ABCF->ab-angle a

Percentage Accurate: 18.7% → 64.1%

Time: 16.4s

Alternatives: 15

Speedup: 18.2×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ \frac{-\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (pow B 2.0) (* (* 4.0 A) C))))
   (/
    (-
     (sqrt
      (*
       (* 2.0 (* t_0 F))
       (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
    t_0)))

C

double code(double A, double B, double C, double F) {
	double t_0 = pow(B, 2.0) - ((4.0 * A) * C);
	return -sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / t_0;
}

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(a, b, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    t_0 = (b ** 2.0d0) - ((4.0d0 * a) * c)
    code = -sqrt(((2.0d0 * (t_0 * f)) * ((a + c) + sqrt((((a - c) ** 2.0d0) + (b ** 2.0d0)))))) / t_0
end function

Java

public static double code(double A, double B, double C, double F) {
	double t_0 = Math.pow(B, 2.0) - ((4.0 * A) * C);
	return -Math.sqrt(((2.0 * (t_0 * F)) * ((A + C) + Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B, 2.0)))))) / t_0;
}

Python

def code(A, B, C, F):
	t_0 = math.pow(B, 2.0) - ((4.0 * A) * C)
	return -math.sqrt(((2.0 * (t_0 * F)) * ((A + C) + math.sqrt((math.pow((A - C), 2.0) + math.pow(B, 2.0)))))) / t_0

Julia

function code(A, B, C, F)
	t_0 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	return Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0))))))) / t_0)
end

MATLAB

function tmp = code(A, B, C, F)
	t_0 = (B ^ 2.0) - ((4.0 * A) * C);
	tmp = -sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((((A - C) ^ 2.0) + (B ^ 2.0)))))) / t_0;
end

Wolfram

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]

Initial Program: 18.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ \frac{-\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (pow B 2.0) (* (* 4.0 A) C))))
   (/
    (-
     (sqrt
      (*
       (* 2.0 (* t_0 F))
       (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
    t_0)))

C

double code(double A, double B, double C, double F) {
	double t_0 = pow(B, 2.0) - ((4.0 * A) * C);
	return -sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / t_0;
}

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(a, b, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    t_0 = (b ** 2.0d0) - ((4.0d0 * a) * c)
    code = -sqrt(((2.0d0 * (t_0 * f)) * ((a + c) + sqrt((((a - c) ** 2.0d0) + (b ** 2.0d0)))))) / t_0
end function

Java

public static double code(double A, double B, double C, double F) {
	double t_0 = Math.pow(B, 2.0) - ((4.0 * A) * C);
	return -Math.sqrt(((2.0 * (t_0 * F)) * ((A + C) + Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B, 2.0)))))) / t_0;
}

Python

def code(A, B, C, F):
	t_0 = math.pow(B, 2.0) - ((4.0 * A) * C)
	return -math.sqrt(((2.0 * (t_0 * F)) * ((A + C) + math.sqrt((math.pow((A - C), 2.0) + math.pow(B, 2.0)))))) / t_0

Julia

function code(A, B, C, F)
	t_0 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	return Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0))))))) / t_0)
end

MATLAB

function tmp = code(A, B, C, F)
	t_0 = (B ^ 2.0) - ((4.0 * A) * C);
	tmp = -sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((((A - C) ^ 2.0) + (B ^ 2.0)))))) / t_0;
end

Wolfram

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]

Alternative 1: 64.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_1 := \frac{B\_m \cdot B\_m}{A}\\ t_2 := \mathsf{fma}\left(t\_1, -0.5, C \cdot 2\right)\\ t_3 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_4 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_5 := -t\_4\\ t_6 := \frac{\sqrt{\left(2 \cdot \left(t\_4 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_5}\\ \mathbf{if}\;t\_6 \leq -\infty:\\ \;\;\;\;\frac{\left(\sqrt{t\_0} \cdot \sqrt{t\_2}\right) \cdot \sqrt{F \cdot 2}}{-t\_0}\\ \mathbf{elif}\;t\_6 \leq -4 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(B\_m, A - C\right) + A\right) + C\right) \cdot 2} \cdot \sqrt{F \cdot t\_3}}{t\_5}\\ \mathbf{elif}\;t\_6 \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, t\_1, C \cdot 2\right) \cdot \left(t\_0 \cdot 2\right)} \cdot \left(-\sqrt{F}\right)}{t\_4}\\ \mathbf{elif}\;t\_6 \leq \infty:\\ \;\;\;\;\frac{\left(-\sqrt{t\_3 \cdot F}\right) \cdot \sqrt{2 \cdot t\_2}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot 2}}{-B\_m} \cdot \sqrt{F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* C A) -4.0 (* B_m B_m)))
        (t_1 (/ (* B_m B_m) A))
        (t_2 (fma t_1 -0.5 (* C 2.0)))
        (t_3 (fma -4.0 (* C A) (* B_m B_m)))
        (t_4 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_5 (- t_4))
        (t_6
         (/
          (sqrt
           (*
            (* 2.0 (* t_4 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          t_5)))
   (if (<= t_6 (- INFINITY))
     (/ (* (* (sqrt t_0) (sqrt t_2)) (sqrt (* F 2.0))) (- t_0))
     (if (<= t_6 -4e-195)
       (/
        (* (sqrt (* (+ (+ (hypot B_m (- A C)) A) C) 2.0)) (sqrt (* F t_3)))
        t_5)
       (if (<= t_6 0.0)
         (/
          (* (sqrt (* (fma -0.5 t_1 (* C 2.0)) (* t_0 2.0))) (- (sqrt F)))
          t_4)
         (if (<= t_6 INFINITY)
           (/ (* (- (sqrt (* t_3 F))) (sqrt (* 2.0 t_2))) t_0)
           (* (/ (sqrt (* (+ (hypot C B_m) C) 2.0)) (- B_m)) (sqrt F))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((C * A), -4.0, (B_m * B_m));
	double t_1 = (B_m * B_m) / A;
	double t_2 = fma(t_1, -0.5, (C * 2.0));
	double t_3 = fma(-4.0, (C * A), (B_m * B_m));
	double t_4 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_5 = -t_4;
	double t_6 = sqrt(((2.0 * (t_4 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_5;
	double tmp;
	if (t_6 <= -((double) INFINITY)) {
		tmp = ((sqrt(t_0) * sqrt(t_2)) * sqrt((F * 2.0))) / -t_0;
	} else if (t_6 <= -4e-195) {
		tmp = (sqrt((((hypot(B_m, (A - C)) + A) + C) * 2.0)) * sqrt((F * t_3))) / t_5;
	} else if (t_6 <= 0.0) {
		tmp = (sqrt((fma(-0.5, t_1, (C * 2.0)) * (t_0 * 2.0))) * -sqrt(F)) / t_4;
	} else if (t_6 <= ((double) INFINITY)) {
		tmp = (-sqrt((t_3 * F)) * sqrt((2.0 * t_2))) / t_0;
	} else {
		tmp = (sqrt(((hypot(C, B_m) + C) * 2.0)) / -B_m) * sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_1 = Float64(Float64(B_m * B_m) / A)
	t_2 = fma(t_1, -0.5, Float64(C * 2.0))
	t_3 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_4 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_5 = Float64(-t_4)
	t_6 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_4 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / t_5)
	tmp = 0.0
	if (t_6 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(sqrt(t_0) * sqrt(t_2)) * sqrt(Float64(F * 2.0))) / Float64(-t_0));
	elseif (t_6 <= -4e-195)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(hypot(B_m, Float64(A - C)) + A) + C) * 2.0)) * sqrt(Float64(F * t_3))) / t_5);
	elseif (t_6 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(fma(-0.5, t_1, Float64(C * 2.0)) * Float64(t_0 * 2.0))) * Float64(-sqrt(F))) / t_4);
	elseif (t_6 <= Inf)
		tmp = Float64(Float64(Float64(-sqrt(Float64(t_3 * F))) * sqrt(Float64(2.0 * t_2))) / t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(hypot(C, B_m) + C) * 2.0)) / Float64(-B_m)) * sqrt(F));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * -0.5 + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = (-t$95$4)}, Block[{t$95$6 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$4 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision]}, If[LessEqual[t$95$6, (-Infinity)], N[(N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$6, -4e-195], N[(N[(N[Sqrt[N[(N[(N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision], If[LessEqual[t$95$6, 0.0], N[(N[(N[Sqrt[N[(N[(-0.5 * t$95$1 + N[(C * 2.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$6, Infinity], N[(N[((-N[Sqrt[N[(t$95$3 * F), $MachinePrecision]], $MachinePrecision]) * N[Sqrt[N[(2.0 * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0

   1. Initial program 3.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\frac{{B}^{2}}{A} \cdot \frac{-1}{2}} + 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{A}}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 19.8

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 19.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied rewrites 19.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \color{blue}{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right)} \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \color{blue}{\left(F \cdot 2\right)}\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      8. sqrt-prod N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{F \cdot 2}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   9. Applied rewrites 29.7%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{F \cdot 2}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       2. pow1/2 N/A

          \[\leadsto \frac{\color{blue}{{\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}^{\frac{1}{2}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       4. *-commutative N/A

          \[\leadsto \frac{{\color{blue}{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right)\right)}}^{\frac{1}{2}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       5. unpow-prod-down N/A

          \[\leadsto \frac{\color{blue}{\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right)\right)}^{\frac{1}{2}}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right)\right)}^{\frac{1}{2}}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   11. Applied rewrites 42.3%

       \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.0000000000000004e-195

   1. Initial program 95.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{-\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \color{blue}{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*r* N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(\left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot 2\right) \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{-\color{blue}{\sqrt{\left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot 2} \cdot \sqrt{\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. pow1/2 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot 2} \cdot \color{blue}{{\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)}^{\frac{1}{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{-\color{blue}{\sqrt{\left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot 2} \cdot {\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)}^{\frac{1}{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied rewrites 96.2%

      \[\leadsto \frac{-\color{blue}{\sqrt{\left(\left(\mathsf{hypot}\left(B, A - C\right) + A\right) + C\right) \cdot 2} \cdot \sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   if -4.0000000000000004e-195 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\frac{{B}^{2}}{A} \cdot \frac{-1}{2}} + 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{A}}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 50.4

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 50.4%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{-\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right) \cdot \left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right) \cdot \color{blue}{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right) \cdot \left(2 \cdot \color{blue}{\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. associate-*r* N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right) \cdot \color{blue}{\left(\left(2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right) \cdot F\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. associate-*r* N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right) \cdot \left(2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right) \cdot F}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 40.6%

      \[\leadsto \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)} \cdot \sqrt{F}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   if -0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 35.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\frac{{B}^{2}}{A} \cdot \frac{-1}{2}} + 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{A}}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 40.8

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 40.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]

   9. Applied rewrites 41.3%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F} \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      12. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{B \cdot B} + {C}^{2}} + C\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{B \cdot B + \color{blue}{C \cdot C}} + C\right) \cdot F} \]

      14. lower-hypot.f64 19.0

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(B, C\right)} + C\right) \cdot F} \]

   5. Applied rewrites 19.0%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(B, C\right) + C\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 29.5%

      \[\leadsto \left(\frac{\sqrt{2}}{-B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \color{blue}{\sqrt{F}} \]
   8. Step-by-step derivation

   9. Applied rewrites 29.5%

      \[\leadsto \frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \color{blue}{\sqrt{F}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -\infty:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}\right) \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -4 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(B, A - C\right) + A\right) + C\right) \cdot 2} \cdot \sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)} \cdot \left(-\sqrt{F}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\left(-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_2 := \mathsf{hypot}\left(C, B\_m\right) + C\\ t_3 := \mathsf{fma}\left(\frac{B\_m \cdot B\_m}{A}, -0.5, C \cdot 2\right)\\ t_4 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_5 := \frac{\sqrt{\left(2 \cdot \left(t\_4 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_4}\\ t_6 := -t\_1\\ \mathbf{if}\;t\_5 \leq -5 \cdot 10^{+172}:\\ \;\;\;\;\frac{\left(\sqrt{t\_1} \cdot \sqrt{t\_3}\right) \cdot \sqrt{F \cdot 2}}{t\_6}\\ \mathbf{elif}\;t\_5 \leq -4 \cdot 10^{-129}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{\left(\left(\mathsf{hypot}\left(A - C, B\_m\right) + C\right) + A\right) \cdot F}{t\_0}}\\ \mathbf{elif}\;t\_5 \leq -1.5 \cdot 10^{-188}:\\ \;\;\;\;\frac{\sqrt{\left(t\_2 \cdot F\right) \cdot 2}}{-B\_m}\\ \mathbf{elif}\;t\_5 \leq 2 \cdot 10^{+159}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(\left(F \cdot 2\right) \cdot t\_3\right)}}{t\_6}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;\frac{\left(-\sqrt{t\_0 \cdot F}\right) \cdot \sqrt{2 \cdot t\_3}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot 2}}{-B\_m} \cdot \sqrt{F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (fma (* C A) -4.0 (* B_m B_m)))
        (t_2 (+ (hypot C B_m) C))
        (t_3 (fma (/ (* B_m B_m) A) -0.5 (* C 2.0)))
        (t_4 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_5
         (/
          (sqrt
           (*
            (* 2.0 (* t_4 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_4)))
        (t_6 (- t_1)))
   (if (<= t_5 -5e+172)
     (/ (* (* (sqrt t_1) (sqrt t_3)) (sqrt (* F 2.0))) t_6)
     (if (<= t_5 -4e-129)
       (* (- (sqrt 2.0)) (sqrt (/ (* (+ (+ (hypot (- A C) B_m) C) A) F) t_0)))
       (if (<= t_5 -1.5e-188)
         (/ (sqrt (* (* t_2 F) 2.0)) (- B_m))
         (if (<= t_5 2e+159)
           (/ (sqrt (* t_0 (* (* F 2.0) t_3))) t_6)
           (if (<= t_5 INFINITY)
             (/ (* (- (sqrt (* t_0 F))) (sqrt (* 2.0 t_3))) t_1)
             (* (/ (sqrt (* t_2 2.0)) (- B_m)) (sqrt F)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = fma((C * A), -4.0, (B_m * B_m));
	double t_2 = hypot(C, B_m) + C;
	double t_3 = fma(((B_m * B_m) / A), -0.5, (C * 2.0));
	double t_4 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_5 = sqrt(((2.0 * (t_4 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_4;
	double t_6 = -t_1;
	double tmp;
	if (t_5 <= -5e+172) {
		tmp = ((sqrt(t_1) * sqrt(t_3)) * sqrt((F * 2.0))) / t_6;
	} else if (t_5 <= -4e-129) {
		tmp = -sqrt(2.0) * sqrt(((((hypot((A - C), B_m) + C) + A) * F) / t_0));
	} else if (t_5 <= -1.5e-188) {
		tmp = sqrt(((t_2 * F) * 2.0)) / -B_m;
	} else if (t_5 <= 2e+159) {
		tmp = sqrt((t_0 * ((F * 2.0) * t_3))) / t_6;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = (-sqrt((t_0 * F)) * sqrt((2.0 * t_3))) / t_1;
	} else {
		tmp = (sqrt((t_2 * 2.0)) / -B_m) * sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_2 = Float64(hypot(C, B_m) + C)
	t_3 = fma(Float64(Float64(B_m * B_m) / A), -0.5, Float64(C * 2.0))
	t_4 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_5 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_4 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_4))
	t_6 = Float64(-t_1)
	tmp = 0.0
	if (t_5 <= -5e+172)
		tmp = Float64(Float64(Float64(sqrt(t_1) * sqrt(t_3)) * sqrt(Float64(F * 2.0))) / t_6);
	elseif (t_5 <= -4e-129)
		tmp = Float64(Float64(-sqrt(2.0)) * sqrt(Float64(Float64(Float64(Float64(hypot(Float64(A - C), B_m) + C) + A) * F) / t_0)));
	elseif (t_5 <= -1.5e-188)
		tmp = Float64(sqrt(Float64(Float64(t_2 * F) * 2.0)) / Float64(-B_m));
	elseif (t_5 <= 2e+159)
		tmp = Float64(sqrt(Float64(t_0 * Float64(Float64(F * 2.0) * t_3))) / t_6);
	elseif (t_5 <= Inf)
		tmp = Float64(Float64(Float64(-sqrt(Float64(t_0 * F))) * sqrt(Float64(2.0 * t_3))) / t_1);
	else
		tmp = Float64(Float64(sqrt(Float64(t_2 * 2.0)) / Float64(-B_m)) * sqrt(F));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5 + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$4 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$4)), $MachinePrecision]}, Block[{t$95$6 = (-t$95$1)}, If[LessEqual[t$95$5, -5e+172], N[(N[(N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$6), $MachinePrecision], If[LessEqual[t$95$5, -4e-129], N[((-N[Sqrt[2.0], $MachinePrecision]) * N[Sqrt[N[(N[(N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] + A), $MachinePrecision] * F), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, -1.5e-188], N[(N[Sqrt[N[(N[(t$95$2 * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], If[LessEqual[t$95$5, 2e+159], N[(N[Sqrt[N[(t$95$0 * N[(N[(F * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$6), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(N[((-N[Sqrt[N[(t$95$0 * F), $MachinePrecision]], $MachinePrecision]) * N[Sqrt[N[(2.0 * t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[N[(t$95$2 * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.0000000000000001e172

   1. Initial program 8.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\frac{{B}^{2}}{A} \cdot \frac{-1}{2}} + 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{A}}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 23.8

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 23.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied rewrites 23.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \color{blue}{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right)} \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \color{blue}{\left(F \cdot 2\right)}\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      8. sqrt-prod N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{F \cdot 2}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   9. Applied rewrites 33.1%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{F \cdot 2}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       2. pow1/2 N/A

          \[\leadsto \frac{\color{blue}{{\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}^{\frac{1}{2}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       4. *-commutative N/A

          \[\leadsto \frac{{\color{blue}{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right)\right)}}^{\frac{1}{2}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       5. unpow-prod-down N/A

          \[\leadsto \frac{\color{blue}{\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right)\right)}^{\frac{1}{2}}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right)\right)}^{\frac{1}{2}}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   11. Applied rewrites 45.1%

       \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if -5.0000000000000001e172 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -3.9999999999999997e-129

   1. Initial program 95.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}} \]

      5. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}} \]

   5. Applied rewrites 96.8%

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{\left(\left(\mathsf{hypot}\left(A - C, B\right) + C\right) + A\right) \cdot F}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}} \]

   if -3.9999999999999997e-129 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.50000000000000008e-188

   1. Initial program 99.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      12. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{B \cdot B} + {C}^{2}} + C\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{B \cdot B + \color{blue}{C \cdot C}} + C\right) \cdot F} \]

      14. lower-hypot.f64 16.0

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(B, C\right)} + C\right) \cdot F} \]

   5. Applied rewrites 16.0%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(B, C\right) + C\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 16.2%

      \[\leadsto \frac{\sqrt{\left(\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F\right) \cdot 2}}{\color{blue}{-B}} \]

   if -1.50000000000000008e-188 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 1.9999999999999999e159

   1. Initial program 15.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\frac{{B}^{2}}{A} \cdot \frac{-1}{2}} + 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{A}}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 52.9

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 52.9%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied rewrites 52.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)} \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right)} \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)} \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \color{blue}{\left(F \cdot 2\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      9. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(C \cdot A\right) \cdot -4 + B \cdot B\right)} \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\left(\color{blue}{-4 \cdot \left(C \cdot A\right)} + B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      11. lift-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   9. Applied rewrites 50.8%

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if 1.9999999999999999e159 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 20.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\frac{{B}^{2}}{A} \cdot \frac{-1}{2}} + 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{A}}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 32.5

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 32.5%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied rewrites 32.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]

   9. Applied rewrites 33.1%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F} \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      12. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{B \cdot B} + {C}^{2}} + C\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{B \cdot B + \color{blue}{C \cdot C}} + C\right) \cdot F} \]

      14. lower-hypot.f64 19.0

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(B, C\right)} + C\right) \cdot F} \]

   5. Applied rewrites 19.0%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(B, C\right) + C\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 29.5%

      \[\leadsto \left(\frac{\sqrt{2}}{-B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \color{blue}{\sqrt{F}} \]
   8. Step-by-step derivation

   9. Applied rewrites 29.5%

      \[\leadsto \frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \color{blue}{\sqrt{F}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -5 \cdot 10^{+172}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}\right) \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -4 \cdot 10^{-129}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{\left(\left(\mathsf{hypot}\left(A - C, B\right) + C\right) + A\right) \cdot F}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1.5 \cdot 10^{-188}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F\right) \cdot 2}}{-B}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 2 \cdot 10^{+159}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\left(-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_2 := \frac{B\_m \cdot B\_m}{A}\\ t_3 := \mathsf{fma}\left(t\_2, -0.5, C \cdot 2\right)\\ t_4 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_5 := \frac{\sqrt{\left(2 \cdot \left(t\_4 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_4}\\ \mathbf{if}\;t\_5 \leq -\infty:\\ \;\;\;\;\frac{\left(\sqrt{t\_0} \cdot \sqrt{t\_3}\right) \cdot \sqrt{F \cdot 2}}{-t\_0}\\ \mathbf{elif}\;t\_5 \leq -4 \cdot 10^{-195}:\\ \;\;\;\;\sqrt{\left(2 \cdot F\right) \cdot t\_1} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(B\_m, A - C\right) + A\right) + C}}{-t\_1}\\ \mathbf{elif}\;t\_5 \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, t\_2, C \cdot 2\right) \cdot \left(t\_0 \cdot 2\right)} \cdot \left(-\sqrt{F}\right)}{t\_4}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;\frac{\left(-\sqrt{t\_1 \cdot F}\right) \cdot \sqrt{2 \cdot t\_3}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot 2}}{-B\_m} \cdot \sqrt{F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* C A) -4.0 (* B_m B_m)))
        (t_1 (fma -4.0 (* C A) (* B_m B_m)))
        (t_2 (/ (* B_m B_m) A))
        (t_3 (fma t_2 -0.5 (* C 2.0)))
        (t_4 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_5
         (/
          (sqrt
           (*
            (* 2.0 (* t_4 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_4))))
   (if (<= t_5 (- INFINITY))
     (/ (* (* (sqrt t_0) (sqrt t_3)) (sqrt (* F 2.0))) (- t_0))
     (if (<= t_5 -4e-195)
       (*
        (sqrt (* (* 2.0 F) t_1))
        (/ (sqrt (+ (+ (hypot B_m (- A C)) A) C)) (- t_1)))
       (if (<= t_5 0.0)
         (/
          (* (sqrt (* (fma -0.5 t_2 (* C 2.0)) (* t_0 2.0))) (- (sqrt F)))
          t_4)
         (if (<= t_5 INFINITY)
           (/ (* (- (sqrt (* t_1 F))) (sqrt (* 2.0 t_3))) t_0)
           (* (/ (sqrt (* (+ (hypot C B_m) C) 2.0)) (- B_m)) (sqrt F))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((C * A), -4.0, (B_m * B_m));
	double t_1 = fma(-4.0, (C * A), (B_m * B_m));
	double t_2 = (B_m * B_m) / A;
	double t_3 = fma(t_2, -0.5, (C * 2.0));
	double t_4 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_5 = sqrt(((2.0 * (t_4 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_4;
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = ((sqrt(t_0) * sqrt(t_3)) * sqrt((F * 2.0))) / -t_0;
	} else if (t_5 <= -4e-195) {
		tmp = sqrt(((2.0 * F) * t_1)) * (sqrt(((hypot(B_m, (A - C)) + A) + C)) / -t_1);
	} else if (t_5 <= 0.0) {
		tmp = (sqrt((fma(-0.5, t_2, (C * 2.0)) * (t_0 * 2.0))) * -sqrt(F)) / t_4;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = (-sqrt((t_1 * F)) * sqrt((2.0 * t_3))) / t_0;
	} else {
		tmp = (sqrt(((hypot(C, B_m) + C) * 2.0)) / -B_m) * sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_1 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_2 = Float64(Float64(B_m * B_m) / A)
	t_3 = fma(t_2, -0.5, Float64(C * 2.0))
	t_4 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_5 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_4 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_4))
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(sqrt(t_0) * sqrt(t_3)) * sqrt(Float64(F * 2.0))) / Float64(-t_0));
	elseif (t_5 <= -4e-195)
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * t_1)) * Float64(sqrt(Float64(Float64(hypot(B_m, Float64(A - C)) + A) + C)) / Float64(-t_1)));
	elseif (t_5 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(fma(-0.5, t_2, Float64(C * 2.0)) * Float64(t_0 * 2.0))) * Float64(-sqrt(F))) / t_4);
	elseif (t_5 <= Inf)
		tmp = Float64(Float64(Float64(-sqrt(Float64(t_1 * F))) * sqrt(Float64(2.0 * t_3))) / t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(hypot(C, B_m) + C) * 2.0)) / Float64(-B_m)) * sqrt(F));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * -0.5 + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$4 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$4)), $MachinePrecision]}, If[LessEqual[t$95$5, (-Infinity)], N[(N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$5, -4e-195], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.0], N[(N[(N[Sqrt[N[(N[(-0.5 * t$95$2 + N[(C * 2.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(N[((-N[Sqrt[N[(t$95$1 * F), $MachinePrecision]], $MachinePrecision]) * N[Sqrt[N[(2.0 * t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0

   1. Initial program 3.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\frac{{B}^{2}}{A} \cdot \frac{-1}{2}} + 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{A}}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 19.8

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 19.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied rewrites 19.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \color{blue}{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right)} \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \color{blue}{\left(F \cdot 2\right)}\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      8. sqrt-prod N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{F \cdot 2}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   9. Applied rewrites 29.7%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{F \cdot 2}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       2. pow1/2 N/A

          \[\leadsto \frac{\color{blue}{{\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}^{\frac{1}{2}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       4. *-commutative N/A

          \[\leadsto \frac{{\color{blue}{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right)\right)}}^{\frac{1}{2}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       5. unpow-prod-down N/A

          \[\leadsto \frac{\color{blue}{\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right)\right)}^{\frac{1}{2}}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right)\right)}^{\frac{1}{2}}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   11. Applied rewrites 42.3%

       \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.0000000000000004e-195

   1. Initial program 95.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(B, A - C\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   if -4.0000000000000004e-195 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\frac{{B}^{2}}{A} \cdot \frac{-1}{2}} + 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{A}}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 50.4

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 50.4%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{-\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right) \cdot \left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right) \cdot \color{blue}{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right) \cdot \left(2 \cdot \color{blue}{\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. associate-*r* N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right) \cdot \color{blue}{\left(\left(2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right) \cdot F\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. associate-*r* N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right) \cdot \left(2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right) \cdot F}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 40.6%

      \[\leadsto \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)} \cdot \sqrt{F}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   if -0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 35.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\frac{{B}^{2}}{A} \cdot \frac{-1}{2}} + 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{A}}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 40.8

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 40.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]

   9. Applied rewrites 41.3%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F} \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      12. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{B \cdot B} + {C}^{2}} + C\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{B \cdot B + \color{blue}{C \cdot C}} + C\right) \cdot F} \]

      14. lower-hypot.f64 19.0

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(B, C\right)} + C\right) \cdot F} \]

   5. Applied rewrites 19.0%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(B, C\right) + C\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 29.5%

      \[\leadsto \left(\frac{\sqrt{2}}{-B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \color{blue}{\sqrt{F}} \]
   8. Step-by-step derivation

   9. Applied rewrites 29.5%

      \[\leadsto \frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \color{blue}{\sqrt{F}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -\infty:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}\right) \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -4 \cdot 10^{-195}:\\ \;\;\;\;\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(B, A - C\right) + A\right) + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)} \cdot \left(-\sqrt{F}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\left(-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_2 := \mathsf{fma}\left(\frac{B\_m \cdot B\_m}{A}, -0.5, C \cdot 2\right)\\ t_3 := \sqrt{F \cdot 2}\\ t_4 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_5 := \frac{\sqrt{\left(2 \cdot \left(t\_4 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_4}\\ t_6 := -t\_0\\ \mathbf{if}\;t\_5 \leq -\infty:\\ \;\;\;\;\frac{\left(\sqrt{t\_0} \cdot \sqrt{t\_2}\right) \cdot t\_3}{t\_6}\\ \mathbf{elif}\;t\_5 \leq -4 \cdot 10^{-195}:\\ \;\;\;\;\sqrt{\left(2 \cdot F\right) \cdot t\_1} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(B\_m, A - C\right) + A\right) + C}}{-t\_1}\\ \mathbf{elif}\;t\_5 \leq 0:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot t\_1} \cdot t\_3}{t\_6}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;\frac{\left(-\sqrt{t\_1 \cdot F}\right) \cdot \sqrt{2 \cdot t\_2}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot 2}}{-B\_m} \cdot \sqrt{F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* C A) -4.0 (* B_m B_m)))
        (t_1 (fma -4.0 (* C A) (* B_m B_m)))
        (t_2 (fma (/ (* B_m B_m) A) -0.5 (* C 2.0)))
        (t_3 (sqrt (* F 2.0)))
        (t_4 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_5
         (/
          (sqrt
           (*
            (* 2.0 (* t_4 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_4)))
        (t_6 (- t_0)))
   (if (<= t_5 (- INFINITY))
     (/ (* (* (sqrt t_0) (sqrt t_2)) t_3) t_6)
     (if (<= t_5 -4e-195)
       (*
        (sqrt (* (* 2.0 F) t_1))
        (/ (sqrt (+ (+ (hypot B_m (- A C)) A) C)) (- t_1)))
       (if (<= t_5 0.0)
         (/ (* (sqrt (* t_2 t_1)) t_3) t_6)
         (if (<= t_5 INFINITY)
           (/ (* (- (sqrt (* t_1 F))) (sqrt (* 2.0 t_2))) t_0)
           (* (/ (sqrt (* (+ (hypot C B_m) C) 2.0)) (- B_m)) (sqrt F))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((C * A), -4.0, (B_m * B_m));
	double t_1 = fma(-4.0, (C * A), (B_m * B_m));
	double t_2 = fma(((B_m * B_m) / A), -0.5, (C * 2.0));
	double t_3 = sqrt((F * 2.0));
	double t_4 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_5 = sqrt(((2.0 * (t_4 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_4;
	double t_6 = -t_0;
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = ((sqrt(t_0) * sqrt(t_2)) * t_3) / t_6;
	} else if (t_5 <= -4e-195) {
		tmp = sqrt(((2.0 * F) * t_1)) * (sqrt(((hypot(B_m, (A - C)) + A) + C)) / -t_1);
	} else if (t_5 <= 0.0) {
		tmp = (sqrt((t_2 * t_1)) * t_3) / t_6;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = (-sqrt((t_1 * F)) * sqrt((2.0 * t_2))) / t_0;
	} else {
		tmp = (sqrt(((hypot(C, B_m) + C) * 2.0)) / -B_m) * sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_1 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_2 = fma(Float64(Float64(B_m * B_m) / A), -0.5, Float64(C * 2.0))
	t_3 = sqrt(Float64(F * 2.0))
	t_4 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_5 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_4 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_4))
	t_6 = Float64(-t_0)
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(sqrt(t_0) * sqrt(t_2)) * t_3) / t_6);
	elseif (t_5 <= -4e-195)
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * t_1)) * Float64(sqrt(Float64(Float64(hypot(B_m, Float64(A - C)) + A) + C)) / Float64(-t_1)));
	elseif (t_5 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(t_2 * t_1)) * t_3) / t_6);
	elseif (t_5 <= Inf)
		tmp = Float64(Float64(Float64(-sqrt(Float64(t_1 * F))) * sqrt(Float64(2.0 * t_2))) / t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(hypot(C, B_m) + C) * 2.0)) / Float64(-B_m)) * sqrt(F));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5 + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$4 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$4)), $MachinePrecision]}, Block[{t$95$6 = (-t$95$0)}, If[LessEqual[t$95$5, (-Infinity)], N[(N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] / t$95$6), $MachinePrecision], If[LessEqual[t$95$5, -4e-195], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.0], N[(N[(N[Sqrt[N[(t$95$2 * t$95$1), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision] / t$95$6), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(N[((-N[Sqrt[N[(t$95$1 * F), $MachinePrecision]], $MachinePrecision]) * N[Sqrt[N[(2.0 * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0

   1. Initial program 3.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\frac{{B}^{2}}{A} \cdot \frac{-1}{2}} + 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{A}}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 19.8

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 19.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied rewrites 19.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \color{blue}{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right)} \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \color{blue}{\left(F \cdot 2\right)}\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      8. sqrt-prod N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{F \cdot 2}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   9. Applied rewrites 29.7%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{F \cdot 2}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       2. pow1/2 N/A

          \[\leadsto \frac{\color{blue}{{\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}^{\frac{1}{2}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       4. *-commutative N/A

          \[\leadsto \frac{{\color{blue}{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right)\right)}}^{\frac{1}{2}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       5. unpow-prod-down N/A

          \[\leadsto \frac{\color{blue}{\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right)\right)}^{\frac{1}{2}}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right)\right)}^{\frac{1}{2}}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   11. Applied rewrites 42.3%

       \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.0000000000000004e-195

   1. Initial program 95.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(B, A - C\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   if -4.0000000000000004e-195 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\frac{{B}^{2}}{A} \cdot \frac{-1}{2}} + 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{A}}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 50.4

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 50.4%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \color{blue}{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right)} \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \color{blue}{\left(F \cdot 2\right)}\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      8. sqrt-prod N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{F \cdot 2}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   9. Applied rewrites 40.6%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{F \cdot 2}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if -0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 35.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\frac{{B}^{2}}{A} \cdot \frac{-1}{2}} + 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{A}}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 40.8

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 40.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]

   9. Applied rewrites 41.3%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F} \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      12. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{B \cdot B} + {C}^{2}} + C\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{B \cdot B + \color{blue}{C \cdot C}} + C\right) \cdot F} \]

      14. lower-hypot.f64 19.0

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(B, C\right)} + C\right) \cdot F} \]

   5. Applied rewrites 19.0%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(B, C\right) + C\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 29.5%

      \[\leadsto \left(\frac{\sqrt{2}}{-B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \color{blue}{\sqrt{F}} \]
   8. Step-by-step derivation

   9. Applied rewrites 29.5%

      \[\leadsto \frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \color{blue}{\sqrt{F}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -\infty:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}\right) \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -4 \cdot 10^{-195}:\\ \;\;\;\;\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(B, A - C\right) + A\right) + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\left(-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(\frac{B\_m \cdot B\_m}{A}, -0.5, C \cdot 2\right)\\ t_2 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_3 := \sqrt{F \cdot 2}\\ t_4 := -t\_0\\ t_5 := \left(4 \cdot A\right) \cdot C\\ t_6 := {B\_m}^{2} - t\_5\\ t_7 := \frac{\sqrt{\left(2 \cdot \left(t\_6 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_6}\\ \mathbf{if}\;t\_7 \leq -\infty:\\ \;\;\;\;\frac{\left(\sqrt{t\_0} \cdot \sqrt{t\_1}\right) \cdot t\_3}{t\_4}\\ \mathbf{elif}\;t\_7 \leq -4 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(B\_m, A - C\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot t\_2\right)}}{\mathsf{fma}\left(-B\_m, B\_m, t\_5\right)}\\ \mathbf{elif}\;t\_7 \leq 0:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot t\_2} \cdot t\_3}{t\_4}\\ \mathbf{elif}\;t\_7 \leq \infty:\\ \;\;\;\;\frac{\left(-\sqrt{t\_2 \cdot F}\right) \cdot \sqrt{2 \cdot t\_1}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot 2}}{-B\_m} \cdot \sqrt{F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* C A) -4.0 (* B_m B_m)))
        (t_1 (fma (/ (* B_m B_m) A) -0.5 (* C 2.0)))
        (t_2 (fma -4.0 (* C A) (* B_m B_m)))
        (t_3 (sqrt (* F 2.0)))
        (t_4 (- t_0))
        (t_5 (* (* 4.0 A) C))
        (t_6 (- (pow B_m 2.0) t_5))
        (t_7
         (/
          (sqrt
           (*
            (* 2.0 (* t_6 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_6))))
   (if (<= t_7 (- INFINITY))
     (/ (* (* (sqrt t_0) (sqrt t_1)) t_3) t_4)
     (if (<= t_7 -4e-195)
       (/
        (sqrt (* (+ (+ (hypot B_m (- A C)) A) C) (* (* 2.0 F) t_2)))
        (fma (- B_m) B_m t_5))
       (if (<= t_7 0.0)
         (/ (* (sqrt (* t_1 t_2)) t_3) t_4)
         (if (<= t_7 INFINITY)
           (/ (* (- (sqrt (* t_2 F))) (sqrt (* 2.0 t_1))) t_0)
           (* (/ (sqrt (* (+ (hypot C B_m) C) 2.0)) (- B_m)) (sqrt F))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((C * A), -4.0, (B_m * B_m));
	double t_1 = fma(((B_m * B_m) / A), -0.5, (C * 2.0));
	double t_2 = fma(-4.0, (C * A), (B_m * B_m));
	double t_3 = sqrt((F * 2.0));
	double t_4 = -t_0;
	double t_5 = (4.0 * A) * C;
	double t_6 = pow(B_m, 2.0) - t_5;
	double t_7 = sqrt(((2.0 * (t_6 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_6;
	double tmp;
	if (t_7 <= -((double) INFINITY)) {
		tmp = ((sqrt(t_0) * sqrt(t_1)) * t_3) / t_4;
	} else if (t_7 <= -4e-195) {
		tmp = sqrt((((hypot(B_m, (A - C)) + A) + C) * ((2.0 * F) * t_2))) / fma(-B_m, B_m, t_5);
	} else if (t_7 <= 0.0) {
		tmp = (sqrt((t_1 * t_2)) * t_3) / t_4;
	} else if (t_7 <= ((double) INFINITY)) {
		tmp = (-sqrt((t_2 * F)) * sqrt((2.0 * t_1))) / t_0;
	} else {
		tmp = (sqrt(((hypot(C, B_m) + C) * 2.0)) / -B_m) * sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_1 = fma(Float64(Float64(B_m * B_m) / A), -0.5, Float64(C * 2.0))
	t_2 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_3 = sqrt(Float64(F * 2.0))
	t_4 = Float64(-t_0)
	t_5 = Float64(Float64(4.0 * A) * C)
	t_6 = Float64((B_m ^ 2.0) - t_5)
	t_7 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_6 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_6))
	tmp = 0.0
	if (t_7 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(sqrt(t_0) * sqrt(t_1)) * t_3) / t_4);
	elseif (t_7 <= -4e-195)
		tmp = Float64(sqrt(Float64(Float64(Float64(hypot(B_m, Float64(A - C)) + A) + C) * Float64(Float64(2.0 * F) * t_2))) / fma(Float64(-B_m), B_m, t_5));
	elseif (t_7 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(t_1 * t_2)) * t_3) / t_4);
	elseif (t_7 <= Inf)
		tmp = Float64(Float64(Float64(-sqrt(Float64(t_2 * F))) * sqrt(Float64(2.0 * t_1))) / t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(hypot(C, B_m) + C) * 2.0)) / Float64(-B_m)) * sqrt(F));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5 + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = (-t$95$0)}, Block[{t$95$5 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$6 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$5), $MachinePrecision]}, Block[{t$95$7 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$6 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$6)), $MachinePrecision]}, If[LessEqual[t$95$7, (-Infinity)], N[(N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$7, -4e-195], N[(N[Sqrt[N[(N[(N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[((-B$95$m) * B$95$m + t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$7, 0.0], N[(N[(N[Sqrt[N[(t$95$1 * t$95$2), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$7, Infinity], N[(N[((-N[Sqrt[N[(t$95$2 * F), $MachinePrecision]], $MachinePrecision]) * N[Sqrt[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0

   1. Initial program 3.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\frac{{B}^{2}}{A} \cdot \frac{-1}{2}} + 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{A}}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 19.8

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 19.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied rewrites 19.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \color{blue}{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right)} \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \color{blue}{\left(F \cdot 2\right)}\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      8. sqrt-prod N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{F \cdot 2}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   9. Applied rewrites 29.7%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{F \cdot 2}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       2. pow1/2 N/A

          \[\leadsto \frac{\color{blue}{{\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}^{\frac{1}{2}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       4. *-commutative N/A

          \[\leadsto \frac{{\color{blue}{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right)\right)}}^{\frac{1}{2}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       5. unpow-prod-down N/A

          \[\leadsto \frac{\color{blue}{\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right)\right)}^{\frac{1}{2}}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right)\right)}^{\frac{1}{2}}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   11. Applied rewrites 42.3%

       \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.0000000000000004e-195

   1. Initial program 95.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(\mathsf{hypot}\left(B, A - C\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{\mathsf{fma}\left(-B, B, \left(4 \cdot A\right) \cdot C\right)}} \]

   if -4.0000000000000004e-195 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\frac{{B}^{2}}{A} \cdot \frac{-1}{2}} + 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{A}}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 50.4

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 50.4%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \color{blue}{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right)} \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \color{blue}{\left(F \cdot 2\right)}\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      8. sqrt-prod N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{F \cdot 2}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   9. Applied rewrites 40.6%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{F \cdot 2}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if -0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 35.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\frac{{B}^{2}}{A} \cdot \frac{-1}{2}} + 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{A}}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 40.8

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 40.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]

   9. Applied rewrites 41.3%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F} \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      12. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{B \cdot B} + {C}^{2}} + C\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{B \cdot B + \color{blue}{C \cdot C}} + C\right) \cdot F} \]

      14. lower-hypot.f64 19.0

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(B, C\right)} + C\right) \cdot F} \]

   5. Applied rewrites 19.0%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(B, C\right) + C\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 29.5%

      \[\leadsto \left(\frac{\sqrt{2}}{-B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \color{blue}{\sqrt{F}} \]
   8. Step-by-step derivation

   9. Applied rewrites 29.5%

      \[\leadsto \frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \color{blue}{\sqrt{F}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -\infty:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}\right) \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -4 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(B, A - C\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{\mathsf{fma}\left(-B, B, \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\left(-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot 2}}{-B\_m} \cdot \sqrt{F}\\ t_1 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_2 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_3 := \mathsf{fma}\left(\frac{B\_m \cdot B\_m}{A}, -0.5, C \cdot 2\right)\\ t_4 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_5 := \frac{\sqrt{\left(2 \cdot \left(t\_4 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_4}\\ t_6 := -t\_1\\ \mathbf{if}\;t\_5 \leq -5 \cdot 10^{+172}:\\ \;\;\;\;\frac{\left(\sqrt{t\_1} \cdot \sqrt{t\_3}\right) \cdot \sqrt{F \cdot 2}}{t\_6}\\ \mathbf{elif}\;t\_5 \leq -1.5 \cdot 10^{-188}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_5 \leq 2 \cdot 10^{+159}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(\left(F \cdot 2\right) \cdot t\_3\right)}}{t\_6}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;\frac{\left(-\sqrt{t\_2 \cdot F}\right) \cdot \sqrt{2 \cdot t\_3}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (/ (sqrt (* (+ (hypot C B_m) C) 2.0)) (- B_m)) (sqrt F)))
        (t_1 (fma (* C A) -4.0 (* B_m B_m)))
        (t_2 (fma -4.0 (* C A) (* B_m B_m)))
        (t_3 (fma (/ (* B_m B_m) A) -0.5 (* C 2.0)))
        (t_4 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_5
         (/
          (sqrt
           (*
            (* 2.0 (* t_4 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_4)))
        (t_6 (- t_1)))
   (if (<= t_5 -5e+172)
     (/ (* (* (sqrt t_1) (sqrt t_3)) (sqrt (* F 2.0))) t_6)
     (if (<= t_5 -1.5e-188)
       t_0
       (if (<= t_5 2e+159)
         (/ (sqrt (* t_2 (* (* F 2.0) t_3))) t_6)
         (if (<= t_5 INFINITY)
           (/ (* (- (sqrt (* t_2 F))) (sqrt (* 2.0 t_3))) t_1)
           t_0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (sqrt(((hypot(C, B_m) + C) * 2.0)) / -B_m) * sqrt(F);
	double t_1 = fma((C * A), -4.0, (B_m * B_m));
	double t_2 = fma(-4.0, (C * A), (B_m * B_m));
	double t_3 = fma(((B_m * B_m) / A), -0.5, (C * 2.0));
	double t_4 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_5 = sqrt(((2.0 * (t_4 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_4;
	double t_6 = -t_1;
	double tmp;
	if (t_5 <= -5e+172) {
		tmp = ((sqrt(t_1) * sqrt(t_3)) * sqrt((F * 2.0))) / t_6;
	} else if (t_5 <= -1.5e-188) {
		tmp = t_0;
	} else if (t_5 <= 2e+159) {
		tmp = sqrt((t_2 * ((F * 2.0) * t_3))) / t_6;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = (-sqrt((t_2 * F)) * sqrt((2.0 * t_3))) / t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(sqrt(Float64(Float64(hypot(C, B_m) + C) * 2.0)) / Float64(-B_m)) * sqrt(F))
	t_1 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_2 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_3 = fma(Float64(Float64(B_m * B_m) / A), -0.5, Float64(C * 2.0))
	t_4 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_5 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_4 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_4))
	t_6 = Float64(-t_1)
	tmp = 0.0
	if (t_5 <= -5e+172)
		tmp = Float64(Float64(Float64(sqrt(t_1) * sqrt(t_3)) * sqrt(Float64(F * 2.0))) / t_6);
	elseif (t_5 <= -1.5e-188)
		tmp = t_0;
	elseif (t_5 <= 2e+159)
		tmp = Float64(sqrt(Float64(t_2 * Float64(Float64(F * 2.0) * t_3))) / t_6);
	elseif (t_5 <= Inf)
		tmp = Float64(Float64(Float64(-sqrt(Float64(t_2 * F))) * sqrt(Float64(2.0 * t_3))) / t_1);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5 + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$4 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$4)), $MachinePrecision]}, Block[{t$95$6 = (-t$95$1)}, If[LessEqual[t$95$5, -5e+172], N[(N[(N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$6), $MachinePrecision], If[LessEqual[t$95$5, -1.5e-188], t$95$0, If[LessEqual[t$95$5, 2e+159], N[(N[Sqrt[N[(t$95$2 * N[(N[(F * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$6), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(N[((-N[Sqrt[N[(t$95$2 * F), $MachinePrecision]], $MachinePrecision]) * N[Sqrt[N[(2.0 * t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$0]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.0000000000000001e172

   1. Initial program 8.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\frac{{B}^{2}}{A} \cdot \frac{-1}{2}} + 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{A}}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 23.8

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 23.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied rewrites 23.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \color{blue}{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right)} \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \color{blue}{\left(F \cdot 2\right)}\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      8. sqrt-prod N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{F \cdot 2}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   9. Applied rewrites 33.1%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{F \cdot 2}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       2. pow1/2 N/A

          \[\leadsto \frac{\color{blue}{{\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}^{\frac{1}{2}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       4. *-commutative N/A

          \[\leadsto \frac{{\color{blue}{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right)\right)}}^{\frac{1}{2}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       5. unpow-prod-down N/A

          \[\leadsto \frac{\color{blue}{\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right)\right)}^{\frac{1}{2}}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right)\right)}^{\frac{1}{2}}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   11. Applied rewrites 45.1%

       \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if -5.0000000000000001e172 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.50000000000000008e-188 or +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 22.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      12. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{B \cdot B} + {C}^{2}} + C\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{B \cdot B + \color{blue}{C \cdot C}} + C\right) \cdot F} \]

      14. lower-hypot.f64 21.4

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(B, C\right)} + C\right) \cdot F} \]

   5. Applied rewrites 21.4%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(B, C\right) + C\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 29.4%

      \[\leadsto \left(\frac{\sqrt{2}}{-B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \color{blue}{\sqrt{F}} \]
   8. Step-by-step derivation

   9. Applied rewrites 29.4%

      \[\leadsto \frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \color{blue}{\sqrt{F}} \]

   if -1.50000000000000008e-188 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 1.9999999999999999e159

   1. Initial program 15.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\frac{{B}^{2}}{A} \cdot \frac{-1}{2}} + 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{A}}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 52.9

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 52.9%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied rewrites 52.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)} \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right)} \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)} \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \color{blue}{\left(F \cdot 2\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      9. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(C \cdot A\right) \cdot -4 + B \cdot B\right)} \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\left(\color{blue}{-4 \cdot \left(C \cdot A\right)} + B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      11. lift-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   9. Applied rewrites 50.8%

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if 1.9999999999999999e159 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 20.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\frac{{B}^{2}}{A} \cdot \frac{-1}{2}} + 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{A}}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 32.5

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 32.5%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied rewrites 32.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]

   9. Applied rewrites 33.1%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F} \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -5 \cdot 10^{+172}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}\right) \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1.5 \cdot 10^{-188}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \sqrt{F}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 2 \cdot 10^{+159}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\left(-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_1 := \sqrt{F \cdot 2}\\ t_2 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_3 := \mathsf{fma}\left(\frac{B\_m \cdot B\_m}{A}, -0.5, C \cdot 2\right)\\ t_4 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_5 := \frac{\sqrt{\left(2 \cdot \left(t\_4 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_4}\\ t_6 := -t\_0\\ \mathbf{if}\;t\_5 \leq -5 \cdot 10^{+172}:\\ \;\;\;\;\frac{\left(\sqrt{t\_0} \cdot \sqrt{t\_3}\right) \cdot t\_1}{t\_6}\\ \mathbf{elif}\;t\_5 \leq -1.5 \cdot 10^{-188}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot F\right) \cdot 2}}{-B\_m}\\ \mathbf{elif}\;t\_5 \leq 2 \cdot 10^{+159}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(\left(F \cdot 2\right) \cdot t\_3\right)}}{t\_6}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;\frac{\left(-\sqrt{t\_2 \cdot F}\right) \cdot \sqrt{2 \cdot t\_3}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* C A) -4.0 (* B_m B_m)))
        (t_1 (sqrt (* F 2.0)))
        (t_2 (fma -4.0 (* C A) (* B_m B_m)))
        (t_3 (fma (/ (* B_m B_m) A) -0.5 (* C 2.0)))
        (t_4 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_5
         (/
          (sqrt
           (*
            (* 2.0 (* t_4 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_4)))
        (t_6 (- t_0)))
   (if (<= t_5 -5e+172)
     (/ (* (* (sqrt t_0) (sqrt t_3)) t_1) t_6)
     (if (<= t_5 -1.5e-188)
       (/ (sqrt (* (* (+ (hypot C B_m) C) F) 2.0)) (- B_m))
       (if (<= t_5 2e+159)
         (/ (sqrt (* t_2 (* (* F 2.0) t_3))) t_6)
         (if (<= t_5 INFINITY)
           (/ (* (- (sqrt (* t_2 F))) (sqrt (* 2.0 t_3))) t_0)
           (/ t_1 (- (sqrt B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((C * A), -4.0, (B_m * B_m));
	double t_1 = sqrt((F * 2.0));
	double t_2 = fma(-4.0, (C * A), (B_m * B_m));
	double t_3 = fma(((B_m * B_m) / A), -0.5, (C * 2.0));
	double t_4 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_5 = sqrt(((2.0 * (t_4 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_4;
	double t_6 = -t_0;
	double tmp;
	if (t_5 <= -5e+172) {
		tmp = ((sqrt(t_0) * sqrt(t_3)) * t_1) / t_6;
	} else if (t_5 <= -1.5e-188) {
		tmp = sqrt((((hypot(C, B_m) + C) * F) * 2.0)) / -B_m;
	} else if (t_5 <= 2e+159) {
		tmp = sqrt((t_2 * ((F * 2.0) * t_3))) / t_6;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = (-sqrt((t_2 * F)) * sqrt((2.0 * t_3))) / t_0;
	} else {
		tmp = t_1 / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_1 = sqrt(Float64(F * 2.0))
	t_2 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_3 = fma(Float64(Float64(B_m * B_m) / A), -0.5, Float64(C * 2.0))
	t_4 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_5 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_4 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_4))
	t_6 = Float64(-t_0)
	tmp = 0.0
	if (t_5 <= -5e+172)
		tmp = Float64(Float64(Float64(sqrt(t_0) * sqrt(t_3)) * t_1) / t_6);
	elseif (t_5 <= -1.5e-188)
		tmp = Float64(sqrt(Float64(Float64(Float64(hypot(C, B_m) + C) * F) * 2.0)) / Float64(-B_m));
	elseif (t_5 <= 2e+159)
		tmp = Float64(sqrt(Float64(t_2 * Float64(Float64(F * 2.0) * t_3))) / t_6);
	elseif (t_5 <= Inf)
		tmp = Float64(Float64(Float64(-sqrt(Float64(t_2 * F))) * sqrt(Float64(2.0 * t_3))) / t_0);
	else
		tmp = Float64(t_1 / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5 + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$4 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$4)), $MachinePrecision]}, Block[{t$95$6 = (-t$95$0)}, If[LessEqual[t$95$5, -5e+172], N[(N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] / t$95$6), $MachinePrecision], If[LessEqual[t$95$5, -1.5e-188], N[(N[Sqrt[N[(N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], If[LessEqual[t$95$5, 2e+159], N[(N[Sqrt[N[(t$95$2 * N[(N[(F * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$6), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(N[((-N[Sqrt[N[(t$95$2 * F), $MachinePrecision]], $MachinePrecision]) * N[Sqrt[N[(2.0 * t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(t$95$1 / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.0000000000000001e172

   1. Initial program 8.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\frac{{B}^{2}}{A} \cdot \frac{-1}{2}} + 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{A}}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 23.8

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 23.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied rewrites 23.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \color{blue}{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right)} \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \color{blue}{\left(F \cdot 2\right)}\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      8. sqrt-prod N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{F \cdot 2}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   9. Applied rewrites 33.1%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{F \cdot 2}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       2. pow1/2 N/A

          \[\leadsto \frac{\color{blue}{{\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}^{\frac{1}{2}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       4. *-commutative N/A

          \[\leadsto \frac{{\color{blue}{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right)\right)}}^{\frac{1}{2}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       5. unpow-prod-down N/A

          \[\leadsto \frac{\color{blue}{\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right)\right)}^{\frac{1}{2}}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right)\right)}^{\frac{1}{2}}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   11. Applied rewrites 45.1%

       \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if -5.0000000000000001e172 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.50000000000000008e-188

   1. Initial program 96.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      12. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{B \cdot B} + {C}^{2}} + C\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{B \cdot B + \color{blue}{C \cdot C}} + C\right) \cdot F} \]

      14. lower-hypot.f64 28.9

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(B, C\right)} + C\right) \cdot F} \]

   5. Applied rewrites 28.9%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(B, C\right) + C\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 29.1%

      \[\leadsto \frac{\sqrt{\left(\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F\right) \cdot 2}}{\color{blue}{-B}} \]

   if -1.50000000000000008e-188 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 1.9999999999999999e159

   1. Initial program 15.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\frac{{B}^{2}}{A} \cdot \frac{-1}{2}} + 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{A}}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 52.9

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 52.9%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied rewrites 52.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)} \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right)} \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)} \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \color{blue}{\left(F \cdot 2\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      9. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(C \cdot A\right) \cdot -4 + B \cdot B\right)} \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\left(\color{blue}{-4 \cdot \left(C \cdot A\right)} + B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      11. lift-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   9. Applied rewrites 50.8%

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if 1.9999999999999999e159 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 20.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\frac{{B}^{2}}{A} \cdot \frac{-1}{2}} + 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{A}}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 32.5

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 32.5%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied rewrites 32.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]

   9. Applied rewrites 33.1%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F} \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

      5. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{F}{B}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

      8. lower-/.f64 18.1

         \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 18.1%

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 18.2%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 26.4%

      \[\leadsto \frac{\sqrt{F \cdot 2}}{\color{blue}{-\sqrt{B}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -5 \cdot 10^{+172}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}\right) \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1.5 \cdot 10^{-188}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F\right) \cdot 2}}{-B}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 2 \cdot 10^{+159}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\left(-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_2 := \sqrt{F \cdot 2}\\ t_3 := \mathsf{fma}\left(\frac{B\_m \cdot B\_m}{A}, -0.5, C \cdot 2\right)\\ t_4 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_5 := \frac{\sqrt{\left(2 \cdot \left(t\_4 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_4}\\ t_6 := -t\_0\\ \mathbf{if}\;t\_5 \leq -5 \cdot 10^{+172}:\\ \;\;\;\;\frac{\left(\sqrt{t\_0} \cdot \sqrt{t\_3}\right) \cdot t\_2}{t\_6}\\ \mathbf{elif}\;t\_5 \leq -1.5 \cdot 10^{-188}:\\ \;\;\;\;\left(-\sqrt{\left(2 \cdot F\right) \cdot t\_1}\right) \cdot \frac{\sqrt{B\_m \cdot \left(\frac{A}{B\_m} + 1\right) + C}}{t\_1}\\ \mathbf{elif}\;t\_5 \leq 2 \cdot 10^{+159}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(\left(F \cdot 2\right) \cdot t\_3\right)}}{t\_6}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;\frac{\left(-\sqrt{t\_1 \cdot F}\right) \cdot \sqrt{2 \cdot t\_3}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* C A) -4.0 (* B_m B_m)))
        (t_1 (fma -4.0 (* C A) (* B_m B_m)))
        (t_2 (sqrt (* F 2.0)))
        (t_3 (fma (/ (* B_m B_m) A) -0.5 (* C 2.0)))
        (t_4 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_5
         (/
          (sqrt
           (*
            (* 2.0 (* t_4 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_4)))
        (t_6 (- t_0)))
   (if (<= t_5 -5e+172)
     (/ (* (* (sqrt t_0) (sqrt t_3)) t_2) t_6)
     (if (<= t_5 -1.5e-188)
       (*
        (- (sqrt (* (* 2.0 F) t_1)))
        (/ (sqrt (+ (* B_m (+ (/ A B_m) 1.0)) C)) t_1))
       (if (<= t_5 2e+159)
         (/ (sqrt (* t_1 (* (* F 2.0) t_3))) t_6)
         (if (<= t_5 INFINITY)
           (/ (* (- (sqrt (* t_1 F))) (sqrt (* 2.0 t_3))) t_0)
           (/ t_2 (- (sqrt B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((C * A), -4.0, (B_m * B_m));
	double t_1 = fma(-4.0, (C * A), (B_m * B_m));
	double t_2 = sqrt((F * 2.0));
	double t_3 = fma(((B_m * B_m) / A), -0.5, (C * 2.0));
	double t_4 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_5 = sqrt(((2.0 * (t_4 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_4;
	double t_6 = -t_0;
	double tmp;
	if (t_5 <= -5e+172) {
		tmp = ((sqrt(t_0) * sqrt(t_3)) * t_2) / t_6;
	} else if (t_5 <= -1.5e-188) {
		tmp = -sqrt(((2.0 * F) * t_1)) * (sqrt(((B_m * ((A / B_m) + 1.0)) + C)) / t_1);
	} else if (t_5 <= 2e+159) {
		tmp = sqrt((t_1 * ((F * 2.0) * t_3))) / t_6;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = (-sqrt((t_1 * F)) * sqrt((2.0 * t_3))) / t_0;
	} else {
		tmp = t_2 / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_1 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_2 = sqrt(Float64(F * 2.0))
	t_3 = fma(Float64(Float64(B_m * B_m) / A), -0.5, Float64(C * 2.0))
	t_4 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_5 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_4 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_4))
	t_6 = Float64(-t_0)
	tmp = 0.0
	if (t_5 <= -5e+172)
		tmp = Float64(Float64(Float64(sqrt(t_0) * sqrt(t_3)) * t_2) / t_6);
	elseif (t_5 <= -1.5e-188)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * F) * t_1))) * Float64(sqrt(Float64(Float64(B_m * Float64(Float64(A / B_m) + 1.0)) + C)) / t_1));
	elseif (t_5 <= 2e+159)
		tmp = Float64(sqrt(Float64(t_1 * Float64(Float64(F * 2.0) * t_3))) / t_6);
	elseif (t_5 <= Inf)
		tmp = Float64(Float64(Float64(-sqrt(Float64(t_1 * F))) * sqrt(Float64(2.0 * t_3))) / t_0);
	else
		tmp = Float64(t_2 / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5 + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$4 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$4)), $MachinePrecision]}, Block[{t$95$6 = (-t$95$0)}, If[LessEqual[t$95$5, -5e+172], N[(N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] / t$95$6), $MachinePrecision], If[LessEqual[t$95$5, -1.5e-188], N[((-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]) * N[(N[Sqrt[N[(N[(B$95$m * N[(N[(A / B$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2e+159], N[(N[Sqrt[N[(t$95$1 * N[(N[(F * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$6), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(N[((-N[Sqrt[N[(t$95$1 * F), $MachinePrecision]], $MachinePrecision]) * N[Sqrt[N[(2.0 * t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(t$95$2 / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.0000000000000001e172

   1. Initial program 8.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\frac{{B}^{2}}{A} \cdot \frac{-1}{2}} + 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{A}}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 23.8

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 23.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied rewrites 23.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \color{blue}{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right)} \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \color{blue}{\left(F \cdot 2\right)}\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      8. sqrt-prod N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{F \cdot 2}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   9. Applied rewrites 33.1%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{F \cdot 2}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       2. pow1/2 N/A

          \[\leadsto \frac{\color{blue}{{\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}^{\frac{1}{2}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       4. *-commutative N/A

          \[\leadsto \frac{{\color{blue}{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right)\right)}}^{\frac{1}{2}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       5. unpow-prod-down N/A

          \[\leadsto \frac{\color{blue}{\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right)\right)}^{\frac{1}{2}}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, C \cdot 2\right)\right)}^{\frac{1}{2}}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   11. Applied rewrites 45.1%

       \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if -5.0000000000000001e172 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.50000000000000008e-188

   1. Initial program 96.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 97.1%

      \[\leadsto \color{blue}{\left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(B, A - C\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in B around inf

      \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{\color{blue}{B \cdot \left(1 + \frac{A}{B}\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{\color{blue}{B \cdot \left(1 + \frac{A}{B}\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{B \cdot \color{blue}{\left(\frac{A}{B} + 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{B \cdot \color{blue}{\left(\frac{A}{B} + 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. lower-/.f64 31.8

         \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{B \cdot \left(\color{blue}{\frac{A}{B}} + 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 31.8%

      \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{\color{blue}{B \cdot \left(\frac{A}{B} + 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if -1.50000000000000008e-188 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 1.9999999999999999e159

   1. Initial program 15.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\frac{{B}^{2}}{A} \cdot \frac{-1}{2}} + 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{A}}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 52.9

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 52.9%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied rewrites 52.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)} \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right)} \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)} \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \color{blue}{\left(F \cdot 2\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      9. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(C \cdot A\right) \cdot -4 + B \cdot B\right)} \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\left(\color{blue}{-4 \cdot \left(C \cdot A\right)} + B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      11. lift-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   9. Applied rewrites 50.8%

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if 1.9999999999999999e159 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 20.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\frac{{B}^{2}}{A} \cdot \frac{-1}{2}} + 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{A}}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 32.5

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 32.5%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied rewrites 32.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]

   9. Applied rewrites 33.1%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F} \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

      5. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{F}{B}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

      8. lower-/.f64 18.1

         \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 18.1%

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 18.2%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 26.4%

      \[\leadsto \frac{\sqrt{F \cdot 2}}{\color{blue}{-\sqrt{B}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -5 \cdot 10^{+172}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}\right) \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1.5 \cdot 10^{-188}:\\ \;\;\;\;\left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{B \cdot \left(\frac{A}{B} + 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 2 \cdot 10^{+159}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\left(-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-87}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C + B\_m\right) \cdot 2}}{B\_m} \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-87)
   (/ (sqrt (* -16.0 (* A (* (* C C) F)))) (- (fma (* C A) -4.0 (* B_m B_m))))
   (* (/ (sqrt (* (+ C B_m) 2.0)) B_m) (- (sqrt F)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-87) {
		tmp = sqrt((-16.0 * (A * ((C * C) * F)))) / -fma((C * A), -4.0, (B_m * B_m));
	} else {
		tmp = (sqrt(((C + B_m) * 2.0)) / B_m) * -sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-87)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(A * Float64(Float64(C * C) * F)))) / Float64(-fma(Float64(C * A), -4.0, Float64(B_m * B_m))));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(C + B_m) * 2.0)) / B_m) * Float64(-sqrt(F)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-87], N[(N[Sqrt[N[(-16.0 * N[(A * N[(N[(C * C), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(C + B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000002e-87

   1. Initial program 16.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\frac{{B}^{2}}{A} \cdot \frac{-1}{2}} + 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{A}}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 25.9

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 25.9%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied rewrites 25.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]

   8. Taylor expanded in A around -inf

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(A \cdot \left({C}^{2} \cdot F\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \left(A \cdot \color{blue}{\left({C}^{2} \cdot F\right)}\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \left(A \cdot \left(\color{blue}{\left(C \cdot C\right)} \cdot F\right)\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      5. lower-*.f64 15.6

         \[\leadsto \frac{\sqrt{-16 \cdot \left(A \cdot \left(\color{blue}{\left(C \cdot C\right)} \cdot F\right)\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   10. Applied rewrites 15.6%

       \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if 1.00000000000000002e-87 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 21.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      12. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{B \cdot B} + {C}^{2}} + C\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{B \cdot B + \color{blue}{C \cdot C}} + C\right) \cdot F} \]

      14. lower-hypot.f64 26.8

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(B, C\right)} + C\right) \cdot F} \]

   5. Applied rewrites 26.8%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(B, C\right) + C\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 39.1%

      \[\leadsto \left(\frac{\sqrt{2}}{-B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \color{blue}{\sqrt{F}} \]
   8. Step-by-step derivation

   9. Applied rewrites 39.2%

      \[\leadsto \frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \color{blue}{\sqrt{F}} \]

   10. Taylor expanded in C around 0

       \[\leadsto \frac{\sqrt{\left(B + C\right) \cdot 2}}{-B} \cdot \sqrt{F} \]
   11. Step-by-step derivation

   12. Applied rewrites 35.4%

       \[\leadsto \frac{\sqrt{\left(C + B\right) \cdot 2}}{-B} \cdot \sqrt{F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-87}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C + B\right) \cdot 2}}{B} \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.3% accurate, 6.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 4.35 \cdot 10^{-41}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(\left(t\_0 \cdot F\right) \cdot 2\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C + B\_m\right) \cdot 2}}{B\_m} \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* C A) -4.0 (* B_m B_m))))
   (if (<= B_m 4.35e-41)
     (/ (sqrt (* (* 2.0 C) (* (* t_0 F) 2.0))) (- t_0))
     (* (/ (sqrt (* (+ C B_m) 2.0)) B_m) (- (sqrt F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((C * A), -4.0, (B_m * B_m));
	double tmp;
	if (B_m <= 4.35e-41) {
		tmp = sqrt(((2.0 * C) * ((t_0 * F) * 2.0))) / -t_0;
	} else {
		tmp = (sqrt(((C + B_m) * 2.0)) / B_m) * -sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 4.35e-41)
		tmp = Float64(sqrt(Float64(Float64(2.0 * C) * Float64(Float64(t_0 * F) * 2.0))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(C + B_m) * 2.0)) / B_m) * Float64(-sqrt(F)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 4.35e-41], N[(N[Sqrt[N[(N[(2.0 * C), $MachinePrecision] * N[(N[(t$95$0 * F), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(C + B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 4.34999999999999992e-41

   1. Initial program 19.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\frac{{B}^{2}}{A} \cdot \frac{-1}{2}} + 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{A}}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 20.8

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 20.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied rewrites 20.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]

   8. Taylor expanded in A around -inf

      \[\leadsto \frac{\sqrt{\color{blue}{\left(2 \cdot C\right)} \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 18.3

         \[\leadsto \frac{\sqrt{\color{blue}{\left(2 \cdot C\right)} \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   10. Applied rewrites 18.3%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(2 \cdot C\right)} \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if 4.34999999999999992e-41 < B

   1. Initial program 18.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      12. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{B \cdot B} + {C}^{2}} + C\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{B \cdot B + \color{blue}{C \cdot C}} + C\right) \cdot F} \]

      14. lower-hypot.f64 45.4

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(B, C\right)} + C\right) \cdot F} \]

   5. Applied rewrites 45.4%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(B, C\right) + C\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.9%

      \[\leadsto \left(\frac{\sqrt{2}}{-B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \color{blue}{\sqrt{F}} \]
   8. Step-by-step derivation

   9. Applied rewrites 67.0%

      \[\leadsto \frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \color{blue}{\sqrt{F}} \]

   10. Taylor expanded in C around 0

       \[\leadsto \frac{\sqrt{\left(B + C\right) \cdot 2}}{-B} \cdot \sqrt{F} \]
   11. Step-by-step derivation

   12. Applied rewrites 62.2%

       \[\leadsto \frac{\sqrt{\left(C + B\right) \cdot 2}}{-B} \cdot \sqrt{F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.35 \cdot 10^{-41}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C + B\right) \cdot 2}}{B} \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.9% accurate, 6.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.3 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot C\right)\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C + B\_m\right) \cdot 2}}{B\_m} \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.3e-42)
   (/
    (* (sqrt (* -8.0 (* A (* C C)))) (sqrt (* F 2.0)))
    (- (fma (* C A) -4.0 (* B_m B_m))))
   (* (/ (sqrt (* (+ C B_m) 2.0)) B_m) (- (sqrt F)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.3e-42) {
		tmp = (sqrt((-8.0 * (A * (C * C)))) * sqrt((F * 2.0))) / -fma((C * A), -4.0, (B_m * B_m));
	} else {
		tmp = (sqrt(((C + B_m) * 2.0)) / B_m) * -sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.3e-42)
		tmp = Float64(Float64(sqrt(Float64(-8.0 * Float64(A * Float64(C * C)))) * sqrt(Float64(F * 2.0))) / Float64(-fma(Float64(C * A), -4.0, Float64(B_m * B_m))));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(C + B_m) * 2.0)) / B_m) * Float64(-sqrt(F)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.3e-42], N[(N[(N[Sqrt[N[(-8.0 * N[(A * N[(C * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(C + B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.3e-42

   1. Initial program 19.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\frac{{B}^{2}}{A} \cdot \frac{-1}{2}} + 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{A}}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 20.8

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 20.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied rewrites 20.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \color{blue}{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right)} \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \color{blue}{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \color{blue}{\left(F \cdot 2\right)}\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      8. sqrt-prod N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{F \cdot 2}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   9. Applied rewrites 17.8%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{F \cdot 2}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   10. Taylor expanded in A around -inf

       \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot {C}^{2}\right)}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]
   11. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\left(\sqrt{-8} \cdot \sqrt{-8}\right)} \cdot \left(A \cdot {C}^{2}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       2. unpow2 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{{\left(\sqrt{-8}\right)}^{2}} \cdot \left(A \cdot {C}^{2}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{{\left(\sqrt{-8}\right)}^{2} \cdot \left(A \cdot {C}^{2}\right)}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       4. unpow2 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\left(\sqrt{-8} \cdot \sqrt{-8}\right)} \cdot \left(A \cdot {C}^{2}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       5. rem-square-sqrt N/A

          \[\leadsto \frac{\sqrt{\color{blue}{-8} \cdot \left(A \cdot {C}^{2}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{-8 \cdot \color{blue}{\left(A \cdot {C}^{2}\right)}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       7. unpow2 N/A

          \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \color{blue}{\left(C \cdot C\right)}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

       8. lower-*.f64 13.9

          \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \color{blue}{\left(C \cdot C\right)}\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   12. Applied rewrites 13.9%

       \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot C\right)\right)}} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if 1.3e-42 < B

   1. Initial program 18.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      12. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{B \cdot B} + {C}^{2}} + C\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{B \cdot B + \color{blue}{C \cdot C}} + C\right) \cdot F} \]

      14. lower-hypot.f64 45.4

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(B, C\right)} + C\right) \cdot F} \]

   5. Applied rewrites 45.4%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(B, C\right) + C\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.9%

      \[\leadsto \left(\frac{\sqrt{2}}{-B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \color{blue}{\sqrt{F}} \]
   8. Step-by-step derivation

   9. Applied rewrites 67.0%

      \[\leadsto \frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \color{blue}{\sqrt{F}} \]

   10. Taylor expanded in C around 0

       \[\leadsto \frac{\sqrt{\left(B + C\right) \cdot 2}}{-B} \cdot \sqrt{F} \]
   11. Step-by-step derivation

   12. Applied rewrites 62.2%

       \[\leadsto \frac{\sqrt{\left(C + B\right) \cdot 2}}{-B} \cdot \sqrt{F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.3 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot C\right)\right)} \cdot \sqrt{F \cdot 2}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C + B\right) \cdot 2}}{B} \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 36.5% accurate, 9.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 1.8 \cdot 10^{+244}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{C} \cdot 2}{-B\_m} \cdot \sqrt{F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 1.8e+244)
   (/ (sqrt (* F 2.0)) (- (sqrt B_m)))
   (* (/ (* (sqrt C) 2.0) (- B_m)) (sqrt F))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 1.8e+244) {
		tmp = sqrt((F * 2.0)) / -sqrt(B_m);
	} else {
		tmp = ((sqrt(C) * 2.0) / -B_m) * sqrt(F);
	}
	return tmp;
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F 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(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= 1.8d+244) then
        tmp = sqrt((f * 2.0d0)) / -sqrt(b_m)
    else
        tmp = ((sqrt(c) * 2.0d0) / -b_m) * sqrt(f)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 1.8e+244) {
		tmp = Math.sqrt((F * 2.0)) / -Math.sqrt(B_m);
	} else {
		tmp = ((Math.sqrt(C) * 2.0) / -B_m) * Math.sqrt(F);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if C <= 1.8e+244:
		tmp = math.sqrt((F * 2.0)) / -math.sqrt(B_m)
	else:
		tmp = ((math.sqrt(C) * 2.0) / -B_m) * math.sqrt(F)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 1.8e+244)
		tmp = Float64(sqrt(Float64(F * 2.0)) / Float64(-sqrt(B_m)));
	else
		tmp = Float64(Float64(Float64(sqrt(C) * 2.0) / Float64(-B_m)) * sqrt(F));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= 1.8e+244)
		tmp = sqrt((F * 2.0)) / -sqrt(B_m);
	else
		tmp = ((sqrt(C) * 2.0) / -B_m) * sqrt(F);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 1.8e+244], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision], N[(N[(N[(N[Sqrt[C], $MachinePrecision] * 2.0), $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 1.8e244

   1. Initial program 20.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

      5. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{F}{B}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

      8. lower-/.f64 15.0

         \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 15.0%

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 15.1%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 19.7%

      \[\leadsto \frac{\sqrt{F \cdot 2}}{\color{blue}{-\sqrt{B}}} \]

   if 1.8e244 < C

   1. Initial program 1.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      12. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{B \cdot B} + {C}^{2}} + C\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{B \cdot B + \color{blue}{C \cdot C}} + C\right) \cdot F} \]

      14. lower-hypot.f64 7.8

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(B, C\right)} + C\right) \cdot F} \]

   5. Applied rewrites 7.8%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(B, C\right) + C\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 12.9%

      \[\leadsto \left(\frac{\sqrt{2}}{-B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \color{blue}{\sqrt{F}} \]
   8. Step-by-step derivation

   9. Applied rewrites 12.9%

      \[\leadsto \frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \color{blue}{\sqrt{F}} \]

   10. Taylor expanded in B around 0

       \[\leadsto \frac{\sqrt{C} \cdot {\left(\sqrt{2}\right)}^{2}}{-B} \cdot \sqrt{F} \]
   11. Step-by-step derivation

   12. Applied rewrites 12.9%

       \[\leadsto \frac{\sqrt{C} \cdot 2}{-B} \cdot \sqrt{F} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 34.6% accurate, 12.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (/ (sqrt (* F 2.0)) (- (sqrt B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return sqrt((F * 2.0)) / -sqrt(B_m);
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F 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(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt((f * 2.0d0)) / -sqrt(b_m)
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt((F * 2.0)) / -Math.sqrt(B_m);
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return math.sqrt((F * 2.0)) / -math.sqrt(B_m)

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(sqrt(Float64(F * 2.0)) / Float64(-sqrt(B_m)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = sqrt((F * 2.0)) / -sqrt(B_m);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 19.3%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   2. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   3. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   5. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(-\color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{F}{B}} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left(-\sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

   8. lower-/.f64 14.2

      \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 14.2%

   \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

7. Applied rewrites 14.3%

   \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
8. Step-by-step derivation

9. Applied rewrites 18.6%

   \[\leadsto \frac{\sqrt{F \cdot 2}}{\color{blue}{-\sqrt{B}}} \]
10. Add Preprocessing

Alternative 14: 34.6% accurate, 12.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \sqrt{\frac{2}{B\_m}} \cdot \left(-\sqrt{F}\right) \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (* (sqrt (/ 2.0 B_m)) (- (sqrt F))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return sqrt((2.0 / B_m)) * -sqrt(F);
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F 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(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt((2.0d0 / b_m)) * -sqrt(f)
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt((2.0 / B_m)) * -Math.sqrt(F);
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return math.sqrt((2.0 / B_m)) * -math.sqrt(F)

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(sqrt(Float64(2.0 / B_m)) * Float64(-sqrt(F)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = sqrt((2.0 / B_m)) * -sqrt(F);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 19.3%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   2. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   3. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   5. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(-\color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{F}{B}} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left(-\sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

   8. lower-/.f64 14.2

      \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 14.2%

   \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

7. Applied rewrites 14.3%

   \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
8. Step-by-step derivation

9. Applied rewrites 18.5%

   \[\leadsto -\sqrt{\frac{2}{B}} \cdot \sqrt{F} \]

10. Final simplification 18.5%

    \[\leadsto \sqrt{\frac{2}{B}} \cdot \left(-\sqrt{F}\right) \]
11. Add Preprocessing

Alternative 15: 26.3% accurate, 18.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -\sqrt{\frac{F + F}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- (sqrt (/ (+ F F) B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -sqrt(((F + F) / B_m));
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F 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(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -sqrt(((f + f) / b_m))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -Math.sqrt(((F + F) / B_m));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -math.sqrt(((F + F) / B_m))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-sqrt(Float64(Float64(F + F) / B_m)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -sqrt(((F + F) / B_m));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := (-N[Sqrt[N[(N[(F + F), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 19.3%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   2. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   3. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   5. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(-\color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{F}{B}} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left(-\sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

   8. lower-/.f64 14.2

      \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 14.2%

   \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

7. Applied rewrites 14.3%

   \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
8. Step-by-step derivation

9. Applied rewrites 14.3%

   \[\leadsto -\sqrt{\frac{F + F}{B}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024354 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))