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

Percentage Accurate: 80.1% → 84.3%

Time: 27.4s

Alternatives: 6

Speedup: N/A×

• 36.6% 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: 80.1% 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: 84.3% accurate, N/A× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y \cdot x, 9, b\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;c\_m \leq 2.8 \cdot 10^{-99}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(4 \cdot z\right) \cdot \left(a \cdot t\right)\right) + b}{z \cdot c\_m}\\ \mathbf{elif}\;c\_m \leq 2.05 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{c\_m}, -4, \frac{t\_1}{\left(t \cdot z\right) \cdot c\_m}\right) \cdot t\\ \mathbf{elif}\;c\_m \leq 1.22 \cdot 10^{+225}:\\ \;\;\;\;\frac{b}{c\_m \cdot z} + \mathsf{fma}\left(\frac{x}{c\_m} \cdot 9, \frac{y}{z}, \frac{a \cdot t}{c\_m} \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{c\_m}, 4, \frac{\frac{\frac{t\_1}{c\_m}}{z}}{-a}\right) \cdot \left(-a\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (fma (* y x) 9.0 b)))
   (*
    c_s
    (if (<= c_m 2.8e-99)
      (/ (+ (- (* (* x 9.0) y) (* (* 4.0 z) (* a t))) b) (* z c_m))
      (if (<= c_m 2.05e+141)
        (* (fma (/ a c_m) -4.0 (/ t_1 (* (* t z) c_m))) t)
        (if (<= c_m 1.22e+225)
          (+
           (/ b (* c_m z))
           (fma (* (/ x c_m) 9.0) (/ y z) (* (/ (* a t) c_m) -4.0)))
          (* (fma (/ t c_m) 4.0 (/ (/ (/ t_1 c_m) z) (- a))) (- a))))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = fma((y * x), 9.0, b);
	double tmp;
	if (c_m <= 2.8e-99) {
		tmp = ((((x * 9.0) * y) - ((4.0 * z) * (a * t))) + b) / (z * c_m);
	} else if (c_m <= 2.05e+141) {
		tmp = fma((a / c_m), -4.0, (t_1 / ((t * z) * c_m))) * t;
	} else if (c_m <= 1.22e+225) {
		tmp = (b / (c_m * z)) + fma(((x / c_m) * 9.0), (y / z), (((a * t) / c_m) * -4.0));
	} else {
		tmp = fma((t / c_m), 4.0, (((t_1 / c_m) / z) / -a)) * -a;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = fma(Float64(y * x), 9.0, b)
	tmp = 0.0
	if (c_m <= 2.8e-99)
		tmp = Float64(Float64(Float64(Float64(Float64(x * 9.0) * y) - Float64(Float64(4.0 * z) * Float64(a * t))) + b) / Float64(z * c_m));
	elseif (c_m <= 2.05e+141)
		tmp = Float64(fma(Float64(a / c_m), -4.0, Float64(t_1 / Float64(Float64(t * z) * c_m))) * t);
	elseif (c_m <= 1.22e+225)
		tmp = Float64(Float64(b / Float64(c_m * z)) + fma(Float64(Float64(x / c_m) * 9.0), Float64(y / z), Float64(Float64(Float64(a * t) / c_m) * -4.0)));
	else
		tmp = Float64(fma(Float64(t / c_m), 4.0, Float64(Float64(Float64(t_1 / c_m) / z) / Float64(-a))) * Float64(-a));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision]}, N[(c$95$s * If[LessEqual[c$95$m, 2.8e-99], N[(N[(N[(N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision] - N[(N[(4.0 * z), $MachinePrecision] * N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[c$95$m, 2.05e+141], N[(N[(N[(a / c$95$m), $MachinePrecision] * -4.0 + N[(t$95$1 / N[(N[(t * z), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[c$95$m, 1.22e+225], N[(N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x / c$95$m), $MachinePrecision] * 9.0), $MachinePrecision] * N[(y / z), $MachinePrecision] + N[(N[(N[(a * t), $MachinePrecision] / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t / c$95$m), $MachinePrecision] * 4.0 + N[(N[(N[(t$95$1 / c$95$m), $MachinePrecision] / z), $MachinePrecision] / (-a)), $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if c < 2.8000000000000001e-99

   1. Initial program 82.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 85.0

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

   4. Applied rewrites 85.0%

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

   if 2.8000000000000001e-99 < c < 2.05000000000000011e141

   1. Initial program 76.9%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 78.8%

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

   if 2.05000000000000011e141 < c < 1.22e225

   1. Initial program 65.0%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 79.6%

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

   6. Taylor expanded in b around 0

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

      1. lower--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. frac-times N/A

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

      9. lower-neg.f64 N/A

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

      10. associate-*r/ N/A

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

   8. Applied rewrites 92.7%

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

   if 1.22e225 < c

   1. Initial program 52.7%

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

   3. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 75.4%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 2.8 \cdot 10^{-99}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(4 \cdot z\right) \cdot \left(a \cdot t\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;c \leq 2.05 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{c}, -4, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{\left(t \cdot z\right) \cdot c}\right) \cdot t\\ \mathbf{elif}\;c \leq 1.22 \cdot 10^{+225}:\\ \;\;\;\;\frac{b}{c \cdot z} + \mathsf{fma}\left(\frac{x}{c} \cdot 9, \frac{y}{z}, \frac{a \cdot t}{c} \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{c}, 4, \frac{\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c}}{z}}{-a}\right) \cdot \left(-a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.6% accurate, N/A× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y \cdot x, 9, b\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq -2.2 \cdot 10^{-149}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{c\_m}, -4, \frac{t\_1}{\left(t \cdot z\right) \cdot c\_m}\right) \cdot t\\ \mathbf{elif}\;a \leq 4.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{b}{c\_m \cdot z} + \mathsf{fma}\left(\frac{x}{c\_m} \cdot 9, \frac{y}{z}, \frac{a \cdot t}{c\_m} \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{c\_m}, 4, \frac{\frac{\frac{t\_1}{c\_m}}{z}}{-a}\right) \cdot \left(-a\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (fma (* y x) 9.0 b)))
   (*
    c_s
    (if (<= a -2.2e-149)
      (* (fma (/ a c_m) -4.0 (/ t_1 (* (* t z) c_m))) t)
      (if (<= a 4.6e-7)
        (+
         (/ b (* c_m z))
         (fma (* (/ x c_m) 9.0) (/ y z) (* (/ (* a t) c_m) -4.0)))
        (* (fma (/ t c_m) 4.0 (/ (/ (/ t_1 c_m) z) (- a))) (- a)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = fma((y * x), 9.0, b);
	double tmp;
	if (a <= -2.2e-149) {
		tmp = fma((a / c_m), -4.0, (t_1 / ((t * z) * c_m))) * t;
	} else if (a <= 4.6e-7) {
		tmp = (b / (c_m * z)) + fma(((x / c_m) * 9.0), (y / z), (((a * t) / c_m) * -4.0));
	} else {
		tmp = fma((t / c_m), 4.0, (((t_1 / c_m) / z) / -a)) * -a;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = fma(Float64(y * x), 9.0, b)
	tmp = 0.0
	if (a <= -2.2e-149)
		tmp = Float64(fma(Float64(a / c_m), -4.0, Float64(t_1 / Float64(Float64(t * z) * c_m))) * t);
	elseif (a <= 4.6e-7)
		tmp = Float64(Float64(b / Float64(c_m * z)) + fma(Float64(Float64(x / c_m) * 9.0), Float64(y / z), Float64(Float64(Float64(a * t) / c_m) * -4.0)));
	else
		tmp = Float64(fma(Float64(t / c_m), 4.0, Float64(Float64(Float64(t_1 / c_m) / z) / Float64(-a))) * Float64(-a));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision]}, N[(c$95$s * If[LessEqual[a, -2.2e-149], N[(N[(N[(a / c$95$m), $MachinePrecision] * -4.0 + N[(t$95$1 / N[(N[(t * z), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[a, 4.6e-7], N[(N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x / c$95$m), $MachinePrecision] * 9.0), $MachinePrecision] * N[(y / z), $MachinePrecision] + N[(N[(N[(a * t), $MachinePrecision] / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t / c$95$m), $MachinePrecision] * 4.0 + N[(N[(N[(t$95$1 / c$95$m), $MachinePrecision] / z), $MachinePrecision] / (-a)), $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if a < -2.1999999999999998e-149

   1. Initial program 79.7%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 83.2%

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

   if -2.1999999999999998e-149 < a < 4.5999999999999999e-7

   1. Initial program 80.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 81.7%

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

   6. Taylor expanded in b around 0

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

      1. lower--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. frac-times N/A

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

      9. lower-neg.f64 N/A

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

      10. associate-*r/ N/A

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

   8. Applied rewrites 90.5%

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

   if 4.5999999999999999e-7 < a

   1. Initial program 71.3%

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

   3. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 85.6%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.2 \cdot 10^{-149}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{c}, -4, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{\left(t \cdot z\right) \cdot c}\right) \cdot t\\ \mathbf{elif}\;a \leq 4.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{b}{c \cdot z} + \mathsf{fma}\left(\frac{x}{c} \cdot 9, \frac{y}{z}, \frac{a \cdot t}{c} \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{c}, 4, \frac{\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c}}{z}}{-a}\right) \cdot \left(-a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.0% accurate, N/A× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;c\_m \leq 1.6 \cdot 10^{-86}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a \cdot t}{b}, -4, \mathsf{fma}\left(\frac{\frac{y \cdot x}{b}}{z}, 9, {z}^{-1}\right)\right)}{-c\_m} \cdot \left(-b\right)\\ \mathbf{elif}\;c\_m \leq 1.22 \cdot 10^{+225}:\\ \;\;\;\;\frac{b}{c\_m \cdot z} + \mathsf{fma}\left(\frac{x}{c\_m} \cdot 9, \frac{y}{z}, \frac{a \cdot t}{c\_m} \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{c\_m}, 4, \frac{\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c\_m}}{z}}{-a}\right) \cdot \left(-a\right)\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= c_m 1.6e-86)
    (*
     (/
      (fma (/ (* a t) b) -4.0 (fma (/ (/ (* y x) b) z) 9.0 (pow z -1.0)))
      (- c_m))
     (- b))
    (if (<= c_m 1.22e+225)
      (+
       (/ b (* c_m z))
       (fma (* (/ x c_m) 9.0) (/ y z) (* (/ (* a t) c_m) -4.0)))
      (*
       (fma (/ t c_m) 4.0 (/ (/ (/ (fma (* y x) 9.0 b) c_m) z) (- a)))
       (- a))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (c_m <= 1.6e-86) {
		tmp = (fma(((a * t) / b), -4.0, fma((((y * x) / b) / z), 9.0, pow(z, -1.0))) / -c_m) * -b;
	} else if (c_m <= 1.22e+225) {
		tmp = (b / (c_m * z)) + fma(((x / c_m) * 9.0), (y / z), (((a * t) / c_m) * -4.0));
	} else {
		tmp = fma((t / c_m), 4.0, (((fma((y * x), 9.0, b) / c_m) / z) / -a)) * -a;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (c_m <= 1.6e-86)
		tmp = Float64(Float64(fma(Float64(Float64(a * t) / b), -4.0, fma(Float64(Float64(Float64(y * x) / b) / z), 9.0, (z ^ -1.0))) / Float64(-c_m)) * Float64(-b));
	elseif (c_m <= 1.22e+225)
		tmp = Float64(Float64(b / Float64(c_m * z)) + fma(Float64(Float64(x / c_m) * 9.0), Float64(y / z), Float64(Float64(Float64(a * t) / c_m) * -4.0)));
	else
		tmp = Float64(fma(Float64(t / c_m), 4.0, Float64(Float64(Float64(fma(Float64(y * x), 9.0, b) / c_m) / z) / Float64(-a))) * Float64(-a));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[c$95$m, 1.6e-86], N[(N[(N[(N[(N[(a * t), $MachinePrecision] / b), $MachinePrecision] * -4.0 + N[(N[(N[(N[(y * x), $MachinePrecision] / b), $MachinePrecision] / z), $MachinePrecision] * 9.0 + N[Power[z, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-c$95$m)), $MachinePrecision] * (-b)), $MachinePrecision], If[LessEqual[c$95$m, 1.22e+225], N[(N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x / c$95$m), $MachinePrecision] * 9.0), $MachinePrecision] * N[(y / z), $MachinePrecision] + N[(N[(N[(a * t), $MachinePrecision] / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t / c$95$m), $MachinePrecision] * 4.0 + N[(N[(N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision] / (-a)), $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if c < 1.60000000000000003e-86

   1. Initial program 81.9%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   8. Applied rewrites 82.2%

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

   if 1.60000000000000003e-86 < c < 1.22e225

   1. Initial program 74.4%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in b around 0

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

      1. lower--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. frac-times N/A

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

      9. lower-neg.f64 N/A

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

      10. associate-*r/ N/A

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

   8. Applied rewrites 89.9%

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

   if 1.22e225 < c

   1. Initial program 52.7%

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

   3. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 75.4%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 1.6 \cdot 10^{-86}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a \cdot t}{b}, -4, \mathsf{fma}\left(\frac{\frac{y \cdot x}{b}}{z}, 9, {z}^{-1}\right)\right)}{-c} \cdot \left(-b\right)\\ \mathbf{elif}\;c \leq 1.22 \cdot 10^{+225}:\\ \;\;\;\;\frac{b}{c \cdot z} + \mathsf{fma}\left(\frac{x}{c} \cdot 9, \frac{y}{z}, \frac{a \cdot t}{c} \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{c}, 4, \frac{\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c}}{z}}{-a}\right) \cdot \left(-a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.7% accurate, N/A× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;c\_m \leq 1.6 \cdot 10^{-86}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a \cdot t}{b}, -4, \mathsf{fma}\left(\frac{\frac{y \cdot x}{b}}{z}, 9, {z}^{-1}\right)\right)}{-c\_m} \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c\_m \cdot z} + \mathsf{fma}\left(\frac{x}{c\_m} \cdot 9, \frac{y}{z}, \frac{a \cdot t}{c\_m} \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= c_m 1.6e-86)
    (*
     (/
      (fma (/ (* a t) b) -4.0 (fma (/ (/ (* y x) b) z) 9.0 (pow z -1.0)))
      (- c_m))
     (- b))
    (+
     (/ b (* c_m z))
     (fma (* (/ x c_m) 9.0) (/ y z) (* (/ (* a t) c_m) -4.0))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (c_m <= 1.6e-86) {
		tmp = (fma(((a * t) / b), -4.0, fma((((y * x) / b) / z), 9.0, pow(z, -1.0))) / -c_m) * -b;
	} else {
		tmp = (b / (c_m * z)) + fma(((x / c_m) * 9.0), (y / z), (((a * t) / c_m) * -4.0));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (c_m <= 1.6e-86)
		tmp = Float64(Float64(fma(Float64(Float64(a * t) / b), -4.0, fma(Float64(Float64(Float64(y * x) / b) / z), 9.0, (z ^ -1.0))) / Float64(-c_m)) * Float64(-b));
	else
		tmp = Float64(Float64(b / Float64(c_m * z)) + fma(Float64(Float64(x / c_m) * 9.0), Float64(y / z), Float64(Float64(Float64(a * t) / c_m) * -4.0)));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[c$95$m, 1.6e-86], N[(N[(N[(N[(N[(a * t), $MachinePrecision] / b), $MachinePrecision] * -4.0 + N[(N[(N[(N[(y * x), $MachinePrecision] / b), $MachinePrecision] / z), $MachinePrecision] * 9.0 + N[Power[z, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-c$95$m)), $MachinePrecision] * (-b)), $MachinePrecision], N[(N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x / c$95$m), $MachinePrecision] * 9.0), $MachinePrecision] * N[(y / z), $MachinePrecision] + N[(N[(N[(a * t), $MachinePrecision] / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if c < 1.60000000000000003e-86

   1. Initial program 81.9%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   8. Applied rewrites 82.2%

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

   if 1.60000000000000003e-86 < c

   1. Initial program 68.5%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 72.7%

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

   6. Taylor expanded in b around 0

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

      1. lower--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. frac-times N/A

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

      9. lower-neg.f64 N/A

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

      10. associate-*r/ N/A

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

   8. Applied rewrites 84.6%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 1.6 \cdot 10^{-86}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a \cdot t}{b}, -4, \mathsf{fma}\left(\frac{\frac{y \cdot x}{b}}{z}, 9, {z}^{-1}\right)\right)}{-c} \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c \cdot z} + \mathsf{fma}\left(\frac{x}{c} \cdot 9, \frac{y}{z}, \frac{a \cdot t}{c} \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.9% accurate, N/A× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{a \cdot t}{c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-122}:\\ \;\;\;\;\frac{b}{c\_m \cdot z} + \mathsf{fma}\left(\frac{x}{c\_m} \cdot 9, \frac{y}{z}, t\_1 \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(t\_1, 4, b \cdot \frac{\frac{-9 \cdot \left(y \cdot x\right)}{c\_m \cdot b} - {c\_m}^{-1}}{z}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (/ (* a t) c_m)))
   (*
    c_s
    (if (<= z -9e-122)
      (+ (/ b (* c_m z)) (fma (* (/ x c_m) 9.0) (/ y z) (* t_1 -4.0)))
      (-
       (fma
        t_1
        4.0
        (* b (/ (- (/ (* -9.0 (* y x)) (* c_m b)) (pow c_m -1.0)) z))))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (a * t) / c_m;
	double tmp;
	if (z <= -9e-122) {
		tmp = (b / (c_m * z)) + fma(((x / c_m) * 9.0), (y / z), (t_1 * -4.0));
	} else {
		tmp = -fma(t_1, 4.0, (b * ((((-9.0 * (y * x)) / (c_m * b)) - pow(c_m, -1.0)) / z)));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(a * t) / c_m)
	tmp = 0.0
	if (z <= -9e-122)
		tmp = Float64(Float64(b / Float64(c_m * z)) + fma(Float64(Float64(x / c_m) * 9.0), Float64(y / z), Float64(t_1 * -4.0)));
	else
		tmp = Float64(-fma(t_1, 4.0, Float64(b * Float64(Float64(Float64(Float64(-9.0 * Float64(y * x)) / Float64(c_m * b)) - (c_m ^ -1.0)) / z))));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(a * t), $MachinePrecision] / c$95$m), $MachinePrecision]}, N[(c$95$s * If[LessEqual[z, -9e-122], N[(N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x / c$95$m), $MachinePrecision] * 9.0), $MachinePrecision] * N[(y / z), $MachinePrecision] + N[(t$95$1 * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(t$95$1 * 4.0 + N[(b * N[(N[(N[(N[(-9.0 * N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * b), $MachinePrecision]), $MachinePrecision] - N[Power[c$95$m, -1.0], $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.99999999999999959e-122

   1. Initial program 71.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in b around 0

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

      1. lower--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. frac-times N/A

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

      9. lower-neg.f64 N/A

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

      10. associate-*r/ N/A

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

   8. Applied rewrites 85.2%

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

   if -8.99999999999999959e-122 < z

   1. Initial program 81.8%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in z around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 86.8%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-122}:\\ \;\;\;\;\frac{b}{c \cdot z} + \mathsf{fma}\left(\frac{x}{c} \cdot 9, \frac{y}{z}, \frac{a \cdot t}{c} \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{a \cdot t}{c}, 4, b \cdot \frac{\frac{-9 \cdot \left(y \cdot x\right)}{c \cdot b} - {c}^{-1}}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.3% accurate, N/A× speedup

Math

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

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (+
   (/ b (* c_m z))
   (fma (* (/ x c_m) 9.0) (/ y z) (* (/ (* a t) c_m) -4.0)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	return c_s * ((b / (c_m * z)) + fma(((x / c_m) * 9.0), (y / z), (((a * t) / c_m) * -4.0)));
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	return Float64(c_s * Float64(Float64(b / Float64(c_m * z)) + fma(Float64(Float64(x / c_m) * 9.0), Float64(y / z), Float64(Float64(Float64(a * t) / c_m) * -4.0))))
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * N[(N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x / c$95$m), $MachinePrecision] * 9.0), $MachinePrecision] * N[(y / z), $MachinePrecision] + N[(N[(N[(a * t), $MachinePrecision] / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.9%

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

3. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

5. Applied rewrites 75.7%

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

6. Taylor expanded in b around 0

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

   1. lower--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. mul-1-neg N/A

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

   5. +-commutative N/A

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

   6. associate-*r/ N/A

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

   7. associate-*r* N/A

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

   8. frac-times N/A

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

   9. lower-neg.f64 N/A

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

   10. associate-*r/ N/A

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

8. Applied rewrites 79.9%

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

9. Final simplification 79.9%

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

Reproduce

herbie shell --seed 2025057 
(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)))