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

Percentage Accurate: 79.5% → 88.4%

Time: 5.6s

Alternatives: 15

Speedup: 0.5×

• 36.9% 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: 79.5% 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: 88.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ t_2 := \frac{\left(t\_1 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+45}:\\ \;\;\;\;\frac{\left(t\_1 - \left(4 \cdot z\right) \cdot \left(a \cdot t\right)\right) + b}{z \cdot c\_m}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\frac{\left(y \cdot x\right) \cdot 9 - \left(\left(\left(4 \cdot z\right) \cdot t\right) \cdot a - b\right)}{z}}{c\_m}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+278}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9, \frac{y}{z}, \frac{-b}{z \cdot x}\right) - \frac{t \cdot a}{x} \cdot -4}{-c\_m} \cdot x\\ \end{array} \end{array} \end{array} \]

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.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y))
        (t_2 (/ (+ (- t_1 (* (* (* z 4.0) t) a)) b) (* z c_m))))
   (*
    c_s
    (if (<= t_2 -5e+45)
      (/ (+ (- t_1 (* (* 4.0 z) (* a t))) b) (* z c_m))
      (if (<= t_2 0.0)
        (/ (/ (- (* (* y x) 9.0) (- (* (* (* 4.0 z) t) a) b)) z) c_m)
        (if (<= t_2 2e+278)
          t_2
          (*
           (/
            (- (fma -9.0 (/ y z) (/ (- b) (* z x))) (* (/ (* t a) x) -4.0))
            (- c_m))
           x)))))))

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);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * 9.0) * y;
	double t_2 = ((t_1 - (((z * 4.0) * t) * a)) + b) / (z * c_m);
	double tmp;
	if (t_2 <= -5e+45) {
		tmp = ((t_1 - ((4.0 * z) * (a * t))) + b) / (z * c_m);
	} else if (t_2 <= 0.0) {
		tmp = ((((y * x) * 9.0) - ((((4.0 * z) * t) * a) - b)) / z) / c_m;
	} else if (t_2 <= 2e+278) {
		tmp = t_2;
	} else {
		tmp = ((fma(-9.0, (y / z), (-b / (z * x))) - (((t * a) / x) * -4.0)) / -c_m) * x;
	}
	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])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(x * 9.0) * y)
	t_2 = Float64(Float64(Float64(t_1 - Float64(Float64(Float64(z * 4.0) * t) * a)) + b) / Float64(z * c_m))
	tmp = 0.0
	if (t_2 <= -5e+45)
		tmp = Float64(Float64(Float64(t_1 - Float64(Float64(4.0 * z) * Float64(a * t))) + b) / Float64(z * c_m));
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(Float64(y * x) * 9.0) - Float64(Float64(Float64(Float64(4.0 * z) * t) * a) - b)) / z) / c_m);
	elseif (t_2 <= 2e+278)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(fma(-9.0, Float64(y / z), Float64(Float64(-b) / Float64(z * x))) - Float64(Float64(Float64(t * a) / x) * -4.0)) / Float64(-c_m)) * x);
	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.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$1 - N[(N[(N[(z * 4.0), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$2, -5e+45], N[(N[(N[(t$95$1 - N[(N[(4.0 * z), $MachinePrecision] * N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(N[(N[(N[(y * x), $MachinePrecision] * 9.0), $MachinePrecision] - N[(N[(N[(N[(4.0 * z), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[t$95$2, 2e+278], t$95$2, N[(N[(N[(N[(-9.0 * N[(y / z), $MachinePrecision] + N[((-b) / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(t * a), $MachinePrecision] / x), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] / (-c$95$m)), $MachinePrecision] * x), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 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)) < -5e45

   1. Initial program 87.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. 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 89.1

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

   4. Applied rewrites 89.1%

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

   if -5e45 < (/.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)) < -0.0

   1. Initial program 72.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 97.2%

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

   if -0.0 < (/.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)) < 1.99999999999999993e278

   1. Initial program 99.4%

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

   if 1.99999999999999993e278 < (/.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 63.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 70.2%

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

   6. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   8. Applied rewrites 84.2%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq -5 \cdot 10^{+45}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(4 \cdot z\right) \cdot \left(a \cdot t\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq 0:\\ \;\;\;\;\frac{\frac{\left(y \cdot x\right) \cdot 9 - \left(\left(\left(4 \cdot z\right) \cdot t\right) \cdot a - b\right)}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq 2 \cdot 10^{+278}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9, \frac{y}{z}, \frac{-b}{z \cdot x}\right) - \frac{t \cdot a}{x} \cdot -4}{-c} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ t_2 := \frac{\left(t\_1 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+45}:\\ \;\;\;\;\frac{\left(t\_1 - \left(4 \cdot z\right) \cdot \left(a \cdot t\right)\right) + b}{z \cdot c\_m}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\frac{\left(y \cdot x\right) \cdot 9 - \left(\left(\left(4 \cdot z\right) \cdot t\right) \cdot a - b\right)}{z}}{c\_m}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+299}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{c\_m}, 4, \frac{\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c\_m}}{z}}{-t}\right) \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c\_m \cdot z} - \frac{t \cdot a}{c\_m} \cdot 4\\ \end{array} \end{array} \end{array} \]

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.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y))
        (t_2 (/ (+ (- t_1 (* (* (* z 4.0) t) a)) b) (* z c_m))))
   (*
    c_s
    (if (<= t_2 -5e+45)
      (/ (+ (- t_1 (* (* 4.0 z) (* a t))) b) (* z c_m))
      (if (<= t_2 0.0)
        (/ (/ (- (* (* y x) 9.0) (- (* (* (* 4.0 z) t) a) b)) z) c_m)
        (if (<= t_2 5e+299)
          t_2
          (if (<= t_2 INFINITY)
            (*
             (fma (/ a c_m) 4.0 (/ (/ (/ (fma (* y x) 9.0 b) c_m) z) (- t)))
             (- t))
            (- (/ b (* c_m z)) (* (/ (* t a) 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);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * 9.0) * y;
	double t_2 = ((t_1 - (((z * 4.0) * t) * a)) + b) / (z * c_m);
	double tmp;
	if (t_2 <= -5e+45) {
		tmp = ((t_1 - ((4.0 * z) * (a * t))) + b) / (z * c_m);
	} else if (t_2 <= 0.0) {
		tmp = ((((y * x) * 9.0) - ((((4.0 * z) * t) * a) - b)) / z) / c_m;
	} else if (t_2 <= 5e+299) {
		tmp = t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = fma((a / c_m), 4.0, (((fma((y * x), 9.0, b) / c_m) / z) / -t)) * -t;
	} else {
		tmp = (b / (c_m * z)) - (((t * a) / 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])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(x * 9.0) * y)
	t_2 = Float64(Float64(Float64(t_1 - Float64(Float64(Float64(z * 4.0) * t) * a)) + b) / Float64(z * c_m))
	tmp = 0.0
	if (t_2 <= -5e+45)
		tmp = Float64(Float64(Float64(t_1 - Float64(Float64(4.0 * z) * Float64(a * t))) + b) / Float64(z * c_m));
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(Float64(y * x) * 9.0) - Float64(Float64(Float64(Float64(4.0 * z) * t) * a) - b)) / z) / c_m);
	elseif (t_2 <= 5e+299)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = Float64(fma(Float64(a / c_m), 4.0, Float64(Float64(Float64(fma(Float64(y * x), 9.0, b) / c_m) / z) / Float64(-t))) * Float64(-t));
	else
		tmp = Float64(Float64(b / Float64(c_m * z)) - Float64(Float64(Float64(t * a) / 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.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$1 - N[(N[(N[(z * 4.0), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$2, -5e+45], N[(N[(N[(t$95$1 - N[(N[(4.0 * z), $MachinePrecision] * N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(N[(N[(N[(y * x), $MachinePrecision] * 9.0), $MachinePrecision] - N[(N[(N[(N[(4.0 * z), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[t$95$2, 5e+299], t$95$2, If[LessEqual[t$95$2, Infinity], N[(N[(N[(a / c$95$m), $MachinePrecision] * 4.0 + N[(N[(N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision] / (-t)), $MachinePrecision]), $MachinePrecision] * (-t)), $MachinePrecision], N[(N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(t * a), $MachinePrecision] / c$95$m), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 5 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)) < -5e45

   1. Initial program 87.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. 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 89.1

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

   4. Applied rewrites 89.1%

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

   if -5e45 < (/.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)) < -0.0

   1. Initial program 72.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 97.2%

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

   if -0.0 < (/.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)) < 5.0000000000000003e299

   1. Initial program 99.4%

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

   if 5.0000000000000003e299 < (/.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)) < +inf.0

   1. Initial program 80.0%

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

   3. Taylor expanded in 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. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 86.6%

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

   if +inf.0 < (/.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 0.0%

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

   3. Taylor expanded in 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 54.1%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - 4 \cdot \color{blue}{\frac{a \cdot t}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - 4 \cdot \frac{\color{blue}{a \cdot t}}{c} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot \color{blue}{t}}{c} \]

      4. *-commutative N/A

         \[\leadsto \frac{b}{c \cdot z} - \frac{a \cdot t}{c} \cdot 4 \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - \frac{a \cdot t}{c} \cdot 4 \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - \frac{a \cdot t}{c} \cdot 4 \]

      7. *-commutative N/A

         \[\leadsto \frac{b}{c \cdot z} - \frac{t \cdot a}{c} \cdot 4 \]

      8. lower-*.f64 54.5

         \[\leadsto \frac{b}{c \cdot z} - \frac{t \cdot a}{c} \cdot 4 \]

   8. Applied rewrites 54.5%

      \[\leadsto \frac{b}{c \cdot z} - \color{blue}{\frac{t \cdot a}{c} \cdot 4} \]
3. Recombined 5 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq -5 \cdot 10^{+45}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(4 \cdot z\right) \cdot \left(a \cdot t\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq 0:\\ \;\;\;\;\frac{\frac{\left(y \cdot x\right) \cdot 9 - \left(\left(\left(4 \cdot z\right) \cdot t\right) \cdot a - b\right)}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{c}, 4, \frac{\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c}}{z}}{-t}\right) \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c \cdot z} - \frac{t \cdot a}{c} \cdot 4\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ t_2 := \frac{\left(t\_1 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+45}:\\ \;\;\;\;\frac{\left(t\_1 - \left(4 \cdot z\right) \cdot \left(a \cdot t\right)\right) + b}{z \cdot c\_m}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\frac{\left(y \cdot x\right) \cdot 9 - \left(\left(\left(4 \cdot z\right) \cdot t\right) \cdot a - b\right)}{z}}{c\_m}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+299}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{c\_m}, -4, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{\left(t \cdot z\right) \cdot c\_m}\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c\_m \cdot z} - \frac{t \cdot a}{c\_m} \cdot 4\\ \end{array} \end{array} \end{array} \]

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.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y))
        (t_2 (/ (+ (- t_1 (* (* (* z 4.0) t) a)) b) (* z c_m))))
   (*
    c_s
    (if (<= t_2 -5e+45)
      (/ (+ (- t_1 (* (* 4.0 z) (* a t))) b) (* z c_m))
      (if (<= t_2 0.0)
        (/ (/ (- (* (* y x) 9.0) (- (* (* (* 4.0 z) t) a) b)) z) c_m)
        (if (<= t_2 5e+299)
          t_2
          (if (<= t_2 INFINITY)
            (* (fma (/ a c_m) -4.0 (/ (fma (* y x) 9.0 b) (* (* t z) c_m))) t)
            (- (/ b (* c_m z)) (* (/ (* t a) 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);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * 9.0) * y;
	double t_2 = ((t_1 - (((z * 4.0) * t) * a)) + b) / (z * c_m);
	double tmp;
	if (t_2 <= -5e+45) {
		tmp = ((t_1 - ((4.0 * z) * (a * t))) + b) / (z * c_m);
	} else if (t_2 <= 0.0) {
		tmp = ((((y * x) * 9.0) - ((((4.0 * z) * t) * a) - b)) / z) / c_m;
	} else if (t_2 <= 5e+299) {
		tmp = t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = fma((a / c_m), -4.0, (fma((y * x), 9.0, b) / ((t * z) * c_m))) * t;
	} else {
		tmp = (b / (c_m * z)) - (((t * a) / 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])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(x * 9.0) * y)
	t_2 = Float64(Float64(Float64(t_1 - Float64(Float64(Float64(z * 4.0) * t) * a)) + b) / Float64(z * c_m))
	tmp = 0.0
	if (t_2 <= -5e+45)
		tmp = Float64(Float64(Float64(t_1 - Float64(Float64(4.0 * z) * Float64(a * t))) + b) / Float64(z * c_m));
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(Float64(y * x) * 9.0) - Float64(Float64(Float64(Float64(4.0 * z) * t) * a) - b)) / z) / c_m);
	elseif (t_2 <= 5e+299)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = Float64(fma(Float64(a / c_m), -4.0, Float64(fma(Float64(y * x), 9.0, b) / Float64(Float64(t * z) * c_m))) * t);
	else
		tmp = Float64(Float64(b / Float64(c_m * z)) - Float64(Float64(Float64(t * a) / 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.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$1 - N[(N[(N[(z * 4.0), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$2, -5e+45], N[(N[(N[(t$95$1 - N[(N[(4.0 * z), $MachinePrecision] * N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(N[(N[(N[(y * x), $MachinePrecision] * 9.0), $MachinePrecision] - N[(N[(N[(N[(4.0 * z), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[t$95$2, 5e+299], t$95$2, If[LessEqual[t$95$2, Infinity], N[(N[(N[(a / c$95$m), $MachinePrecision] * -4.0 + N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], N[(N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(t * a), $MachinePrecision] / c$95$m), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 5 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)) < -5e45

   1. Initial program 87.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. 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 89.1

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

   4. Applied rewrites 89.1%

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

   if -5e45 < (/.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)) < -0.0

   1. Initial program 72.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 97.2%

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

   if -0.0 < (/.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)) < 5.0000000000000003e299

   1. Initial program 99.4%

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

   if 5.0000000000000003e299 < (/.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)) < +inf.0

   1. Initial program 80.0%

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

   3. Taylor expanded in 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 82.1%

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

   if +inf.0 < (/.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 0.0%

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

   3. Taylor expanded in 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 54.1%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - 4 \cdot \color{blue}{\frac{a \cdot t}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - 4 \cdot \frac{\color{blue}{a \cdot t}}{c} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot \color{blue}{t}}{c} \]

      4. *-commutative N/A

         \[\leadsto \frac{b}{c \cdot z} - \frac{a \cdot t}{c} \cdot 4 \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - \frac{a \cdot t}{c} \cdot 4 \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - \frac{a \cdot t}{c} \cdot 4 \]

      7. *-commutative N/A

         \[\leadsto \frac{b}{c \cdot z} - \frac{t \cdot a}{c} \cdot 4 \]

      8. lower-*.f64 54.5

         \[\leadsto \frac{b}{c \cdot z} - \frac{t \cdot a}{c} \cdot 4 \]

   8. Applied rewrites 54.5%

      \[\leadsto \frac{b}{c \cdot z} - \color{blue}{\frac{t \cdot a}{c} \cdot 4} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 4: 88.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ t_2 := \frac{\left(t\_1 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+45}:\\ \;\;\;\;\frac{\left(t\_1 - \left(4 \cdot z\right) \cdot \left(a \cdot t\right)\right) + b}{z \cdot c\_m}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\frac{\left(y \cdot x\right) \cdot 9 - \left(\left(\left(4 \cdot z\right) \cdot t\right) \cdot a - b\right)}{z}}{c\_m}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+278}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9, \frac{y}{z}, \frac{-b}{z \cdot x}\right) - \left(a \cdot \frac{t}{x}\right) \cdot -4}{-c\_m} \cdot x\\ \end{array} \end{array} \end{array} \]

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.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y))
        (t_2 (/ (+ (- t_1 (* (* (* z 4.0) t) a)) b) (* z c_m))))
   (*
    c_s
    (if (<= t_2 -5e+45)
      (/ (+ (- t_1 (* (* 4.0 z) (* a t))) b) (* z c_m))
      (if (<= t_2 0.0)
        (/ (/ (- (* (* y x) 9.0) (- (* (* (* 4.0 z) t) a) b)) z) c_m)
        (if (<= t_2 2e+278)
          t_2
          (*
           (/
            (- (fma -9.0 (/ y z) (/ (- b) (* z x))) (* (* a (/ t x)) -4.0))
            (- c_m))
           x)))))))

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);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * 9.0) * y;
	double t_2 = ((t_1 - (((z * 4.0) * t) * a)) + b) / (z * c_m);
	double tmp;
	if (t_2 <= -5e+45) {
		tmp = ((t_1 - ((4.0 * z) * (a * t))) + b) / (z * c_m);
	} else if (t_2 <= 0.0) {
		tmp = ((((y * x) * 9.0) - ((((4.0 * z) * t) * a) - b)) / z) / c_m;
	} else if (t_2 <= 2e+278) {
		tmp = t_2;
	} else {
		tmp = ((fma(-9.0, (y / z), (-b / (z * x))) - ((a * (t / x)) * -4.0)) / -c_m) * x;
	}
	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])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(x * 9.0) * y)
	t_2 = Float64(Float64(Float64(t_1 - Float64(Float64(Float64(z * 4.0) * t) * a)) + b) / Float64(z * c_m))
	tmp = 0.0
	if (t_2 <= -5e+45)
		tmp = Float64(Float64(Float64(t_1 - Float64(Float64(4.0 * z) * Float64(a * t))) + b) / Float64(z * c_m));
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(Float64(y * x) * 9.0) - Float64(Float64(Float64(Float64(4.0 * z) * t) * a) - b)) / z) / c_m);
	elseif (t_2 <= 2e+278)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(fma(-9.0, Float64(y / z), Float64(Float64(-b) / Float64(z * x))) - Float64(Float64(a * Float64(t / x)) * -4.0)) / Float64(-c_m)) * x);
	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.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$1 - N[(N[(N[(z * 4.0), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$2, -5e+45], N[(N[(N[(t$95$1 - N[(N[(4.0 * z), $MachinePrecision] * N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(N[(N[(N[(y * x), $MachinePrecision] * 9.0), $MachinePrecision] - N[(N[(N[(N[(4.0 * z), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[t$95$2, 2e+278], t$95$2, N[(N[(N[(N[(-9.0 * N[(y / z), $MachinePrecision] + N[((-b) / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a * N[(t / x), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] / (-c$95$m)), $MachinePrecision] * x), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 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)) < -5e45

   1. Initial program 87.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. 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 89.1

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

   4. Applied rewrites 89.1%

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

   if -5e45 < (/.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)) < -0.0

   1. Initial program 72.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 97.2%

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

   if -0.0 < (/.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)) < 1.99999999999999993e278

   1. Initial program 99.4%

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

   if 1.99999999999999993e278 < (/.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 63.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 70.2%

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

   6. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   8. Applied rewrites 84.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-/.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 82.9

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

   10. Applied rewrites 82.9%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq -5 \cdot 10^{+45}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(4 \cdot z\right) \cdot \left(a \cdot t\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq 0:\\ \;\;\;\;\frac{\frac{\left(y \cdot x\right) \cdot 9 - \left(\left(\left(4 \cdot z\right) \cdot t\right) \cdot a - b\right)}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq 2 \cdot 10^{+278}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9, \frac{y}{z}, \frac{-b}{z \cdot x}\right) - \left(a \cdot \frac{t}{x}\right) \cdot -4}{-c} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ t_2 := \frac{\left(t\_1 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c\_m}\\ t_3 := \frac{\left(t\_1 - \left(4 \cdot z\right) \cdot \left(a \cdot t\right)\right) + b}{z \cdot c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+45}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\frac{\left(y \cdot x\right) \cdot 9 - \left(\left(\left(4 \cdot z\right) \cdot t\right) \cdot a - b\right)}{z}}{c\_m}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c\_m \cdot z} - \frac{t \cdot a}{c\_m} \cdot 4\\ \end{array} \end{array} \end{array} \]

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.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y))
        (t_2 (/ (+ (- t_1 (* (* (* z 4.0) t) a)) b) (* z c_m)))
        (t_3 (/ (+ (- t_1 (* (* 4.0 z) (* a t))) b) (* z c_m))))
   (*
    c_s
    (if (<= t_2 -5e+45)
      t_3
      (if (<= t_2 0.0)
        (/ (/ (- (* (* y x) 9.0) (- (* (* (* 4.0 z) t) a) b)) z) c_m)
        (if (<= t_2 INFINITY)
          t_3
          (- (/ b (* c_m z)) (* (/ (* t a) 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);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * 9.0) * y;
	double t_2 = ((t_1 - (((z * 4.0) * t) * a)) + b) / (z * c_m);
	double t_3 = ((t_1 - ((4.0 * z) * (a * t))) + b) / (z * c_m);
	double tmp;
	if (t_2 <= -5e+45) {
		tmp = t_3;
	} else if (t_2 <= 0.0) {
		tmp = ((((y * x) * 9.0) - ((((4.0 * z) * t) * a) - b)) / z) / c_m;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = (b / (c_m * z)) - (((t * a) / c_m) * 4.0);
	}
	return c_s * tmp;
}

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * 9.0) * y;
	double t_2 = ((t_1 - (((z * 4.0) * t) * a)) + b) / (z * c_m);
	double t_3 = ((t_1 - ((4.0 * z) * (a * t))) + b) / (z * c_m);
	double tmp;
	if (t_2 <= -5e+45) {
		tmp = t_3;
	} else if (t_2 <= 0.0) {
		tmp = ((((y * x) * 9.0) - ((((4.0 * z) * t) * a) - b)) / z) / c_m;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else {
		tmp = (b / (c_m * z)) - (((t * a) / c_m) * 4.0);
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	t_1 = (x * 9.0) * y
	t_2 = ((t_1 - (((z * 4.0) * t) * a)) + b) / (z * c_m)
	t_3 = ((t_1 - ((4.0 * z) * (a * t))) + b) / (z * c_m)
	tmp = 0
	if t_2 <= -5e+45:
		tmp = t_3
	elif t_2 <= 0.0:
		tmp = ((((y * x) * 9.0) - ((((4.0 * z) * t) * a) - b)) / z) / c_m
	elif t_2 <= math.inf:
		tmp = t_3
	else:
		tmp = (b / (c_m * z)) - (((t * a) / 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])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(x * 9.0) * y)
	t_2 = Float64(Float64(Float64(t_1 - Float64(Float64(Float64(z * 4.0) * t) * a)) + b) / Float64(z * c_m))
	t_3 = Float64(Float64(Float64(t_1 - Float64(Float64(4.0 * z) * Float64(a * t))) + b) / Float64(z * c_m))
	tmp = 0.0
	if (t_2 <= -5e+45)
		tmp = t_3;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(Float64(y * x) * 9.0) - Float64(Float64(Float64(Float64(4.0 * z) * t) * a) - b)) / z) / c_m);
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(Float64(b / Float64(c_m * z)) - Float64(Float64(Float64(t * a) / c_m) * 4.0));
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	t_1 = (x * 9.0) * y;
	t_2 = ((t_1 - (((z * 4.0) * t) * a)) + b) / (z * c_m);
	t_3 = ((t_1 - ((4.0 * z) * (a * t))) + b) / (z * c_m);
	tmp = 0.0;
	if (t_2 <= -5e+45)
		tmp = t_3;
	elseif (t_2 <= 0.0)
		tmp = ((((y * x) * 9.0) - ((((4.0 * z) * t) * a) - b)) / z) / c_m;
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = (b / (c_m * z)) - (((t * a) / c_m) * 4.0);
	end
	tmp_2 = 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.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$1 - N[(N[(N[(z * 4.0), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$1 - N[(N[(4.0 * z), $MachinePrecision] * N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$2, -5e+45], t$95$3, If[LessEqual[t$95$2, 0.0], N[(N[(N[(N[(N[(y * x), $MachinePrecision] * 9.0), $MachinePrecision] - N[(N[(N[(N[(4.0 * z), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$3, N[(N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(t * a), $MachinePrecision] / c$95$m), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 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)) < -5e45 or -0.0 < (/.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)) < +inf.0

   1. Initial program 88.2%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 89.7

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

   4. Applied rewrites 89.7%

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

   if -5e45 < (/.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)) < -0.0

   1. Initial program 72.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 97.2%

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

   if +inf.0 < (/.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 0.0%

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

   3. Taylor expanded in 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 54.1%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - 4 \cdot \color{blue}{\frac{a \cdot t}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - 4 \cdot \frac{\color{blue}{a \cdot t}}{c} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot \color{blue}{t}}{c} \]

      4. *-commutative N/A

         \[\leadsto \frac{b}{c \cdot z} - \frac{a \cdot t}{c} \cdot 4 \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - \frac{a \cdot t}{c} \cdot 4 \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - \frac{a \cdot t}{c} \cdot 4 \]

      7. *-commutative N/A

         \[\leadsto \frac{b}{c \cdot z} - \frac{t \cdot a}{c} \cdot 4 \]

      8. lower-*.f64 54.5

         \[\leadsto \frac{b}{c \cdot z} - \frac{t \cdot a}{c} \cdot 4 \]

   8. Applied rewrites 54.5%

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

Alternative 6: 84.4% accurate, 0.5× speedup

Math

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

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.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)))
   (*
    c_s
    (if (<= (/ (+ (- t_1 (* (* (* z 4.0) t) a)) b) (* z c_m)) INFINITY)
      (/ (+ (- t_1 (* (* 4.0 z) (* a t))) b) (* z c_m))
      (- (/ b (* c_m z)) (* (/ (* t a) 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);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if ((((t_1 - (((z * 4.0) * t) * a)) + b) / (z * c_m)) <= ((double) INFINITY)) {
		tmp = ((t_1 - ((4.0 * z) * (a * t))) + b) / (z * c_m);
	} else {
		tmp = (b / (c_m * z)) - (((t * a) / c_m) * 4.0);
	}
	return c_s * tmp;
}

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if ((((t_1 - (((z * 4.0) * t) * a)) + b) / (z * c_m)) <= Double.POSITIVE_INFINITY) {
		tmp = ((t_1 - ((4.0 * z) * (a * t))) + b) / (z * c_m);
	} else {
		tmp = (b / (c_m * z)) - (((t * a) / c_m) * 4.0);
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	t_1 = (x * 9.0) * y
	tmp = 0
	if (((t_1 - (((z * 4.0) * t) * a)) + b) / (z * c_m)) <= math.inf:
		tmp = ((t_1 - ((4.0 * z) * (a * t))) + b) / (z * c_m)
	else:
		tmp = (b / (c_m * z)) - (((t * a) / 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])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (Float64(Float64(Float64(t_1 - Float64(Float64(Float64(z * 4.0) * t) * a)) + b) / Float64(z * c_m)) <= Inf)
		tmp = Float64(Float64(Float64(t_1 - Float64(Float64(4.0 * z) * Float64(a * t))) + b) / Float64(z * c_m));
	else
		tmp = Float64(Float64(b / Float64(c_m * z)) - Float64(Float64(Float64(t * a) / c_m) * 4.0));
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	t_1 = (x * 9.0) * y;
	tmp = 0.0;
	if ((((t_1 - (((z * 4.0) * t) * a)) + b) / (z * c_m)) <= Inf)
		tmp = ((t_1 - ((4.0 * z) * (a * t))) + b) / (z * c_m);
	else
		tmp = (b / (c_m * z)) - (((t * a) / c_m) * 4.0);
	end
	tmp_2 = 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.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, N[(c$95$s * If[LessEqual[N[(N[(N[(t$95$1 - N[(N[(N[(z * 4.0), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(t$95$1 - N[(N[(4.0 * z), $MachinePrecision] * N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(t * a), $MachinePrecision] / c$95$m), $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)) < +inf.0

   1. Initial program 85.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. 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 86.3

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

   4. Applied rewrites 86.3%

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

   if +inf.0 < (/.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 0.0%

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

   3. Taylor expanded in 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 54.1%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - 4 \cdot \color{blue}{\frac{a \cdot t}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - 4 \cdot \frac{\color{blue}{a \cdot t}}{c} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot \color{blue}{t}}{c} \]

      4. *-commutative N/A

         \[\leadsto \frac{b}{c \cdot z} - \frac{a \cdot t}{c} \cdot 4 \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - \frac{a \cdot t}{c} \cdot 4 \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - \frac{a \cdot t}{c} \cdot 4 \]

      7. *-commutative N/A

         \[\leadsto \frac{b}{c \cdot z} - \frac{t \cdot a}{c} \cdot 4 \]

      8. lower-*.f64 54.5

         \[\leadsto \frac{b}{c \cdot z} - \frac{t \cdot a}{c} \cdot 4 \]

   8. Applied rewrites 54.5%

      \[\leadsto \frac{b}{c \cdot z} - \color{blue}{\frac{t \cdot a}{c} \cdot 4} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 75.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+270}:\\ \;\;\;\;\frac{\left(\frac{y}{c\_m} \cdot 9\right) \cdot x}{z}\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{+58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z \cdot c\_m}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+241}:\\ \;\;\;\;\frac{b}{c\_m \cdot z} - \frac{t \cdot a}{c\_m} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot -9}{-c\_m} \cdot x\\ \end{array} \end{array} \end{array} \]

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.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)))
   (*
    c_s
    (if (<= t_1 -2e+270)
      (/ (* (* (/ y c_m) 9.0) x) z)
      (if (<= t_1 -4e+58)
        (/ (fma (* 9.0 x) y b) (* z c_m))
        (if (<= t_1 5e+241)
          (- (/ b (* c_m z)) (* (/ (* t a) c_m) 4.0))
          (* (/ (* (/ y z) -9.0) (- c_m)) x)))))))

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);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if (t_1 <= -2e+270) {
		tmp = (((y / c_m) * 9.0) * x) / z;
	} else if (t_1 <= -4e+58) {
		tmp = fma((9.0 * x), y, b) / (z * c_m);
	} else if (t_1 <= 5e+241) {
		tmp = (b / (c_m * z)) - (((t * a) / c_m) * 4.0);
	} else {
		tmp = (((y / z) * -9.0) / -c_m) * x;
	}
	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])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_1 <= -2e+270)
		tmp = Float64(Float64(Float64(Float64(y / c_m) * 9.0) * x) / z);
	elseif (t_1 <= -4e+58)
		tmp = Float64(fma(Float64(9.0 * x), y, b) / Float64(z * c_m));
	elseif (t_1 <= 5e+241)
		tmp = Float64(Float64(b / Float64(c_m * z)) - Float64(Float64(Float64(t * a) / c_m) * 4.0));
	else
		tmp = Float64(Float64(Float64(Float64(y / z) * -9.0) / Float64(-c_m)) * x);
	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.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -2e+270], N[(N[(N[(N[(y / c$95$m), $MachinePrecision] * 9.0), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, -4e+58], N[(N[(N[(9.0 * x), $MachinePrecision] * y + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+241], N[(N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(t * a), $MachinePrecision] / c$95$m), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y / z), $MachinePrecision] * -9.0), $MachinePrecision] / (-c$95$m)), $MachinePrecision] * x), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -2.0000000000000001e270

   1. Initial program 61.8%

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

   3. Taylor expanded in 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 89.1%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 68.4

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

   8. Applied rewrites 68.4%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lift-/.f64 75.3

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

   11. Applied rewrites 75.3%

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

   if -2.0000000000000001e270 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -3.99999999999999978e58

   1. Initial program 91.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 83.0

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

   5. Applied rewrites 83.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 83.0

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

   7. Applied rewrites 83.0%

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

   if -3.99999999999999978e58 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 5.00000000000000025e241

   1. Initial program 81.1%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 67.5%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - 4 \cdot \color{blue}{\frac{a \cdot t}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - 4 \cdot \frac{\color{blue}{a \cdot t}}{c} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot \color{blue}{t}}{c} \]

      4. *-commutative N/A

         \[\leadsto \frac{b}{c \cdot z} - \frac{a \cdot t}{c} \cdot 4 \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - \frac{a \cdot t}{c} \cdot 4 \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - \frac{a \cdot t}{c} \cdot 4 \]

      7. *-commutative N/A

         \[\leadsto \frac{b}{c \cdot z} - \frac{t \cdot a}{c} \cdot 4 \]

      8. lower-*.f64 79.4

         \[\leadsto \frac{b}{c \cdot z} - \frac{t \cdot a}{c} \cdot 4 \]

   8. Applied rewrites 79.4%

      \[\leadsto \frac{b}{c \cdot z} - \color{blue}{\frac{t \cdot a}{c} \cdot 4} \]

   if 5.00000000000000025e241 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 75.7%

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

   3. Taylor expanded in 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 68.2%

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

   6. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   8. Applied rewrites 85.9%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lift-/.f64 86.0

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

   11. Applied rewrites 86.0%

       \[\leadsto \left(-\frac{\frac{y}{z} \cdot -9}{c}\right) \cdot x \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y \leq -2 \cdot 10^{+270}:\\ \;\;\;\;\frac{\left(\frac{y}{c} \cdot 9\right) \cdot x}{z}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \leq -4 \cdot 10^{+58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z \cdot c}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \leq 5 \cdot 10^{+241}:\\ \;\;\;\;\frac{b}{c \cdot z} - \frac{t \cdot a}{c} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot -9}{-c} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+58}:\\ \;\;\;\;\frac{x}{c\_m} \cdot \frac{\mathsf{fma}\left(y, 9, \frac{b}{x}\right)}{z}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+241}:\\ \;\;\;\;\frac{b}{c\_m \cdot z} - \frac{t \cdot a}{c\_m} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot -9}{-c\_m} \cdot x\\ \end{array} \end{array} \end{array} \]

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.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)))
   (*
    c_s
    (if (<= t_1 -4e+58)
      (* (/ x c_m) (/ (fma y 9.0 (/ b x)) z))
      (if (<= t_1 5e+241)
        (- (/ b (* c_m z)) (* (/ (* t a) c_m) 4.0))
        (* (/ (* (/ y z) -9.0) (- c_m)) x))))))

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);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if (t_1 <= -4e+58) {
		tmp = (x / c_m) * (fma(y, 9.0, (b / x)) / z);
	} else if (t_1 <= 5e+241) {
		tmp = (b / (c_m * z)) - (((t * a) / c_m) * 4.0);
	} else {
		tmp = (((y / z) * -9.0) / -c_m) * x;
	}
	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])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_1 <= -4e+58)
		tmp = Float64(Float64(x / c_m) * Float64(fma(y, 9.0, Float64(b / x)) / z));
	elseif (t_1 <= 5e+241)
		tmp = Float64(Float64(b / Float64(c_m * z)) - Float64(Float64(Float64(t * a) / c_m) * 4.0));
	else
		tmp = Float64(Float64(Float64(Float64(y / z) * -9.0) / Float64(-c_m)) * x);
	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.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -4e+58], N[(N[(x / c$95$m), $MachinePrecision] * N[(N[(y * 9.0 + N[(b / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+241], N[(N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(t * a), $MachinePrecision] / c$95$m), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y / z), $MachinePrecision] * -9.0), $MachinePrecision] / (-c$95$m)), $MachinePrecision] * x), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -3.99999999999999978e58

   1. Initial program 78.3%

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

   3. Taylor expanded in 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 77.0%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 65.9

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

   8. Applied rewrites 65.9%

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

   9. Taylor expanded in c around 0

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

       1. times-frac N/A

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

       2. div-add N/A

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

       3. associate-*r/ N/A

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

       4. associate-/r* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. associate-*r/ N/A

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

       8. associate-/r* N/A

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

       9. div-add N/A

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

       10. lower-/.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. lower-/.f64 73.8

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

   11. Applied rewrites 73.8%

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

   if -3.99999999999999978e58 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 5.00000000000000025e241

   1. Initial program 81.1%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 67.5%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - 4 \cdot \color{blue}{\frac{a \cdot t}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - 4 \cdot \frac{\color{blue}{a \cdot t}}{c} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot \color{blue}{t}}{c} \]

      4. *-commutative N/A

         \[\leadsto \frac{b}{c \cdot z} - \frac{a \cdot t}{c} \cdot 4 \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - \frac{a \cdot t}{c} \cdot 4 \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{b}{c \cdot z} - \frac{a \cdot t}{c} \cdot 4 \]

      7. *-commutative N/A

         \[\leadsto \frac{b}{c \cdot z} - \frac{t \cdot a}{c} \cdot 4 \]

      8. lower-*.f64 79.4

         \[\leadsto \frac{b}{c \cdot z} - \frac{t \cdot a}{c} \cdot 4 \]

   8. Applied rewrites 79.4%

      \[\leadsto \frac{b}{c \cdot z} - \color{blue}{\frac{t \cdot a}{c} \cdot 4} \]

   if 5.00000000000000025e241 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 75.7%

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

   3. Taylor expanded in 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 68.2%

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

   6. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   8. Applied rewrites 85.9%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lift-/.f64 86.0

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

   11. Applied rewrites 86.0%

       \[\leadsto \left(-\frac{\frac{y}{z} \cdot -9}{c}\right) \cdot x \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y \leq -4 \cdot 10^{+58}:\\ \;\;\;\;\frac{x}{c} \cdot \frac{\mathsf{fma}\left(y, 9, \frac{b}{x}\right)}{z}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \leq 5 \cdot 10^{+241}:\\ \;\;\;\;\frac{b}{c \cdot z} - \frac{t \cdot a}{c} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot -9}{-c} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+55}:\\ \;\;\;\;-4 \cdot \frac{a \cdot t}{c\_m}\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{-136}:\\ \;\;\;\;\left(x \cdot \frac{y}{c\_m \cdot z}\right) \cdot 9\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+36}:\\ \;\;\;\;\frac{b}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t \cdot a\right) \cdot -4}{c\_m}\\ \end{array} \end{array} \]

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.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= z -3.1e+55)
    (* -4.0 (/ (* a t) c_m))
    (if (<= z -1.55e-136)
      (* (* x (/ y (* c_m z))) 9.0)
      (if (<= z 3.8e+36) (/ b (* z c_m)) (/ (* (* t a) -4.0) c_m))))))

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);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (z <= -3.1e+55) {
		tmp = -4.0 * ((a * t) / c_m);
	} else if (z <= -1.55e-136) {
		tmp = (x * (y / (c_m * z))) * 9.0;
	} else if (z <= 3.8e+36) {
		tmp = b / (z * c_m);
	} else {
		tmp = ((t * a) * -4.0) / c_m;
	}
	return c_s * tmp;
}

Fortran

c\_m =     private
c\_s =     private
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c_s, x, y, z, t, a, b, c_m)
use fmin_fmax_functions
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: tmp
    if (z <= (-3.1d+55)) then
        tmp = (-4.0d0) * ((a * t) / c_m)
    else if (z <= (-1.55d-136)) then
        tmp = (x * (y / (c_m * z))) * 9.0d0
    else if (z <= 3.8d+36) then
        tmp = b / (z * c_m)
    else
        tmp = ((t * a) * (-4.0d0)) / c_m
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (z <= -3.1e+55) {
		tmp = -4.0 * ((a * t) / c_m);
	} else if (z <= -1.55e-136) {
		tmp = (x * (y / (c_m * z))) * 9.0;
	} else if (z <= 3.8e+36) {
		tmp = b / (z * c_m);
	} else {
		tmp = ((t * a) * -4.0) / c_m;
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if z <= -3.1e+55:
		tmp = -4.0 * ((a * t) / c_m)
	elif z <= -1.55e-136:
		tmp = (x * (y / (c_m * z))) * 9.0
	elif z <= 3.8e+36:
		tmp = b / (z * c_m)
	else:
		tmp = ((t * a) * -4.0) / c_m
	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])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (z <= -3.1e+55)
		tmp = Float64(-4.0 * Float64(Float64(a * t) / c_m));
	elseif (z <= -1.55e-136)
		tmp = Float64(Float64(x * Float64(y / Float64(c_m * z))) * 9.0);
	elseif (z <= 3.8e+36)
		tmp = Float64(b / Float64(z * c_m));
	else
		tmp = Float64(Float64(Float64(t * a) * -4.0) / c_m);
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if (z <= -3.1e+55)
		tmp = -4.0 * ((a * t) / c_m);
	elseif (z <= -1.55e-136)
		tmp = (x * (y / (c_m * z))) * 9.0;
	elseif (z <= 3.8e+36)
		tmp = b / (z * c_m);
	else
		tmp = ((t * a) * -4.0) / c_m;
	end
	tmp_2 = 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.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[z, -3.1e+55], N[(-4.0 * N[(N[(a * t), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.55e-136], N[(N[(x * N[(y / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 9.0), $MachinePrecision], If[LessEqual[z, 3.8e+36], N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t * a), $MachinePrecision] * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if z < -3.09999999999999994e55

   1. Initial program 57.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\frac{a \cdot t}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto -4 \cdot \frac{a \cdot t}{\color{blue}{c}} \]

      3. lower-*.f64 56.0

         \[\leadsto -4 \cdot \frac{a \cdot t}{c} \]

   5. Applied rewrites 56.0%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]

   if -3.09999999999999994e55 < z < -1.55e-136

   1. Initial program 87.7%

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

   3. Taylor expanded in 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 84.4%

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

   6. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   8. Applied rewrites 87.6%

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

   9. Taylor expanded in x around inf

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

       1. associate-+l- N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

       7. associate-/l/ N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. associate-/l* N/A

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

       11. lower-*.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lower-*.f64 62.4

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

   11. Applied rewrites 62.4%

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

   if -1.55e-136 < z < 3.80000000000000025e36

   1. Initial program 96.3%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 58.5%

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

   if 3.80000000000000025e36 < z

   1. Initial program 62.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 73.1%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

         \[\leadsto \frac{\left(t \cdot a\right) \cdot -4}{c} \]

      4. lower-*.f64 66.8

         \[\leadsto \frac{\left(t \cdot a\right) \cdot -4}{c} \]

   7. Applied rewrites 66.8%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+55}:\\ \;\;\;\;-4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{-136}:\\ \;\;\;\;\left(x \cdot \frac{y}{c \cdot z}\right) \cdot 9\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+36}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t \cdot a\right) \cdot -4}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+55}:\\ \;\;\;\;-4 \cdot \frac{a \cdot t}{c\_m}\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{-136}:\\ \;\;\;\;\left(\frac{y}{c\_m \cdot z} \cdot 9\right) \cdot x\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+36}:\\ \;\;\;\;\frac{b}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t \cdot a\right) \cdot -4}{c\_m}\\ \end{array} \end{array} \]

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.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= z -3.1e+55)
    (* -4.0 (/ (* a t) c_m))
    (if (<= z -1.55e-136)
      (* (* (/ y (* c_m z)) 9.0) x)
      (if (<= z 3.8e+36) (/ b (* z c_m)) (/ (* (* t a) -4.0) c_m))))))

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);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (z <= -3.1e+55) {
		tmp = -4.0 * ((a * t) / c_m);
	} else if (z <= -1.55e-136) {
		tmp = ((y / (c_m * z)) * 9.0) * x;
	} else if (z <= 3.8e+36) {
		tmp = b / (z * c_m);
	} else {
		tmp = ((t * a) * -4.0) / c_m;
	}
	return c_s * tmp;
}

Fortran

c\_m =     private
c\_s =     private
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c_s, x, y, z, t, a, b, c_m)
use fmin_fmax_functions
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: tmp
    if (z <= (-3.1d+55)) then
        tmp = (-4.0d0) * ((a * t) / c_m)
    else if (z <= (-1.55d-136)) then
        tmp = ((y / (c_m * z)) * 9.0d0) * x
    else if (z <= 3.8d+36) then
        tmp = b / (z * c_m)
    else
        tmp = ((t * a) * (-4.0d0)) / c_m
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (z <= -3.1e+55) {
		tmp = -4.0 * ((a * t) / c_m);
	} else if (z <= -1.55e-136) {
		tmp = ((y / (c_m * z)) * 9.0) * x;
	} else if (z <= 3.8e+36) {
		tmp = b / (z * c_m);
	} else {
		tmp = ((t * a) * -4.0) / c_m;
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if z <= -3.1e+55:
		tmp = -4.0 * ((a * t) / c_m)
	elif z <= -1.55e-136:
		tmp = ((y / (c_m * z)) * 9.0) * x
	elif z <= 3.8e+36:
		tmp = b / (z * c_m)
	else:
		tmp = ((t * a) * -4.0) / c_m
	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])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (z <= -3.1e+55)
		tmp = Float64(-4.0 * Float64(Float64(a * t) / c_m));
	elseif (z <= -1.55e-136)
		tmp = Float64(Float64(Float64(y / Float64(c_m * z)) * 9.0) * x);
	elseif (z <= 3.8e+36)
		tmp = Float64(b / Float64(z * c_m));
	else
		tmp = Float64(Float64(Float64(t * a) * -4.0) / c_m);
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if (z <= -3.1e+55)
		tmp = -4.0 * ((a * t) / c_m);
	elseif (z <= -1.55e-136)
		tmp = ((y / (c_m * z)) * 9.0) * x;
	elseif (z <= 3.8e+36)
		tmp = b / (z * c_m);
	else
		tmp = ((t * a) * -4.0) / c_m;
	end
	tmp_2 = 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.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[z, -3.1e+55], N[(-4.0 * N[(N[(a * t), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.55e-136], N[(N[(N[(y / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision] * 9.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 3.8e+36], N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t * a), $MachinePrecision] * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if z < -3.09999999999999994e55

   1. Initial program 57.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\frac{a \cdot t}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto -4 \cdot \frac{a \cdot t}{\color{blue}{c}} \]

      3. lower-*.f64 56.0

         \[\leadsto -4 \cdot \frac{a \cdot t}{c} \]

   5. Applied rewrites 56.0%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]

   if -3.09999999999999994e55 < z < -1.55e-136

   1. Initial program 87.7%

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

   3. Taylor expanded in 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 84.4%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 62.3

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

   8. Applied rewrites 62.3%

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

   if -1.55e-136 < z < 3.80000000000000025e36

   1. Initial program 96.3%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 58.5%

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

   if 3.80000000000000025e36 < z

   1. Initial program 62.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 73.1%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

         \[\leadsto \frac{\left(t \cdot a\right) \cdot -4}{c} \]

      4. lower-*.f64 66.8

         \[\leadsto \frac{\left(t \cdot a\right) \cdot -4}{c} \]

   7. Applied rewrites 66.8%

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

Alternative 11: 68.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{+149}:\\ \;\;\;\;\left(\frac{a}{c\_m} \cdot -4\right) \cdot t\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t \cdot a\right) \cdot -4}{c\_m}\\ \end{array} \end{array} \]

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.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= z -2.15e+149)
    (* (* (/ a c_m) -4.0) t)
    (if (<= z 5.2e+36)
      (/ (fma (* y x) 9.0 b) (* z c_m))
      (/ (* (* t a) -4.0) c_m)))))

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);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (z <= -2.15e+149) {
		tmp = ((a / c_m) * -4.0) * t;
	} else if (z <= 5.2e+36) {
		tmp = fma((y * x), 9.0, b) / (z * c_m);
	} else {
		tmp = ((t * a) * -4.0) / c_m;
	}
	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])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (z <= -2.15e+149)
		tmp = Float64(Float64(Float64(a / c_m) * -4.0) * t);
	elseif (z <= 5.2e+36)
		tmp = Float64(fma(Float64(y * x), 9.0, b) / Float64(z * c_m));
	else
		tmp = Float64(Float64(Float64(t * a) * -4.0) / c_m);
	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.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[z, -2.15e+149], N[(N[(N[(a / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[z, 5.2e+36], N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t * a), $MachinePrecision] * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -2.15000000000000008e149

   1. Initial program 57.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 76.7%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{a}{c} \cdot -4\right) \cdot t \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{a}{c} \cdot -4\right) \cdot t \]

      3. lift-/.f64 71.5

         \[\leadsto \left(\frac{a}{c} \cdot -4\right) \cdot t \]

   8. Applied rewrites 71.5%

      \[\leadsto \left(\frac{a}{c} \cdot -4\right) \cdot t \]

   if -2.15000000000000008e149 < z < 5.2000000000000003e36

   1. Initial program 89.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 78.6

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

   5. Applied rewrites 78.6%

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

   if 5.2000000000000003e36 < z

   1. Initial program 62.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 73.1%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

         \[\leadsto \frac{\left(t \cdot a\right) \cdot -4}{c} \]

      4. lower-*.f64 66.8

         \[\leadsto \frac{\left(t \cdot a\right) \cdot -4}{c} \]

   7. Applied rewrites 66.8%

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

Alternative 12: 68.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{+149}:\\ \;\;\;\;\left(\frac{a}{c\_m} \cdot -4\right) \cdot t\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t \cdot a\right) \cdot -4}{c\_m}\\ \end{array} \end{array} \]

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.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= z -2.15e+149)
    (* (* (/ a c_m) -4.0) t)
    (if (<= z 5.2e+36)
      (/ (fma (* 9.0 x) y b) (* z c_m))
      (/ (* (* t a) -4.0) c_m)))))

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);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (z <= -2.15e+149) {
		tmp = ((a / c_m) * -4.0) * t;
	} else if (z <= 5.2e+36) {
		tmp = fma((9.0 * x), y, b) / (z * c_m);
	} else {
		tmp = ((t * a) * -4.0) / c_m;
	}
	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])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (z <= -2.15e+149)
		tmp = Float64(Float64(Float64(a / c_m) * -4.0) * t);
	elseif (z <= 5.2e+36)
		tmp = Float64(fma(Float64(9.0 * x), y, b) / Float64(z * c_m));
	else
		tmp = Float64(Float64(Float64(t * a) * -4.0) / c_m);
	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.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[z, -2.15e+149], N[(N[(N[(a / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[z, 5.2e+36], N[(N[(N[(9.0 * x), $MachinePrecision] * y + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t * a), $MachinePrecision] * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -2.15000000000000008e149

   1. Initial program 57.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 76.7%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{a}{c} \cdot -4\right) \cdot t \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{a}{c} \cdot -4\right) \cdot t \]

      3. lift-/.f64 71.5

         \[\leadsto \left(\frac{a}{c} \cdot -4\right) \cdot t \]

   8. Applied rewrites 71.5%

      \[\leadsto \left(\frac{a}{c} \cdot -4\right) \cdot t \]

   if -2.15000000000000008e149 < z < 5.2000000000000003e36

   1. Initial program 89.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 78.6

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

   5. Applied rewrites 78.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 78.6

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

   7. Applied rewrites 78.6%

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

   if 5.2000000000000003e36 < z

   1. Initial program 62.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 73.1%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

         \[\leadsto \frac{\left(t \cdot a\right) \cdot -4}{c} \]

      4. lower-*.f64 66.8

         \[\leadsto \frac{\left(t \cdot a\right) \cdot -4}{c} \]

   7. Applied rewrites 66.8%

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

Alternative 13: 50.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-7} \lor \neg \left(z \leq 3.8 \cdot 10^{+36}\right):\\ \;\;\;\;-4 \cdot \frac{a \cdot t}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{z \cdot c\_m}\\ \end{array} \end{array} \]

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.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (or (<= z -1.1e-7) (not (<= z 3.8e+36)))
    (* -4.0 (/ (* a t) c_m))
    (/ b (* z c_m)))))

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);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if ((z <= -1.1e-7) || !(z <= 3.8e+36)) {
		tmp = -4.0 * ((a * t) / c_m);
	} else {
		tmp = b / (z * c_m);
	}
	return c_s * tmp;
}

Fortran

c\_m =     private
c\_s =     private
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c_s, x, y, z, t, a, b, c_m)
use fmin_fmax_functions
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: tmp
    if ((z <= (-1.1d-7)) .or. (.not. (z <= 3.8d+36))) then
        tmp = (-4.0d0) * ((a * t) / c_m)
    else
        tmp = b / (z * c_m)
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if ((z <= -1.1e-7) || !(z <= 3.8e+36)) {
		tmp = -4.0 * ((a * t) / c_m);
	} else {
		tmp = b / (z * c_m);
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if (z <= -1.1e-7) or not (z <= 3.8e+36):
		tmp = -4.0 * ((a * t) / c_m)
	else:
		tmp = b / (z * c_m)
	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])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if ((z <= -1.1e-7) || !(z <= 3.8e+36))
		tmp = Float64(-4.0 * Float64(Float64(a * t) / c_m));
	else
		tmp = Float64(b / Float64(z * c_m));
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if ((z <= -1.1e-7) || ~((z <= 3.8e+36)))
		tmp = -4.0 * ((a * t) / c_m);
	else
		tmp = b / (z * c_m);
	end
	tmp_2 = 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.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[Or[LessEqual[z, -1.1e-7], N[Not[LessEqual[z, 3.8e+36]], $MachinePrecision]], N[(-4.0 * N[(N[(a * t), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision], N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.1000000000000001e-7 or 3.80000000000000025e36 < z

   1. Initial program 61.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\frac{a \cdot t}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto -4 \cdot \frac{a \cdot t}{\color{blue}{c}} \]

      3. lower-*.f64 57.8

         \[\leadsto -4 \cdot \frac{a \cdot t}{c} \]

   5. Applied rewrites 57.8%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]

   if -1.1000000000000001e-7 < z < 3.80000000000000025e36

   1. Initial program 95.7%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 54.9%

      \[\leadsto \frac{\color{blue}{b}}{z \cdot c} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-7} \lor \neg \left(z \leq 3.8 \cdot 10^{+36}\right):\\ \;\;\;\;-4 \cdot \frac{a \cdot t}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-7}:\\ \;\;\;\;-4 \cdot \frac{a \cdot t}{c\_m}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+36}:\\ \;\;\;\;\frac{b}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t \cdot a\right) \cdot -4}{c\_m}\\ \end{array} \end{array} \]

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.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= z -1.1e-7)
    (* -4.0 (/ (* a t) c_m))
    (if (<= z 3.8e+36) (/ b (* z c_m)) (/ (* (* t a) -4.0) c_m)))))

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);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (z <= -1.1e-7) {
		tmp = -4.0 * ((a * t) / c_m);
	} else if (z <= 3.8e+36) {
		tmp = b / (z * c_m);
	} else {
		tmp = ((t * a) * -4.0) / c_m;
	}
	return c_s * tmp;
}

Fortran

c\_m =     private
c\_s =     private
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c_s, x, y, z, t, a, b, c_m)
use fmin_fmax_functions
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: tmp
    if (z <= (-1.1d-7)) then
        tmp = (-4.0d0) * ((a * t) / c_m)
    else if (z <= 3.8d+36) then
        tmp = b / (z * c_m)
    else
        tmp = ((t * a) * (-4.0d0)) / c_m
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (z <= -1.1e-7) {
		tmp = -4.0 * ((a * t) / c_m);
	} else if (z <= 3.8e+36) {
		tmp = b / (z * c_m);
	} else {
		tmp = ((t * a) * -4.0) / c_m;
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if z <= -1.1e-7:
		tmp = -4.0 * ((a * t) / c_m)
	elif z <= 3.8e+36:
		tmp = b / (z * c_m)
	else:
		tmp = ((t * a) * -4.0) / c_m
	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])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (z <= -1.1e-7)
		tmp = Float64(-4.0 * Float64(Float64(a * t) / c_m));
	elseif (z <= 3.8e+36)
		tmp = Float64(b / Float64(z * c_m));
	else
		tmp = Float64(Float64(Float64(t * a) * -4.0) / c_m);
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if (z <= -1.1e-7)
		tmp = -4.0 * ((a * t) / c_m);
	elseif (z <= 3.8e+36)
		tmp = b / (z * c_m);
	else
		tmp = ((t * a) * -4.0) / c_m;
	end
	tmp_2 = 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.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[z, -1.1e-7], N[(-4.0 * N[(N[(a * t), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.8e+36], N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t * a), $MachinePrecision] * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1.1000000000000001e-7

   1. Initial program 59.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\frac{a \cdot t}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto -4 \cdot \frac{a \cdot t}{\color{blue}{c}} \]

      3. lower-*.f64 51.8

         \[\leadsto -4 \cdot \frac{a \cdot t}{c} \]

   5. Applied rewrites 51.8%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]

   if -1.1000000000000001e-7 < z < 3.80000000000000025e36

   1. Initial program 95.7%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 54.9%

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

   if 3.80000000000000025e36 < z

   1. Initial program 62.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 73.1%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

         \[\leadsto \frac{\left(t \cdot a\right) \cdot -4}{c} \]

      4. lower-*.f64 66.8

         \[\leadsto \frac{\left(t \cdot a\right) \cdot -4}{c} \]

   7. Applied rewrites 66.8%

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

Alternative 15: 35.0% accurate, 2.8× speedup

Math

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

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.
(FPCore (c_s x y z t a b c_m) :precision binary64 (* c_s (/ b (* z c_m))))

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);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	return c_s * (b / (z * c_m));
}

Fortran

c\_m =     private
c\_s =     private
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c_s, x, y, z, t, a, b, c_m)
use fmin_fmax_functions
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    code = c_s * (b / (z * c_m))
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	return c_s * (b / (z * c_m));
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	return c_s * (b / (z * c_m))

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])
function code(c_s, x, y, z, t, a, b, c_m)
	return Float64(c_s * Float64(b / Float64(z * c_m)))
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp = code(c_s, x, y, z, t, a, b, c_m)
	tmp = c_s * (b / (z * c_m));
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.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.9%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 38.6%

   \[\leadsto \frac{\color{blue}{b}}{z \cdot c} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025085 
(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)))