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

Percentage Accurate: 79.4% → 85.2%

Time: 6.1s

Alternatives: 8

Speedup: N/A×

• 37.4% 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: 79.4% 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: 85.2% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+55} \lor \neg \left(z \leq 1.15 \cdot 10^{+35}\right):\\ \;\;\;\;\left(-1 \cdot y\right) \cdot \mathsf{fma}\left(\frac{\frac{b}{c \cdot z}}{y} - \frac{\frac{a \cdot t}{c} \cdot 4}{y}, -1, \frac{x}{c \cdot z} \cdot -9\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \end{array} \end{array} \]

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 (or (<= z -1.3e+55) (not (<= z 1.15e+35)))
   (*
    (* -1.0 y)
    (fma
     (- (/ (/ b (* c z)) y) (/ (* (/ (* a t) c) 4.0) y))
     -1.0
     (* (/ x (* c z)) -9.0)))
   (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) 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 <= -1.3e+55) || !(z <= 1.15e+35)) {
		tmp = (-1.0 * y) * fma((((b / (c * z)) / y) - ((((a * t) / c) * 4.0) / y)), -1.0, ((x / (c * z)) * -9.0));
	} else {
		tmp = ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + 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 <= -1.3e+55) || !(z <= 1.15e+35))
		tmp = Float64(Float64(-1.0 * y) * fma(Float64(Float64(Float64(b / Float64(c * z)) / y) - Float64(Float64(Float64(Float64(a * t) / c) * 4.0) / y)), -1.0, Float64(Float64(x / Float64(c * z)) * -9.0)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x * 9.0) * y) - Float64(Float64(Float64(z * 4.0) * t) * a)) + b) / Float64(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[Or[LessEqual[z, -1.3e+55], N[Not[LessEqual[z, 1.15e+35]], $MachinePrecision]], N[(N[(-1.0 * y), $MachinePrecision] * N[(N[(N[(N[(b / N[(c * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] - N[(N[(N[(N[(a * t), $MachinePrecision] / c), $MachinePrecision] * 4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(N[(x / N[(c * z), $MachinePrecision]), $MachinePrecision] * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3e55 or 1.1499999999999999e35 < z

   1. Initial program 53.0%

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

   3. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 78.7%

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

   if -1.3e55 < z < 1.1499999999999999e35

   1. Initial program 94.5%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+55} \lor \neg \left(z \leq 1.15 \cdot 10^{+35}\right):\\ \;\;\;\;\left(-1 \cdot y\right) \cdot \mathsf{fma}\left(\frac{\frac{b}{c \cdot z}}{y} - \frac{\frac{a \cdot t}{c} \cdot 4}{y}, -1, \frac{x}{c \cdot z} \cdot -9\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.7% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+50} \lor \neg \left(z \leq 0.00046\right):\\ \;\;\;\;\left(-1 \cdot y\right) \cdot \mathsf{fma}\left(\frac{\frac{b}{c \cdot z}}{y} - \frac{\frac{a \cdot t}{c} \cdot 4}{y}, -1, \frac{x}{c \cdot z} \cdot -9\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x, \frac{b - \left(\left(t \cdot z\right) \cdot a\right) \cdot 4}{y}\right) \cdot y}{z \cdot c}\\ \end{array} \end{array} \]

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 (or (<= z -1.25e+50) (not (<= z 0.00046)))
   (*
    (* -1.0 y)
    (fma
     (- (/ (/ b (* c z)) y) (/ (* (/ (* a t) c) 4.0) y))
     -1.0
     (* (/ x (* c z)) -9.0)))
   (/ (* (fma 9.0 x (/ (- b (* (* (* t z) a) 4.0)) y)) y) (* 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 <= -1.25e+50) || !(z <= 0.00046)) {
		tmp = (-1.0 * y) * fma((((b / (c * z)) / y) - ((((a * t) / c) * 4.0) / y)), -1.0, ((x / (c * z)) * -9.0));
	} else {
		tmp = (fma(9.0, x, ((b - (((t * z) * a) * 4.0)) / y)) * y) / (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 <= -1.25e+50) || !(z <= 0.00046))
		tmp = Float64(Float64(-1.0 * y) * fma(Float64(Float64(Float64(b / Float64(c * z)) / y) - Float64(Float64(Float64(Float64(a * t) / c) * 4.0) / y)), -1.0, Float64(Float64(x / Float64(c * z)) * -9.0)));
	else
		tmp = Float64(Float64(fma(9.0, x, Float64(Float64(b - Float64(Float64(Float64(t * z) * a) * 4.0)) / y)) * y) / Float64(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[Or[LessEqual[z, -1.25e+50], N[Not[LessEqual[z, 0.00046]], $MachinePrecision]], N[(N[(-1.0 * y), $MachinePrecision] * N[(N[(N[(N[(b / N[(c * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] - N[(N[(N[(N[(a * t), $MachinePrecision] / c), $MachinePrecision] * 4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(N[(x / N[(c * z), $MachinePrecision]), $MachinePrecision] * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(9.0 * x + N[(N[(b - N[(N[(N[(t * z), $MachinePrecision] * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.25e50 or 4.6000000000000001e-4 < z

   1. Initial program 55.8%

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

   3. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 79.9%

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

   if -1.25e50 < z < 4.6000000000000001e-4

   1. Initial program 94.3%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 92.2%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+50} \lor \neg \left(z \leq 0.00046\right):\\ \;\;\;\;\left(-1 \cdot y\right) \cdot \mathsf{fma}\left(\frac{\frac{b}{c \cdot z}}{y} - \frac{\frac{a \cdot t}{c} \cdot 4}{y}, -1, \frac{x}{c \cdot z} \cdot -9\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x, \frac{b - \left(\left(t \cdot z\right) \cdot a\right) \cdot 4}{y}\right) \cdot y}{z \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.3% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-242} \lor \neg \left(z \leq 1.2 \cdot 10^{+33}\right):\\ \;\;\;\;\left(-1 \cdot y\right) \cdot \mathsf{fma}\left(\frac{\frac{b}{c \cdot z}}{y} - \frac{\frac{a \cdot t}{c} \cdot 4}{y}, -1, \frac{x}{c \cdot z} \cdot -9\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{c}, -4, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{\left(c \cdot z\right) \cdot a}\right) \cdot a\\ \end{array} \end{array} \]

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 (or (<= z -9e-242) (not (<= z 1.2e+33)))
   (*
    (* -1.0 y)
    (fma
     (- (/ (/ b (* c z)) y) (/ (* (/ (* a t) c) 4.0) y))
     -1.0
     (* (/ x (* c z)) -9.0)))
   (* (fma (/ t c) -4.0 (/ (fma (* y x) 9.0 b) (* (* c z) a))) a)))

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 <= -9e-242) || !(z <= 1.2e+33)) {
		tmp = (-1.0 * y) * fma((((b / (c * z)) / y) - ((((a * t) / c) * 4.0) / y)), -1.0, ((x / (c * z)) * -9.0));
	} else {
		tmp = fma((t / c), -4.0, (fma((y * x), 9.0, b) / ((c * z) * a))) * a;
	}
	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 <= -9e-242) || !(z <= 1.2e+33))
		tmp = Float64(Float64(-1.0 * y) * fma(Float64(Float64(Float64(b / Float64(c * z)) / y) - Float64(Float64(Float64(Float64(a * t) / c) * 4.0) / y)), -1.0, Float64(Float64(x / Float64(c * z)) * -9.0)));
	else
		tmp = Float64(fma(Float64(t / c), -4.0, Float64(fma(Float64(y * x), 9.0, b) / Float64(Float64(c * z) * a))) * a);
	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[Or[LessEqual[z, -9e-242], N[Not[LessEqual[z, 1.2e+33]], $MachinePrecision]], N[(N[(-1.0 * y), $MachinePrecision] * N[(N[(N[(N[(b / N[(c * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] - N[(N[(N[(N[(a * t), $MachinePrecision] / c), $MachinePrecision] * 4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(N[(x / N[(c * z), $MachinePrecision]), $MachinePrecision] * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t / c), $MachinePrecision] * -4.0 + N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / N[(N[(c * z), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.9999999999999997e-242 or 1.2e33 < z

   1. Initial program 68.0%

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

   3. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 81.5%

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

   if -8.9999999999999997e-242 < z < 1.2e33

   1. Initial program 94.5%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 77.9%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-242} \lor \neg \left(z \leq 1.2 \cdot 10^{+33}\right):\\ \;\;\;\;\left(-1 \cdot y\right) \cdot \mathsf{fma}\left(\frac{\frac{b}{c \cdot z}}{y} - \frac{\frac{a \cdot t}{c} \cdot 4}{y}, -1, \frac{x}{c \cdot z} \cdot -9\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{c}, -4, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{\left(c \cdot z\right) \cdot a}\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.2% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{a \cdot t}{c}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-34}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{y}{c \cdot z}, 9, \frac{b}{\left(z \cdot x\right) \cdot c}\right) - \frac{t\_1}{x} \cdot 4\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot y\right) \cdot \mathsf{fma}\left(\frac{\frac{b}{c \cdot z}}{y} - \frac{t\_1 \cdot 4}{y}, -1, \frac{x}{c \cdot z} \cdot -9\right)\\ \end{array} \end{array} \]

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
 (let* ((t_1 (/ (* a t) c)))
   (if (<= x -1e-34)
     (* (- (fma (/ y (* c z)) 9.0 (/ b (* (* z x) c))) (* (/ t_1 x) 4.0)) x)
     (*
      (* -1.0 y)
      (fma
       (- (/ (/ b (* c z)) y) (/ (* t_1 4.0) y))
       -1.0
       (* (/ x (* c z)) -9.0))))))

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 = (a * t) / c;
	double tmp;
	if (x <= -1e-34) {
		tmp = (fma((y / (c * z)), 9.0, (b / ((z * x) * c))) - ((t_1 / x) * 4.0)) * x;
	} else {
		tmp = (-1.0 * y) * fma((((b / (c * z)) / y) - ((t_1 * 4.0) / y)), -1.0, ((x / (c * z)) * -9.0));
	}
	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(a * t) / c)
	tmp = 0.0
	if (x <= -1e-34)
		tmp = Float64(Float64(fma(Float64(y / Float64(c * z)), 9.0, Float64(b / Float64(Float64(z * x) * c))) - Float64(Float64(t_1 / x) * 4.0)) * x);
	else
		tmp = Float64(Float64(-1.0 * y) * fma(Float64(Float64(Float64(b / Float64(c * z)) / y) - Float64(Float64(t_1 * 4.0) / y)), -1.0, Float64(Float64(x / Float64(c * z)) * -9.0)));
	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[(a * t), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[x, -1e-34], N[(N[(N[(N[(y / N[(c * z), $MachinePrecision]), $MachinePrecision] * 9.0 + N[(b / N[(N[(z * x), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$1 / x), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(-1.0 * y), $MachinePrecision] * N[(N[(N[(N[(b / N[(c * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] - N[(N[(t$95$1 * 4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(N[(x / N[(c * z), $MachinePrecision]), $MachinePrecision] * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.99999999999999928e-35

   1. Initial program 66.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 81.1%

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

   if -9.99999999999999928e-35 < x

   1. Initial program 80.7%

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

   3. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 76.7%

      \[\leadsto \color{blue}{\left(-1 \cdot y\right) \cdot \mathsf{fma}\left(\frac{\frac{b}{c \cdot z}}{y} - \frac{\frac{a \cdot t}{c} \cdot 4}{y}, -1, \frac{x}{c \cdot z} \cdot -9\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 72.4% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{a \cdot t}{c}\\ \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq 5 \cdot 10^{+273}:\\ \;\;\;\;\left(-1 \cdot y\right) \cdot \mathsf{fma}\left(\frac{\frac{b}{c \cdot z}}{y} - \frac{t\_1 \cdot 4}{y}, -1, \frac{x}{c \cdot z} \cdot -9\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot y\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{x}{c}, 9, \frac{\frac{b}{c}}{y}\right)}{z}, -1, \frac{t\_1}{y} \cdot 4\right)\\ \end{array} \end{array} \]

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
 (let* ((t_1 (/ (* a t) c)))
   (if (<= (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 5e+273)
     (*
      (* -1.0 y)
      (fma
       (- (/ (/ b (* c z)) y) (/ (* t_1 4.0) y))
       -1.0
       (* (/ x (* c z)) -9.0)))
     (*
      (* -1.0 y)
      (fma (/ (fma (/ x c) 9.0 (/ (/ b c) y)) z) -1.0 (* (/ t_1 y) 4.0))))))

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 = (a * t) / c;
	double tmp;
	if ((((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c)) <= 5e+273) {
		tmp = (-1.0 * y) * fma((((b / (c * z)) / y) - ((t_1 * 4.0) / y)), -1.0, ((x / (c * z)) * -9.0));
	} else {
		tmp = (-1.0 * y) * fma((fma((x / c), 9.0, ((b / c) / y)) / z), -1.0, ((t_1 / y) * 4.0));
	}
	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(a * t) / c)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(x * 9.0) * y) - Float64(Float64(Float64(z * 4.0) * t) * a)) + b) / Float64(z * c)) <= 5e+273)
		tmp = Float64(Float64(-1.0 * y) * fma(Float64(Float64(Float64(b / Float64(c * z)) / y) - Float64(Float64(t_1 * 4.0) / y)), -1.0, Float64(Float64(x / Float64(c * z)) * -9.0)));
	else
		tmp = Float64(Float64(-1.0 * y) * fma(Float64(fma(Float64(x / c), 9.0, Float64(Float64(b / c) / y)) / z), -1.0, Float64(Float64(t_1 / y) * 4.0)));
	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[(a * t), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[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], 5e+273], N[(N[(-1.0 * y), $MachinePrecision] * N[(N[(N[(N[(b / N[(c * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] - N[(N[(t$95$1 * 4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(N[(x / N[(c * z), $MachinePrecision]), $MachinePrecision] * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * y), $MachinePrecision] * N[(N[(N[(N[(x / c), $MachinePrecision] * 9.0 + N[(N[(b / c), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * -1.0 + N[(N[(t$95$1 / y), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < 4.99999999999999961e273

   1. Initial program 90.3%

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

   3. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 77.4%

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

   if 4.99999999999999961e273 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c))

   1. Initial program 53.3%

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

   3. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 70.9%

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

   6. Taylor expanded in z around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 78.0%

      \[\leadsto \left(-1 \cdot y\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{x}{c}, 9, \frac{\frac{b}{c}}{y}\right)}{z}, \color{blue}{-1}, \frac{\frac{a \cdot t}{c}}{y} \cdot 4\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 68.8% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{-118}:\\ \;\;\;\;\left(-1 \cdot y\right) \cdot \left(\mathsf{fma}\left(\frac{x}{\left(t \cdot z\right) \cdot c}, -9, \mathsf{fma}\left(\frac{\frac{a}{c}}{y}, 4, \frac{b}{\left(\left(t \cdot y\right) \cdot z\right) \cdot c} \cdot -1\right)\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot y\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{x}{c}, 9, \frac{\frac{b}{c}}{y}\right)}{z}, -1, \frac{\frac{a \cdot t}{c}}{y} \cdot 4\right)\\ \end{array} \end{array} \]

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 (<= a -1.45e-118)
   (*
    (* -1.0 y)
    (*
     (fma
      (/ x (* (* t z) c))
      -9.0
      (fma (/ (/ a c) y) 4.0 (* (/ b (* (* (* t y) z) c)) -1.0)))
     t))
   (*
    (* -1.0 y)
    (fma
     (/ (fma (/ x c) 9.0 (/ (/ b c) y)) z)
     -1.0
     (* (/ (/ (* a t) c) y) 4.0)))))

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 (a <= -1.45e-118) {
		tmp = (-1.0 * y) * (fma((x / ((t * z) * c)), -9.0, fma(((a / c) / y), 4.0, ((b / (((t * y) * z) * c)) * -1.0))) * t);
	} else {
		tmp = (-1.0 * y) * fma((fma((x / c), 9.0, ((b / c) / y)) / z), -1.0, ((((a * t) / c) / y) * 4.0));
	}
	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 (a <= -1.45e-118)
		tmp = Float64(Float64(-1.0 * y) * Float64(fma(Float64(x / Float64(Float64(t * z) * c)), -9.0, fma(Float64(Float64(a / c) / y), 4.0, Float64(Float64(b / Float64(Float64(Float64(t * y) * z) * c)) * -1.0))) * t));
	else
		tmp = Float64(Float64(-1.0 * y) * fma(Float64(fma(Float64(x / c), 9.0, Float64(Float64(b / c) / y)) / z), -1.0, Float64(Float64(Float64(Float64(a * t) / c) / y) * 4.0)));
	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[a, -1.45e-118], N[(N[(-1.0 * y), $MachinePrecision] * N[(N[(N[(x / N[(N[(t * z), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] * -9.0 + N[(N[(N[(a / c), $MachinePrecision] / y), $MachinePrecision] * 4.0 + N[(N[(b / N[(N[(N[(t * y), $MachinePrecision] * z), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * y), $MachinePrecision] * N[(N[(N[(N[(x / c), $MachinePrecision] * 9.0 + N[(N[(b / c), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * -1.0 + N[(N[(N[(N[(a * t), $MachinePrecision] / c), $MachinePrecision] / y), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.4499999999999999e-118

   1. Initial program 72.4%

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

   3. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 67.2%

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

   if -1.4499999999999999e-118 < a

   1. Initial program 78.8%

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

   3. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 76.5%

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

   6. Taylor expanded in z around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 74.1%

      \[\leadsto \left(-1 \cdot y\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{x}{c}, 9, \frac{\frac{b}{c}}{y}\right)}{z}, \color{blue}{-1}, \frac{\frac{a \cdot t}{c}}{y} \cdot 4\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 61.9% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-104} \lor \neg \left(t\_1 \leq 5 \cdot 10^{-135}\right):\\ \;\;\;\;\left(-1 \cdot y\right) \cdot \left(\mathsf{fma}\left(\frac{x}{\left(t \cdot z\right) \cdot c}, -9, \mathsf{fma}\left(\frac{\frac{a}{c}}{y}, 4, \frac{b}{\left(\left(t \cdot y\right) \cdot z\right) \cdot c} \cdot -1\right)\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot t\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{9 \cdot x}{c} \cdot \frac{y}{z}, t, t \cdot \frac{b}{c \cdot z}\right)}{t \cdot t}, -1, \frac{a}{c} \cdot 4\right)\\ \end{array} \end{array} \]

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
 (let* ((t_1 (* (* x 9.0) y)))
   (if (or (<= t_1 -1e-104) (not (<= t_1 5e-135)))
     (*
      (* -1.0 y)
      (*
       (fma
        (/ x (* (* t z) c))
        -9.0
        (fma (/ (/ a c) y) 4.0 (* (/ b (* (* (* t y) z) c)) -1.0)))
       t))
     (*
      (* -1.0 t)
      (fma
       (/ (fma (* (/ (* 9.0 x) c) (/ y z)) t (* t (/ b (* c z)))) (* t t))
       -1.0
       (* (/ a c) 4.0))))))

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-104) || !(t_1 <= 5e-135)) {
		tmp = (-1.0 * y) * (fma((x / ((t * z) * c)), -9.0, fma(((a / c) / y), 4.0, ((b / (((t * y) * z) * c)) * -1.0))) * t);
	} else {
		tmp = (-1.0 * t) * fma((fma((((9.0 * x) / c) * (y / z)), t, (t * (b / (c * z)))) / (t * t)), -1.0, ((a / c) * 4.0));
	}
	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-104) || !(t_1 <= 5e-135))
		tmp = Float64(Float64(-1.0 * y) * Float64(fma(Float64(x / Float64(Float64(t * z) * c)), -9.0, fma(Float64(Float64(a / c) / y), 4.0, Float64(Float64(b / Float64(Float64(Float64(t * y) * z) * c)) * -1.0))) * t));
	else
		tmp = Float64(Float64(-1.0 * t) * fma(Float64(fma(Float64(Float64(Float64(9.0 * x) / c) * Float64(y / z)), t, Float64(t * Float64(b / Float64(c * z)))) / Float64(t * t)), -1.0, Float64(Float64(a / c) * 4.0)));
	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[Or[LessEqual[t$95$1, -1e-104], N[Not[LessEqual[t$95$1, 5e-135]], $MachinePrecision]], N[(N[(-1.0 * y), $MachinePrecision] * N[(N[(N[(x / N[(N[(t * z), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] * -9.0 + N[(N[(N[(a / c), $MachinePrecision] / y), $MachinePrecision] * 4.0 + N[(N[(b / N[(N[(N[(t * y), $MachinePrecision] * z), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * t), $MachinePrecision] * N[(N[(N[(N[(N[(N[(9.0 * x), $MachinePrecision] / c), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision] * t + N[(t * N[(b / N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(N[(a / c), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -9.99999999999999927e-105 or 5.0000000000000002e-135 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 78.0%

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

   3. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 77.1%

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

   6. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 60.7%

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

   if -9.99999999999999927e-105 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 5.0000000000000002e-135

   1. Initial program 73.4%

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

   3. Taylor expanded in t around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\left(-1 \cdot t\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{9 \cdot x}{c} \cdot \frac{y}{z}, t, t \cdot \frac{b}{c \cdot z}\right)}{t \cdot t}, -1, \frac{a}{c} \cdot 4\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y \leq -1 \cdot 10^{-104} \lor \neg \left(\left(x \cdot 9\right) \cdot y \leq 5 \cdot 10^{-135}\right):\\ \;\;\;\;\left(-1 \cdot y\right) \cdot \left(\mathsf{fma}\left(\frac{x}{\left(t \cdot z\right) \cdot c}, -9, \mathsf{fma}\left(\frac{\frac{a}{c}}{y}, 4, \frac{b}{\left(\left(t \cdot y\right) \cdot z\right) \cdot c} \cdot -1\right)\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot t\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{9 \cdot x}{c} \cdot \frac{y}{z}, t, t \cdot \frac{b}{c \cdot z}\right)}{t \cdot t}, -1, \frac{a}{c} \cdot 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.7% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \left(-1 \cdot t\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{9 \cdot x}{c} \cdot \frac{y}{z}, t, t \cdot \frac{b}{c \cdot z}\right)}{t \cdot t}, -1, \frac{a}{c} \cdot 4\right) \end{array} \]

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
 (*
  (* -1.0 t)
  (fma
   (/ (fma (* (/ (* 9.0 x) c) (/ y z)) t (* t (/ b (* c z)))) (* t t))
   -1.0
   (* (/ a c) 4.0))))

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 (-1.0 * t) * fma((fma((((9.0 * x) / c) * (y / z)), t, (t * (b / (c * z)))) / (t * t)), -1.0, ((a / c) * 4.0));
}

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(Float64(-1.0 * t) * fma(Float64(fma(Float64(Float64(Float64(9.0 * x) / c) * Float64(y / z)), t, Float64(t * Float64(b / Float64(c * z)))) / Float64(t * t)), -1.0, Float64(Float64(a / c) * 4.0)))
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[(N[(-1.0 * t), $MachinePrecision] * N[(N[(N[(N[(N[(N[(9.0 * x), $MachinePrecision] / c), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision] * t + N[(t * N[(b / N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(N[(a / c), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.7%

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

3. Taylor expanded in t around -inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 53.0%

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

Reproduce

herbie shell --seed 2025065 
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, J"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) -220031348160821/200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 0) (/ (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) z) c) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 365902434742109/31250000000000000000000000000000000000000000000000000000000000000000) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 28768236795461370000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (+ (* (* 9 (/ y c)) (/ x z)) (/ b (* c z))) (* 4 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 138385150424563190000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (- (+ (* 9 (* (/ y (* c z)) x)) (/ b (* c z))) (* 4 (/ (* a t) c)))))))))

  (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))