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

Percentage Accurate: 80.4% → 94.4%

Time: 9.9s

Alternatives: 17

Speedup: 0.9×

• 37.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
}

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)
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
    code = ((((x * 9.0d0) * y) - (((z * 4.0d0) * t) * a)) + b) / (z * c)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
}

Python

def code(x, y, z, t, a, b, c):
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c)

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(Float64(x * 9.0) * y) - Float64(Float64(Float64(z * 4.0) * t) * a)) + b) / Float64(z * c))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision] - N[(N[(N[(z * 4.0), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]

Initial Program: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
}

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)
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
    code = ((((x * 9.0d0) * y) - (((z * 4.0d0) * t) * a)) + b) / (z * c)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
}

Python

def code(x, y, z, t, a, b, c):
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c)

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(Float64(x * 9.0) * y) - Float64(Float64(Float64(z * 4.0) * t) * a)) + b) / Float64(z * c))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision] - N[(N[(N[(z * 4.0), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]

Alternative 1: 94.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(9 \cdot y, x, b\right)\\ \mathbf{if}\;z \leq -1.5 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot -4, \frac{t}{c}, \frac{\frac{t\_1}{z}}{c}\right)\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{\frac{c}{\mathsf{fma}\left(\left(-4 \cdot z\right) \cdot t, a, t\_1\right)} \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(y, \frac{9 \cdot x}{z}, \frac{b}{z}\right)\right)}{c}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* 9.0 y) x b)))
   (if (<= z -1.5e-24)
     (fma (* a -4.0) (/ t c) (/ (/ t_1 z) c))
     (if (<= z 2.5e-20)
       (/ 1.0 (* (/ c (fma (* (* -4.0 z) t) a t_1)) z))
       (/ (fma -4.0 (* a t) (fma y (/ (* 9.0 x) z) (/ b z))) c)))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma((9.0 * y), x, b);
	double tmp;
	if (z <= -1.5e-24) {
		tmp = fma((a * -4.0), (t / c), ((t_1 / z) / c));
	} else if (z <= 2.5e-20) {
		tmp = 1.0 / ((c / fma(((-4.0 * z) * t), a, t_1)) * z);
	} else {
		tmp = fma(-4.0, (a * t), fma(y, ((9.0 * x) / z), (b / z))) / c;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(9.0 * y), x, b)
	tmp = 0.0
	if (z <= -1.5e-24)
		tmp = fma(Float64(a * -4.0), Float64(t / c), Float64(Float64(t_1 / z) / c));
	elseif (z <= 2.5e-20)
		tmp = Float64(1.0 / Float64(Float64(c / fma(Float64(Float64(-4.0 * z) * t), a, t_1)) * z));
	else
		tmp = Float64(fma(-4.0, Float64(a * t), fma(y, Float64(Float64(9.0 * x) / z), Float64(b / z))) / c);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(9.0 * y), $MachinePrecision] * x + b), $MachinePrecision]}, If[LessEqual[z, -1.5e-24], N[(N[(a * -4.0), $MachinePrecision] * N[(t / c), $MachinePrecision] + N[(N[(t$95$1 / z), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.5e-20], N[(1.0 / N[(N[(c / N[(N[(N[(-4.0 * z), $MachinePrecision] * t), $MachinePrecision] * a + t$95$1), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * N[(a * t), $MachinePrecision] + N[(y * N[(N[(9.0 * x), $MachinePrecision] / z), $MachinePrecision] + N[(b / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.49999999999999998e-24

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b}{z \cdot c} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b}{z}}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      7. mult-flip N/A

         \[\leadsto \color{blue}{\frac{b}{z} \cdot \frac{1}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      8. inv-pow N/A

         \[\leadsto \frac{b}{z} \cdot \color{blue}{{c}^{-1}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{b}{z}}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      11. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      13. lower-/.f64 73.9

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \color{blue}{\frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}}\right) \]

   3. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \frac{\mathsf{fma}\left(y \cdot x, 9, -4 \cdot \left(z \cdot \left(a \cdot t\right)\right)\right)}{c \cdot z}\right)} \]

   4. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right) + \left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{\color{blue}{c}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      7. lower-/.f64 84.1

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

   6. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{\color{blue}{c}} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right) + \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c} \]

      3. div-add N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right)}{c} + \color{blue}{\frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right)}{c} + \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\left(-4 \cdot a\right) \cdot t}{c} + \frac{\mathsf{fma}\left(\color{blue}{9}, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c} \]

      6. associate-/l* N/A

         \[\leadsto \left(-4 \cdot a\right) \cdot \frac{t}{c} + \frac{\color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}}{c} \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, \color{blue}{\frac{t}{c}}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a \cdot -4, \frac{\color{blue}{t}}{c}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot -4, \frac{\color{blue}{t}}{c}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot -4, \frac{t}{\color{blue}{c}}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

      11. lower-/.f64 81.2

          \[\leadsto \mathsf{fma}\left(a \cdot -4, \frac{t}{c}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

   8. Applied rewrites 82.9%

      \[\leadsto \mathsf{fma}\left(a \cdot -4, \color{blue}{\frac{t}{c}}, \frac{\frac{\mathsf{fma}\left(9 \cdot y, x, b\right)}{z}}{c}\right) \]

   if -1.49999999999999998e-24 < z < 2.4999999999999999e-20

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b}{z \cdot c} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b}{z}}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      7. mult-flip N/A

         \[\leadsto \color{blue}{\frac{b}{z} \cdot \frac{1}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      8. inv-pow N/A

         \[\leadsto \frac{b}{z} \cdot \color{blue}{{c}^{-1}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{b}{z}}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      11. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      13. lower-/.f64 73.9

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \color{blue}{\frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}}\right) \]

   3. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \frac{\mathsf{fma}\left(y \cdot x, 9, -4 \cdot \left(z \cdot \left(a \cdot t\right)\right)\right)}{c \cdot z}\right)} \]

   4. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right) + \left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{\color{blue}{c}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      7. lower-/.f64 84.1

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

   6. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c}} \]

   8. Applied rewrites 80.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{c}{\mathsf{fma}\left(\left(-4 \cdot z\right) \cdot t, a, \mathsf{fma}\left(9 \cdot y, x, b\right)\right)} \cdot z}} \]

   if 2.4999999999999999e-20 < z

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b}{z \cdot c} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b}{z}}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      7. mult-flip N/A

         \[\leadsto \color{blue}{\frac{b}{z} \cdot \frac{1}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      8. inv-pow N/A

         \[\leadsto \frac{b}{z} \cdot \color{blue}{{c}^{-1}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{b}{z}}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      11. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      13. lower-/.f64 73.9

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \color{blue}{\frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}}\right) \]

   3. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \frac{\mathsf{fma}\left(y \cdot x, 9, -4 \cdot \left(z \cdot \left(a \cdot t\right)\right)\right)}{c \cdot z}\right)} \]

   4. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right) + \left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{\color{blue}{c}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      7. lower-/.f64 84.1

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

   6. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{9 \cdot \left(x \cdot y\right)}{z} + \frac{b}{z}\right)}{c} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{9 \cdot \left(x \cdot y\right)}{z} + \frac{b}{z}\right)}{c} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{\left(9 \cdot x\right) \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{\left(9 \cdot x\right) \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{y \cdot \left(9 \cdot x\right)}{z} + \frac{b}{z}\right)}{c} \]

      8. associate-/l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, y \cdot \frac{9 \cdot x}{z} + \frac{b}{z}\right)}{c} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(y, \frac{9 \cdot x}{z}, \frac{b}{z}\right)\right)}{c} \]

      10. lower-/.f64 84.4

          \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(y, \frac{9 \cdot x}{z}, \frac{b}{z}\right)\right)}{c} \]

   8. Applied rewrites 84.4%

      \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(y, \frac{9 \cdot x}{z}, \frac{b}{z}\right)\right)}{c} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 92.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot -4, \frac{t}{c}, \frac{\frac{\mathsf{fma}\left(9 \cdot y, x, b\right)}{z}}{c}\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b - a \cdot \left(t \cdot \left(4 \cdot z\right)\right)\right)}{c}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(y, \frac{9 \cdot x}{z}, \frac{b}{z}\right)\right)}{c}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (if (<= z -1.5e-24)
   (fma (* a -4.0) (/ t c) (/ (/ (fma (* 9.0 y) x b) z) c))
   (if (<= z 4e-19)
     (/ (/ (fma (* y x) 9.0 (- b (* a (* t (* 4.0 z))))) c) z)
     (/ (fma -4.0 (* a t) (fma y (/ (* 9.0 x) z) (/ b z))) c))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (z <= -1.5e-24) {
		tmp = fma((a * -4.0), (t / c), ((fma((9.0 * y), x, b) / z) / c));
	} else if (z <= 4e-19) {
		tmp = (fma((y * x), 9.0, (b - (a * (t * (4.0 * z))))) / c) / z;
	} else {
		tmp = fma(-4.0, (a * t), fma(y, ((9.0 * x) / z), (b / z))) / c;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (z <= -1.5e-24)
		tmp = fma(Float64(a * -4.0), Float64(t / c), Float64(Float64(fma(Float64(9.0 * y), x, b) / z) / c));
	elseif (z <= 4e-19)
		tmp = Float64(Float64(fma(Float64(y * x), 9.0, Float64(b - Float64(a * Float64(t * Float64(4.0 * z))))) / c) / z);
	else
		tmp = Float64(fma(-4.0, Float64(a * t), fma(y, Float64(Float64(9.0 * x) / z), Float64(b / z))) / c);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[z, -1.5e-24], N[(N[(a * -4.0), $MachinePrecision] * N[(t / c), $MachinePrecision] + N[(N[(N[(N[(9.0 * y), $MachinePrecision] * x + b), $MachinePrecision] / z), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4e-19], N[(N[(N[(N[(y * x), $MachinePrecision] * 9.0 + N[(b - N[(a * N[(t * N[(4.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] / z), $MachinePrecision], N[(N[(-4.0 * N[(a * t), $MachinePrecision] + N[(y * N[(N[(9.0 * x), $MachinePrecision] / z), $MachinePrecision] + N[(b / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.49999999999999998e-24

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b}{z \cdot c} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b}{z}}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      7. mult-flip N/A

         \[\leadsto \color{blue}{\frac{b}{z} \cdot \frac{1}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      8. inv-pow N/A

         \[\leadsto \frac{b}{z} \cdot \color{blue}{{c}^{-1}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{b}{z}}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      11. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      13. lower-/.f64 73.9

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \color{blue}{\frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}}\right) \]

   3. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \frac{\mathsf{fma}\left(y \cdot x, 9, -4 \cdot \left(z \cdot \left(a \cdot t\right)\right)\right)}{c \cdot z}\right)} \]

   4. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right) + \left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{\color{blue}{c}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      7. lower-/.f64 84.1

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

   6. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{\color{blue}{c}} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right) + \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c} \]

      3. div-add N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right)}{c} + \color{blue}{\frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right)}{c} + \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\left(-4 \cdot a\right) \cdot t}{c} + \frac{\mathsf{fma}\left(\color{blue}{9}, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c} \]

      6. associate-/l* N/A

         \[\leadsto \left(-4 \cdot a\right) \cdot \frac{t}{c} + \frac{\color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}}{c} \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, \color{blue}{\frac{t}{c}}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a \cdot -4, \frac{\color{blue}{t}}{c}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot -4, \frac{\color{blue}{t}}{c}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot -4, \frac{t}{\color{blue}{c}}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

      11. lower-/.f64 81.2

          \[\leadsto \mathsf{fma}\left(a \cdot -4, \frac{t}{c}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

   8. Applied rewrites 82.9%

      \[\leadsto \mathsf{fma}\left(a \cdot -4, \color{blue}{\frac{t}{c}}, \frac{\frac{\mathsf{fma}\left(9 \cdot y, x, b\right)}{z}}{c}\right) \]

   if -1.49999999999999998e-24 < z < 3.9999999999999999e-19

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{c \cdot z}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

   3. Applied rewrites 80.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b - a \cdot \left(t \cdot \left(4 \cdot z\right)\right)\right)}{c}}{z}} \]

   if 3.9999999999999999e-19 < z

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b}{z \cdot c} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b}{z}}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      7. mult-flip N/A

         \[\leadsto \color{blue}{\frac{b}{z} \cdot \frac{1}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      8. inv-pow N/A

         \[\leadsto \frac{b}{z} \cdot \color{blue}{{c}^{-1}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{b}{z}}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      11. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      13. lower-/.f64 73.9

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \color{blue}{\frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}}\right) \]

   3. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \frac{\mathsf{fma}\left(y \cdot x, 9, -4 \cdot \left(z \cdot \left(a \cdot t\right)\right)\right)}{c \cdot z}\right)} \]

   4. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right) + \left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{\color{blue}{c}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      7. lower-/.f64 84.1

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

   6. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{9 \cdot \left(x \cdot y\right)}{z} + \frac{b}{z}\right)}{c} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{9 \cdot \left(x \cdot y\right)}{z} + \frac{b}{z}\right)}{c} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{\left(9 \cdot x\right) \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{\left(9 \cdot x\right) \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{y \cdot \left(9 \cdot x\right)}{z} + \frac{b}{z}\right)}{c} \]

      8. associate-/l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, y \cdot \frac{9 \cdot x}{z} + \frac{b}{z}\right)}{c} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(y, \frac{9 \cdot x}{z}, \frac{b}{z}\right)\right)}{c} \]

      10. lower-/.f64 84.4

          \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(y, \frac{9 \cdot x}{z}, \frac{b}{z}\right)\right)}{c} \]

   8. Applied rewrites 84.4%

      \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(y, \frac{9 \cdot x}{z}, \frac{b}{z}\right)\right)}{c} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 92.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot -4, \frac{t}{c}, \frac{\frac{\mathsf{fma}\left(9 \cdot y, x, b\right)}{z}}{c}\right)\\ \mathbf{elif}\;z \leq 10^{-18}:\\ \;\;\;\;\frac{b - \mathsf{fma}\left(a \cdot \left(4 \cdot z\right), t, -9 \cdot \left(x \cdot y\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(y, \frac{9 \cdot x}{z}, \frac{b}{z}\right)\right)}{c}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (if (<= z -1.95e-19)
   (fma (* a -4.0) (/ t c) (/ (/ (fma (* 9.0 y) x b) z) c))
   (if (<= z 1e-18)
     (/ (- b (fma (* a (* 4.0 z)) t (* -9.0 (* x y)))) (* z c))
     (/ (fma -4.0 (* a t) (fma y (/ (* 9.0 x) z) (/ b z))) c))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (z <= -1.95e-19) {
		tmp = fma((a * -4.0), (t / c), ((fma((9.0 * y), x, b) / z) / c));
	} else if (z <= 1e-18) {
		tmp = (b - fma((a * (4.0 * z)), t, (-9.0 * (x * y)))) / (z * c);
	} else {
		tmp = fma(-4.0, (a * t), fma(y, ((9.0 * x) / z), (b / z))) / c;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (z <= -1.95e-19)
		tmp = fma(Float64(a * -4.0), Float64(t / c), Float64(Float64(fma(Float64(9.0 * y), x, b) / z) / c));
	elseif (z <= 1e-18)
		tmp = Float64(Float64(b - fma(Float64(a * Float64(4.0 * z)), t, Float64(-9.0 * Float64(x * y)))) / Float64(z * c));
	else
		tmp = Float64(fma(-4.0, Float64(a * t), fma(y, Float64(Float64(9.0 * x) / z), Float64(b / z))) / c);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[z, -1.95e-19], N[(N[(a * -4.0), $MachinePrecision] * N[(t / c), $MachinePrecision] + N[(N[(N[(N[(9.0 * y), $MachinePrecision] * x + b), $MachinePrecision] / z), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1e-18], N[(N[(b - N[(N[(a * N[(4.0 * z), $MachinePrecision]), $MachinePrecision] * t + N[(-9.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * N[(a * t), $MachinePrecision] + N[(y * N[(N[(9.0 * x), $MachinePrecision] / z), $MachinePrecision] + N[(b / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.94999999999999998e-19

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b}{z \cdot c} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b}{z}}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      7. mult-flip N/A

         \[\leadsto \color{blue}{\frac{b}{z} \cdot \frac{1}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      8. inv-pow N/A

         \[\leadsto \frac{b}{z} \cdot \color{blue}{{c}^{-1}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{b}{z}}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      11. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      13. lower-/.f64 73.9

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \color{blue}{\frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}}\right) \]

   3. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \frac{\mathsf{fma}\left(y \cdot x, 9, -4 \cdot \left(z \cdot \left(a \cdot t\right)\right)\right)}{c \cdot z}\right)} \]

   4. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right) + \left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{\color{blue}{c}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      7. lower-/.f64 84.1

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

   6. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{\color{blue}{c}} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right) + \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c} \]

      3. div-add N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right)}{c} + \color{blue}{\frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right)}{c} + \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\left(-4 \cdot a\right) \cdot t}{c} + \frac{\mathsf{fma}\left(\color{blue}{9}, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c} \]

      6. associate-/l* N/A

         \[\leadsto \left(-4 \cdot a\right) \cdot \frac{t}{c} + \frac{\color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}}{c} \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, \color{blue}{\frac{t}{c}}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a \cdot -4, \frac{\color{blue}{t}}{c}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot -4, \frac{\color{blue}{t}}{c}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot -4, \frac{t}{\color{blue}{c}}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

      11. lower-/.f64 81.2

          \[\leadsto \mathsf{fma}\left(a \cdot -4, \frac{t}{c}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

   8. Applied rewrites 82.9%

      \[\leadsto \mathsf{fma}\left(a \cdot -4, \color{blue}{\frac{t}{c}}, \frac{\frac{\mathsf{fma}\left(9 \cdot y, x, b\right)}{z}}{c}\right) \]

   if -1.94999999999999998e-19 < z < 1.0000000000000001e-18

   1. Initial program 80.4%

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

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      3. add-flip N/A

         \[\leadsto \frac{\color{blue}{b - \left(\mathsf{neg}\left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right)\right)}}{z \cdot c} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{b - \left(\mathsf{neg}\left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right)\right)}}{z \cdot c} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{b - \left(\mathsf{neg}\left(\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}\right)\right)}{z \cdot c} \]

      6. sub-negate-rev N/A

         \[\leadsto \frac{b - \color{blue}{\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - \left(x \cdot 9\right) \cdot y\right)}}{z \cdot c} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{b - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - \color{blue}{\left(x \cdot 9\right) \cdot y}\right)}{z \cdot c} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{b - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - \color{blue}{\left(x \cdot 9\right)} \cdot y\right)}{z \cdot c} \]

      9. *-commutative N/A

         \[\leadsto \frac{b - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - \color{blue}{\left(9 \cdot x\right)} \cdot y\right)}{z \cdot c} \]

      10. associate-*l* N/A

          \[\leadsto \frac{b - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - \color{blue}{9 \cdot \left(x \cdot y\right)}\right)}{z \cdot c} \]

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

          \[\leadsto \frac{b - \color{blue}{\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a + \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}}{z \cdot c} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{b - \left(\color{blue}{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a} + \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      13. *-commutative N/A

          \[\leadsto \frac{b - \left(\color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)} + \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{b - \left(a \cdot \color{blue}{\left(\left(z \cdot 4\right) \cdot t\right)} + \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      15. associate-*r* N/A

          \[\leadsto \frac{b - \left(\color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t} + \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{b - \color{blue}{\mathsf{fma}\left(a \cdot \left(z \cdot 4\right), t, \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}}{z \cdot c} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{b - \mathsf{fma}\left(\color{blue}{a \cdot \left(z \cdot 4\right)}, t, \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{b - \mathsf{fma}\left(a \cdot \color{blue}{\left(z \cdot 4\right)}, t, \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      19. *-commutative N/A

          \[\leadsto \frac{b - \mathsf{fma}\left(a \cdot \color{blue}{\left(4 \cdot z\right)}, t, \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      20. lower-*.f64 N/A

          \[\leadsto \frac{b - \mathsf{fma}\left(a \cdot \color{blue}{\left(4 \cdot z\right)}, t, \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      21. lower-*.f64 N/A

          \[\leadsto \frac{b - \mathsf{fma}\left(a \cdot \left(4 \cdot z\right), t, \color{blue}{\left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)}\right)}{z \cdot c} \]

      22. metadata-eval N/A

          \[\leadsto \frac{b - \mathsf{fma}\left(a \cdot \left(4 \cdot z\right), t, \color{blue}{-9} \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      23. lower-*.f64 80.2

          \[\leadsto \frac{b - \mathsf{fma}\left(a \cdot \left(4 \cdot z\right), t, -9 \cdot \color{blue}{\left(x \cdot y\right)}\right)}{z \cdot c} \]

   3. Applied rewrites 80.2%

      \[\leadsto \frac{\color{blue}{b - \mathsf{fma}\left(a \cdot \left(4 \cdot z\right), t, -9 \cdot \left(x \cdot y\right)\right)}}{z \cdot c} \]

   if 1.0000000000000001e-18 < z

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b}{z \cdot c} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b}{z}}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      7. mult-flip N/A

         \[\leadsto \color{blue}{\frac{b}{z} \cdot \frac{1}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      8. inv-pow N/A

         \[\leadsto \frac{b}{z} \cdot \color{blue}{{c}^{-1}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{b}{z}}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      11. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      13. lower-/.f64 73.9

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \color{blue}{\frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}}\right) \]

   3. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \frac{\mathsf{fma}\left(y \cdot x, 9, -4 \cdot \left(z \cdot \left(a \cdot t\right)\right)\right)}{c \cdot z}\right)} \]

   4. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right) + \left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{\color{blue}{c}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      7. lower-/.f64 84.1

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

   6. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{9 \cdot \left(x \cdot y\right)}{z} + \frac{b}{z}\right)}{c} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{9 \cdot \left(x \cdot y\right)}{z} + \frac{b}{z}\right)}{c} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{\left(9 \cdot x\right) \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{\left(9 \cdot x\right) \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{y \cdot \left(9 \cdot x\right)}{z} + \frac{b}{z}\right)}{c} \]

      8. associate-/l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, y \cdot \frac{9 \cdot x}{z} + \frac{b}{z}\right)}{c} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(y, \frac{9 \cdot x}{z}, \frac{b}{z}\right)\right)}{c} \]

      10. lower-/.f64 84.4

          \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(y, \frac{9 \cdot x}{z}, \frac{b}{z}\right)\right)}{c} \]

   8. Applied rewrites 84.4%

      \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(y, \frac{9 \cdot x}{z}, \frac{b}{z}\right)\right)}{c} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 92.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(y, \frac{9 \cdot x}{z}, \frac{b}{z}\right)\right)}{c}\\ \mathbf{if}\;z \leq -0.002:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 10^{-18}:\\ \;\;\;\;\frac{b - \mathsf{fma}\left(a \cdot \left(4 \cdot z\right), t, -9 \cdot \left(x \cdot y\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (fma -4.0 (* a t) (fma y (/ (* 9.0 x) z) (/ b z))) c)))
   (if (<= z -0.002)
     t_1
     (if (<= z 1e-18)
       (/ (- b (fma (* a (* 4.0 z)) t (* -9.0 (* x y)))) (* z c))
       t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(-4.0, (a * t), fma(y, ((9.0 * x) / z), (b / z))) / c;
	double tmp;
	if (z <= -0.002) {
		tmp = t_1;
	} else if (z <= 1e-18) {
		tmp = (b - fma((a * (4.0 * z)), t, (-9.0 * (x * y)))) / (z * c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(fma(-4.0, Float64(a * t), fma(y, Float64(Float64(9.0 * x) / z), Float64(b / z))) / c)
	tmp = 0.0
	if (z <= -0.002)
		tmp = t_1;
	elseif (z <= 1e-18)
		tmp = Float64(Float64(b - fma(Float64(a * Float64(4.0 * z)), t, Float64(-9.0 * Float64(x * y)))) / Float64(z * c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(-4.0 * N[(a * t), $MachinePrecision] + N[(y * N[(N[(9.0 * x), $MachinePrecision] / z), $MachinePrecision] + N[(b / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[z, -0.002], t$95$1, If[LessEqual[z, 1e-18], N[(N[(b - N[(N[(a * N[(4.0 * z), $MachinePrecision]), $MachinePrecision] * t + N[(-9.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2e-3 or 1.0000000000000001e-18 < z

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b}{z \cdot c} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b}{z}}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      7. mult-flip N/A

         \[\leadsto \color{blue}{\frac{b}{z} \cdot \frac{1}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      8. inv-pow N/A

         \[\leadsto \frac{b}{z} \cdot \color{blue}{{c}^{-1}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{b}{z}}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      11. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      13. lower-/.f64 73.9

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \color{blue}{\frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}}\right) \]

   3. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \frac{\mathsf{fma}\left(y \cdot x, 9, -4 \cdot \left(z \cdot \left(a \cdot t\right)\right)\right)}{c \cdot z}\right)} \]

   4. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right) + \left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{\color{blue}{c}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      7. lower-/.f64 84.1

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

   6. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{9 \cdot \left(x \cdot y\right)}{z} + \frac{b}{z}\right)}{c} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{9 \cdot \left(x \cdot y\right)}{z} + \frac{b}{z}\right)}{c} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{\left(9 \cdot x\right) \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{\left(9 \cdot x\right) \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{y \cdot \left(9 \cdot x\right)}{z} + \frac{b}{z}\right)}{c} \]

      8. associate-/l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, y \cdot \frac{9 \cdot x}{z} + \frac{b}{z}\right)}{c} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(y, \frac{9 \cdot x}{z}, \frac{b}{z}\right)\right)}{c} \]

      10. lower-/.f64 84.4

          \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(y, \frac{9 \cdot x}{z}, \frac{b}{z}\right)\right)}{c} \]

   8. Applied rewrites 84.4%

      \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(y, \frac{9 \cdot x}{z}, \frac{b}{z}\right)\right)}{c} \]

   if -2e-3 < z < 1.0000000000000001e-18

   1. Initial program 80.4%

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

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      3. add-flip N/A

         \[\leadsto \frac{\color{blue}{b - \left(\mathsf{neg}\left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right)\right)}}{z \cdot c} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{b - \left(\mathsf{neg}\left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right)\right)}}{z \cdot c} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{b - \left(\mathsf{neg}\left(\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}\right)\right)}{z \cdot c} \]

      6. sub-negate-rev N/A

         \[\leadsto \frac{b - \color{blue}{\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - \left(x \cdot 9\right) \cdot y\right)}}{z \cdot c} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{b - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - \color{blue}{\left(x \cdot 9\right) \cdot y}\right)}{z \cdot c} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{b - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - \color{blue}{\left(x \cdot 9\right)} \cdot y\right)}{z \cdot c} \]

      9. *-commutative N/A

         \[\leadsto \frac{b - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - \color{blue}{\left(9 \cdot x\right)} \cdot y\right)}{z \cdot c} \]

      10. associate-*l* N/A

          \[\leadsto \frac{b - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - \color{blue}{9 \cdot \left(x \cdot y\right)}\right)}{z \cdot c} \]

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

          \[\leadsto \frac{b - \color{blue}{\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a + \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}}{z \cdot c} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{b - \left(\color{blue}{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a} + \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      13. *-commutative N/A

          \[\leadsto \frac{b - \left(\color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)} + \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{b - \left(a \cdot \color{blue}{\left(\left(z \cdot 4\right) \cdot t\right)} + \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      15. associate-*r* N/A

          \[\leadsto \frac{b - \left(\color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t} + \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{b - \color{blue}{\mathsf{fma}\left(a \cdot \left(z \cdot 4\right), t, \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}}{z \cdot c} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{b - \mathsf{fma}\left(\color{blue}{a \cdot \left(z \cdot 4\right)}, t, \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{b - \mathsf{fma}\left(a \cdot \color{blue}{\left(z \cdot 4\right)}, t, \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      19. *-commutative N/A

          \[\leadsto \frac{b - \mathsf{fma}\left(a \cdot \color{blue}{\left(4 \cdot z\right)}, t, \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      20. lower-*.f64 N/A

          \[\leadsto \frac{b - \mathsf{fma}\left(a \cdot \color{blue}{\left(4 \cdot z\right)}, t, \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      21. lower-*.f64 N/A

          \[\leadsto \frac{b - \mathsf{fma}\left(a \cdot \left(4 \cdot z\right), t, \color{blue}{\left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)}\right)}{z \cdot c} \]

      22. metadata-eval N/A

          \[\leadsto \frac{b - \mathsf{fma}\left(a \cdot \left(4 \cdot z\right), t, \color{blue}{-9} \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      23. lower-*.f64 80.2

          \[\leadsto \frac{b - \mathsf{fma}\left(a \cdot \left(4 \cdot z\right), t, -9 \cdot \color{blue}{\left(x \cdot y\right)}\right)}{z \cdot c} \]

   3. Applied rewrites 80.2%

      \[\leadsto \frac{\color{blue}{b - \mathsf{fma}\left(a \cdot \left(4 \cdot z\right), t, -9 \cdot \left(x \cdot y\right)\right)}}{z \cdot c} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} \mathbf{if}\;c \leq 9.2 \cdot 10^{+164}:\\ \;\;\;\;\mathsf{fma}\left(-4 \cdot t, a, \frac{\mathsf{fma}\left(9 \cdot y, x, b\right)}{z}\right) \cdot \frac{1}{c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot -4, \frac{t}{c}, \frac{b}{c \cdot z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (if (<= c 9.2e+164)
   (* (fma (* -4.0 t) a (/ (fma (* 9.0 y) x b) z)) (/ 1.0 c))
   (fma (* a -4.0) (/ t c) (/ b (* c z)))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (c <= 9.2e+164) {
		tmp = fma((-4.0 * t), a, (fma((9.0 * y), x, b) / z)) * (1.0 / c);
	} else {
		tmp = fma((a * -4.0), (t / c), (b / (c * z)));
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (c <= 9.2e+164)
		tmp = Float64(fma(Float64(-4.0 * t), a, Float64(fma(Float64(9.0 * y), x, b) / z)) * Float64(1.0 / c));
	else
		tmp = fma(Float64(a * -4.0), Float64(t / c), Float64(b / Float64(c * z)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[c, 9.2e+164], N[(N[(N[(-4.0 * t), $MachinePrecision] * a + N[(N[(N[(9.0 * y), $MachinePrecision] * x + b), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] * N[(1.0 / c), $MachinePrecision]), $MachinePrecision], N[(N[(a * -4.0), $MachinePrecision] * N[(t / c), $MachinePrecision] + N[(b / N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 9.1999999999999998e164

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b}{z \cdot c} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b}{z}}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      7. mult-flip N/A

         \[\leadsto \color{blue}{\frac{b}{z} \cdot \frac{1}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      8. inv-pow N/A

         \[\leadsto \frac{b}{z} \cdot \color{blue}{{c}^{-1}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{b}{z}}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      11. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      13. lower-/.f64 73.9

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \color{blue}{\frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}}\right) \]

   3. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \frac{\mathsf{fma}\left(y \cdot x, 9, -4 \cdot \left(z \cdot \left(a \cdot t\right)\right)\right)}{c \cdot z}\right)} \]

   4. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right) + \left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{\color{blue}{c}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      7. lower-/.f64 84.1

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

   6. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{\color{blue}{c}} \]

      2. mult-flip N/A

         \[\leadsto \mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right) \cdot \color{blue}{\frac{1}{c}} \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right) \cdot \color{blue}{\frac{1}{c}} \]

   8. Applied rewrites 86.5%

      \[\leadsto \mathsf{fma}\left(-4 \cdot t, a, \frac{\mathsf{fma}\left(9 \cdot y, x, b\right)}{z}\right) \cdot \color{blue}{\frac{1}{c}} \]

   if 9.1999999999999998e164 < c

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b}{z \cdot c} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b}{z}}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      7. mult-flip N/A

         \[\leadsto \color{blue}{\frac{b}{z} \cdot \frac{1}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      8. inv-pow N/A

         \[\leadsto \frac{b}{z} \cdot \color{blue}{{c}^{-1}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{b}{z}}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      11. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      13. lower-/.f64 73.9

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \color{blue}{\frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}}\right) \]

   3. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \frac{\mathsf{fma}\left(y \cdot x, 9, -4 \cdot \left(z \cdot \left(a \cdot t\right)\right)\right)}{c \cdot z}\right)} \]

   4. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right) + \left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{\color{blue}{c}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      7. lower-/.f64 84.1

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

   6. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{\color{blue}{c}} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right) + \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c} \]

      3. div-add N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right)}{c} + \color{blue}{\frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right)}{c} + \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\left(-4 \cdot a\right) \cdot t}{c} + \frac{\mathsf{fma}\left(\color{blue}{9}, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c} \]

      6. associate-/l* N/A

         \[\leadsto \left(-4 \cdot a\right) \cdot \frac{t}{c} + \frac{\color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}}{c} \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, \color{blue}{\frac{t}{c}}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a \cdot -4, \frac{\color{blue}{t}}{c}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot -4, \frac{\color{blue}{t}}{c}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot -4, \frac{t}{\color{blue}{c}}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

      11. lower-/.f64 81.2

          \[\leadsto \mathsf{fma}\left(a \cdot -4, \frac{t}{c}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

   8. Applied rewrites 82.9%

      \[\leadsto \mathsf{fma}\left(a \cdot -4, \color{blue}{\frac{t}{c}}, \frac{\frac{\mathsf{fma}\left(9 \cdot y, x, b\right)}{z}}{c}\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(a \cdot -4, \frac{t}{c}, \frac{b}{c \cdot z}\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. lower-*.f64 63.0

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

   11. Applied rewrites 63.0%

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

Alternative 6: 76.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{y \cdot \frac{9 \cdot x}{z}}{c}\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+66}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z}\right)}{c}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+192}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot -4, \frac{t}{c}, \frac{b}{c \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{9 \cdot x}{c} \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)))
   (if (<= t_1 (- INFINITY))
     (/ (* y (/ (* 9.0 x) z)) c)
     (if (<= t_1 -1e+66)
       (/ (fma -4.0 (* a t) (* 9.0 (/ (* x y) z))) c)
       (if (<= t_1 4e+192)
         (fma (* a -4.0) (/ t c) (/ b (* c z)))
         (/ (* (/ (* 9.0 x) c) y) z))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (y * ((9.0 * x) / z)) / c;
	} else if (t_1 <= -1e+66) {
		tmp = fma(-4.0, (a * t), (9.0 * ((x * y) / z))) / c;
	} else if (t_1 <= 4e+192) {
		tmp = fma((a * -4.0), (t / c), (b / (c * z)));
	} else {
		tmp = (((9.0 * x) / c) * y) / z;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(y * Float64(Float64(9.0 * x) / z)) / c);
	elseif (t_1 <= -1e+66)
		tmp = Float64(fma(-4.0, Float64(a * t), Float64(9.0 * Float64(Float64(x * y) / z))) / c);
	elseif (t_1 <= 4e+192)
		tmp = fma(Float64(a * -4.0), Float64(t / c), Float64(b / Float64(c * z)));
	else
		tmp = Float64(Float64(Float64(Float64(9.0 * x) / c) * y) / z);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(y * N[(N[(9.0 * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t$95$1, -1e+66], N[(N[(-4.0 * N[(a * t), $MachinePrecision] + N[(9.0 * N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t$95$1, 4e+192], N[(N[(a * -4.0), $MachinePrecision] * N[(t / c), $MachinePrecision] + N[(b / N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(9.0 * x), $MachinePrecision] / c), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b}{z \cdot c} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b}{z}}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      7. mult-flip N/A

         \[\leadsto \color{blue}{\frac{b}{z} \cdot \frac{1}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      8. inv-pow N/A

         \[\leadsto \frac{b}{z} \cdot \color{blue}{{c}^{-1}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{b}{z}}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      11. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      13. lower-/.f64 73.9

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \color{blue}{\frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}}\right) \]

   3. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \frac{\mathsf{fma}\left(y \cdot x, 9, -4 \cdot \left(z \cdot \left(a \cdot t\right)\right)\right)}{c \cdot z}\right)} \]

   4. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right) + \left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{\color{blue}{c}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      7. lower-/.f64 84.1

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

   6. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c}} \]

   7. Taylor expanded in x around inf

      \[\leadsto \frac{9 \cdot \frac{x \cdot y}{z}}{c} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{z}}{c} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{z}}{c} \]

      3. lower-*.f64 34.3

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{z}}{c} \]

   9. Applied rewrites 34.3%

      \[\leadsto \frac{9 \cdot \frac{x \cdot y}{z}}{c} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{9 \cdot \frac{x \cdot y}{z}}{c} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{9 \cdot \frac{x \cdot y}{z}}{c} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{9 \cdot \frac{x \cdot y}{z}}{c} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{9 \cdot \frac{x \cdot y}{z}}{c} \]

       5. associate-*r/ N/A

          \[\leadsto \frac{\frac{9 \cdot \left(x \cdot y\right)}{z}}{c} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\frac{9 \cdot \left(x \cdot y\right)}{z}}{c} \]

       7. *-commutative N/A

          \[\leadsto \frac{\frac{9 \cdot \left(y \cdot x\right)}{z}}{c} \]

       8. associate-*l* N/A

          \[\leadsto \frac{\frac{\left(9 \cdot y\right) \cdot x}{z}}{c} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(9 \cdot y\right) \cdot x}{z}}{c} \]

       10. *-commutative N/A

           \[\leadsto \frac{\frac{x \cdot \left(9 \cdot y\right)}{z}}{c} \]

       11. lift-*.f64 N/A

           \[\leadsto \frac{\frac{x \cdot \left(9 \cdot y\right)}{z}}{c} \]

       12. associate-*r* N/A

           \[\leadsto \frac{\frac{\left(x \cdot 9\right) \cdot y}{z}}{c} \]

       13. *-commutative N/A

           \[\leadsto \frac{\frac{y \cdot \left(x \cdot 9\right)}{z}}{c} \]

       14. associate-/l* N/A

           \[\leadsto \frac{y \cdot \frac{x \cdot 9}{z}}{c} \]

       15. lower-*.f64 N/A

           \[\leadsto \frac{y \cdot \frac{x \cdot 9}{z}}{c} \]

       16. lower-/.f64 N/A

           \[\leadsto \frac{y \cdot \frac{x \cdot 9}{z}}{c} \]

       17. *-commutative N/A

           \[\leadsto \frac{y \cdot \frac{9 \cdot x}{z}}{c} \]

       18. lower-*.f64 35.6

           \[\leadsto \frac{y \cdot \frac{9 \cdot x}{z}}{c} \]

   11. Applied rewrites 35.6%

       \[\leadsto \frac{y \cdot \frac{9 \cdot x}{z}}{c} \]

   if -inf.0 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -9.99999999999999945e65

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b}{z \cdot c} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b}{z}}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      7. mult-flip N/A

         \[\leadsto \color{blue}{\frac{b}{z} \cdot \frac{1}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      8. inv-pow N/A

         \[\leadsto \frac{b}{z} \cdot \color{blue}{{c}^{-1}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{b}{z}}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      11. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      13. lower-/.f64 73.9

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \color{blue}{\frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}}\right) \]

   3. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \frac{\mathsf{fma}\left(y \cdot x, 9, -4 \cdot \left(z \cdot \left(a \cdot t\right)\right)\right)}{c \cdot z}\right)} \]

   4. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right) + \left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{\color{blue}{c}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      7. lower-/.f64 84.1

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

   6. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c}} \]

   7. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z}\right)}{c} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z}\right)}{c} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z}\right)}{c} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z}\right)}{c} \]

      5. lower-*.f64 63.7

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z}\right)}{c} \]

   9. Applied rewrites 63.7%

      \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z}\right)}{c} \]

   if -9.99999999999999945e65 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 4.00000000000000016e192

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b}{z \cdot c} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b}{z}}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      7. mult-flip N/A

         \[\leadsto \color{blue}{\frac{b}{z} \cdot \frac{1}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      8. inv-pow N/A

         \[\leadsto \frac{b}{z} \cdot \color{blue}{{c}^{-1}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{b}{z}}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      11. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      13. lower-/.f64 73.9

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \color{blue}{\frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}}\right) \]

   3. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \frac{\mathsf{fma}\left(y \cdot x, 9, -4 \cdot \left(z \cdot \left(a \cdot t\right)\right)\right)}{c \cdot z}\right)} \]

   4. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right) + \left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{\color{blue}{c}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      7. lower-/.f64 84.1

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

   6. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{\color{blue}{c}} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right) + \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c} \]

      3. div-add N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right)}{c} + \color{blue}{\frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right)}{c} + \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\left(-4 \cdot a\right) \cdot t}{c} + \frac{\mathsf{fma}\left(\color{blue}{9}, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c} \]

      6. associate-/l* N/A

         \[\leadsto \left(-4 \cdot a\right) \cdot \frac{t}{c} + \frac{\color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}}{c} \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, \color{blue}{\frac{t}{c}}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a \cdot -4, \frac{\color{blue}{t}}{c}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot -4, \frac{\color{blue}{t}}{c}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot -4, \frac{t}{\color{blue}{c}}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

      11. lower-/.f64 81.2

          \[\leadsto \mathsf{fma}\left(a \cdot -4, \frac{t}{c}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

   8. Applied rewrites 82.9%

      \[\leadsto \mathsf{fma}\left(a \cdot -4, \color{blue}{\frac{t}{c}}, \frac{\frac{\mathsf{fma}\left(9 \cdot y, x, b\right)}{z}}{c}\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(a \cdot -4, \frac{t}{c}, \frac{b}{c \cdot z}\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. lower-*.f64 63.0

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

   11. Applied rewrites 63.0%

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

   if 4.00000000000000016e192 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{c \cdot z}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

   3. Applied rewrites 80.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b - a \cdot \left(t \cdot \left(4 \cdot z\right)\right)\right)}{c}}{z}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{9 \cdot \frac{x \cdot y}{c}}}{z} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\frac{x \cdot y}{c}}}{z} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{\color{blue}{c}}}{z} \]

      3. lower-*.f64 35.8

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{c}}{z} \]

   6. Applied rewrites 35.8%

      \[\leadsto \frac{\color{blue}{9 \cdot \frac{x \cdot y}{c}}}{z} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\frac{x \cdot y}{c}}}{z} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{\color{blue}{c}}}{z} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{c}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \frac{9 \cdot \left(x \cdot \color{blue}{\frac{y}{c}}\right)}{z} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\left(9 \cdot x\right) \cdot \color{blue}{\frac{y}{c}}}{z} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(9 \cdot x\right) \cdot \frac{\color{blue}{y}}{c}}{z} \]

      7. associate-/l* N/A

         \[\leadsto \frac{\frac{\left(9 \cdot x\right) \cdot y}{\color{blue}{c}}}{z} \]

      8. associate-*l/ N/A

         \[\leadsto \frac{\frac{9 \cdot x}{c} \cdot \color{blue}{y}}{z} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{\frac{9 \cdot x}{c} \cdot y}{z} \]

      10. lower-*.f64 37.1

          \[\leadsto \frac{\frac{9 \cdot x}{c} \cdot \color{blue}{y}}{z} \]

   8. Applied rewrites 37.1%

      \[\leadsto \frac{\frac{9 \cdot x}{c} \cdot \color{blue}{y}}{z} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 76.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+94}:\\ \;\;\;\;\frac{x \cdot \frac{9 \cdot y}{c}}{z}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+192}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot -4, \frac{t}{c}, \frac{b}{c \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{9 \cdot x}{c} \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)))
   (if (<= t_1 -5e+94)
     (/ (* x (/ (* 9.0 y) c)) z)
     (if (<= t_1 4e+192)
       (fma (* a -4.0) (/ t c) (/ b (* c z)))
       (/ (* (/ (* 9.0 x) c) y) z)))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if (t_1 <= -5e+94) {
		tmp = (x * ((9.0 * y) / c)) / z;
	} else if (t_1 <= 4e+192) {
		tmp = fma((a * -4.0), (t / c), (b / (c * z)));
	} else {
		tmp = (((9.0 * x) / c) * y) / z;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_1 <= -5e+94)
		tmp = Float64(Float64(x * Float64(Float64(9.0 * y) / c)) / z);
	elseif (t_1 <= 4e+192)
		tmp = fma(Float64(a * -4.0), Float64(t / c), Float64(b / Float64(c * z)));
	else
		tmp = Float64(Float64(Float64(Float64(9.0 * x) / c) * y) / z);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+94], N[(N[(x * N[(N[(9.0 * y), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 4e+192], N[(N[(a * -4.0), $MachinePrecision] * N[(t / c), $MachinePrecision] + N[(b / N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(9.0 * x), $MachinePrecision] / c), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -5.0000000000000001e94

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{c \cdot z}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

   3. Applied rewrites 80.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b - a \cdot \left(t \cdot \left(4 \cdot z\right)\right)\right)}{c}}{z}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{9 \cdot \frac{x \cdot y}{c}}}{z} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\frac{x \cdot y}{c}}}{z} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{\color{blue}{c}}}{z} \]

      3. lower-*.f64 35.8

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{c}}{z} \]

   6. Applied rewrites 35.8%

      \[\leadsto \frac{\color{blue}{9 \cdot \frac{x \cdot y}{c}}}{z} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\frac{x \cdot y}{c}}}{z} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{\color{blue}{c}}}{z} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{9 \cdot \left(x \cdot y\right)}{\color{blue}{c}}}{z} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{\left(x \cdot y\right) \cdot 9}{c}}{z} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(x \cdot y\right) \cdot 9}{c}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{\left(y \cdot x\right) \cdot 9}{c}}{z} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\frac{y \cdot \left(x \cdot 9\right)}{c}}{z} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\left(x \cdot 9\right) \cdot y}{c}}{z} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\frac{x \cdot \left(9 \cdot y\right)}{c}}{z} \]

      10. associate-/l* N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{9 \cdot y}{c}}}{z} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{9 \cdot y}{c}}}{z} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x \cdot \frac{9 \cdot y}{\color{blue}{c}}}{z} \]

      13. lower-*.f64 37.1

          \[\leadsto \frac{x \cdot \frac{9 \cdot y}{c}}{z} \]

   8. Applied rewrites 37.1%

      \[\leadsto \frac{x \cdot \color{blue}{\frac{9 \cdot y}{c}}}{z} \]

   if -5.0000000000000001e94 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 4.00000000000000016e192

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b}{z \cdot c} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b}{z}}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      7. mult-flip N/A

         \[\leadsto \color{blue}{\frac{b}{z} \cdot \frac{1}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      8. inv-pow N/A

         \[\leadsto \frac{b}{z} \cdot \color{blue}{{c}^{-1}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{b}{z}}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      11. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      13. lower-/.f64 73.9

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \color{blue}{\frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}}\right) \]

   3. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \frac{\mathsf{fma}\left(y \cdot x, 9, -4 \cdot \left(z \cdot \left(a \cdot t\right)\right)\right)}{c \cdot z}\right)} \]

   4. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right) + \left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{\color{blue}{c}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      7. lower-/.f64 84.1

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

   6. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{\color{blue}{c}} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right) + \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c} \]

      3. div-add N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right)}{c} + \color{blue}{\frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right)}{c} + \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\left(-4 \cdot a\right) \cdot t}{c} + \frac{\mathsf{fma}\left(\color{blue}{9}, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c} \]

      6. associate-/l* N/A

         \[\leadsto \left(-4 \cdot a\right) \cdot \frac{t}{c} + \frac{\color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}}{c} \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4 \cdot a, \color{blue}{\frac{t}{c}}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a \cdot -4, \frac{\color{blue}{t}}{c}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot -4, \frac{\color{blue}{t}}{c}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot -4, \frac{t}{\color{blue}{c}}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

      11. lower-/.f64 81.2

          \[\leadsto \mathsf{fma}\left(a \cdot -4, \frac{t}{c}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right) \]

   8. Applied rewrites 82.9%

      \[\leadsto \mathsf{fma}\left(a \cdot -4, \color{blue}{\frac{t}{c}}, \frac{\frac{\mathsf{fma}\left(9 \cdot y, x, b\right)}{z}}{c}\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(a \cdot -4, \frac{t}{c}, \frac{b}{c \cdot z}\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. lower-*.f64 63.0

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

   11. Applied rewrites 63.0%

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

   if 4.00000000000000016e192 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{c \cdot z}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

   3. Applied rewrites 80.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b - a \cdot \left(t \cdot \left(4 \cdot z\right)\right)\right)}{c}}{z}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{9 \cdot \frac{x \cdot y}{c}}}{z} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\frac{x \cdot y}{c}}}{z} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{\color{blue}{c}}}{z} \]

      3. lower-*.f64 35.8

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{c}}{z} \]

   6. Applied rewrites 35.8%

      \[\leadsto \frac{\color{blue}{9 \cdot \frac{x \cdot y}{c}}}{z} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\frac{x \cdot y}{c}}}{z} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{\color{blue}{c}}}{z} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{c}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \frac{9 \cdot \left(x \cdot \color{blue}{\frac{y}{c}}\right)}{z} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\left(9 \cdot x\right) \cdot \color{blue}{\frac{y}{c}}}{z} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(9 \cdot x\right) \cdot \frac{\color{blue}{y}}{c}}{z} \]

      7. associate-/l* N/A

         \[\leadsto \frac{\frac{\left(9 \cdot x\right) \cdot y}{\color{blue}{c}}}{z} \]

      8. associate-*l/ N/A

         \[\leadsto \frac{\frac{9 \cdot x}{c} \cdot \color{blue}{y}}{z} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{\frac{9 \cdot x}{c} \cdot y}{z} \]

      10. lower-*.f64 37.1

          \[\leadsto \frac{\frac{9 \cdot x}{c} \cdot \color{blue}{y}}{z} \]

   8. Applied rewrites 37.1%

      \[\leadsto \frac{\frac{9 \cdot x}{c} \cdot \color{blue}{y}}{z} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 76.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+94}:\\ \;\;\;\;\frac{x \cdot \frac{9 \cdot y}{c}}{z}\\ \mathbf{elif}\;t\_1 \leq 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{a \cdot t}{c}, \frac{b}{c \cdot z}\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+279}:\\ \;\;\;\;\frac{b - -9 \cdot \left(x \cdot y\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{9 \cdot x}{c} \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)))
   (if (<= t_1 -5e+94)
     (/ (* x (/ (* 9.0 y) c)) z)
     (if (<= t_1 1e+39)
       (fma -4.0 (/ (* a t) c) (/ b (* c z)))
       (if (<= t_1 1e+279)
         (/ (- b (* -9.0 (* x y))) (* z c))
         (/ (* (/ (* 9.0 x) c) y) z))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if (t_1 <= -5e+94) {
		tmp = (x * ((9.0 * y) / c)) / z;
	} else if (t_1 <= 1e+39) {
		tmp = fma(-4.0, ((a * t) / c), (b / (c * z)));
	} else if (t_1 <= 1e+279) {
		tmp = (b - (-9.0 * (x * y))) / (z * c);
	} else {
		tmp = (((9.0 * x) / c) * y) / z;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_1 <= -5e+94)
		tmp = Float64(Float64(x * Float64(Float64(9.0 * y) / c)) / z);
	elseif (t_1 <= 1e+39)
		tmp = fma(-4.0, Float64(Float64(a * t) / c), Float64(b / Float64(c * z)));
	elseif (t_1 <= 1e+279)
		tmp = Float64(Float64(b - Float64(-9.0 * Float64(x * y))) / Float64(z * c));
	else
		tmp = Float64(Float64(Float64(Float64(9.0 * x) / c) * y) / z);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+94], N[(N[(x * N[(N[(9.0 * y), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 1e+39], N[(-4.0 * N[(N[(a * t), $MachinePrecision] / c), $MachinePrecision] + N[(b / N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+279], N[(N[(b - N[(-9.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(9.0 * x), $MachinePrecision] / c), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -5.0000000000000001e94

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{c \cdot z}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

   3. Applied rewrites 80.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b - a \cdot \left(t \cdot \left(4 \cdot z\right)\right)\right)}{c}}{z}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{9 \cdot \frac{x \cdot y}{c}}}{z} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\frac{x \cdot y}{c}}}{z} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{\color{blue}{c}}}{z} \]

      3. lower-*.f64 35.8

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{c}}{z} \]

   6. Applied rewrites 35.8%

      \[\leadsto \frac{\color{blue}{9 \cdot \frac{x \cdot y}{c}}}{z} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\frac{x \cdot y}{c}}}{z} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{\color{blue}{c}}}{z} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{9 \cdot \left(x \cdot y\right)}{\color{blue}{c}}}{z} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{\left(x \cdot y\right) \cdot 9}{c}}{z} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(x \cdot y\right) \cdot 9}{c}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{\left(y \cdot x\right) \cdot 9}{c}}{z} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\frac{y \cdot \left(x \cdot 9\right)}{c}}{z} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\left(x \cdot 9\right) \cdot y}{c}}{z} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\frac{x \cdot \left(9 \cdot y\right)}{c}}{z} \]

      10. associate-/l* N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{9 \cdot y}{c}}}{z} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{9 \cdot y}{c}}}{z} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x \cdot \frac{9 \cdot y}{\color{blue}{c}}}{z} \]

      13. lower-*.f64 37.1

          \[\leadsto \frac{x \cdot \frac{9 \cdot y}{c}}{z} \]

   8. Applied rewrites 37.1%

      \[\leadsto \frac{x \cdot \color{blue}{\frac{9 \cdot y}{c}}}{z} \]

   if -5.0000000000000001e94 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.9999999999999994e38

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b}{z \cdot c} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b}{z}}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      7. mult-flip N/A

         \[\leadsto \color{blue}{\frac{b}{z} \cdot \frac{1}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      8. inv-pow N/A

         \[\leadsto \frac{b}{z} \cdot \color{blue}{{c}^{-1}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{b}{z}}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      11. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      13. lower-/.f64 73.9

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \color{blue}{\frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}}\right) \]

   3. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \frac{\mathsf{fma}\left(y \cdot x, 9, -4 \cdot \left(z \cdot \left(a \cdot t\right)\right)\right)}{c \cdot z}\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c} + \frac{b}{c \cdot z}} \]
   5. Step-by-step derivation

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 62.1

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

   6. Applied rewrites 62.1%

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

   if 9.9999999999999994e38 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.00000000000000006e279

   1. Initial program 80.4%

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

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      3. add-flip N/A

         \[\leadsto \frac{\color{blue}{b - \left(\mathsf{neg}\left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right)\right)}}{z \cdot c} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{b - \left(\mathsf{neg}\left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right)\right)}}{z \cdot c} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{b - \left(\mathsf{neg}\left(\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}\right)\right)}{z \cdot c} \]

      6. sub-negate-rev N/A

         \[\leadsto \frac{b - \color{blue}{\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - \left(x \cdot 9\right) \cdot y\right)}}{z \cdot c} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{b - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - \color{blue}{\left(x \cdot 9\right) \cdot y}\right)}{z \cdot c} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{b - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - \color{blue}{\left(x \cdot 9\right)} \cdot y\right)}{z \cdot c} \]

      9. *-commutative N/A

         \[\leadsto \frac{b - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - \color{blue}{\left(9 \cdot x\right)} \cdot y\right)}{z \cdot c} \]

      10. associate-*l* N/A

          \[\leadsto \frac{b - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - \color{blue}{9 \cdot \left(x \cdot y\right)}\right)}{z \cdot c} \]

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

          \[\leadsto \frac{b - \color{blue}{\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a + \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}}{z \cdot c} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{b - \left(\color{blue}{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a} + \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      13. *-commutative N/A

          \[\leadsto \frac{b - \left(\color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)} + \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{b - \left(a \cdot \color{blue}{\left(\left(z \cdot 4\right) \cdot t\right)} + \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      15. associate-*r* N/A

          \[\leadsto \frac{b - \left(\color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t} + \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{b - \color{blue}{\mathsf{fma}\left(a \cdot \left(z \cdot 4\right), t, \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}}{z \cdot c} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{b - \mathsf{fma}\left(\color{blue}{a \cdot \left(z \cdot 4\right)}, t, \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{b - \mathsf{fma}\left(a \cdot \color{blue}{\left(z \cdot 4\right)}, t, \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      19. *-commutative N/A

          \[\leadsto \frac{b - \mathsf{fma}\left(a \cdot \color{blue}{\left(4 \cdot z\right)}, t, \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      20. lower-*.f64 N/A

          \[\leadsto \frac{b - \mathsf{fma}\left(a \cdot \color{blue}{\left(4 \cdot z\right)}, t, \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      21. lower-*.f64 N/A

          \[\leadsto \frac{b - \mathsf{fma}\left(a \cdot \left(4 \cdot z\right), t, \color{blue}{\left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)}\right)}{z \cdot c} \]

      22. metadata-eval N/A

          \[\leadsto \frac{b - \mathsf{fma}\left(a \cdot \left(4 \cdot z\right), t, \color{blue}{-9} \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      23. lower-*.f64 80.2

          \[\leadsto \frac{b - \mathsf{fma}\left(a \cdot \left(4 \cdot z\right), t, -9 \cdot \color{blue}{\left(x \cdot y\right)}\right)}{z \cdot c} \]

   3. Applied rewrites 80.2%

      \[\leadsto \frac{\color{blue}{b - \mathsf{fma}\left(a \cdot \left(4 \cdot z\right), t, -9 \cdot \left(x \cdot y\right)\right)}}{z \cdot c} \]

   4. Taylor expanded in z around 0

      \[\leadsto \frac{\color{blue}{b - -9 \cdot \left(x \cdot y\right)}}{z \cdot c} \]
   5. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{b - \color{blue}{-9 \cdot \left(x \cdot y\right)}}{z \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{b - -9 \cdot \color{blue}{\left(x \cdot y\right)}}{z \cdot c} \]

      3. lower-*.f64 60.6

         \[\leadsto \frac{b - -9 \cdot \left(x \cdot \color{blue}{y}\right)}{z \cdot c} \]

   6. Applied rewrites 60.6%

      \[\leadsto \frac{\color{blue}{b - -9 \cdot \left(x \cdot y\right)}}{z \cdot c} \]

   if 1.00000000000000006e279 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{c \cdot z}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

   3. Applied rewrites 80.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b - a \cdot \left(t \cdot \left(4 \cdot z\right)\right)\right)}{c}}{z}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{9 \cdot \frac{x \cdot y}{c}}}{z} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\frac{x \cdot y}{c}}}{z} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{\color{blue}{c}}}{z} \]

      3. lower-*.f64 35.8

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{c}}{z} \]

   6. Applied rewrites 35.8%

      \[\leadsto \frac{\color{blue}{9 \cdot \frac{x \cdot y}{c}}}{z} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\frac{x \cdot y}{c}}}{z} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{\color{blue}{c}}}{z} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{c}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \frac{9 \cdot \left(x \cdot \color{blue}{\frac{y}{c}}\right)}{z} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\left(9 \cdot x\right) \cdot \color{blue}{\frac{y}{c}}}{z} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(9 \cdot x\right) \cdot \frac{\color{blue}{y}}{c}}{z} \]

      7. associate-/l* N/A

         \[\leadsto \frac{\frac{\left(9 \cdot x\right) \cdot y}{\color{blue}{c}}}{z} \]

      8. associate-*l/ N/A

         \[\leadsto \frac{\frac{9 \cdot x}{c} \cdot \color{blue}{y}}{z} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{\frac{9 \cdot x}{c} \cdot y}{z} \]

      10. lower-*.f64 37.1

          \[\leadsto \frac{\frac{9 \cdot x}{c} \cdot \color{blue}{y}}{z} \]

   8. Applied rewrites 37.1%

      \[\leadsto \frac{\frac{9 \cdot x}{c} \cdot \color{blue}{y}}{z} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 76.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+94}:\\ \;\;\;\;\frac{x \cdot \frac{9 \cdot y}{c}}{z}\\ \mathbf{elif}\;t\_1 \leq 10^{+39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{c}\\ \mathbf{elif}\;t\_1 \leq 10^{+279}:\\ \;\;\;\;\frac{b - -9 \cdot \left(x \cdot y\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{9 \cdot x}{c} \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)))
   (if (<= t_1 -5e+94)
     (/ (* x (/ (* 9.0 y) c)) z)
     (if (<= t_1 1e+39)
       (/ (fma -4.0 (* a t) (/ b z)) c)
       (if (<= t_1 1e+279)
         (/ (- b (* -9.0 (* x y))) (* z c))
         (/ (* (/ (* 9.0 x) c) y) z))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if (t_1 <= -5e+94) {
		tmp = (x * ((9.0 * y) / c)) / z;
	} else if (t_1 <= 1e+39) {
		tmp = fma(-4.0, (a * t), (b / z)) / c;
	} else if (t_1 <= 1e+279) {
		tmp = (b - (-9.0 * (x * y))) / (z * c);
	} else {
		tmp = (((9.0 * x) / c) * y) / z;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_1 <= -5e+94)
		tmp = Float64(Float64(x * Float64(Float64(9.0 * y) / c)) / z);
	elseif (t_1 <= 1e+39)
		tmp = Float64(fma(-4.0, Float64(a * t), Float64(b / z)) / c);
	elseif (t_1 <= 1e+279)
		tmp = Float64(Float64(b - Float64(-9.0 * Float64(x * y))) / Float64(z * c));
	else
		tmp = Float64(Float64(Float64(Float64(9.0 * x) / c) * y) / z);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+94], N[(N[(x * N[(N[(9.0 * y), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 1e+39], N[(N[(-4.0 * N[(a * t), $MachinePrecision] + N[(b / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t$95$1, 1e+279], N[(N[(b - N[(-9.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(9.0 * x), $MachinePrecision] / c), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -5.0000000000000001e94

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{c \cdot z}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

   3. Applied rewrites 80.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b - a \cdot \left(t \cdot \left(4 \cdot z\right)\right)\right)}{c}}{z}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{9 \cdot \frac{x \cdot y}{c}}}{z} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\frac{x \cdot y}{c}}}{z} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{\color{blue}{c}}}{z} \]

      3. lower-*.f64 35.8

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{c}}{z} \]

   6. Applied rewrites 35.8%

      \[\leadsto \frac{\color{blue}{9 \cdot \frac{x \cdot y}{c}}}{z} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\frac{x \cdot y}{c}}}{z} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{\color{blue}{c}}}{z} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{9 \cdot \left(x \cdot y\right)}{\color{blue}{c}}}{z} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{\left(x \cdot y\right) \cdot 9}{c}}{z} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(x \cdot y\right) \cdot 9}{c}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{\left(y \cdot x\right) \cdot 9}{c}}{z} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\frac{y \cdot \left(x \cdot 9\right)}{c}}{z} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\left(x \cdot 9\right) \cdot y}{c}}{z} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\frac{x \cdot \left(9 \cdot y\right)}{c}}{z} \]

      10. associate-/l* N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{9 \cdot y}{c}}}{z} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{9 \cdot y}{c}}}{z} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x \cdot \frac{9 \cdot y}{\color{blue}{c}}}{z} \]

      13. lower-*.f64 37.1

          \[\leadsto \frac{x \cdot \frac{9 \cdot y}{c}}{z} \]

   8. Applied rewrites 37.1%

      \[\leadsto \frac{x \cdot \color{blue}{\frac{9 \cdot y}{c}}}{z} \]

   if -5.0000000000000001e94 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.9999999999999994e38

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b}{z \cdot c} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b}{z}}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      7. mult-flip N/A

         \[\leadsto \color{blue}{\frac{b}{z} \cdot \frac{1}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      8. inv-pow N/A

         \[\leadsto \frac{b}{z} \cdot \color{blue}{{c}^{-1}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{b}{z}}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      11. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      13. lower-/.f64 73.9

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \color{blue}{\frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}}\right) \]

   3. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \frac{\mathsf{fma}\left(y \cdot x, 9, -4 \cdot \left(z \cdot \left(a \cdot t\right)\right)\right)}{c \cdot z}\right)} \]

   4. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right) + \left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{\color{blue}{c}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      7. lower-/.f64 84.1

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

   6. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{c} \]
   8. Step-by-step derivation

      1. lower-/.f64 62.9

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

   9. Applied rewrites 62.9%

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

   if 9.9999999999999994e38 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.00000000000000006e279

   1. Initial program 80.4%

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

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      3. add-flip N/A

         \[\leadsto \frac{\color{blue}{b - \left(\mathsf{neg}\left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right)\right)}}{z \cdot c} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{b - \left(\mathsf{neg}\left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right)\right)}}{z \cdot c} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{b - \left(\mathsf{neg}\left(\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}\right)\right)}{z \cdot c} \]

      6. sub-negate-rev N/A

         \[\leadsto \frac{b - \color{blue}{\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - \left(x \cdot 9\right) \cdot y\right)}}{z \cdot c} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{b - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - \color{blue}{\left(x \cdot 9\right) \cdot y}\right)}{z \cdot c} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{b - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - \color{blue}{\left(x \cdot 9\right)} \cdot y\right)}{z \cdot c} \]

      9. *-commutative N/A

         \[\leadsto \frac{b - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - \color{blue}{\left(9 \cdot x\right)} \cdot y\right)}{z \cdot c} \]

      10. associate-*l* N/A

          \[\leadsto \frac{b - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - \color{blue}{9 \cdot \left(x \cdot y\right)}\right)}{z \cdot c} \]

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

          \[\leadsto \frac{b - \color{blue}{\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a + \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}}{z \cdot c} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{b - \left(\color{blue}{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a} + \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      13. *-commutative N/A

          \[\leadsto \frac{b - \left(\color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)} + \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{b - \left(a \cdot \color{blue}{\left(\left(z \cdot 4\right) \cdot t\right)} + \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      15. associate-*r* N/A

          \[\leadsto \frac{b - \left(\color{blue}{\left(a \cdot \left(z \cdot 4\right)\right) \cdot t} + \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{b - \color{blue}{\mathsf{fma}\left(a \cdot \left(z \cdot 4\right), t, \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}}{z \cdot c} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{b - \mathsf{fma}\left(\color{blue}{a \cdot \left(z \cdot 4\right)}, t, \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{b - \mathsf{fma}\left(a \cdot \color{blue}{\left(z \cdot 4\right)}, t, \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      19. *-commutative N/A

          \[\leadsto \frac{b - \mathsf{fma}\left(a \cdot \color{blue}{\left(4 \cdot z\right)}, t, \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      20. lower-*.f64 N/A

          \[\leadsto \frac{b - \mathsf{fma}\left(a \cdot \color{blue}{\left(4 \cdot z\right)}, t, \left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      21. lower-*.f64 N/A

          \[\leadsto \frac{b - \mathsf{fma}\left(a \cdot \left(4 \cdot z\right), t, \color{blue}{\left(\mathsf{neg}\left(9\right)\right) \cdot \left(x \cdot y\right)}\right)}{z \cdot c} \]

      22. metadata-eval N/A

          \[\leadsto \frac{b - \mathsf{fma}\left(a \cdot \left(4 \cdot z\right), t, \color{blue}{-9} \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      23. lower-*.f64 80.2

          \[\leadsto \frac{b - \mathsf{fma}\left(a \cdot \left(4 \cdot z\right), t, -9 \cdot \color{blue}{\left(x \cdot y\right)}\right)}{z \cdot c} \]

   3. Applied rewrites 80.2%

      \[\leadsto \frac{\color{blue}{b - \mathsf{fma}\left(a \cdot \left(4 \cdot z\right), t, -9 \cdot \left(x \cdot y\right)\right)}}{z \cdot c} \]

   4. Taylor expanded in z around 0

      \[\leadsto \frac{\color{blue}{b - -9 \cdot \left(x \cdot y\right)}}{z \cdot c} \]
   5. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{b - \color{blue}{-9 \cdot \left(x \cdot y\right)}}{z \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{b - -9 \cdot \color{blue}{\left(x \cdot y\right)}}{z \cdot c} \]

      3. lower-*.f64 60.6

         \[\leadsto \frac{b - -9 \cdot \left(x \cdot \color{blue}{y}\right)}{z \cdot c} \]

   6. Applied rewrites 60.6%

      \[\leadsto \frac{\color{blue}{b - -9 \cdot \left(x \cdot y\right)}}{z \cdot c} \]

   if 1.00000000000000006e279 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{c \cdot z}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

   3. Applied rewrites 80.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b - a \cdot \left(t \cdot \left(4 \cdot z\right)\right)\right)}{c}}{z}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{9 \cdot \frac{x \cdot y}{c}}}{z} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\frac{x \cdot y}{c}}}{z} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{\color{blue}{c}}}{z} \]

      3. lower-*.f64 35.8

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{c}}{z} \]

   6. Applied rewrites 35.8%

      \[\leadsto \frac{\color{blue}{9 \cdot \frac{x \cdot y}{c}}}{z} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\frac{x \cdot y}{c}}}{z} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{\color{blue}{c}}}{z} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{c}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \frac{9 \cdot \left(x \cdot \color{blue}{\frac{y}{c}}\right)}{z} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\left(9 \cdot x\right) \cdot \color{blue}{\frac{y}{c}}}{z} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(9 \cdot x\right) \cdot \frac{\color{blue}{y}}{c}}{z} \]

      7. associate-/l* N/A

         \[\leadsto \frac{\frac{\left(9 \cdot x\right) \cdot y}{\color{blue}{c}}}{z} \]

      8. associate-*l/ N/A

         \[\leadsto \frac{\frac{9 \cdot x}{c} \cdot \color{blue}{y}}{z} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{\frac{9 \cdot x}{c} \cdot y}{z} \]

      10. lower-*.f64 37.1

          \[\leadsto \frac{\frac{9 \cdot x}{c} \cdot \color{blue}{y}}{z} \]

   8. Applied rewrites 37.1%

      \[\leadsto \frac{\frac{9 \cdot x}{c} \cdot \color{blue}{y}}{z} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 10: 75.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+94}:\\ \;\;\;\;\frac{x \cdot \frac{9 \cdot y}{c}}{z}\\ \mathbf{elif}\;t\_1 \leq 10^{+39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{c}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+192}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{9 \cdot x}{c} \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)))
   (if (<= t_1 -5e+94)
     (/ (* x (/ (* 9.0 y) c)) z)
     (if (<= t_1 1e+39)
       (/ (fma -4.0 (* a t) (/ b z)) c)
       (if (<= t_1 4e+192)
         (/ (/ (fma (* y x) 9.0 b) c) z)
         (/ (* (/ (* 9.0 x) c) y) z))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if (t_1 <= -5e+94) {
		tmp = (x * ((9.0 * y) / c)) / z;
	} else if (t_1 <= 1e+39) {
		tmp = fma(-4.0, (a * t), (b / z)) / c;
	} else if (t_1 <= 4e+192) {
		tmp = (fma((y * x), 9.0, b) / c) / z;
	} else {
		tmp = (((9.0 * x) / c) * y) / z;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_1 <= -5e+94)
		tmp = Float64(Float64(x * Float64(Float64(9.0 * y) / c)) / z);
	elseif (t_1 <= 1e+39)
		tmp = Float64(fma(-4.0, Float64(a * t), Float64(b / z)) / c);
	elseif (t_1 <= 4e+192)
		tmp = Float64(Float64(fma(Float64(y * x), 9.0, b) / c) / z);
	else
		tmp = Float64(Float64(Float64(Float64(9.0 * x) / c) * y) / z);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+94], N[(N[(x * N[(N[(9.0 * y), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 1e+39], N[(N[(-4.0 * N[(a * t), $MachinePrecision] + N[(b / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t$95$1, 4e+192], N[(N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / c), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(9.0 * x), $MachinePrecision] / c), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -5.0000000000000001e94

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{c \cdot z}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

   3. Applied rewrites 80.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b - a \cdot \left(t \cdot \left(4 \cdot z\right)\right)\right)}{c}}{z}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{9 \cdot \frac{x \cdot y}{c}}}{z} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\frac{x \cdot y}{c}}}{z} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{\color{blue}{c}}}{z} \]

      3. lower-*.f64 35.8

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{c}}{z} \]

   6. Applied rewrites 35.8%

      \[\leadsto \frac{\color{blue}{9 \cdot \frac{x \cdot y}{c}}}{z} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\frac{x \cdot y}{c}}}{z} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{\color{blue}{c}}}{z} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{9 \cdot \left(x \cdot y\right)}{\color{blue}{c}}}{z} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{\left(x \cdot y\right) \cdot 9}{c}}{z} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(x \cdot y\right) \cdot 9}{c}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{\left(y \cdot x\right) \cdot 9}{c}}{z} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\frac{y \cdot \left(x \cdot 9\right)}{c}}{z} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\left(x \cdot 9\right) \cdot y}{c}}{z} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\frac{x \cdot \left(9 \cdot y\right)}{c}}{z} \]

      10. associate-/l* N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{9 \cdot y}{c}}}{z} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{9 \cdot y}{c}}}{z} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x \cdot \frac{9 \cdot y}{\color{blue}{c}}}{z} \]

      13. lower-*.f64 37.1

          \[\leadsto \frac{x \cdot \frac{9 \cdot y}{c}}{z} \]

   8. Applied rewrites 37.1%

      \[\leadsto \frac{x \cdot \color{blue}{\frac{9 \cdot y}{c}}}{z} \]

   if -5.0000000000000001e94 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.9999999999999994e38

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b}{z \cdot c} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b}{z}}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      7. mult-flip N/A

         \[\leadsto \color{blue}{\frac{b}{z} \cdot \frac{1}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      8. inv-pow N/A

         \[\leadsto \frac{b}{z} \cdot \color{blue}{{c}^{-1}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{b}{z}}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      11. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      13. lower-/.f64 73.9

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \color{blue}{\frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}}\right) \]

   3. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \frac{\mathsf{fma}\left(y \cdot x, 9, -4 \cdot \left(z \cdot \left(a \cdot t\right)\right)\right)}{c \cdot z}\right)} \]

   4. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right) + \left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{\color{blue}{c}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      7. lower-/.f64 84.1

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

   6. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{c} \]
   8. Step-by-step derivation

      1. lower-/.f64 62.9

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

   9. Applied rewrites 62.9%

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

   if 9.9999999999999994e38 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 4.00000000000000016e192

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{c \cdot z}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

   3. Applied rewrites 80.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b - a \cdot \left(t \cdot \left(4 \cdot z\right)\right)\right)}{c}}{z}} \]

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 61.0%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(y \cdot x, 9, \color{blue}{b}\right)}{c}}{z} \]

   if 4.00000000000000016e192 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{c \cdot z}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

   3. Applied rewrites 80.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b - a \cdot \left(t \cdot \left(4 \cdot z\right)\right)\right)}{c}}{z}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{9 \cdot \frac{x \cdot y}{c}}}{z} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\frac{x \cdot y}{c}}}{z} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{\color{blue}{c}}}{z} \]

      3. lower-*.f64 35.8

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{c}}{z} \]

   6. Applied rewrites 35.8%

      \[\leadsto \frac{\color{blue}{9 \cdot \frac{x \cdot y}{c}}}{z} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\frac{x \cdot y}{c}}}{z} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{\color{blue}{c}}}{z} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{c}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \frac{9 \cdot \left(x \cdot \color{blue}{\frac{y}{c}}\right)}{z} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\left(9 \cdot x\right) \cdot \color{blue}{\frac{y}{c}}}{z} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(9 \cdot x\right) \cdot \frac{\color{blue}{y}}{c}}{z} \]

      7. associate-/l* N/A

         \[\leadsto \frac{\frac{\left(9 \cdot x\right) \cdot y}{\color{blue}{c}}}{z} \]

      8. associate-*l/ N/A

         \[\leadsto \frac{\frac{9 \cdot x}{c} \cdot \color{blue}{y}}{z} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{\frac{9 \cdot x}{c} \cdot y}{z} \]

      10. lower-*.f64 37.1

          \[\leadsto \frac{\frac{9 \cdot x}{c} \cdot \color{blue}{y}}{z} \]

   8. Applied rewrites 37.1%

      \[\leadsto \frac{\frac{9 \cdot x}{c} \cdot \color{blue}{y}}{z} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 11: 74.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ t_2 := \frac{x \cdot \frac{9 \cdot y}{c}}{z}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+94}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)) (t_2 (/ (* x (/ (* 9.0 y) c)) z)))
   (if (<= t_1 -5e+94)
     t_2
     (if (<= t_1 1e+39) (/ (fma -4.0 (* a t) (/ b z)) c) t_2))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (x * 9.0) * y;
	double t_2 = (x * ((9.0 * y) / c)) / z;
	double tmp;
	if (t_1 <= -5e+94) {
		tmp = t_2;
	} else if (t_1 <= 1e+39) {
		tmp = fma(-4.0, (a * t), (b / z)) / c;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(x * 9.0) * y)
	t_2 = Float64(Float64(x * Float64(Float64(9.0 * y) / c)) / z)
	tmp = 0.0
	if (t_1 <= -5e+94)
		tmp = t_2;
	elseif (t_1 <= 1e+39)
		tmp = Float64(fma(-4.0, Float64(a * t), Float64(b / z)) / c);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(N[(9.0 * y), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+94], t$95$2, If[LessEqual[t$95$1, 1e+39], N[(N[(-4.0 * N[(a * t), $MachinePrecision] + N[(b / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -5.0000000000000001e94 or 9.9999999999999994e38 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{c \cdot z}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

   3. Applied rewrites 80.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b - a \cdot \left(t \cdot \left(4 \cdot z\right)\right)\right)}{c}}{z}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{9 \cdot \frac{x \cdot y}{c}}}{z} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\frac{x \cdot y}{c}}}{z} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{\color{blue}{c}}}{z} \]

      3. lower-*.f64 35.8

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{c}}{z} \]

   6. Applied rewrites 35.8%

      \[\leadsto \frac{\color{blue}{9 \cdot \frac{x \cdot y}{c}}}{z} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\frac{x \cdot y}{c}}}{z} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{\color{blue}{c}}}{z} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{9 \cdot \left(x \cdot y\right)}{\color{blue}{c}}}{z} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{\left(x \cdot y\right) \cdot 9}{c}}{z} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(x \cdot y\right) \cdot 9}{c}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{\left(y \cdot x\right) \cdot 9}{c}}{z} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\frac{y \cdot \left(x \cdot 9\right)}{c}}{z} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\left(x \cdot 9\right) \cdot y}{c}}{z} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\frac{x \cdot \left(9 \cdot y\right)}{c}}{z} \]

      10. associate-/l* N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{9 \cdot y}{c}}}{z} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{9 \cdot y}{c}}}{z} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x \cdot \frac{9 \cdot y}{\color{blue}{c}}}{z} \]

      13. lower-*.f64 37.1

          \[\leadsto \frac{x \cdot \frac{9 \cdot y}{c}}{z} \]

   8. Applied rewrites 37.1%

      \[\leadsto \frac{x \cdot \color{blue}{\frac{9 \cdot y}{c}}}{z} \]

   if -5.0000000000000001e94 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.9999999999999994e38

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b}{z \cdot c} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b}{z}}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      7. mult-flip N/A

         \[\leadsto \color{blue}{\frac{b}{z} \cdot \frac{1}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      8. inv-pow N/A

         \[\leadsto \frac{b}{z} \cdot \color{blue}{{c}^{-1}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{b}{z}}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      11. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      13. lower-/.f64 73.9

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \color{blue}{\frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}}\right) \]

   3. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \frac{\mathsf{fma}\left(y \cdot x, 9, -4 \cdot \left(z \cdot \left(a \cdot t\right)\right)\right)}{c \cdot z}\right)} \]

   4. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right) + \left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{\color{blue}{c}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      7. lower-/.f64 84.1

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

   6. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{c} \]
   8. Step-by-step derivation

      1. lower-/.f64 62.9

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

   9. Applied rewrites 62.9%

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

Alternative 12: 54.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ t_2 := \frac{x \cdot \frac{9 \cdot y}{c}}{z}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+64}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-232}:\\ \;\;\;\;\frac{b}{c \cdot z}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;-4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+32}:\\ \;\;\;\;\frac{1}{\frac{z}{b} \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)) (t_2 (/ (* x (/ (* 9.0 y) c)) z)))
   (if (<= t_1 -1e+64)
     t_2
     (if (<= t_1 -5e-232)
       (/ b (* c z))
       (if (<= t_1 2e-11)
         (* -4.0 (/ (* a t) c))
         (if (<= t_1 5e+32) (/ 1.0 (* (/ z b) c)) t_2))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (x * 9.0) * y;
	double t_2 = (x * ((9.0 * y) / c)) / z;
	double tmp;
	if (t_1 <= -1e+64) {
		tmp = t_2;
	} else if (t_1 <= -5e-232) {
		tmp = b / (c * z);
	} else if (t_1 <= 2e-11) {
		tmp = -4.0 * ((a * t) / c);
	} else if (t_1 <= 5e+32) {
		tmp = 1.0 / ((z / b) * c);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c)
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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x * 9.0d0) * y
    t_2 = (x * ((9.0d0 * y) / c)) / z
    if (t_1 <= (-1d+64)) then
        tmp = t_2
    else if (t_1 <= (-5d-232)) then
        tmp = b / (c * z)
    else if (t_1 <= 2d-11) then
        tmp = (-4.0d0) * ((a * t) / c)
    else if (t_1 <= 5d+32) then
        tmp = 1.0d0 / ((z / b) * c)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b && b < c;
public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (x * 9.0) * y;
	double t_2 = (x * ((9.0 * y) / c)) / z;
	double tmp;
	if (t_1 <= -1e+64) {
		tmp = t_2;
	} else if (t_1 <= -5e-232) {
		tmp = b / (c * z);
	} else if (t_1 <= 2e-11) {
		tmp = -4.0 * ((a * t) / c);
	} else if (t_1 <= 5e+32) {
		tmp = 1.0 / ((z / b) * c);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y, z, t, a, b, c] = sort([x, y, z, t, a, b, c])
def code(x, y, z, t, a, b, c):
	t_1 = (x * 9.0) * y
	t_2 = (x * ((9.0 * y) / c)) / z
	tmp = 0
	if t_1 <= -1e+64:
		tmp = t_2
	elif t_1 <= -5e-232:
		tmp = b / (c * z)
	elif t_1 <= 2e-11:
		tmp = -4.0 * ((a * t) / c)
	elif t_1 <= 5e+32:
		tmp = 1.0 / ((z / b) * c)
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(x * 9.0) * y)
	t_2 = Float64(Float64(x * Float64(Float64(9.0 * y) / c)) / z)
	tmp = 0.0
	if (t_1 <= -1e+64)
		tmp = t_2;
	elseif (t_1 <= -5e-232)
		tmp = Float64(b / Float64(c * z));
	elseif (t_1 <= 2e-11)
		tmp = Float64(-4.0 * Float64(Float64(a * t) / c));
	elseif (t_1 <= 5e+32)
		tmp = Float64(1.0 / Float64(Float64(z / b) * c));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b, c = num2cell(sort([x, y, z, t, a, b, c])){:}
function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (x * 9.0) * y;
	t_2 = (x * ((9.0 * y) / c)) / z;
	tmp = 0.0;
	if (t_1 <= -1e+64)
		tmp = t_2;
	elseif (t_1 <= -5e-232)
		tmp = b / (c * z);
	elseif (t_1 <= 2e-11)
		tmp = -4.0 * ((a * t) / c);
	elseif (t_1 <= 5e+32)
		tmp = 1.0 / ((z / b) * c);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(N[(9.0 * y), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+64], t$95$2, If[LessEqual[t$95$1, -5e-232], N[(b / N[(c * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-11], N[(-4.0 * N[(N[(a * t), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+32], N[(1.0 / N[(N[(z / b), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -1.00000000000000002e64 or 4.9999999999999997e32 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{c \cdot z}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

   3. Applied rewrites 80.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b - a \cdot \left(t \cdot \left(4 \cdot z\right)\right)\right)}{c}}{z}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{9 \cdot \frac{x \cdot y}{c}}}{z} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\frac{x \cdot y}{c}}}{z} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{\color{blue}{c}}}{z} \]

      3. lower-*.f64 35.8

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{c}}{z} \]

   6. Applied rewrites 35.8%

      \[\leadsto \frac{\color{blue}{9 \cdot \frac{x \cdot y}{c}}}{z} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\frac{x \cdot y}{c}}}{z} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{\color{blue}{c}}}{z} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{9 \cdot \left(x \cdot y\right)}{\color{blue}{c}}}{z} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{\left(x \cdot y\right) \cdot 9}{c}}{z} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(x \cdot y\right) \cdot 9}{c}}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{\left(y \cdot x\right) \cdot 9}{c}}{z} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\frac{y \cdot \left(x \cdot 9\right)}{c}}{z} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\left(x \cdot 9\right) \cdot y}{c}}{z} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\frac{x \cdot \left(9 \cdot y\right)}{c}}{z} \]

      10. associate-/l* N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{9 \cdot y}{c}}}{z} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{9 \cdot y}{c}}}{z} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x \cdot \frac{9 \cdot y}{\color{blue}{c}}}{z} \]

      13. lower-*.f64 37.1

          \[\leadsto \frac{x \cdot \frac{9 \cdot y}{c}}{z} \]

   8. Applied rewrites 37.1%

      \[\leadsto \frac{x \cdot \color{blue}{\frac{9 \cdot y}{c}}}{z} \]

   if -1.00000000000000002e64 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.9999999999999999e-232

   1. Initial program 80.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 34.9

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

   4. Applied rewrites 34.9%

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

   if -4.9999999999999999e-232 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.99999999999999988e-11

   1. Initial program 80.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\frac{a \cdot t}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto -4 \cdot \frac{a \cdot t}{\color{blue}{c}} \]

      3. lower-*.f64 38.1

         \[\leadsto -4 \cdot \frac{a \cdot t}{c} \]

   4. Applied rewrites 38.1%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]

   if 1.99999999999999988e-11 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 4.9999999999999997e32

   1. Initial program 80.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 34.9

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

   4. Applied rewrites 34.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l/ N/A

         \[\leadsto \frac{\frac{b}{z}}{\color{blue}{c}} \]

      5. div-flip N/A

         \[\leadsto \frac{\frac{1}{\frac{z}{b}}}{c} \]

      6. associate-/l/ N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{z}{b} \cdot c}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{z}{b} \cdot c}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{z}{b} \cdot \color{blue}{c}} \]

      9. lower-/.f64 32.9

         \[\leadsto \frac{1}{\frac{z}{b} \cdot c} \]

   6. Applied rewrites 32.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{z}{b} \cdot c}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 52.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+66}:\\ \;\;\;\;\frac{y \cdot \frac{9 \cdot x}{z}}{c}\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-232}:\\ \;\;\;\;\frac{b}{c \cdot z}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;-4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+32}:\\ \;\;\;\;\frac{1}{\frac{z}{b} \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{9 \cdot \frac{x \cdot y}{c}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)))
   (if (<= t_1 -1e+66)
     (/ (* y (/ (* 9.0 x) z)) c)
     (if (<= t_1 -5e-232)
       (/ b (* c z))
       (if (<= t_1 2e-11)
         (* -4.0 (/ (* a t) c))
         (if (<= t_1 5e+32)
           (/ 1.0 (* (/ z b) c))
           (/ (* 9.0 (/ (* x y) c)) z)))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if (t_1 <= -1e+66) {
		tmp = (y * ((9.0 * x) / z)) / c;
	} else if (t_1 <= -5e-232) {
		tmp = b / (c * z);
	} else if (t_1 <= 2e-11) {
		tmp = -4.0 * ((a * t) / c);
	} else if (t_1 <= 5e+32) {
		tmp = 1.0 / ((z / b) * c);
	} else {
		tmp = (9.0 * ((x * y) / c)) / z;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c)
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) :: t_1
    real(8) :: tmp
    t_1 = (x * 9.0d0) * y
    if (t_1 <= (-1d+66)) then
        tmp = (y * ((9.0d0 * x) / z)) / c
    else if (t_1 <= (-5d-232)) then
        tmp = b / (c * z)
    else if (t_1 <= 2d-11) then
        tmp = (-4.0d0) * ((a * t) / c)
    else if (t_1 <= 5d+32) then
        tmp = 1.0d0 / ((z / b) * c)
    else
        tmp = (9.0d0 * ((x * y) / c)) / z
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b && b < c;
public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if (t_1 <= -1e+66) {
		tmp = (y * ((9.0 * x) / z)) / c;
	} else if (t_1 <= -5e-232) {
		tmp = b / (c * z);
	} else if (t_1 <= 2e-11) {
		tmp = -4.0 * ((a * t) / c);
	} else if (t_1 <= 5e+32) {
		tmp = 1.0 / ((z / b) * c);
	} else {
		tmp = (9.0 * ((x * y) / c)) / z;
	}
	return tmp;
}

Python

[x, y, z, t, a, b, c] = sort([x, y, z, t, a, b, c])
def code(x, y, z, t, a, b, c):
	t_1 = (x * 9.0) * y
	tmp = 0
	if t_1 <= -1e+66:
		tmp = (y * ((9.0 * x) / z)) / c
	elif t_1 <= -5e-232:
		tmp = b / (c * z)
	elif t_1 <= 2e-11:
		tmp = -4.0 * ((a * t) / c)
	elif t_1 <= 5e+32:
		tmp = 1.0 / ((z / b) * c)
	else:
		tmp = (9.0 * ((x * y) / c)) / z
	return tmp

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_1 <= -1e+66)
		tmp = Float64(Float64(y * Float64(Float64(9.0 * x) / z)) / c);
	elseif (t_1 <= -5e-232)
		tmp = Float64(b / Float64(c * z));
	elseif (t_1 <= 2e-11)
		tmp = Float64(-4.0 * Float64(Float64(a * t) / c));
	elseif (t_1 <= 5e+32)
		tmp = Float64(1.0 / Float64(Float64(z / b) * c));
	else
		tmp = Float64(Float64(9.0 * Float64(Float64(x * y) / c)) / z);
	end
	return tmp
end

MATLAB

x, y, z, t, a, b, c = num2cell(sort([x, y, z, t, a, b, c])){:}
function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (x * 9.0) * y;
	tmp = 0.0;
	if (t_1 <= -1e+66)
		tmp = (y * ((9.0 * x) / z)) / c;
	elseif (t_1 <= -5e-232)
		tmp = b / (c * z);
	elseif (t_1 <= 2e-11)
		tmp = -4.0 * ((a * t) / c);
	elseif (t_1 <= 5e+32)
		tmp = 1.0 / ((z / b) * c);
	else
		tmp = (9.0 * ((x * y) / c)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+66], N[(N[(y * N[(N[(9.0 * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t$95$1, -5e-232], N[(b / N[(c * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-11], N[(-4.0 * N[(N[(a * t), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+32], N[(1.0 / N[(N[(z / b), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], N[(N[(9.0 * N[(N[(x * y), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -9.99999999999999945e65

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b}{z \cdot c} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b}{z}}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      7. mult-flip N/A

         \[\leadsto \color{blue}{\frac{b}{z} \cdot \frac{1}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      8. inv-pow N/A

         \[\leadsto \frac{b}{z} \cdot \color{blue}{{c}^{-1}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{b}{z}}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      11. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      13. lower-/.f64 73.9

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \color{blue}{\frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}}\right) \]

   3. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \frac{\mathsf{fma}\left(y \cdot x, 9, -4 \cdot \left(z \cdot \left(a \cdot t\right)\right)\right)}{c \cdot z}\right)} \]

   4. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right) + \left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{\color{blue}{c}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      7. lower-/.f64 84.1

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

   6. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c}} \]

   7. Taylor expanded in x around inf

      \[\leadsto \frac{9 \cdot \frac{x \cdot y}{z}}{c} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{z}}{c} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{z}}{c} \]

      3. lower-*.f64 34.3

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{z}}{c} \]

   9. Applied rewrites 34.3%

      \[\leadsto \frac{9 \cdot \frac{x \cdot y}{z}}{c} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{9 \cdot \frac{x \cdot y}{z}}{c} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{9 \cdot \frac{x \cdot y}{z}}{c} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{9 \cdot \frac{x \cdot y}{z}}{c} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{9 \cdot \frac{x \cdot y}{z}}{c} \]

       5. associate-*r/ N/A

          \[\leadsto \frac{\frac{9 \cdot \left(x \cdot y\right)}{z}}{c} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\frac{9 \cdot \left(x \cdot y\right)}{z}}{c} \]

       7. *-commutative N/A

          \[\leadsto \frac{\frac{9 \cdot \left(y \cdot x\right)}{z}}{c} \]

       8. associate-*l* N/A

          \[\leadsto \frac{\frac{\left(9 \cdot y\right) \cdot x}{z}}{c} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(9 \cdot y\right) \cdot x}{z}}{c} \]

       10. *-commutative N/A

           \[\leadsto \frac{\frac{x \cdot \left(9 \cdot y\right)}{z}}{c} \]

       11. lift-*.f64 N/A

           \[\leadsto \frac{\frac{x \cdot \left(9 \cdot y\right)}{z}}{c} \]

       12. associate-*r* N/A

           \[\leadsto \frac{\frac{\left(x \cdot 9\right) \cdot y}{z}}{c} \]

       13. *-commutative N/A

           \[\leadsto \frac{\frac{y \cdot \left(x \cdot 9\right)}{z}}{c} \]

       14. associate-/l* N/A

           \[\leadsto \frac{y \cdot \frac{x \cdot 9}{z}}{c} \]

       15. lower-*.f64 N/A

           \[\leadsto \frac{y \cdot \frac{x \cdot 9}{z}}{c} \]

       16. lower-/.f64 N/A

           \[\leadsto \frac{y \cdot \frac{x \cdot 9}{z}}{c} \]

       17. *-commutative N/A

           \[\leadsto \frac{y \cdot \frac{9 \cdot x}{z}}{c} \]

       18. lower-*.f64 35.6

           \[\leadsto \frac{y \cdot \frac{9 \cdot x}{z}}{c} \]

   11. Applied rewrites 35.6%

       \[\leadsto \frac{y \cdot \frac{9 \cdot x}{z}}{c} \]

   if -9.99999999999999945e65 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.9999999999999999e-232

   1. Initial program 80.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 34.9

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

   4. Applied rewrites 34.9%

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

   if -4.9999999999999999e-232 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.99999999999999988e-11

   1. Initial program 80.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\frac{a \cdot t}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto -4 \cdot \frac{a \cdot t}{\color{blue}{c}} \]

      3. lower-*.f64 38.1

         \[\leadsto -4 \cdot \frac{a \cdot t}{c} \]

   4. Applied rewrites 38.1%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]

   if 1.99999999999999988e-11 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 4.9999999999999997e32

   1. Initial program 80.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 34.9

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

   4. Applied rewrites 34.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l/ N/A

         \[\leadsto \frac{\frac{b}{z}}{\color{blue}{c}} \]

      5. div-flip N/A

         \[\leadsto \frac{\frac{1}{\frac{z}{b}}}{c} \]

      6. associate-/l/ N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{z}{b} \cdot c}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{z}{b} \cdot c}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{z}{b} \cdot \color{blue}{c}} \]

      9. lower-/.f64 32.9

         \[\leadsto \frac{1}{\frac{z}{b} \cdot c} \]

   6. Applied rewrites 32.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{z}{b} \cdot c}} \]

   if 4.9999999999999997e32 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{c \cdot z}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

   3. Applied rewrites 80.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b - a \cdot \left(t \cdot \left(4 \cdot z\right)\right)\right)}{c}}{z}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{9 \cdot \frac{x \cdot y}{c}}}{z} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\frac{x \cdot y}{c}}}{z} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{\color{blue}{c}}}{z} \]

      3. lower-*.f64 35.8

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{c}}{z} \]

   6. Applied rewrites 35.8%

      \[\leadsto \frac{\color{blue}{9 \cdot \frac{x \cdot y}{c}}}{z} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 14: 52.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+66}:\\ \;\;\;\;\frac{y \cdot \frac{9 \cdot x}{z}}{c}\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-232}:\\ \;\;\;\;\frac{b}{c \cdot z}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;-4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;t\_1 \leq 10^{+39}:\\ \;\;\;\;\frac{1}{\frac{z}{b} \cdot c}\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \frac{x \cdot y}{c \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)))
   (if (<= t_1 -1e+66)
     (/ (* y (/ (* 9.0 x) z)) c)
     (if (<= t_1 -5e-232)
       (/ b (* c z))
       (if (<= t_1 2e-11)
         (* -4.0 (/ (* a t) c))
         (if (<= t_1 1e+39)
           (/ 1.0 (* (/ z b) c))
           (* 9.0 (/ (* x y) (* c z)))))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if (t_1 <= -1e+66) {
		tmp = (y * ((9.0 * x) / z)) / c;
	} else if (t_1 <= -5e-232) {
		tmp = b / (c * z);
	} else if (t_1 <= 2e-11) {
		tmp = -4.0 * ((a * t) / c);
	} else if (t_1 <= 1e+39) {
		tmp = 1.0 / ((z / b) * c);
	} else {
		tmp = 9.0 * ((x * y) / (c * z));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c)
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) :: t_1
    real(8) :: tmp
    t_1 = (x * 9.0d0) * y
    if (t_1 <= (-1d+66)) then
        tmp = (y * ((9.0d0 * x) / z)) / c
    else if (t_1 <= (-5d-232)) then
        tmp = b / (c * z)
    else if (t_1 <= 2d-11) then
        tmp = (-4.0d0) * ((a * t) / c)
    else if (t_1 <= 1d+39) then
        tmp = 1.0d0 / ((z / b) * c)
    else
        tmp = 9.0d0 * ((x * y) / (c * z))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b && b < c;
public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if (t_1 <= -1e+66) {
		tmp = (y * ((9.0 * x) / z)) / c;
	} else if (t_1 <= -5e-232) {
		tmp = b / (c * z);
	} else if (t_1 <= 2e-11) {
		tmp = -4.0 * ((a * t) / c);
	} else if (t_1 <= 1e+39) {
		tmp = 1.0 / ((z / b) * c);
	} else {
		tmp = 9.0 * ((x * y) / (c * z));
	}
	return tmp;
}

Python

[x, y, z, t, a, b, c] = sort([x, y, z, t, a, b, c])
def code(x, y, z, t, a, b, c):
	t_1 = (x * 9.0) * y
	tmp = 0
	if t_1 <= -1e+66:
		tmp = (y * ((9.0 * x) / z)) / c
	elif t_1 <= -5e-232:
		tmp = b / (c * z)
	elif t_1 <= 2e-11:
		tmp = -4.0 * ((a * t) / c)
	elif t_1 <= 1e+39:
		tmp = 1.0 / ((z / b) * c)
	else:
		tmp = 9.0 * ((x * y) / (c * z))
	return tmp

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_1 <= -1e+66)
		tmp = Float64(Float64(y * Float64(Float64(9.0 * x) / z)) / c);
	elseif (t_1 <= -5e-232)
		tmp = Float64(b / Float64(c * z));
	elseif (t_1 <= 2e-11)
		tmp = Float64(-4.0 * Float64(Float64(a * t) / c));
	elseif (t_1 <= 1e+39)
		tmp = Float64(1.0 / Float64(Float64(z / b) * c));
	else
		tmp = Float64(9.0 * Float64(Float64(x * y) / Float64(c * z)));
	end
	return tmp
end

MATLAB

x, y, z, t, a, b, c = num2cell(sort([x, y, z, t, a, b, c])){:}
function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (x * 9.0) * y;
	tmp = 0.0;
	if (t_1 <= -1e+66)
		tmp = (y * ((9.0 * x) / z)) / c;
	elseif (t_1 <= -5e-232)
		tmp = b / (c * z);
	elseif (t_1 <= 2e-11)
		tmp = -4.0 * ((a * t) / c);
	elseif (t_1 <= 1e+39)
		tmp = 1.0 / ((z / b) * c);
	else
		tmp = 9.0 * ((x * y) / (c * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+66], N[(N[(y * N[(N[(9.0 * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t$95$1, -5e-232], N[(b / N[(c * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-11], N[(-4.0 * N[(N[(a * t), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+39], N[(1.0 / N[(N[(z / b), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], N[(9.0 * N[(N[(x * y), $MachinePrecision] / N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -9.99999999999999945e65

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z \cdot c} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b}{z \cdot c} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b}{z}}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      7. mult-flip N/A

         \[\leadsto \color{blue}{\frac{b}{z} \cdot \frac{1}{c}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      8. inv-pow N/A

         \[\leadsto \frac{b}{z} \cdot \color{blue}{{c}^{-1}} + \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{b}{z}}, {c}^{-1}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      11. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \color{blue}{\frac{1}{c}}, \frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}\right) \]

      13. lower-/.f64 73.9

          \[\leadsto \mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \color{blue}{\frac{\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z \cdot c}}\right) \]

   3. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{z}, \frac{1}{c}, \frac{\mathsf{fma}\left(y \cdot x, 9, -4 \cdot \left(z \cdot \left(a \cdot t\right)\right)\right)}{c \cdot z}\right)} \]

   4. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{-4 \cdot \left(a \cdot t\right) + \left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{\color{blue}{c}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, 9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)}{c} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

      7. lower-/.f64 84.1

         \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c} \]

   6. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, a \cdot t, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c}} \]

   7. Taylor expanded in x around inf

      \[\leadsto \frac{9 \cdot \frac{x \cdot y}{z}}{c} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{z}}{c} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{z}}{c} \]

      3. lower-*.f64 34.3

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{z}}{c} \]

   9. Applied rewrites 34.3%

      \[\leadsto \frac{9 \cdot \frac{x \cdot y}{z}}{c} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{9 \cdot \frac{x \cdot y}{z}}{c} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{9 \cdot \frac{x \cdot y}{z}}{c} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{9 \cdot \frac{x \cdot y}{z}}{c} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{9 \cdot \frac{x \cdot y}{z}}{c} \]

       5. associate-*r/ N/A

          \[\leadsto \frac{\frac{9 \cdot \left(x \cdot y\right)}{z}}{c} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\frac{9 \cdot \left(x \cdot y\right)}{z}}{c} \]

       7. *-commutative N/A

          \[\leadsto \frac{\frac{9 \cdot \left(y \cdot x\right)}{z}}{c} \]

       8. associate-*l* N/A

          \[\leadsto \frac{\frac{\left(9 \cdot y\right) \cdot x}{z}}{c} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(9 \cdot y\right) \cdot x}{z}}{c} \]

       10. *-commutative N/A

           \[\leadsto \frac{\frac{x \cdot \left(9 \cdot y\right)}{z}}{c} \]

       11. lift-*.f64 N/A

           \[\leadsto \frac{\frac{x \cdot \left(9 \cdot y\right)}{z}}{c} \]

       12. associate-*r* N/A

           \[\leadsto \frac{\frac{\left(x \cdot 9\right) \cdot y}{z}}{c} \]

       13. *-commutative N/A

           \[\leadsto \frac{\frac{y \cdot \left(x \cdot 9\right)}{z}}{c} \]

       14. associate-/l* N/A

           \[\leadsto \frac{y \cdot \frac{x \cdot 9}{z}}{c} \]

       15. lower-*.f64 N/A

           \[\leadsto \frac{y \cdot \frac{x \cdot 9}{z}}{c} \]

       16. lower-/.f64 N/A

           \[\leadsto \frac{y \cdot \frac{x \cdot 9}{z}}{c} \]

       17. *-commutative N/A

           \[\leadsto \frac{y \cdot \frac{9 \cdot x}{z}}{c} \]

       18. lower-*.f64 35.6

           \[\leadsto \frac{y \cdot \frac{9 \cdot x}{z}}{c} \]

   11. Applied rewrites 35.6%

       \[\leadsto \frac{y \cdot \frac{9 \cdot x}{z}}{c} \]

   if -9.99999999999999945e65 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.9999999999999999e-232

   1. Initial program 80.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 34.9

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

   4. Applied rewrites 34.9%

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

   if -4.9999999999999999e-232 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.99999999999999988e-11

   1. Initial program 80.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\frac{a \cdot t}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto -4 \cdot \frac{a \cdot t}{\color{blue}{c}} \]

      3. lower-*.f64 38.1

         \[\leadsto -4 \cdot \frac{a \cdot t}{c} \]

   4. Applied rewrites 38.1%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]

   if 1.99999999999999988e-11 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.9999999999999994e38

   1. Initial program 80.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 34.9

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

   4. Applied rewrites 34.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l/ N/A

         \[\leadsto \frac{\frac{b}{z}}{\color{blue}{c}} \]

      5. div-flip N/A

         \[\leadsto \frac{\frac{1}{\frac{z}{b}}}{c} \]

      6. associate-/l/ N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{z}{b} \cdot c}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{z}{b} \cdot c}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{z}{b} \cdot \color{blue}{c}} \]

      9. lower-/.f64 32.9

         \[\leadsto \frac{1}{\frac{z}{b} \cdot c} \]

   6. Applied rewrites 32.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{z}{b} \cdot c}} \]

   if 9.9999999999999994e38 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 80.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 9 \cdot \color{blue}{\frac{x \cdot y}{c \cdot z}} \]

      2. lower-/.f64 N/A

         \[\leadsto 9 \cdot \frac{x \cdot y}{\color{blue}{c \cdot z}} \]

      3. lower-*.f64 N/A

         \[\leadsto 9 \cdot \frac{x \cdot y}{\color{blue}{c} \cdot z} \]

      4. lower-*.f64 36.4

         \[\leadsto 9 \cdot \frac{x \cdot y}{c \cdot \color{blue}{z}} \]

   4. Applied rewrites 36.4%

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 15: 52.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ t_2 := 9 \cdot \frac{x \cdot y}{c \cdot z}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+64}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-232}:\\ \;\;\;\;\frac{b}{c \cdot z}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;-4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;t\_1 \leq 10^{+39}:\\ \;\;\;\;\frac{1}{\frac{z}{b} \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)) (t_2 (* 9.0 (/ (* x y) (* c z)))))
   (if (<= t_1 -1e+64)
     t_2
     (if (<= t_1 -5e-232)
       (/ b (* c z))
       (if (<= t_1 2e-11)
         (* -4.0 (/ (* a t) c))
         (if (<= t_1 1e+39) (/ 1.0 (* (/ z b) c)) t_2))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (x * 9.0) * y;
	double t_2 = 9.0 * ((x * y) / (c * z));
	double tmp;
	if (t_1 <= -1e+64) {
		tmp = t_2;
	} else if (t_1 <= -5e-232) {
		tmp = b / (c * z);
	} else if (t_1 <= 2e-11) {
		tmp = -4.0 * ((a * t) / c);
	} else if (t_1 <= 1e+39) {
		tmp = 1.0 / ((z / b) * c);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c)
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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x * 9.0d0) * y
    t_2 = 9.0d0 * ((x * y) / (c * z))
    if (t_1 <= (-1d+64)) then
        tmp = t_2
    else if (t_1 <= (-5d-232)) then
        tmp = b / (c * z)
    else if (t_1 <= 2d-11) then
        tmp = (-4.0d0) * ((a * t) / c)
    else if (t_1 <= 1d+39) then
        tmp = 1.0d0 / ((z / b) * c)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b && b < c;
public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (x * 9.0) * y;
	double t_2 = 9.0 * ((x * y) / (c * z));
	double tmp;
	if (t_1 <= -1e+64) {
		tmp = t_2;
	} else if (t_1 <= -5e-232) {
		tmp = b / (c * z);
	} else if (t_1 <= 2e-11) {
		tmp = -4.0 * ((a * t) / c);
	} else if (t_1 <= 1e+39) {
		tmp = 1.0 / ((z / b) * c);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y, z, t, a, b, c] = sort([x, y, z, t, a, b, c])
def code(x, y, z, t, a, b, c):
	t_1 = (x * 9.0) * y
	t_2 = 9.0 * ((x * y) / (c * z))
	tmp = 0
	if t_1 <= -1e+64:
		tmp = t_2
	elif t_1 <= -5e-232:
		tmp = b / (c * z)
	elif t_1 <= 2e-11:
		tmp = -4.0 * ((a * t) / c)
	elif t_1 <= 1e+39:
		tmp = 1.0 / ((z / b) * c)
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(x * 9.0) * y)
	t_2 = Float64(9.0 * Float64(Float64(x * y) / Float64(c * z)))
	tmp = 0.0
	if (t_1 <= -1e+64)
		tmp = t_2;
	elseif (t_1 <= -5e-232)
		tmp = Float64(b / Float64(c * z));
	elseif (t_1 <= 2e-11)
		tmp = Float64(-4.0 * Float64(Float64(a * t) / c));
	elseif (t_1 <= 1e+39)
		tmp = Float64(1.0 / Float64(Float64(z / b) * c));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b, c = num2cell(sort([x, y, z, t, a, b, c])){:}
function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (x * 9.0) * y;
	t_2 = 9.0 * ((x * y) / (c * z));
	tmp = 0.0;
	if (t_1 <= -1e+64)
		tmp = t_2;
	elseif (t_1 <= -5e-232)
		tmp = b / (c * z);
	elseif (t_1 <= 2e-11)
		tmp = -4.0 * ((a * t) / c);
	elseif (t_1 <= 1e+39)
		tmp = 1.0 / ((z / b) * c);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(9.0 * N[(N[(x * y), $MachinePrecision] / N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+64], t$95$2, If[LessEqual[t$95$1, -5e-232], N[(b / N[(c * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-11], N[(-4.0 * N[(N[(a * t), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+39], N[(1.0 / N[(N[(z / b), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -1.00000000000000002e64 or 9.9999999999999994e38 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 80.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 9 \cdot \color{blue}{\frac{x \cdot y}{c \cdot z}} \]

      2. lower-/.f64 N/A

         \[\leadsto 9 \cdot \frac{x \cdot y}{\color{blue}{c \cdot z}} \]

      3. lower-*.f64 N/A

         \[\leadsto 9 \cdot \frac{x \cdot y}{\color{blue}{c} \cdot z} \]

      4. lower-*.f64 36.4

         \[\leadsto 9 \cdot \frac{x \cdot y}{c \cdot \color{blue}{z}} \]

   4. Applied rewrites 36.4%

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]

   if -1.00000000000000002e64 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.9999999999999999e-232

   1. Initial program 80.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 34.9

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

   4. Applied rewrites 34.9%

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

   if -4.9999999999999999e-232 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.99999999999999988e-11

   1. Initial program 80.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\frac{a \cdot t}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto -4 \cdot \frac{a \cdot t}{\color{blue}{c}} \]

      3. lower-*.f64 38.1

         \[\leadsto -4 \cdot \frac{a \cdot t}{c} \]

   4. Applied rewrites 38.1%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]

   if 1.99999999999999988e-11 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.9999999999999994e38

   1. Initial program 80.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 34.9

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

   4. Applied rewrites 34.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l/ N/A

         \[\leadsto \frac{\frac{b}{z}}{\color{blue}{c}} \]

      5. div-flip N/A

         \[\leadsto \frac{\frac{1}{\frac{z}{b}}}{c} \]

      6. associate-/l/ N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{z}{b} \cdot c}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{z}{b} \cdot c}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{z}{b} \cdot \color{blue}{c}} \]

      9. lower-/.f64 32.9

         \[\leadsto \frac{1}{\frac{z}{b} \cdot c} \]

   6. Applied rewrites 32.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{z}{b} \cdot c}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 16: 48.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := -4 \cdot \frac{a \cdot t}{c}\\ \mathbf{if}\;t \leq -1.04 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-96}:\\ \;\;\;\;\frac{b}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* -4.0 (/ (* a t) c))))
   (if (<= t -1.04e+27) t_1 (if (<= t 7.2e-96) (/ b (* c z)) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = -4.0 * ((a * t) / c);
	double tmp;
	if (t <= -1.04e+27) {
		tmp = t_1;
	} else if (t <= 7.2e-96) {
		tmp = b / (c * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c)
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) :: t_1
    real(8) :: tmp
    t_1 = (-4.0d0) * ((a * t) / c)
    if (t <= (-1.04d+27)) then
        tmp = t_1
    else if (t <= 7.2d-96) then
        tmp = b / (c * z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b && b < c;
public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = -4.0 * ((a * t) / c);
	double tmp;
	if (t <= -1.04e+27) {
		tmp = t_1;
	} else if (t <= 7.2e-96) {
		tmp = b / (c * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a, b, c] = sort([x, y, z, t, a, b, c])
def code(x, y, z, t, a, b, c):
	t_1 = -4.0 * ((a * t) / c)
	tmp = 0
	if t <= -1.04e+27:
		tmp = t_1
	elif t <= 7.2e-96:
		tmp = b / (c * z)
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(-4.0 * Float64(Float64(a * t) / c))
	tmp = 0.0
	if (t <= -1.04e+27)
		tmp = t_1;
	elseif (t <= 7.2e-96)
		tmp = Float64(b / Float64(c * z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b, c = num2cell(sort([x, y, z, t, a, b, c])){:}
function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = -4.0 * ((a * t) / c);
	tmp = 0.0;
	if (t <= -1.04e+27)
		tmp = t_1;
	elseif (t <= 7.2e-96)
		tmp = b / (c * z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(-4.0 * N[(N[(a * t), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.04e+27], t$95$1, If[LessEqual[t, 7.2e-96], N[(b / N[(c * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.04000000000000001e27 or 7.20000000000000016e-96 < t

   1. Initial program 80.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\frac{a \cdot t}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto -4 \cdot \frac{a \cdot t}{\color{blue}{c}} \]

      3. lower-*.f64 38.1

         \[\leadsto -4 \cdot \frac{a \cdot t}{c} \]

   4. Applied rewrites 38.1%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]

   if -1.04000000000000001e27 < t < 7.20000000000000016e-96

   1. Initial program 80.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 34.9

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

   4. Applied rewrites 34.9%

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 34.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \frac{b}{c \cdot z} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c) :precision binary64 (/ b (* c z)))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	return b / (c * z);
}

Fortran

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c)
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
    code = b / (c * z)
end function

Java

assert x < y && y < z && z < t && t < a && a < b && b < c;
public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return b / (c * z);
}

Python

[x, y, z, t, a, b, c] = sort([x, y, z, t, a, b, c])
def code(x, y, z, t, a, b, c):
	return b / (c * z)

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	return Float64(b / Float64(c * z))
end

MATLAB

x, y, z, t, a, b, c = num2cell(sort([x, y, z, t, a, b, c])){:}
function tmp = code(x, y, z, t, a, b, c)
	tmp = b / (c * z);
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := N[(b / N[(c * z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.4%

   \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 34.9

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

4. Applied rewrites 34.9%

   \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025150 
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, J"
  :precision binary64
  (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))