
       t_2
       (if (<= t_3 4e-137)
         (* (* (* (* y x) z) t) 18.0)
         (if (<= t_3 1e+98) t_2 t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = fma(c, b, ((k * j) * -27.0));
	double t_2 = fma(b, c, (fma(i, x, (t * a)) * -4.0));
	double t_3 = (j * 27.0) * k;
	double tmp;
	if (t_3 <= -2e+147) {
		tmp = t_1;
	} else if (t_3 <= 5e-207) {
		tmp = t_2;
	} else if (t_3 <= 4e-137) {
		tmp = (((y * x) * z) * t) * 18.0;
	} else if (t_3 <= 1e+98) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = fma(c, b, Float64(Float64(k * j) * -27.0))
	t_2 = fma(b, c, Float64(fma(i, x, Float64(t * a)) * -4.0))
	t_3 = Float64(Float64(j * 27.0) * k)
	tmp = 0.0
	if (t_3 <= -2e+147)
		tmp = t_1;
	elseif (t_3 <= 5e-207)
		tmp = t_2;
	elseif (t_3 <= 4e-137)
		tmp = Float64(Float64(Float64(Float64(y * x) * z) * t) * 18.0);
	elseif (t_3 <= 1e+98)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(c * b + N[(N[(k * j), $MachinePrecision] * -27.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * c + N[(N[(i * x + N[(t * a), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+147], t$95$1, If[LessEqual[t$95$3, 5e-207], t$95$2, If[LessEqual[t$95$3, 4e-137], N[(N[(N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] * 18.0), $MachinePrecision], If[LessEqual[t$95$3, 1e+98], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -2e147 or 9.99999999999999998e97 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 81.0%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   4. Applied rewrites 83.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(z \cdot y\right) \cdot x, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right)} \]

   6. Applied rewrites 84.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(\left(y \cdot z\right) \cdot x, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right)} \]

   7. Taylor expanded in j around inf

      \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{-27 \cdot \left(j \cdot k\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(j \cdot k\right) \cdot \color{blue}{-27}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(j \cdot k\right) \cdot \color{blue}{-27}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right) \]

      4. lift-*.f64 67.8

         \[\leadsto \mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right) \]

   9. Applied rewrites 67.8%

      \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(k \cdot j\right) \cdot -27}\right) \]

   if -2e147 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 5.00000000000000014e-207 or 3.99999999999999991e-137 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 9.99999999999999998e97

   1. Initial program 87.4%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in j around 0

      \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + \color{blue}{\left(b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(18 \cdot t\right) \cdot \left(x \cdot \left(y \cdot z\right)\right) + \left(\color{blue}{b \cdot c} - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \color{blue}{x \cdot \left(y \cdot z\right)}, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \color{blue}{x} \cdot \left(y \cdot z\right), b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(y \cdot z\right) \cdot \color{blue}{x}, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(y \cdot z\right) \cdot \color{blue}{x}, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      12. distribute-lft-out N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \left(a \cdot t + i \cdot x\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \left(a \cdot t + i \cdot x\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \mathsf{fma}\left(a, t, i \cdot x\right)\right) \]

      15. lower-*.f64 82.5

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \mathsf{fma}\left(a, t, i \cdot x\right)\right) \]

   5. Applied rewrites 82.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \mathsf{fma}\left(a, t, i \cdot x\right)\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto b \cdot c - \color{blue}{4 \cdot \left(a \cdot t + i \cdot x\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto b \cdot c + \left(\mathsf{neg}\left(4\right)\right) \cdot \color{blue}{\left(a \cdot t + i \cdot x\right)} \]

      2. metadata-eval N/A

         \[\leadsto b \cdot c + -4 \cdot \left(a \cdot t + \color{blue}{i} \cdot x\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, c, -4 \cdot \left(a \cdot t + i \cdot x\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, c, \left(a \cdot t + i \cdot x\right) \cdot -4\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, c, \left(a \cdot t + i \cdot x\right) \cdot -4\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, c, \left(i \cdot x + a \cdot t\right) \cdot -4\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(i, x, a \cdot t\right) \cdot -4\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(i, x, t \cdot a\right) \cdot -4\right) \]

      9. lower-*.f64 70.4

         \[\leadsto \mathsf{fma}\left(b, c, \mathsf{fma}\left(i, x, t \cdot a\right) \cdot -4\right) \]

   8. Applied rewrites 70.4%

      \[\leadsto \mathsf{fma}\left(b, \color{blue}{c}, \mathsf{fma}\left(i, x, t \cdot a\right) \cdot -4\right) \]

   if 5.00000000000000014e-207 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 3.99999999999999991e-137

   1. Initial program 85.0%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   4. Applied rewrites 85.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(z \cdot y\right) \cdot x, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right)} \]

   6. Applied rewrites 87.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(\left(y \cdot z\right) \cdot x, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(y \cdot z\right)} \cdot x, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(y \cdot z\right) \cdot x}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{x \cdot \left(y \cdot z\right)}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(x \cdot y\right) \cdot z}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(x \cdot y\right) \cdot z}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      6. lower-*.f64 90.5

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(x \cdot y\right)} \cdot z, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

   8. Applied rewrites 90.5%

      \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(x \cdot y\right) \cdot z}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot \color{blue}{18} \]

       4. lower-*.f64 N/A

          \[\leadsto \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot \color{blue}{18} \]

       5. *-commutative N/A

          \[\leadsto \left(\left(x \cdot \left(y \cdot z\right)\right) \cdot t\right) \cdot 18 \]

       6. lower-*.f64 N/A

          \[\leadsto \left(\left(x \cdot \left(y \cdot z\right)\right) \cdot t\right) \cdot 18 \]

       7. associate-*r* N/A

          \[\leadsto \left(\left(\left(x \cdot y\right) \cdot z\right) \cdot t\right) \cdot 18 \]

       8. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(x \cdot y\right) \cdot z\right) \cdot t\right) \cdot 18 \]

       9. *-commutative N/A

          \[\leadsto \left(\left(\left(y \cdot x\right) \cdot z\right) \cdot t\right) \cdot 18 \]

       10. lower-*.f64 29.8

           \[\leadsto \left(\left(\left(y \cdot x\right) \cdot z\right) \cdot t\right) \cdot 18 \]

   11. Applied rewrites 29.8%

       \[\leadsto \color{blue}{\left(\left(\left(y \cdot x\right) \cdot z\right) \cdot t\right) \cdot 18} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 53.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right)\\ t_2 := \left(j \cdot 27\right) \cdot k\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-192}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right)\\ \mathbf{elif}\;t\_2 \leq 10^{-99}:\\ \;\;\;\;\left(\left(\left(z \cdot y\right) \cdot 18\right) \cdot t\right) \cdot x\\ \mathbf{elif}\;t\_2 \leq 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(i \cdot -4, x, b \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (fma c b (* (* k j) -27.0))) (t_2 (* (* j 27.0) k)))
   (if (<= t_2 -1e+68)
     t_1
     (if (<= t_2 1e-192)
       (fma c b (* (* i x) -4.0))
       (if (<= t_2 1e-99)
         (* (* (* (* z y) 18.0) t) x)
         (if (<= t_2 1e+60) (fma (* i -4.0) x (* b c)) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = fma(c, b, ((k * j) * -27.0));
	double t_2 = (j * 27.0) * k;
	double tmp;
	if (t_2 <= -1e+68) {
		tmp = t_1;
	} else if (t_2 <= 1e-192) {
		tmp = fma(c, b, ((i * x) * -4.0));
	} else if (t_2 <= 1e-99) {
		tmp = (((z * y) * 18.0) * t) * x;
	} else if (t_2 <= 1e+60) {
		tmp = fma((i * -4.0), x, (b * c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = fma(c, b, Float64(Float64(k * j) * -27.0))
	t_2 = Float64(Float64(j * 27.0) * k)
	tmp = 0.0
	if (t_2 <= -1e+68)
		tmp = t_1;
	elseif (t_2 <= 1e-192)
		tmp = fma(c, b, Float64(Float64(i * x) * -4.0));
	elseif (t_2 <= 1e-99)
		tmp = Float64(Float64(Float64(Float64(z * y) * 18.0) * t) * x);
	elseif (t_2 <= 1e+60)
		tmp = fma(Float64(i * -4.0), x, Float64(b * c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(c * b + N[(N[(k * j), $MachinePrecision] * -27.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+68], t$95$1, If[LessEqual[t$95$2, 1e-192], N[(c * b + N[(N[(i * x), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-99], N[(N[(N[(N[(z * y), $MachinePrecision] * 18.0), $MachinePrecision] * t), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$2, 1e+60], N[(N[(i * -4.0), $MachinePrecision] * x + N[(b * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -9.99999999999999953e67 or 9.9999999999999995e59 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 82.1%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   4. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(z \cdot y\right) \cdot x, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right)} \]

   6. Applied rewrites 85.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(\left(y \cdot z\right) \cdot x, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right)} \]

   7. Taylor expanded in j around inf

      \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{-27 \cdot \left(j \cdot k\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(j \cdot k\right) \cdot \color{blue}{-27}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(j \cdot k\right) \cdot \color{blue}{-27}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right) \]

      4. lift-*.f64 62.4

         \[\leadsto \mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right) \]

   9. Applied rewrites 62.4%

      \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(k \cdot j\right) \cdot -27}\right) \]

   if -9.99999999999999953e67 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 1.0000000000000001e-192

   1. Initial program 87.2%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   4. Applied rewrites 89.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(z \cdot y\right) \cdot x, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right)} \]

   6. Applied rewrites 89.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(\left(y \cdot z\right) \cdot x, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right)} \]

   7. Taylor expanded in i around inf

      \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{-4 \cdot \left(i \cdot x\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot \color{blue}{-4}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot \color{blue}{-4}\right) \]

      3. lift-*.f64 50.7

         \[\leadsto \mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right) \]

   9. Applied rewrites 50.7%

      \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(i \cdot x\right) \cdot -4}\right) \]

   if 1.0000000000000001e-192 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 1e-99

   1. Initial program 85.7%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) - 4 \cdot i\right) \cdot \color{blue}{x} \]

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

         \[\leadsto \left(18 \cdot \left(t \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot i\right) \cdot x \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(18 \cdot t\right) \cdot \left(y \cdot z\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot i\right) \cdot x \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(18 \cdot t\right) \cdot \left(y \cdot z\right) + -4 \cdot i\right) \cdot x \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, y \cdot z, -4 \cdot i\right) \cdot x \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, y \cdot z, -4 \cdot i\right) \cdot x \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, z \cdot y, -4 \cdot i\right) \cdot x \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, z \cdot y, -4 \cdot i\right) \cdot x \]

      10. lower-*.f64 52.6

          \[\leadsto \mathsf{fma}\left(18 \cdot t, z \cdot y, -4 \cdot i\right) \cdot x \]

   5. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(18 \cdot t, z \cdot y, -4 \cdot i\right) \cdot x} \]

   6. Taylor expanded in t around inf

      \[\leadsto \left(t \cdot \left(-4 \cdot \frac{i}{t} + 18 \cdot \left(y \cdot z\right)\right)\right) \cdot x \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(-4 \cdot \frac{i}{t} + 18 \cdot \left(y \cdot z\right)\right) \cdot t\right) \cdot x \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(-4 \cdot \frac{i}{t} + 18 \cdot \left(y \cdot z\right)\right) \cdot t\right) \cdot x \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{i}{t} \cdot -4 + 18 \cdot \left(y \cdot z\right)\right) \cdot t\right) \cdot x \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{i}{t}, -4, 18 \cdot \left(y \cdot z\right)\right) \cdot t\right) \cdot x \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{i}{t}, -4, 18 \cdot \left(y \cdot z\right)\right) \cdot t\right) \cdot x \]

      6. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{i}{t}, -4, \left(y \cdot z\right) \cdot 18\right) \cdot t\right) \cdot x \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{i}{t}, -4, \left(y \cdot z\right) \cdot 18\right) \cdot t\right) \cdot x \]

      8. lower-*.f64 46.3

         \[\leadsto \left(\mathsf{fma}\left(\frac{i}{t}, -4, \left(y \cdot z\right) \cdot 18\right) \cdot t\right) \cdot x \]

   8. Applied rewrites 46.3%

      \[\leadsto \left(\mathsf{fma}\left(\frac{i}{t}, -4, \left(y \cdot z\right) \cdot 18\right) \cdot t\right) \cdot x \]

   9. Taylor expanded in y around inf

      \[\leadsto \left(\left(18 \cdot \left(y \cdot z\right)\right) \cdot t\right) \cdot x \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\left(y \cdot z\right) \cdot 18\right) \cdot t\right) \cdot x \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(y \cdot z\right) \cdot 18\right) \cdot t\right) \cdot x \]

       3. *-commutative N/A

          \[\leadsto \left(\left(\left(z \cdot y\right) \cdot 18\right) \cdot t\right) \cdot x \]

       4. lift-*.f64 29.0

          \[\leadsto \left(\left(\left(z \cdot y\right) \cdot 18\right) \cdot t\right) \cdot x \]

   11. Applied rewrites 29.0%

       \[\leadsto \left(\left(\left(z \cdot y\right) \cdot 18\right) \cdot t\right) \cdot x \]

   if 1e-99 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 9.9999999999999995e59

   1. Initial program 88.8%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in j around 0

      \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + \color{blue}{\left(b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(18 \cdot t\right) \cdot \left(x \cdot \left(y \cdot z\right)\right) + \left(\color{blue}{b \cdot c} - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \color{blue}{x \cdot \left(y \cdot z\right)}, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \color{blue}{x} \cdot \left(y \cdot z\right), b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(y \cdot z\right) \cdot \color{blue}{x}, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(y \cdot z\right) \cdot \color{blue}{x}, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      12. distribute-lft-out N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \left(a \cdot t + i \cdot x\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \left(a \cdot t + i \cdot x\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \mathsf{fma}\left(a, t, i \cdot x\right)\right) \]

      15. lower-*.f64 80.6

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \mathsf{fma}\left(a, t, i \cdot x\right)\right) \]

   5. Applied rewrites 80.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \mathsf{fma}\left(a, t, i \cdot x\right)\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto b \cdot c - \color{blue}{4 \cdot \left(i \cdot x\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto b \cdot c + \left(\mathsf{neg}\left(4\right)\right) \cdot \color{blue}{\left(i \cdot x\right)} \]

      2. metadata-eval N/A

         \[\leadsto b \cdot c + -4 \cdot \left(i \cdot x\right) \]

      3. +-commutative N/A

         \[\leadsto -4 \cdot \left(i \cdot x\right) + b \cdot \color{blue}{c} \]

      4. associate-*r* N/A

         \[\leadsto \left(-4 \cdot i\right) \cdot x + b \cdot c \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot i, x, b \cdot c\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(i \cdot -4, x, b \cdot c\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(i \cdot -4, x, b \cdot c\right) \]

      8. lower-*.f64 48.1

         \[\leadsto \mathsf{fma}\left(i \cdot -4, x, b \cdot c\right) \]

   8. Applied rewrites 48.1%

      \[\leadsto \mathsf{fma}\left(i \cdot -4, \color{blue}{x}, b \cdot c\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 53.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right)\\ t_2 := \mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right)\\ t_3 := \left(j \cdot 27\right) \cdot k\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{+68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 10^{-251}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{-137}:\\ \;\;\;\;\left(\left(\left(y \cdot x\right) \cdot z\right) \cdot t\right) \cdot 18\\ \mathbf{elif}\;t\_3 \leq 10^{+60}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (fma c b (* (* k j) -27.0)))
        (t_2 (fma c b (* (* i x) -4.0)))
        (t_3 (* (* j 27.0) k)))
   (if (<= t_3 -1e+68)
     t_1
     (if (<= t_3 1e-251)
       t_2
       (if (<= t_3 4e-137)
         (* (* (* (* y x) z) t) 18.0)
         (if (<= t_3 1e+60) t_2 t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = fma(c, b, ((k * j) * -27.0));
	double t_2 = fma(c, b, ((i * x) * -4.0));
	double t_3 = (j * 27.0) * k;
	double tmp;
	if (t_3 <= -1e+68) {
		tmp = t_1;
	} else if (t_3 <= 1e-251) {
		tmp = t_2;
	} else if (t_3 <= 4e-137) {
		tmp = (((y * x) * z) * t) * 18.0;
	} else if (t_3 <= 1e+60) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = fma(c, b, Float64(Float64(k * j) * -27.0))
	t_2 = fma(c, b, Float64(Float64(i * x) * -4.0))
	t_3 = Float64(Float64(j * 27.0) * k)
	tmp = 0.0
	if (t_3 <= -1e+68)
		tmp = t_1;
	elseif (t_3 <= 1e-251)
		tmp = t_2;
	elseif (t_3 <= 4e-137)
		tmp = Float64(Float64(Float64(Float64(y * x) * z) * t) * 18.0);
	elseif (t_3 <= 1e+60)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(c * b + N[(N[(k * j), $MachinePrecision] * -27.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c * b + N[(N[(i * x), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[t$95$3, -1e+68], t$95$1, If[LessEqual[t$95$3, 1e-251], t$95$2, If[LessEqual[t$95$3, 4e-137], N[(N[(N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] * 18.0), $MachinePrecision], If[LessEqual[t$95$3, 1e+60], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -9.99999999999999953e67 or 9.9999999999999995e59 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 82.1%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   4. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(z \cdot y\right) \cdot x, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right)} \]

   6. Applied rewrites 85.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(\left(y \cdot z\right) \cdot x, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right)} \]

   7. Taylor expanded in j around inf

      \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{-27 \cdot \left(j \cdot k\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(j \cdot k\right) \cdot \color{blue}{-27}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(j \cdot k\right) \cdot \color{blue}{-27}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right) \]

      4. lift-*.f64 62.4

         \[\leadsto \mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right) \]

   9. Applied rewrites 62.4%

      \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(k \cdot j\right) \cdot -27}\right) \]

   if -9.99999999999999953e67 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 1.00000000000000002e-251 or 3.99999999999999991e-137 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 9.9999999999999995e59

   1. Initial program 87.5%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   4. Applied rewrites 89.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(z \cdot y\right) \cdot x, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right)} \]

   6. Applied rewrites 90.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(\left(y \cdot z\right) \cdot x, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right)} \]

   7. Taylor expanded in i around inf

      \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{-4 \cdot \left(i \cdot x\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot \color{blue}{-4}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot \color{blue}{-4}\right) \]

      3. lift-*.f64 50.2

         \[\leadsto \mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right) \]

   9. Applied rewrites 50.2%

      \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(i \cdot x\right) \cdot -4}\right) \]

   if 1.00000000000000002e-251 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 3.99999999999999991e-137

   1. Initial program 86.5%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   4. Applied rewrites 86.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(z \cdot y\right) \cdot x, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right)} \]

   6. Applied rewrites 88.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(\left(y \cdot z\right) \cdot x, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(y \cdot z\right)} \cdot x, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(y \cdot z\right) \cdot x}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{x \cdot \left(y \cdot z\right)}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(x \cdot y\right) \cdot z}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(x \cdot y\right) \cdot z}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      6. lower-*.f64 91.0

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(x \cdot y\right)} \cdot z, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

   8. Applied rewrites 91.0%

      \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(x \cdot y\right) \cdot z}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot \color{blue}{18} \]

       4. lower-*.f64 N/A

          \[\leadsto \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot \color{blue}{18} \]

       5. *-commutative N/A

          \[\leadsto \left(\left(x \cdot \left(y \cdot z\right)\right) \cdot t\right) \cdot 18 \]

       6. lower-*.f64 N/A

          \[\leadsto \left(\left(x \cdot \left(y \cdot z\right)\right) \cdot t\right) \cdot 18 \]

       7. associate-*r* N/A

          \[\leadsto \left(\left(\left(x \cdot y\right) \cdot z\right) \cdot t\right) \cdot 18 \]

       8. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(x \cdot y\right) \cdot z\right) \cdot t\right) \cdot 18 \]

       9. *-commutative N/A

          \[\leadsto \left(\left(\left(y \cdot x\right) \cdot z\right) \cdot t\right) \cdot 18 \]

       10. lower-*.f64 29.2

           \[\leadsto \left(\left(\left(y \cdot x\right) \cdot z\right) \cdot t\right) \cdot 18 \]

   11. Applied rewrites 29.2%

       \[\leadsto \color{blue}{\left(\left(\left(y \cdot x\right) \cdot z\right) \cdot t\right) \cdot 18} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 79.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4 \cdot i, x, \left(k \cdot j\right) \cdot 27\right)\right)\\ t_2 := \left(j \cdot 27\right) \cdot k\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+147}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \mathsf{fma}\left(a, t, i \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (fma c b (- (fma (* 4.0 i) x (* (* k j) 27.0)))))
        (t_2 (* (* j 27.0) k)))
   (if (<= t_2 -2e+147)
     t_1
     (if (<= t_2 1e+18)
       (fma (* 18.0 t) (* (* z y) x) (- (* c b) (* 4.0 (fma a t (* i x)))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = fma(c, b, -fma((4.0 * i), x, ((k * j) * 27.0)));
	double t_2 = (j * 27.0) * k;
	double tmp;
	if (t_2 <= -2e+147) {
		tmp = t_1;
	} else if (t_2 <= 1e+18) {
		tmp = fma((18.0 * t), ((z * y) * x), ((c * b) - (4.0 * fma(a, t, (i * x)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = fma(c, b, Float64(-fma(Float64(4.0 * i), x, Float64(Float64(k * j) * 27.0))))
	t_2 = Float64(Float64(j * 27.0) * k)
	tmp = 0.0
	if (t_2 <= -2e+147)
		tmp = t_1;
	elseif (t_2 <= 1e+18)
		tmp = fma(Float64(18.0 * t), Float64(Float64(z * y) * x), Float64(Float64(c * b) - Float64(4.0 * fma(a, t, Float64(i * x)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(c * b + (-N[(N[(4.0 * i), $MachinePrecision] * x + N[(N[(k * j), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]}, Block[{t$95$2 = N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+147], t$95$1, If[LessEqual[t$95$2, 1e+18], N[(N[(18.0 * t), $MachinePrecision] * N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision] + N[(N[(c * b), $MachinePrecision] - N[(4.0 * N[(a * t + N[(i * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -2e147 or 1e18 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 81.9%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   4. Applied rewrites 84.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(z \cdot y\right) \cdot x, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right)} \]

   6. Applied rewrites 85.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(\left(y \cdot z\right) \cdot x, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right)} \]

   7. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{-1 \cdot \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, -\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(c, b, -\left(\left(4 \cdot i\right) \cdot x + 27 \cdot \left(j \cdot k\right)\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4 \cdot i, x, 27 \cdot \left(j \cdot k\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4 \cdot i, x, 27 \cdot \left(j \cdot k\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4 \cdot i, x, \left(j \cdot k\right) \cdot 27\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4 \cdot i, x, \left(j \cdot k\right) \cdot 27\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4 \cdot i, x, \left(k \cdot j\right) \cdot 27\right)\right) \]

      9. lift-*.f64 74.3

         \[\leadsto \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4 \cdot i, x, \left(k \cdot j\right) \cdot 27\right)\right) \]

   9. Applied rewrites 74.3%

      \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{-\mathsf{fma}\left(4 \cdot i, x, \left(k \cdot j\right) \cdot 27\right)}\right) \]

   if -2e147 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 1e18

   1. Initial program 87.3%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in j around 0

      \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + \color{blue}{\left(b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(18 \cdot t\right) \cdot \left(x \cdot \left(y \cdot z\right)\right) + \left(\color{blue}{b \cdot c} - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \color{blue}{x \cdot \left(y \cdot z\right)}, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \color{blue}{x} \cdot \left(y \cdot z\right), b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(y \cdot z\right) \cdot \color{blue}{x}, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(y \cdot z\right) \cdot \color{blue}{x}, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      12. distribute-lft-out N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \left(a \cdot t + i \cdot x\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \left(a \cdot t + i \cdot x\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \mathsf{fma}\left(a, t, i \cdot x\right)\right) \]

      15. lower-*.f64 83.3

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \mathsf{fma}\left(a, t, i \cdot x\right)\right) \]

   5. Applied rewrites 83.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \mathsf{fma}\left(a, t, i \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 35.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(j \cdot 27\right) \cdot k\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+147}:\\ \;\;\;\;-27 \cdot \left(k \cdot j\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-99}:\\ \;\;\;\;-4 \cdot \left(a \cdot t\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+51}:\\ \;\;\;\;c \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(-27 \cdot k\right) \cdot j\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* (* j 27.0) k)))
   (if (<= t_1 -2e+147)
     (* -27.0 (* k j))
     (if (<= t_1 1e-99)
       (* -4.0 (* a t))
       (if (<= t_1 2e+51) (* c b) (* (* -27.0 k) j))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = (j * 27.0) * k;
	double tmp;
	if (t_1 <= -2e+147) {
		tmp = -27.0 * (k * j);
	} else if (t_1 <= 1e-99) {
		tmp = -4.0 * (a * t);
	} else if (t_1 <= 2e+51) {
		tmp = c * b;
	} else {
		tmp = (-27.0 * k) * j;
	}
	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, b, c, i, j, k)
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), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (j * 27.0d0) * k
    if (t_1 <= (-2d+147)) then
        tmp = (-27.0d0) * (k * j)
    else if (t_1 <= 1d-99) then
        tmp = (-4.0d0) * (a * t)
    else if (t_1 <= 2d+51) then
        tmp = c * b
    else
        tmp = ((-27.0d0) * k) * j
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = (j * 27.0) * k;
	double tmp;
	if (t_1 <= -2e+147) {
		tmp = -27.0 * (k * j);
	} else if (t_1 <= 1e-99) {
		tmp = -4.0 * (a * t);
	} else if (t_1 <= 2e+51) {
		tmp = c * b;
	} else {
		tmp = (-27.0 * k) * j;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j, k):
	t_1 = (j * 27.0) * k
	tmp = 0
	if t_1 <= -2e+147:
		tmp = -27.0 * (k * j)
	elif t_1 <= 1e-99:
		tmp = -4.0 * (a * t)
	elif t_1 <= 2e+51:
		tmp = c * b
	else:
		tmp = (-27.0 * k) * j
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(Float64(j * 27.0) * k)
	tmp = 0.0
	if (t_1 <= -2e+147)
		tmp = Float64(-27.0 * Float64(k * j));
	elseif (t_1 <= 1e-99)
		tmp = Float64(-4.0 * Float64(a * t));
	elseif (t_1 <= 2e+51)
		tmp = Float64(c * b);
	else
		tmp = Float64(Float64(-27.0 * k) * j);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j, k)
	t_1 = (j * 27.0) * k;
	tmp = 0.0;
	if (t_1 <= -2e+147)
		tmp = -27.0 * (k * j);
	elseif (t_1 <= 1e-99)
		tmp = -4.0 * (a * t);
	elseif (t_1 <= 2e+51)
		tmp = c * b;
	else
		tmp = (-27.0 * k) * j;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+147], N[(-27.0 * N[(k * j), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-99], N[(-4.0 * N[(a * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+51], N[(c * b), $MachinePrecision], N[(N[(-27.0 * k), $MachinePrecision] * j), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -2e147

   1. Initial program 79.0%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{-27 \cdot \left(j \cdot k\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -27 \cdot \color{blue}{\left(j \cdot k\right)} \]

      2. *-commutative N/A

         \[\leadsto -27 \cdot \left(k \cdot \color{blue}{j}\right) \]

      3. lower-*.f64 60.3

         \[\leadsto -27 \cdot \left(k \cdot \color{blue}{j}\right) \]

   5. Applied rewrites 60.3%

      \[\leadsto \color{blue}{-27 \cdot \left(k \cdot j\right)} \]

   if -2e147 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 1e-99

   1. Initial program 87.0%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\left(a \cdot t\right)} \]

      2. lower-*.f64 25.0

         \[\leadsto -4 \cdot \left(a \cdot \color{blue}{t}\right) \]

   5. Applied rewrites 25.0%

      \[\leadsto \color{blue}{-4 \cdot \left(a \cdot t\right)} \]

   if 1e-99 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 2e51

   1. Initial program 88.9%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

         \[\leadsto c \cdot \color{blue}{b} \]

      2. lower-*.f64 26.6

         \[\leadsto c \cdot \color{blue}{b} \]

   5. Applied rewrites 26.6%

      \[\leadsto \color{blue}{c \cdot b} \]

   if 2e51 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 83.1%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{-27 \cdot \left(j \cdot k\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -27 \cdot \color{blue}{\left(j \cdot k\right)} \]

      2. *-commutative N/A

         \[\leadsto -27 \cdot \left(k \cdot \color{blue}{j}\right) \]

      3. lower-*.f64 48.9

         \[\leadsto -27 \cdot \left(k \cdot \color{blue}{j}\right) \]

   5. Applied rewrites 48.9%

      \[\leadsto \color{blue}{-27 \cdot \left(k \cdot j\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -27 \cdot \color{blue}{\left(k \cdot j\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto -27 \cdot \left(k \cdot \color{blue}{j}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(-27 \cdot k\right) \cdot \color{blue}{j} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(-27 \cdot k\right) \cdot \color{blue}{j} \]

      5. lower-*.f64 48.9

         \[\leadsto \left(-27 \cdot k\right) \cdot j \]

   7. Applied rewrites 48.9%

      \[\leadsto \left(-27 \cdot k\right) \cdot \color{blue}{j} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 11: 35.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -27 \cdot \left(k \cdot j\right)\\ t_2 := \left(j \cdot 27\right) \cdot k\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+147}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-99}:\\ \;\;\;\;-4 \cdot \left(a \cdot t\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+51}:\\ \;\;\;\;c \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* -27.0 (* k j))) (t_2 (* (* j 27.0) k)))
   (if (<= t_2 -2e+147)
     t_1
     (if (<= t_2 1e-99) (* -4.0 (* a t)) (if (<= t_2 2e+51) (* c b) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = -27.0 * (k * j);
	double t_2 = (j * 27.0) * k;
	double tmp;
	if (t_2 <= -2e+147) {
		tmp = t_1;
	} else if (t_2 <= 1e-99) {
		tmp = -4.0 * (a * t);
	} else if (t_2 <= 2e+51) {
		tmp = c * b;
	} else {
		tmp = t_1;
	}
	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, b, c, i, j, k)
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), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (-27.0d0) * (k * j)
    t_2 = (j * 27.0d0) * k
    if (t_2 <= (-2d+147)) then
        tmp = t_1
    else if (t_2 <= 1d-99) then
        tmp = (-4.0d0) * (a * t)
    else if (t_2 <= 2d+51) then
        tmp = c * b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = -27.0 * (k * j);
	double t_2 = (j * 27.0) * k;
	double tmp;
	if (t_2 <= -2e+147) {
		tmp = t_1;
	} else if (t_2 <= 1e-99) {
		tmp = -4.0 * (a * t);
	} else if (t_2 <= 2e+51) {
		tmp = c * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j, k):
	t_1 = -27.0 * (k * j)
	t_2 = (j * 27.0) * k
	tmp = 0
	if t_2 <= -2e+147:
		tmp = t_1
	elif t_2 <= 1e-99:
		tmp = -4.0 * (a * t)
	elif t_2 <= 2e+51:
		tmp = c * b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(-27.0 * Float64(k * j))
	t_2 = Float64(Float64(j * 27.0) * k)
	tmp = 0.0
	if (t_2 <= -2e+147)
		tmp = t_1;
	elseif (t_2 <= 1e-99)
		tmp = Float64(-4.0 * Float64(a * t));
	elseif (t_2 <= 2e+51)
		tmp = Float64(c * b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j, k)
	t_1 = -27.0 * (k * j);
	t_2 = (j * 27.0) * k;
	tmp = 0.0;
	if (t_2 <= -2e+147)
		tmp = t_1;
	elseif (t_2 <= 1e-99)
		tmp = -4.0 * (a * t);
	elseif (t_2 <= 2e+51)
		tmp = c * b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(-27.0 * N[(k * j), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+147], t$95$1, If[LessEqual[t$95$2, 1e-99], N[(-4.0 * N[(a * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+51], N[(c * b), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -2e147 or 2e51 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 81.4%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{-27 \cdot \left(j \cdot k\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -27 \cdot \color{blue}{\left(j \cdot k\right)} \]

      2. *-commutative N/A

         \[\leadsto -27 \cdot \left(k \cdot \color{blue}{j}\right) \]

      3. lower-*.f64 53.6

         \[\leadsto -27 \cdot \left(k \cdot \color{blue}{j}\right) \]

   5. Applied rewrites 53.6%

      \[\leadsto \color{blue}{-27 \cdot \left(k \cdot j\right)} \]

   if -2e147 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 1e-99

   1. Initial program 87.0%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\left(a \cdot t\right)} \]

      2. lower-*.f64 25.0

         \[\leadsto -4 \cdot \left(a \cdot \color{blue}{t}\right) \]

   5. Applied rewrites 25.0%

      \[\leadsto \color{blue}{-4 \cdot \left(a \cdot t\right)} \]

   if 1e-99 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 2e51

   1. Initial program 88.9%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

         \[\leadsto c \cdot \color{blue}{b} \]

      2. lower-*.f64 26.6

         \[\leadsto c \cdot \color{blue}{b} \]

   5. Applied rewrites 26.6%

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

Alternative 12: 55.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right)\\ t_2 := \left(j \cdot 27\right) \cdot k\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (fma c b (* (* k j) -27.0))) (t_2 (* (* j 27.0) k)))
   (if (<= t_2 -1e+68)
     t_1
     (if (<= t_2 1e+60) (fma c b (* (* i x) -4.0)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = fma(c, b, ((k * j) * -27.0));
	double t_2 = (j * 27.0) * k;
	double tmp;
	if (t_2 <= -1e+68) {
		tmp = t_1;
	} else if (t_2 <= 1e+60) {
		tmp = fma(c, b, ((i * x) * -4.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = fma(c, b, Float64(Float64(k * j) * -27.0))
	t_2 = Float64(Float64(j * 27.0) * k)
	tmp = 0.0
	if (t_2 <= -1e+68)
		tmp = t_1;
	elseif (t_2 <= 1e+60)
		tmp = fma(c, b, Float64(Float64(i * x) * -4.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(c * b + N[(N[(k * j), $MachinePrecision] * -27.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+68], t$95$1, If[LessEqual[t$95$2, 1e+60], N[(c * b + N[(N[(i * x), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -9.99999999999999953e67 or 9.9999999999999995e59 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 82.1%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   4. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(z \cdot y\right) \cdot x, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right)} \]

   6. Applied rewrites 85.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(\left(y \cdot z\right) \cdot x, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right)} \]

   7. Taylor expanded in j around inf

      \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{-27 \cdot \left(j \cdot k\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(j \cdot k\right) \cdot \color{blue}{-27}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(j \cdot k\right) \cdot \color{blue}{-27}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right) \]

      4. lift-*.f64 62.4

         \[\leadsto \mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right) \]

   9. Applied rewrites 62.4%

      \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(k \cdot j\right) \cdot -27}\right) \]

   if -9.99999999999999953e67 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 9.9999999999999995e59

   1. Initial program 87.4%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   4. Applied rewrites 89.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(z \cdot y\right) \cdot x, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right)} \]

   6. Applied rewrites 90.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(\left(y \cdot z\right) \cdot x, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right)} \]

   7. Taylor expanded in i around inf

      \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{-4 \cdot \left(i \cdot x\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot \color{blue}{-4}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot \color{blue}{-4}\right) \]

      3. lift-*.f64 50.4

         \[\leadsto \mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right) \]

   9. Applied rewrites 50.4%

      \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(i \cdot x\right) \cdot -4}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 52.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -27 \cdot \left(k \cdot j\right)\\ t_2 := \left(j \cdot 27\right) \cdot k\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+262}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+195}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* -27.0 (* k j))) (t_2 (* (* j 27.0) k)))
   (if (<= t_2 -2e+262)
     t_1
     (if (<= t_2 5e+195) (fma c b (* (* i x) -4.0)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = -27.0 * (k * j);
	double t_2 = (j * 27.0) * k;
	double tmp;
	if (t_2 <= -2e+262) {
		tmp = t_1;
	} else if (t_2 <= 5e+195) {
		tmp = fma(c, b, ((i * x) * -4.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(-27.0 * Float64(k * j))
	t_2 = Float64(Float64(j * 27.0) * k)
	tmp = 0.0
	if (t_2 <= -2e+262)
		tmp = t_1;
	elseif (t_2 <= 5e+195)
		tmp = fma(c, b, Float64(Float64(i * x) * -4.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(-27.0 * N[(k * j), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+262], t$95$1, If[LessEqual[t$95$2, 5e+195], N[(c * b + N[(N[(i * x), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -2e262 or 4.9999999999999998e195 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 76.3%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{-27 \cdot \left(j \cdot k\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -27 \cdot \color{blue}{\left(j \cdot k\right)} \]

      2. *-commutative N/A

         \[\leadsto -27 \cdot \left(k \cdot \color{blue}{j}\right) \]

      3. lower-*.f64 72.3

         \[\leadsto -27 \cdot \left(k \cdot \color{blue}{j}\right) \]

   5. Applied rewrites 72.3%

      \[\leadsto \color{blue}{-27 \cdot \left(k \cdot j\right)} \]

   if -2e262 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 4.9999999999999998e195

   1. Initial program 87.5%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   4. Applied rewrites 89.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(z \cdot y\right) \cdot x, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right)} \]

   6. Applied rewrites 90.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(\left(y \cdot z\right) \cdot x, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right)} \]

   7. Taylor expanded in i around inf

      \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{-4 \cdot \left(i \cdot x\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot \color{blue}{-4}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot \color{blue}{-4}\right) \]

      3. lift-*.f64 47.3

         \[\leadsto \mathsf{fma}\left(c, b, \left(i \cdot x\right) \cdot -4\right) \]

   9. Applied rewrites 47.3%

      \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{\left(i \cdot x\right) \cdot -4}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 52.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -27 \cdot \left(k \cdot j\right)\\ t_2 := \left(j \cdot 27\right) \cdot k\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+262}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+195}:\\ \;\;\;\;\mathsf{fma}\left(i \cdot -4, x, b \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* -27.0 (* k j))) (t_2 (* (* j 27.0) k)))
   (if (<= t_2 -2e+262)
     t_1
     (if (<= t_2 5e+195) (fma (* i -4.0) x (* b c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = -27.0 * (k * j);
	double t_2 = (j * 27.0) * k;
	double tmp;
	if (t_2 <= -2e+262) {
		tmp = t_1;
	} else if (t_2 <= 5e+195) {
		tmp = fma((i * -4.0), x, (b * c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(-27.0 * Float64(k * j))
	t_2 = Float64(Float64(j * 27.0) * k)
	tmp = 0.0
	if (t_2 <= -2e+262)
		tmp = t_1;
	elseif (t_2 <= 5e+195)
		tmp = fma(Float64(i * -4.0), x, Float64(b * c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(-27.0 * N[(k * j), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+262], t$95$1, If[LessEqual[t$95$2, 5e+195], N[(N[(i * -4.0), $MachinePrecision] * x + N[(b * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 j #s(literal 27 binary64)) k) < -2e262 or 4.9999999999999998e195 < (*.f64 (*.f64 j #s(literal 27 binary64)) k)

   1. Initial program 76.3%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{-27 \cdot \left(j \cdot k\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -27 \cdot \color{blue}{\left(j \cdot k\right)} \]

      2. *-commutative N/A

         \[\leadsto -27 \cdot \left(k \cdot \color{blue}{j}\right) \]

      3. lower-*.f64 72.3

         \[\leadsto -27 \cdot \left(k \cdot \color{blue}{j}\right) \]

   5. Applied rewrites 72.3%

      \[\leadsto \color{blue}{-27 \cdot \left(k \cdot j\right)} \]

   if -2e262 < (*.f64 (*.f64 j #s(literal 27 binary64)) k) < 4.9999999999999998e195

   1. Initial program 87.5%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in j around 0

      \[\leadsto \color{blue}{\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + b \cdot c\right) - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto 18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) + \color{blue}{\left(b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(18 \cdot t\right) \cdot \left(x \cdot \left(y \cdot z\right)\right) + \left(\color{blue}{b \cdot c} - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \color{blue}{x \cdot \left(y \cdot z\right)}, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \color{blue}{x} \cdot \left(y \cdot z\right), b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(y \cdot z\right) \cdot \color{blue}{x}, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(y \cdot z\right) \cdot \color{blue}{x}, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, b \cdot c - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - \left(4 \cdot \left(a \cdot t\right) + 4 \cdot \left(i \cdot x\right)\right)\right) \]

      12. distribute-lft-out N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \left(a \cdot t + i \cdot x\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \left(a \cdot t + i \cdot x\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \mathsf{fma}\left(a, t, i \cdot x\right)\right) \]

      15. lower-*.f64 79.0

          \[\leadsto \mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \mathsf{fma}\left(a, t, i \cdot x\right)\right) \]

   5. Applied rewrites 79.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(18 \cdot t, \left(z \cdot y\right) \cdot x, c \cdot b - 4 \cdot \mathsf{fma}\left(a, t, i \cdot x\right)\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto b \cdot c - \color{blue}{4 \cdot \left(i \cdot x\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto b \cdot c + \left(\mathsf{neg}\left(4\right)\right) \cdot \color{blue}{\left(i \cdot x\right)} \]

      2. metadata-eval N/A

         \[\leadsto b \cdot c + -4 \cdot \left(i \cdot x\right) \]

      3. +-commutative N/A

         \[\leadsto -4 \cdot \left(i \cdot x\right) + b \cdot \color{blue}{c} \]

      4. associate-*r* N/A

         \[\leadsto \left(-4 \cdot i\right) \cdot x + b \cdot c \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot i, x, b \cdot c\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(i \cdot -4, x, b \cdot c\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(i \cdot -4, x, b \cdot c\right) \]

      8. lower-*.f64 47.3

         \[\leadsto \mathsf{fma}\left(i \cdot -4, x, b \cdot c\right) \]

   8. Applied rewrites 47.3%

      \[\leadsto \mathsf{fma}\left(i \cdot -4, \color{blue}{x}, b \cdot c\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 72.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\left(y \cdot x\right) \cdot z, 18, -4 \cdot a\right) \cdot t\\ \mathbf{if}\;t \leq -2.65 \cdot 10^{+122}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(c, b, -\mathsf{fma}\left(4 \cdot i, x, \left(k \cdot j\right) \cdot 27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* (fma (* (* y x) z) 18.0 (* -4.0 a)) t)))
   (if (<= t -2.65e+122)
     t_1
     (if (<= t 7.8e+62)
       (fma c b (- (fma (* 4.0 i) x (* (* k j) 27.0))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = fma(((y * x) * z), 18.0, (-4.0 * a)) * t;
	double tmp;
	if (t <= -2.65e+122) {
		tmp = t_1;
	} else if (t <= 7.8e+62) {
		tmp = fma(c, b, -fma((4.0 * i), x, ((k * j) * 27.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(fma(Float64(Float64(y * x) * z), 18.0, Float64(-4.0 * a)) * t)
	tmp = 0.0
	if (t <= -2.65e+122)
		tmp = t_1;
	elseif (t <= 7.8e+62)
		tmp = fma(c, b, Float64(-fma(Float64(4.0 * i), x, Float64(Float64(k * j) * 27.0))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision] * 18.0 + N[(-4.0 * a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, -2.65e+122], t$95$1, If[LessEqual[t, 7.8e+62], N[(c * b + (-N[(N[(4.0 * i), $MachinePrecision] * x + N[(N[(k * j), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.65e122 or 7.8e62 < t

   1. Initial program 81.7%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   4. Applied rewrites 88.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(z \cdot y\right) \cdot x, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right)} \]

   6. Applied rewrites 89.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(\left(y \cdot z\right) \cdot x, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(y \cdot z\right)} \cdot x, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(y \cdot z\right) \cdot x}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{x \cdot \left(y \cdot z\right)}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(x \cdot y\right) \cdot z}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(x \cdot y\right) \cdot z}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      6. lower-*.f64 89.3

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(x \cdot y\right)} \cdot z, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

   8. Applied rewrites 89.3%

      \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(x \cdot y\right) \cdot z}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-4 \cdot a + 18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto t \cdot \left(-4 \cdot a + 18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto t \cdot \left(-4 \cdot a + 18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \left(-4 \cdot a + 18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot \color{blue}{t} \]

   11. Applied rewrites 70.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y \cdot x\right) \cdot z, 18, -4 \cdot a\right) \cdot t} \]

   if -2.65e122 < t < 7.8e62

   1. Initial program 87.2%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   4. Applied rewrites 87.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(z \cdot y\right) \cdot x, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right)} \]

   6. Applied rewrites 87.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(\left(y \cdot z\right) \cdot x, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right)} \]

   7. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{-1 \cdot \left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{neg}\left(\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, -\left(4 \cdot \left(i \cdot x\right) + 27 \cdot \left(j \cdot k\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(c, b, -\left(\left(4 \cdot i\right) \cdot x + 27 \cdot \left(j \cdot k\right)\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4 \cdot i, x, 27 \cdot \left(j \cdot k\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4 \cdot i, x, 27 \cdot \left(j \cdot k\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4 \cdot i, x, \left(j \cdot k\right) \cdot 27\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4 \cdot i, x, \left(j \cdot k\right) \cdot 27\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4 \cdot i, x, \left(k \cdot j\right) \cdot 27\right)\right) \]

      9. lift-*.f64 73.8

         \[\leadsto \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4 \cdot i, x, \left(k \cdot j\right) \cdot 27\right)\right) \]

   9. Applied rewrites 73.8%

      \[\leadsto \mathsf{fma}\left(c, b, \color{blue}{-\mathsf{fma}\left(4 \cdot i, x, \left(k \cdot j\right) \cdot 27\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 72.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\left(y \cdot x\right) \cdot z, 18, -4 \cdot a\right) \cdot t\\ \mathbf{if}\;t \leq -2.65 \cdot 10^{+122}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{+62}:\\ \;\;\;\;b \cdot c - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* (fma (* (* y x) z) 18.0 (* -4.0 a)) t)))
   (if (<= t -2.65e+122)
     t_1
     (if (<= t 7.8e+62) (- (* b c) (fma (* 4.0 x) i (* (* k j) 27.0))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = fma(((y * x) * z), 18.0, (-4.0 * a)) * t;
	double tmp;
	if (t <= -2.65e+122) {
		tmp = t_1;
	} else if (t <= 7.8e+62) {
		tmp = (b * c) - fma((4.0 * x), i, ((k * j) * 27.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(fma(Float64(Float64(y * x) * z), 18.0, Float64(-4.0 * a)) * t)
	tmp = 0.0
	if (t <= -2.65e+122)
		tmp = t_1;
	elseif (t <= 7.8e+62)
		tmp = Float64(Float64(b * c) - fma(Float64(4.0 * x), i, Float64(Float64(k * j) * 27.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision] * 18.0 + N[(-4.0 * a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, -2.65e+122], t$95$1, If[LessEqual[t, 7.8e+62], N[(N[(b * c), $MachinePrecision] - N[(N[(4.0 * x), $MachinePrecision] * i + N[(N[(k * j), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.65e122 or 7.8e62 < t

   1. Initial program 81.7%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   4. Applied rewrites 88.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(z \cdot y\right) \cdot x, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right)} \]

   6. Applied rewrites 89.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, b, \mathsf{fma}\left(\left(y \cdot z\right) \cdot x, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(y \cdot z\right)} \cdot x, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(y \cdot z\right) \cdot x}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{x \cdot \left(y \cdot z\right)}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(x \cdot y\right) \cdot z}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(x \cdot y\right) \cdot z}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

      6. lower-*.f64 89.3

         \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(x \cdot y\right)} \cdot z, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

   8. Applied rewrites 89.3%

      \[\leadsto \mathsf{fma}\left(c, b, \mathsf{fma}\left(\color{blue}{\left(x \cdot y\right) \cdot z}, 18, a \cdot -4\right) \cdot t - \mathsf{fma}\left(i \cdot x, 4, \left(j \cdot k\right) \cdot 27\right)\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-4 \cdot a + 18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto t \cdot \left(-4 \cdot a + 18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto t \cdot \left(-4 \cdot a + 18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \left(-4 \cdot a + 18 \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) \cdot \color{blue}{t} \]

   11. Applied rewrites 70.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y \cdot x\right) \cdot z, 18, -4 \cdot a\right) \cdot t} \]

   if -2.65e122 < t < 7.8e62

   1. Initial program 87.2%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   4. Applied rewrites 87.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(z \cdot y\right) \cdot x, 18, -4 \cdot a\right), t, c \cdot b\right) - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right)} \]

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{b \cdot c} - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 73.1

         \[\leadsto b \cdot \color{blue}{c} - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right) \]

   7. Applied rewrites 73.1%

      \[\leadsto \color{blue}{b \cdot c} - \mathsf{fma}\left(4 \cdot x, i, \left(k \cdot j\right) \cdot 27\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 36.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot c \leq -4 \cdot 10^{+162}:\\ \;\;\;\;c \cdot b\\ \mathbf{elif}\;b \cdot c \leq 4 \cdot 10^{+110}:\\ \;\;\;\;-27 \cdot \left(k \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j k)
 :precision binary64
 (if (<= (* b c) -4e+162)
   (* c b)
   (if (<= (* b c) 4e+110) (* -27.0 (* k j)) (* c b))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double tmp;
	if ((b * c) <= -4e+162) {
		tmp = c * b;
	} else if ((b * c) <= 4e+110) {
		tmp = -27.0 * (k * j);
	} else {
		tmp = c * b;
	}
	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, b, c, i, j, k)
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), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((b * c) <= (-4d+162)) then
        tmp = c * b
    else if ((b * c) <= 4d+110) then
        tmp = (-27.0d0) * (k * j)
    else
        tmp = c * b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double tmp;
	if ((b * c) <= -4e+162) {
		tmp = c * b;
	} else if ((b * c) <= 4e+110) {
		tmp = -27.0 * (k * j);
	} else {
		tmp = c * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j, k):
	tmp = 0
	if (b * c) <= -4e+162:
		tmp = c * b
	elif (b * c) <= 4e+110:
		tmp = -27.0 * (k * j)
	else:
		tmp = c * b
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j, k)
	tmp = 0.0
	if (Float64(b * c) <= -4e+162)
		tmp = Float64(c * b);
	elseif (Float64(b * c) <= 4e+110)
		tmp = Float64(-27.0 * Float64(k * j));
	else
		tmp = Float64(c * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j, k)
	tmp = 0.0;
	if ((b * c) <= -4e+162)
		tmp = c * b;
	elseif ((b * c) <= 4e+110)
		tmp = -27.0 * (k * j);
	else
		tmp = c * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := If[LessEqual[N[(b * c), $MachinePrecision], -4e+162], N[(c * b), $MachinePrecision], If[LessEqual[N[(b * c), $MachinePrecision], 4e+110], N[(-27.0 * N[(k * j), $MachinePrecision]), $MachinePrecision], N[(c * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b c) < -3.9999999999999998e162 or 4.0000000000000001e110 < (*.f64 b c)

   1. Initial program 81.9%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

         \[\leadsto c \cdot \color{blue}{b} \]

      2. lower-*.f64 60.0

         \[\leadsto c \cdot \color{blue}{b} \]

   5. Applied rewrites 60.0%

      \[\leadsto \color{blue}{c \cdot b} \]

   if -3.9999999999999998e162 < (*.f64 b c) < 4.0000000000000001e110

   1. Initial program 86.7%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{-27 \cdot \left(j \cdot k\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -27 \cdot \color{blue}{\left(j \cdot k\right)} \]

      2. *-commutative N/A

         \[\leadsto -27 \cdot \left(k \cdot \color{blue}{j}\right) \]

      3. lower-*.f64 27.1

         \[\leadsto -27 \cdot \left(k \cdot \color{blue}{j}\right) \]

   5. Applied rewrites 27.1%

      \[\leadsto \color{blue}{-27 \cdot \left(k \cdot j\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 32.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -4 \cdot \left(a \cdot t\right)\\ \mathbf{if}\;a \leq -2.6 \cdot 10^{+65}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -1.15 \cdot 10^{-171}:\\ \;\;\;\;\left(-4 \cdot i\right) \cdot x\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+95}:\\ \;\;\;\;\left(-27 \cdot k\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* -4.0 (* a t))))
   (if (<= a -2.6e+65)
     t_1
     (if (<= a -1.15e-171)
       (* (* -4.0 i) x)
       (if (<= a 1.2e+95) (* (* -27.0 k) j) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = -4.0 * (a * t);
	double tmp;
	if (a <= -2.6e+65) {
		tmp = t_1;
	} else if (a <= -1.15e-171) {
		tmp = (-4.0 * i) * x;
	} else if (a <= 1.2e+95) {
		tmp = (-27.0 * k) * j;
	} else {
		tmp = t_1;
	}
	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, b, c, i, j, k)
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), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-4.0d0) * (a * t)
    if (a <= (-2.6d+65)) then
        tmp = t_1
    else if (a <= (-1.15d-171)) then
        tmp = ((-4.0d0) * i) * x
    else if (a <= 1.2d+95) then
        tmp = ((-27.0d0) * k) * j
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = -4.0 * (a * t);
	double tmp;
	if (a <= -2.6e+65) {
		tmp = t_1;
	} else if (a <= -1.15e-171) {
		tmp = (-4.0 * i) * x;
	} else if (a <= 1.2e+95) {
		tmp = (-27.0 * k) * j;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j, k):
	t_1 = -4.0 * (a * t)
	tmp = 0
	if a <= -2.6e+65:
		tmp = t_1
	elif a <= -1.15e-171:
		tmp = (-4.0 * i) * x
	elif a <= 1.2e+95:
		tmp = (-27.0 * k) * j
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(-4.0 * Float64(a * t))
	tmp = 0.0
	if (a <= -2.6e+65)
		tmp = t_1;
	elseif (a <= -1.15e-171)
		tmp = Float64(Float64(-4.0 * i) * x);
	elseif (a <= 1.2e+95)
		tmp = Float64(Float64(-27.0 * k) * j);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j, k)
	t_1 = -4.0 * (a * t);
	tmp = 0.0;
	if (a <= -2.6e+65)
		tmp = t_1;
	elseif (a <= -1.15e-171)
		tmp = (-4.0 * i) * x;
	elseif (a <= 1.2e+95)
		tmp = (-27.0 * k) * j;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(-4.0 * N[(a * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.6e+65], t$95$1, If[LessEqual[a, -1.15e-171], N[(N[(-4.0 * i), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[a, 1.2e+95], N[(N[(-27.0 * k), $MachinePrecision] * j), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.60000000000000003e65 or 1.2e95 < a

   1. Initial program 81.6%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\left(a \cdot t\right)} \]

      2. lower-*.f64 43.3

         \[\leadsto -4 \cdot \left(a \cdot \color{blue}{t}\right) \]

   5. Applied rewrites 43.3%

      \[\leadsto \color{blue}{-4 \cdot \left(a \cdot t\right)} \]

   if -2.60000000000000003e65 < a < -1.14999999999999989e-171

   1. Initial program 87.1%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

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

      1. associate-*r* N/A

         \[\leadsto \left(-4 \cdot i\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-4 \cdot i\right) \cdot \color{blue}{x} \]

      3. lower-*.f64 23.7

         \[\leadsto \left(-4 \cdot i\right) \cdot x \]

   5. Applied rewrites 23.7%

      \[\leadsto \color{blue}{\left(-4 \cdot i\right) \cdot x} \]

   if -1.14999999999999989e-171 < a < 1.2e95

   1. Initial program 87.5%

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{-27 \cdot \left(j \cdot k\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -27 \cdot \color{blue}{\left(j \cdot k\right)} \]

      2. *-commutative N/A

         \[\leadsto -27 \cdot \left(k \cdot \color{blue}{j}\right) \]

      3. lower-*.f64 26.1

         \[\leadsto -27 \cdot \left(k \cdot \color{blue}{j}\right) \]

   5. Applied rewrites 26.1%

      \[\leadsto \color{blue}{-27 \cdot \left(k \cdot j\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -27 \cdot \color{blue}{\left(k \cdot j\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto -27 \cdot \left(k \cdot \color{blue}{j}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(-27 \cdot k\right) \cdot \color{blue}{j} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(-27 \cdot k\right) \cdot \color{blue}{j} \]

      5. lower-*.f64 26.1

         \[\leadsto \left(-27 \cdot k\right) \cdot j \]

   7. Applied rewrites 26.1%

      \[\leadsto \left(-27 \cdot k\right) \cdot \color{blue}{j} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 23.6% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ c \cdot b \end{array} \]

FPCore

(FPCore (x y z t a b c i j k) :precision binary64 (* c b))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	return c * b;
}

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, b, c, i, j, k)
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), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    code = c * b
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	return c * b;
}

Python

def code(x, y, z, t, a, b, c, i, j, k):
	return c * b

Julia

function code(x, y, z, t, a, b, c, i, j, k)
	return Float64(c * b)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j, k)
	tmp = c * b;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := N[(c * b), $MachinePrecision]

Derivation

1. Initial program 85.3%

   \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
2. Add Preprocessing

3. Taylor expanded in b around inf

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

   1. *-commutative N/A

      \[\leadsto c \cdot \color{blue}{b} \]

   2. lower-*.f64 23.6

      \[\leadsto c \cdot \color{blue}{b} \]

5. Applied rewrites 23.6%

   \[\leadsto \color{blue}{c \cdot b} \]
6. Add Preprocessing

Developer Target 1: 89.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a \cdot t + i \cdot x\right) \cdot 4\\ t_2 := \left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - t\_1\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{if}\;t < -1.6210815397541398 \cdot 10^{-69}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < 165.68027943805222:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - t\_1\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* (+ (* a t) (* i x)) 4.0))
        (t_2
         (-
          (- (* (* 18.0 t) (* (* x y) z)) t_1)
          (- (* (* k j) 27.0) (* c b)))))
   (if (< t -1.6210815397541398e-69)
     t_2
     (if (< t 165.68027943805222)
       (+ (- (* (* 18.0 y) (* x (* z t))) t_1) (- (* c b) (* 27.0 (* k j))))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = ((a * t) + (i * x)) * 4.0;
	double t_2 = (((18.0 * t) * ((x * y) * z)) - t_1) - (((k * j) * 27.0) - (c * b));
	double tmp;
	if (t < -1.6210815397541398e-69) {
		tmp = t_2;
	} else if (t < 165.68027943805222) {
		tmp = (((18.0 * y) * (x * (z * t))) - t_1) + ((c * b) - (27.0 * (k * j)));
	} else {
		tmp = t_2;
	}
	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, b, c, i, j, k)
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), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((a * t) + (i * x)) * 4.0d0
    t_2 = (((18.0d0 * t) * ((x * y) * z)) - t_1) - (((k * j) * 27.0d0) - (c * b))
    if (t < (-1.6210815397541398d-69)) then
        tmp = t_2
    else if (t < 165.68027943805222d0) then
        tmp = (((18.0d0 * y) * (x * (z * t))) - t_1) + ((c * b) - (27.0d0 * (k * j)))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = ((a * t) + (i * x)) * 4.0;
	double t_2 = (((18.0 * t) * ((x * y) * z)) - t_1) - (((k * j) * 27.0) - (c * b));
	double tmp;
	if (t < -1.6210815397541398e-69) {
		tmp = t_2;
	} else if (t < 165.68027943805222) {
		tmp = (((18.0 * y) * (x * (z * t))) - t_1) + ((c * b) - (27.0 * (k * j)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j, k):
	t_1 = ((a * t) + (i * x)) * 4.0
	t_2 = (((18.0 * t) * ((x * y) * z)) - t_1) - (((k * j) * 27.0) - (c * b))
	tmp = 0
	if t < -1.6210815397541398e-69:
		tmp = t_2
	elif t < 165.68027943805222:
		tmp = (((18.0 * y) * (x * (z * t))) - t_1) + ((c * b) - (27.0 * (k * j)))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(Float64(Float64(a * t) + Float64(i * x)) * 4.0)
	t_2 = Float64(Float64(Float64(Float64(18.0 * t) * Float64(Float64(x * y) * z)) - t_1) - Float64(Float64(Float64(k * j) * 27.0) - Float64(c * b)))
	tmp = 0.0
	if (t < -1.6210815397541398e-69)
		tmp = t_2;
	elseif (t < 165.68027943805222)
		tmp = Float64(Float64(Float64(Float64(18.0 * y) * Float64(x * Float64(z * t))) - t_1) + Float64(Float64(c * b) - Float64(27.0 * Float64(k * j))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j, k)
	t_1 = ((a * t) + (i * x)) * 4.0;
	t_2 = (((18.0 * t) * ((x * y) * z)) - t_1) - (((k * j) * 27.0) - (c * b));
	tmp = 0.0;
	if (t < -1.6210815397541398e-69)
		tmp = t_2;
	elseif (t < 165.68027943805222)
		tmp = (((18.0 * y) * (x * (z * t))) - t_1) + ((c * b) - (27.0 * (k * j)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(N[(a * t), $MachinePrecision] + N[(i * x), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(18.0 * t), $MachinePrecision] * N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] - N[(N[(N[(k * j), $MachinePrecision] * 27.0), $MachinePrecision] - N[(c * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -1.6210815397541398e-69], t$95$2, If[Less[t, 165.68027943805222], N[(N[(N[(N[(18.0 * y), $MachinePrecision] * N[(x * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] + N[(N[(c * b), $MachinePrecision] - N[(27.0 * N[(k * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2025091 
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, E"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -8105407698770699/5000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (- (* (* 18 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b))) (if (< t 8284013971902611/50000000000000) (+ (- (* (* 18 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4)) (- (* c b) (* 27 (* k j)))) (- (- (* (* 18 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b))))))

  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))