ABCF->ab-angle a

Percentage Accurate: 18.9% → 63.4%

Time: 12.5s

Alternatives: 20

Speedup: 16.9×

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: 18.9% 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: 63.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C \cdot 2\right)}\\ t_1 := \mathsf{hypot}\left(A - C, B\_m\right)\\ t_2 := \mathsf{fma}\left(C \cdot -4, A, B\_m \cdot B\_m\right)\\ t_3 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ t_4 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_5 := -t\_4\\ t_6 := \frac{\sqrt{\left(2 \cdot \left(t\_4 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_5}\\ t_7 := \left(2 \cdot F\right) \cdot t\_3\\ \mathbf{if}\;t\_6 \leq -2 \cdot 10^{+255}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right) \cdot 2} \cdot \sqrt{F}\right) \cdot \frac{t\_0}{-t\_2}\\ \mathbf{elif}\;t\_6 \leq -2 \cdot 10^{-216}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(t\_3 \cdot 2, F \cdot t\_1, \left(C + A\right) \cdot t\_7\right)}}{t\_5}\\ \mathbf{elif}\;t\_6 \leq 0:\\ \;\;\;\;\frac{\left(t\_0 \cdot \sqrt{F \cdot 2}\right) \cdot \sqrt{t\_2}}{t\_5}\\ \mathbf{elif}\;t\_6 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_7} \cdot \sqrt{\left(t\_1 + A\right) + C}}{t\_5}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{e^{\log 2 \cdot 0.5}}{B\_m} \cdot \sqrt{\mathsf{hypot}\left(C, B\_m\right) + C}\right) \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

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 (sqrt (fma -0.5 (/ (* B_m B_m) A) (* C 2.0))))
        (t_1 (hypot (- A C) B_m))
        (t_2 (fma (* C -4.0) A (* B_m B_m)))
        (t_3 (fma (* -4.0 A) C (* 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))
        (t_7 (* (* 2.0 F) t_3)))
   (if (<= t_6 -2e+255)
     (*
      (* (sqrt (* (fma (* -4.0 C) A (* B_m B_m)) 2.0)) (sqrt F))
      (/ t_0 (- t_2)))
     (if (<= t_6 -2e-216)
       (/ (sqrt (fma (* t_3 2.0) (* F t_1) (* (+ C A) t_7))) t_5)
       (if (<= t_6 0.0)
         (/ (* (* t_0 (sqrt (* F 2.0))) (sqrt t_2)) t_5)
         (if (<= t_6 INFINITY)
           (/ (* (sqrt t_7) (sqrt (+ (+ t_1 A) C))) t_5)
           (*
            (* (/ (exp (* (log 2.0) 0.5)) B_m) (sqrt (+ (hypot C B_m) C)))
            (- (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 = sqrt(fma(-0.5, ((B_m * B_m) / A), (C * 2.0)));
	double t_1 = hypot((A - C), B_m);
	double t_2 = fma((C * -4.0), A, (B_m * B_m));
	double t_3 = fma((-4.0 * A), C, (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 t_7 = (2.0 * F) * t_3;
	double tmp;
	if (t_6 <= -2e+255) {
		tmp = (sqrt((fma((-4.0 * C), A, (B_m * B_m)) * 2.0)) * sqrt(F)) * (t_0 / -t_2);
	} else if (t_6 <= -2e-216) {
		tmp = sqrt(fma((t_3 * 2.0), (F * t_1), ((C + A) * t_7))) / t_5;
	} else if (t_6 <= 0.0) {
		tmp = ((t_0 * sqrt((F * 2.0))) * sqrt(t_2)) / t_5;
	} else if (t_6 <= ((double) INFINITY)) {
		tmp = (sqrt(t_7) * sqrt(((t_1 + A) + C))) / t_5;
	} else {
		tmp = ((exp((log(2.0) * 0.5)) / B_m) * sqrt((hypot(C, B_m) + C))) * -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 = sqrt(fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(C * 2.0)))
	t_1 = hypot(Float64(A - C), B_m)
	t_2 = fma(Float64(C * -4.0), A, Float64(B_m * B_m))
	t_3 = fma(Float64(-4.0 * A), C, 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)
	t_7 = Float64(Float64(2.0 * F) * t_3)
	tmp = 0.0
	if (t_6 <= -2e+255)
		tmp = Float64(Float64(sqrt(Float64(fma(Float64(-4.0 * C), A, Float64(B_m * B_m)) * 2.0)) * sqrt(F)) * Float64(t_0 / Float64(-t_2)));
	elseif (t_6 <= -2e-216)
		tmp = Float64(sqrt(fma(Float64(t_3 * 2.0), Float64(F * t_1), Float64(Float64(C + A) * t_7))) / t_5);
	elseif (t_6 <= 0.0)
		tmp = Float64(Float64(Float64(t_0 * sqrt(Float64(F * 2.0))) * sqrt(t_2)) / t_5);
	elseif (t_6 <= Inf)
		tmp = Float64(Float64(sqrt(t_7) * sqrt(Float64(Float64(t_1 + A) + C))) / t_5);
	else
		tmp = Float64(Float64(Float64(exp(Float64(log(2.0) * 0.5)) / B_m) * sqrt(Float64(hypot(C, B_m) + C))) * 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[Sqrt[N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(N[(C * -4.0), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-4.0 * A), $MachinePrecision] * C + 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]}, Block[{t$95$7 = N[(N[(2.0 * F), $MachinePrecision] * t$95$3), $MachinePrecision]}, If[LessEqual[t$95$6, -2e+255], N[(N[(N[Sqrt[N[(N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / (-t$95$2)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, -2e-216], N[(N[Sqrt[N[(N[(t$95$3 * 2.0), $MachinePrecision] * N[(F * t$95$1), $MachinePrecision] + N[(N[(C + A), $MachinePrecision] * t$95$7), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision], If[LessEqual[t$95$6, 0.0], N[(N[(N[(t$95$0 * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision], If[LessEqual[t$95$6, Infinity], N[(N[(N[Sqrt[t$95$7], $MachinePrecision] * N[Sqrt[N[(N[(t$95$1 + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision], N[(N[(N[(N[Exp[N[(N[Log[2.0], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision]), $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))) < -1.99999999999999998e255

   1. Initial program 5.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 18.5

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

   5. Applied rewrites 18.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 25.7%

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

   9. Applied rewrites 25.7%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lift-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. metadata-eval N/A

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

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

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

       13. lift-*.f64 N/A

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

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

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

       15. lift-*.f64 N/A

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

       16. lower--.f64 N/A

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

       17. lift-*.f64 N/A

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

       18. pow2 N/A

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

       19. lift-pow.f64 N/A

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

   11. Applied rewrites 40.2%

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

   if -1.99999999999999998e255 < (/.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))) < -2.0000000000000001e-216

   1. Initial program 98.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

   4. Applied rewrites 98.3%

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

   if -2.0000000000000001e-216 < (/.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 3.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 30.2

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

   5. Applied rewrites 30.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 24.8%

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

   9. Applied rewrites 29.7%

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

   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 42.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 86.3%

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

   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. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 21.5

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

   5. Applied rewrites 21.5%

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

   7. Applied rewrites 30.0%

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

   9. Applied rewrites 30.1%

      \[\leadsto -\left(\frac{e^{\log 2 \cdot 0.5}}{B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \sqrt{F} \]
3. Recombined 5 regimes into one program.

4. Final simplification 47.6%

   \[\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 -2 \cdot 10^{+255}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot 2} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot -4, 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^{-216}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right) \cdot 2, F \cdot \mathsf{hypot}\left(A - C, B\right), \left(C + A\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right)\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{\left(\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)} \cdot \sqrt{F \cdot 2}\right) \cdot \sqrt{\mathsf{fma}\left(C \cdot -4, 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 \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot \sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{e^{\log 2 \cdot 0.5}}{B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(C, B\_m\right) + C\\ t_1 := -\sqrt{2}\\ t_2 := \mathsf{fma}\left(C \cdot -4, A, B\_m \cdot B\_m\right)\\ t_3 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(t\_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_3}\\ t_5 := -t\_2\\ t_6 := \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C \cdot 2\right)}}{t\_5}\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right) \cdot 2} \cdot \sqrt{F}\right) \cdot t\_6\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-99}:\\ \;\;\;\;\sqrt{\frac{\left(\left(\mathsf{hypot}\left(A - C, B\_m\right) + C\right) + A\right) \cdot F}{\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)}} \cdot t\_1\\ \mathbf{elif}\;t\_4 \leq -2 \cdot 10^{-216}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot F\right) \cdot 2}}{-B\_m}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\sqrt{F \cdot 2} \cdot \left(\sqrt{t\_2} \cdot t\_6\right)\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot t\_2} \cdot \frac{\sqrt{C} \cdot \sqrt{2}}{t\_5}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 \cdot \frac{\sqrt{t\_0}}{B\_m}\right) \cdot \sqrt{F}\\ \end{array} \end{array} \]

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 (+ (hypot C B_m) C))
        (t_1 (- (sqrt 2.0)))
        (t_2 (fma (* C -4.0) A (* B_m B_m)))
        (t_3 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* t_3 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_3)))
        (t_5 (- t_2))
        (t_6 (/ (sqrt (fma -0.5 (/ (* B_m B_m) A) (* C 2.0))) t_5)))
   (if (<= t_4 -2e+154)
     (* (* (sqrt (* (fma (* -4.0 C) A (* B_m B_m)) 2.0)) (sqrt F)) t_6)
     (if (<= t_4 -1e-99)
       (*
        (sqrt
         (/
          (* (+ (+ (hypot (- A C) B_m) C) A) F)
          (fma (* C A) -4.0 (* B_m B_m))))
        t_1)
       (if (<= t_4 -2e-216)
         (/ (sqrt (* (* t_0 F) 2.0)) (- B_m))
         (if (<= t_4 0.0)
           (* (sqrt (* F 2.0)) (* (sqrt t_2) t_6))
           (if (<= t_4 INFINITY)
             (* (sqrt (* (* F 2.0) t_2)) (/ (* (sqrt C) (sqrt 2.0)) t_5))
             (* (* t_1 (/ (sqrt t_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 = hypot(C, B_m) + C;
	double t_1 = -sqrt(2.0);
	double t_2 = fma((C * -4.0), A, (B_m * B_m));
	double t_3 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_4 = sqrt(((2.0 * (t_3 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_3;
	double t_5 = -t_2;
	double t_6 = sqrt(fma(-0.5, ((B_m * B_m) / A), (C * 2.0))) / t_5;
	double tmp;
	if (t_4 <= -2e+154) {
		tmp = (sqrt((fma((-4.0 * C), A, (B_m * B_m)) * 2.0)) * sqrt(F)) * t_6;
	} else if (t_4 <= -1e-99) {
		tmp = sqrt(((((hypot((A - C), B_m) + C) + A) * F) / fma((C * A), -4.0, (B_m * B_m)))) * t_1;
	} else if (t_4 <= -2e-216) {
		tmp = sqrt(((t_0 * F) * 2.0)) / -B_m;
	} else if (t_4 <= 0.0) {
		tmp = sqrt((F * 2.0)) * (sqrt(t_2) * t_6);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt(((F * 2.0) * t_2)) * ((sqrt(C) * sqrt(2.0)) / t_5);
	} else {
		tmp = (t_1 * (sqrt(t_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 = Float64(hypot(C, B_m) + C)
	t_1 = Float64(-sqrt(2.0))
	t_2 = fma(Float64(C * -4.0), A, Float64(B_m * B_m))
	t_3 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_3 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_3))
	t_5 = Float64(-t_2)
	t_6 = Float64(sqrt(fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(C * 2.0))) / t_5)
	tmp = 0.0
	if (t_4 <= -2e+154)
		tmp = Float64(Float64(sqrt(Float64(fma(Float64(-4.0 * C), A, Float64(B_m * B_m)) * 2.0)) * sqrt(F)) * t_6);
	elseif (t_4 <= -1e-99)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(hypot(Float64(A - C), B_m) + C) + A) * F) / fma(Float64(C * A), -4.0, Float64(B_m * B_m)))) * t_1);
	elseif (t_4 <= -2e-216)
		tmp = Float64(sqrt(Float64(Float64(t_0 * F) * 2.0)) / Float64(-B_m));
	elseif (t_4 <= 0.0)
		tmp = Float64(sqrt(Float64(F * 2.0)) * Float64(sqrt(t_2) * t_6));
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(F * 2.0) * t_2)) * Float64(Float64(sqrt(C) * sqrt(2.0)) / t_5));
	else
		tmp = Float64(Float64(t_1 * Float64(sqrt(t_0) / 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[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$2 = N[(N[(C * -4.0), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$3 * 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$3)), $MachinePrecision]}, Block[{t$95$5 = (-t$95$2)}, Block[{t$95$6 = N[(N[Sqrt[N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision]}, If[LessEqual[t$95$4, -2e+154], N[(N[(N[Sqrt[N[(N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * t$95$6), $MachinePrecision], If[LessEqual[t$95$4, -1e-99], N[(N[Sqrt[N[(N[(N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] + A), $MachinePrecision] * F), $MachinePrecision] / N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$4, -2e-216], N[(N[Sqrt[N[(N[(t$95$0 * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[t$95$2], $MachinePrecision] * t$95$6), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[C], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(N[Sqrt[t$95$0], $MachinePrecision] / B$95$m), $MachinePrecision]), $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))) < -2.00000000000000007e154

   1. Initial program 18.0%

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

   3. Taylor expanded in A around -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 22.9

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

   5. Applied rewrites 22.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 29.1%

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

   9. Applied rewrites 29.1%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lift-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. metadata-eval N/A

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

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

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

       13. lift-*.f64 N/A

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

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

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

       15. lift-*.f64 N/A

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

       16. lower--.f64 N/A

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

       17. lift-*.f64 N/A

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

       18. pow2 N/A

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

       19. lift-pow.f64 N/A

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

   11. Applied rewrites 41.7%

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

   if -2.00000000000000007e154 < (/.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))) < -1e-99

   1. Initial program 97.5%

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

   3. Taylor expanded in 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. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 97.2%

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

   if -1e-99 < (/.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))) < -2.0000000000000001e-216

   1. Initial program 98.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 47.6

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

   5. Applied rewrites 47.6%

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

   7. Applied rewrites 47.6%

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

   if -2.0000000000000001e-216 < (/.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 3.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 30.2

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

   5. Applied rewrites 30.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 24.8%

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

   9. Applied rewrites 29.6%

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

   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 42.9%

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

   3. Taylor expanded in A around -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 26.1

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

   5. Applied rewrites 26.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 30.7%

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

   9. Applied rewrites 30.8%

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

   10. Taylor expanded in A around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-sqrt.f64 30.9

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

   12. Applied rewrites 30.9%

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

   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. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 21.5

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

   5. Applied rewrites 21.5%

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

   7. Applied rewrites 30.0%

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

   9. Applied rewrites 30.1%

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

4. Final simplification 39.5%

   \[\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 -2 \cdot 10^{+154}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot 2} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot -4, 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 \cdot 10^{-99}:\\ \;\;\;\;\sqrt{\frac{\left(\left(\mathsf{hypot}\left(A - C, B\right) + C\right) + A\right) \cdot F}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \cdot \left(-\sqrt{2}\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^{-216}:\\ \;\;\;\;\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 0:\\ \;\;\;\;\sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\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:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \frac{\sqrt{C} \cdot \sqrt{2}}{-\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\sqrt{2}\right) \cdot \frac{\sqrt{\mathsf{hypot}\left(C, B\right) + C}}{B}\right) \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_0}\\ t_2 := \mathsf{fma}\left(C \cdot -4, A, B\_m \cdot B\_m\right)\\ t_3 := \sqrt{\left(F \cdot 2\right) \cdot t\_2}\\ t_4 := -t\_2\\ t_5 := \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C \cdot 2\right)}}{t\_4}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right) \cdot 2} \cdot \sqrt{F}\right) \cdot t\_5\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-65}:\\ \;\;\;\;\left(-t\_3\right) \cdot \frac{\sqrt{2 \cdot C}}{t\_2}\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-216}:\\ \;\;\;\;t\_3 \cdot \frac{\sqrt{B\_m \cdot \left(1 + \frac{A + C}{B\_m}\right)}}{t\_4}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{F \cdot 2} \cdot \left(\sqrt{t\_2} \cdot t\_5\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_3 \cdot \frac{\sqrt{C} \cdot \sqrt{2}}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-\sqrt{2}}{B\_m} \cdot \sqrt{B\_m + C}\right) \cdot \sqrt{F}\\ \end{array} \end{array} \]

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 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_1
         (/
          (sqrt
           (*
            (* 2.0 (* t_0 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_0)))
        (t_2 (fma (* C -4.0) A (* B_m B_m)))
        (t_3 (sqrt (* (* F 2.0) t_2)))
        (t_4 (- t_2))
        (t_5 (/ (sqrt (fma -0.5 (/ (* B_m B_m) A) (* C 2.0))) t_4)))
   (if (<= t_1 (- INFINITY))
     (* (* (sqrt (* (fma (* -4.0 C) A (* B_m B_m)) 2.0)) (sqrt F)) t_5)
     (if (<= t_1 -2e-65)
       (* (- t_3) (/ (sqrt (* 2.0 C)) t_2))
       (if (<= t_1 -2e-216)
         (* t_3 (/ (sqrt (* B_m (+ 1.0 (/ (+ A C) B_m)))) t_4))
         (if (<= t_1 0.0)
           (* (sqrt (* F 2.0)) (* (sqrt t_2) t_5))
           (if (<= t_1 INFINITY)
             (* t_3 (/ (* (sqrt C) (sqrt 2.0)) t_4))
             (* (* (/ (- (sqrt 2.0)) B_m) (sqrt (+ B_m C))) (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 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_0;
	double t_2 = fma((C * -4.0), A, (B_m * B_m));
	double t_3 = sqrt(((F * 2.0) * t_2));
	double t_4 = -t_2;
	double t_5 = sqrt(fma(-0.5, ((B_m * B_m) / A), (C * 2.0))) / t_4;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (sqrt((fma((-4.0 * C), A, (B_m * B_m)) * 2.0)) * sqrt(F)) * t_5;
	} else if (t_1 <= -2e-65) {
		tmp = -t_3 * (sqrt((2.0 * C)) / t_2);
	} else if (t_1 <= -2e-216) {
		tmp = t_3 * (sqrt((B_m * (1.0 + ((A + C) / B_m)))) / t_4);
	} else if (t_1 <= 0.0) {
		tmp = sqrt((F * 2.0)) * (sqrt(t_2) * t_5);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_3 * ((sqrt(C) * sqrt(2.0)) / t_4);
	} else {
		tmp = ((-sqrt(2.0) / B_m) * sqrt((B_m + C))) * 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 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_0))
	t_2 = fma(Float64(C * -4.0), A, Float64(B_m * B_m))
	t_3 = sqrt(Float64(Float64(F * 2.0) * t_2))
	t_4 = Float64(-t_2)
	t_5 = Float64(sqrt(fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(C * 2.0))) / t_4)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(fma(Float64(-4.0 * C), A, Float64(B_m * B_m)) * 2.0)) * sqrt(F)) * t_5);
	elseif (t_1 <= -2e-65)
		tmp = Float64(Float64(-t_3) * Float64(sqrt(Float64(2.0 * C)) / t_2));
	elseif (t_1 <= -2e-216)
		tmp = Float64(t_3 * Float64(sqrt(Float64(B_m * Float64(1.0 + Float64(Float64(A + C) / B_m)))) / t_4));
	elseif (t_1 <= 0.0)
		tmp = Float64(sqrt(Float64(F * 2.0)) * Float64(sqrt(t_2) * t_5));
	elseif (t_1 <= Inf)
		tmp = Float64(t_3 * Float64(Float64(sqrt(C) * sqrt(2.0)) / t_4));
	else
		tmp = Float64(Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(B_m + C))) * 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[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]}, Block[{t$95$2 = N[(N[(C * -4.0), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = (-t$95$2)}, Block[{t$95$5 = N[(N[Sqrt[N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[Sqrt[N[(N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision], If[LessEqual[t$95$1, -2e-65], N[((-t$95$3) * N[(N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e-216], N[(t$95$3 * N[(N[Sqrt[N[(B$95$m * N[(1.0 + N[(N[(A + C), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[t$95$2], $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(t$95$3 * N[(N[(N[Sqrt[C], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision]), $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))) < -inf.0

   1. Initial program 3.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 19.0

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

   5. Applied rewrites 19.0%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 26.3%

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

   9. Applied rewrites 26.3%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lift-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. metadata-eval N/A

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

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

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

       13. lift-*.f64 N/A

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

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

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

       15. lift-*.f64 N/A

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

       16. lower--.f64 N/A

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

       17. lift-*.f64 N/A

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

       18. pow2 N/A

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

       19. lift-pow.f64 N/A

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

   11. Applied rewrites 41.2%

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

   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))) < -1.99999999999999985e-65

   1. Initial program 97.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 22.7

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

   5. Applied rewrites 22.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 22.7%

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

   9. Applied rewrites 22.8%

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

   10. Taylor expanded in A around inf

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

   12. Applied rewrites 39.2%

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

   if -1.99999999999999985e-65 < (/.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))) < -2.0000000000000001e-216

   1. Initial program 99.0%

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

   3. Taylor expanded in A around -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 3.0

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

   5. Applied rewrites 3.0%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 3.0%

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

   9. Applied rewrites 3.0%

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

   10. Taylor expanded in B around inf

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. div-add-rev N/A

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

       4. lower-/.f64 N/A

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

       5. lower-+.f64 52.5

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

   12. Applied rewrites 52.5%

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

   if -2.0000000000000001e-216 < (/.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 3.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 30.2

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

   5. Applied rewrites 30.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 24.8%

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

   9. Applied rewrites 29.6%

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

   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 42.9%

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

   3. Taylor expanded in A around -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 26.1

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

   5. Applied rewrites 26.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 30.7%

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

   9. Applied rewrites 30.8%

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

   10. Taylor expanded in A around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-sqrt.f64 30.9

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

   12. Applied rewrites 30.9%

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

   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. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 21.5

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

   5. Applied rewrites 21.5%

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

   7. Applied rewrites 30.0%

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

   8. Taylor expanded in C around 0

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

   10. Applied rewrites 27.6%

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

4. Final simplification 32.9%

   \[\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:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot 2} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot -4, 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^{-65}:\\ \;\;\;\;\left(-\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \cdot \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(C \cdot -4, 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^{-216}:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \frac{\sqrt{B \cdot \left(1 + \frac{A + C}{B}\right)}}{-\mathsf{fma}\left(C \cdot -4, 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:\\ \;\;\;\;\sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\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:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \frac{\sqrt{C} \cdot \sqrt{2}}{-\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-\sqrt{2}}{B} \cdot \sqrt{B + C}\right) \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot -4, A, B\_m \cdot B\_m\right)\\ t_1 := \sqrt{\left(F \cdot 2\right) \cdot t\_0}\\ t_2 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_2}\\ t_4 := -t\_0\\ t_5 := \sqrt{F \cdot 2} \cdot \left(\sqrt{t\_0} \cdot \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C \cdot 2\right)}}{t\_4}\right)\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{+195}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-65}:\\ \;\;\;\;\left(-t\_1\right) \cdot \frac{\sqrt{2 \cdot C}}{t\_0}\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-216}:\\ \;\;\;\;t\_1 \cdot \frac{\sqrt{B\_m \cdot \left(1 + \frac{A + C}{B\_m}\right)}}{t\_4}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_1 \cdot \frac{\sqrt{C} \cdot \sqrt{2}}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-\sqrt{2}}{B\_m} \cdot \sqrt{B\_m + C}\right) \cdot \sqrt{F}\\ \end{array} \end{array} \]

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 -4.0) A (* B_m B_m)))
        (t_1 (sqrt (* (* F 2.0) t_0)))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* t_2 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_2)))
        (t_4 (- t_0))
        (t_5
         (*
          (sqrt (* F 2.0))
          (*
           (sqrt t_0)
           (/ (sqrt (fma -0.5 (/ (* B_m B_m) A) (* C 2.0))) t_4)))))
   (if (<= t_3 -1e+195)
     t_5
     (if (<= t_3 -2e-65)
       (* (- t_1) (/ (sqrt (* 2.0 C)) t_0))
       (if (<= t_3 -2e-216)
         (* t_1 (/ (sqrt (* B_m (+ 1.0 (/ (+ A C) B_m)))) t_4))
         (if (<= t_3 0.0)
           t_5
           (if (<= t_3 INFINITY)
             (* t_1 (/ (* (sqrt C) (sqrt 2.0)) t_4))
             (* (* (/ (- (sqrt 2.0)) B_m) (sqrt (+ B_m C))) (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 * -4.0), A, (B_m * B_m));
	double t_1 = sqrt(((F * 2.0) * t_0));
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_3 = sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_2;
	double t_4 = -t_0;
	double t_5 = sqrt((F * 2.0)) * (sqrt(t_0) * (sqrt(fma(-0.5, ((B_m * B_m) / A), (C * 2.0))) / t_4));
	double tmp;
	if (t_3 <= -1e+195) {
		tmp = t_5;
	} else if (t_3 <= -2e-65) {
		tmp = -t_1 * (sqrt((2.0 * C)) / t_0);
	} else if (t_3 <= -2e-216) {
		tmp = t_1 * (sqrt((B_m * (1.0 + ((A + C) / B_m)))) / t_4);
	} else if (t_3 <= 0.0) {
		tmp = t_5;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_1 * ((sqrt(C) * sqrt(2.0)) / t_4);
	} else {
		tmp = ((-sqrt(2.0) / B_m) * sqrt((B_m + C))) * 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 * -4.0), A, Float64(B_m * B_m))
	t_1 = sqrt(Float64(Float64(F * 2.0) * t_0))
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_2))
	t_4 = Float64(-t_0)
	t_5 = Float64(sqrt(Float64(F * 2.0)) * Float64(sqrt(t_0) * Float64(sqrt(fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(C * 2.0))) / t_4)))
	tmp = 0.0
	if (t_3 <= -1e+195)
		tmp = t_5;
	elseif (t_3 <= -2e-65)
		tmp = Float64(Float64(-t_1) * Float64(sqrt(Float64(2.0 * C)) / t_0));
	elseif (t_3 <= -2e-216)
		tmp = Float64(t_1 * Float64(sqrt(Float64(B_m * Float64(1.0 + Float64(Float64(A + C) / B_m)))) / t_4));
	elseif (t_3 <= 0.0)
		tmp = t_5;
	elseif (t_3 <= Inf)
		tmp = Float64(t_1 * Float64(Float64(sqrt(C) * sqrt(2.0)) / t_4));
	else
		tmp = Float64(Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(B_m + C))) * 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 * -4.0), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$2 * 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$2)), $MachinePrecision]}, Block[{t$95$4 = (-t$95$0)}, Block[{t$95$5 = N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] * N[(N[Sqrt[N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1e+195], t$95$5, If[LessEqual[t$95$3, -2e-65], N[((-t$95$1) * N[(N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -2e-216], N[(t$95$1 * N[(N[Sqrt[N[(B$95$m * N[(1.0 + N[(N[(A + C), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], t$95$5, If[LessEqual[t$95$3, Infinity], N[(t$95$1 * N[(N[(N[Sqrt[C], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision]), $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))) < -9.99999999999999977e194 or -2.0000000000000001e-216 < (/.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 5.9%

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

   3. Taylor expanded in A around -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 25.1

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

   5. Applied rewrites 25.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 26.2%

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

   9. Applied rewrites 35.9%

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

   if -9.99999999999999977e194 < (/.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))) < -1.99999999999999985e-65

   1. Initial program 97.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 20.7

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

   5. Applied rewrites 20.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 20.8%

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

   9. Applied rewrites 20.9%

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

   10. Taylor expanded in A around inf

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

   12. Applied rewrites 38.5%

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

   if -1.99999999999999985e-65 < (/.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))) < -2.0000000000000001e-216

   1. Initial program 99.0%

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

   3. Taylor expanded in A around -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 3.0

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

   5. Applied rewrites 3.0%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 3.0%

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

   9. Applied rewrites 3.0%

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

   10. Taylor expanded in B around inf

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. div-add-rev N/A

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

       4. lower-/.f64 N/A

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

       5. lower-+.f64 52.5

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

   12. Applied rewrites 52.5%

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

   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 42.9%

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

   3. Taylor expanded in A around -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 26.1

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

   5. Applied rewrites 26.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 30.7%

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

   9. Applied rewrites 30.8%

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

   10. Taylor expanded in A around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-sqrt.f64 30.9

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

   12. Applied rewrites 30.9%

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

   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. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 21.5

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

   5. Applied rewrites 21.5%

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

   7. Applied rewrites 30.0%

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

   8. Taylor expanded in C around 0

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

   10. Applied rewrites 27.6%

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

4. Final simplification 32.9%

   \[\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 -1 \cdot 10^{+195}:\\ \;\;\;\;\sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\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^{-65}:\\ \;\;\;\;\left(-\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \cdot \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(C \cdot -4, 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^{-216}:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \frac{\sqrt{B \cdot \left(1 + \frac{A + C}{B}\right)}}{-\mathsf{fma}\left(C \cdot -4, 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:\\ \;\;\;\;\sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\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:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \frac{\sqrt{C} \cdot \sqrt{2}}{-\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-\sqrt{2}}{B} \cdot \sqrt{B + C}\right) \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C \cdot 2\right)}\\ t_1 := \mathsf{hypot}\left(A - C, B\_m\right)\\ t_2 := \mathsf{fma}\left(C \cdot -4, A, B\_m \cdot B\_m\right)\\ t_3 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ t_4 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_5 := -t\_4\\ t_6 := \frac{\sqrt{\left(2 \cdot \left(t\_4 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_5}\\ t_7 := \left(2 \cdot F\right) \cdot t\_3\\ \mathbf{if}\;t\_6 \leq -2 \cdot 10^{+255}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right) \cdot 2} \cdot \sqrt{F}\right) \cdot \frac{t\_0}{-t\_2}\\ \mathbf{elif}\;t\_6 \leq -2 \cdot 10^{-216}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(t\_3 \cdot 2, F \cdot t\_1, \left(C + A\right) \cdot t\_7\right)}}{t\_5}\\ \mathbf{elif}\;t\_6 \leq 0:\\ \;\;\;\;\frac{\left(t\_0 \cdot \sqrt{F \cdot 2}\right) \cdot \sqrt{t\_2}}{t\_5}\\ \mathbf{elif}\;t\_6 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_7} \cdot \sqrt{\left(t\_1 + A\right) + C}}{t\_5}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\sqrt{2}\right) \cdot \frac{\sqrt{\mathsf{hypot}\left(C, B\_m\right) + C}}{B\_m}\right) \cdot \sqrt{F}\\ \end{array} \end{array} \]

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 (sqrt (fma -0.5 (/ (* B_m B_m) A) (* C 2.0))))
        (t_1 (hypot (- A C) B_m))
        (t_2 (fma (* C -4.0) A (* B_m B_m)))
        (t_3 (fma (* -4.0 A) C (* 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))
        (t_7 (* (* 2.0 F) t_3)))
   (if (<= t_6 -2e+255)
     (*
      (* (sqrt (* (fma (* -4.0 C) A (* B_m B_m)) 2.0)) (sqrt F))
      (/ t_0 (- t_2)))
     (if (<= t_6 -2e-216)
       (/ (sqrt (fma (* t_3 2.0) (* F t_1) (* (+ C A) t_7))) t_5)
       (if (<= t_6 0.0)
         (/ (* (* t_0 (sqrt (* F 2.0))) (sqrt t_2)) t_5)
         (if (<= t_6 INFINITY)
           (/ (* (sqrt t_7) (sqrt (+ (+ t_1 A) C))) t_5)
           (*
            (* (- (sqrt 2.0)) (/ (sqrt (+ (hypot C B_m) C)) 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 = sqrt(fma(-0.5, ((B_m * B_m) / A), (C * 2.0)));
	double t_1 = hypot((A - C), B_m);
	double t_2 = fma((C * -4.0), A, (B_m * B_m));
	double t_3 = fma((-4.0 * A), C, (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 t_7 = (2.0 * F) * t_3;
	double tmp;
	if (t_6 <= -2e+255) {
		tmp = (sqrt((fma((-4.0 * C), A, (B_m * B_m)) * 2.0)) * sqrt(F)) * (t_0 / -t_2);
	} else if (t_6 <= -2e-216) {
		tmp = sqrt(fma((t_3 * 2.0), (F * t_1), ((C + A) * t_7))) / t_5;
	} else if (t_6 <= 0.0) {
		tmp = ((t_0 * sqrt((F * 2.0))) * sqrt(t_2)) / t_5;
	} else if (t_6 <= ((double) INFINITY)) {
		tmp = (sqrt(t_7) * sqrt(((t_1 + A) + C))) / t_5;
	} else {
		tmp = (-sqrt(2.0) * (sqrt((hypot(C, B_m) + C)) / 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 = sqrt(fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(C * 2.0)))
	t_1 = hypot(Float64(A - C), B_m)
	t_2 = fma(Float64(C * -4.0), A, Float64(B_m * B_m))
	t_3 = fma(Float64(-4.0 * A), C, 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)
	t_7 = Float64(Float64(2.0 * F) * t_3)
	tmp = 0.0
	if (t_6 <= -2e+255)
		tmp = Float64(Float64(sqrt(Float64(fma(Float64(-4.0 * C), A, Float64(B_m * B_m)) * 2.0)) * sqrt(F)) * Float64(t_0 / Float64(-t_2)));
	elseif (t_6 <= -2e-216)
		tmp = Float64(sqrt(fma(Float64(t_3 * 2.0), Float64(F * t_1), Float64(Float64(C + A) * t_7))) / t_5);
	elseif (t_6 <= 0.0)
		tmp = Float64(Float64(Float64(t_0 * sqrt(Float64(F * 2.0))) * sqrt(t_2)) / t_5);
	elseif (t_6 <= Inf)
		tmp = Float64(Float64(sqrt(t_7) * sqrt(Float64(Float64(t_1 + A) + C))) / t_5);
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) * Float64(sqrt(Float64(hypot(C, B_m) + C)) / 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[Sqrt[N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(N[(C * -4.0), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-4.0 * A), $MachinePrecision] * C + 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]}, Block[{t$95$7 = N[(N[(2.0 * F), $MachinePrecision] * t$95$3), $MachinePrecision]}, If[LessEqual[t$95$6, -2e+255], N[(N[(N[Sqrt[N[(N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / (-t$95$2)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, -2e-216], N[(N[Sqrt[N[(N[(t$95$3 * 2.0), $MachinePrecision] * N[(F * t$95$1), $MachinePrecision] + N[(N[(C + A), $MachinePrecision] * t$95$7), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision], If[LessEqual[t$95$6, 0.0], N[(N[(N[(t$95$0 * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision], If[LessEqual[t$95$6, Infinity], N[(N[(N[Sqrt[t$95$7], $MachinePrecision] * N[Sqrt[N[(N[(t$95$1 + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) * N[(N[Sqrt[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $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))) < -1.99999999999999998e255

   1. Initial program 5.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 18.5

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

   5. Applied rewrites 18.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 25.7%

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

   9. Applied rewrites 25.7%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lift-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. metadata-eval N/A

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

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

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

       13. lift-*.f64 N/A

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

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

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

       15. lift-*.f64 N/A

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

       16. lower--.f64 N/A

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

       17. lift-*.f64 N/A

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

       18. pow2 N/A

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

       19. lift-pow.f64 N/A

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

   11. Applied rewrites 40.2%

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

   if -1.99999999999999998e255 < (/.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))) < -2.0000000000000001e-216

   1. Initial program 98.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

   4. Applied rewrites 98.3%

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

   if -2.0000000000000001e-216 < (/.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 3.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 30.2

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

   5. Applied rewrites 30.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 24.8%

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

   9. Applied rewrites 29.7%

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

   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 42.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 86.3%

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

   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. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 21.5

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

   5. Applied rewrites 21.5%

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

   7. Applied rewrites 30.0%

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

   9. Applied rewrites 30.1%

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

4. Final simplification 47.6%

   \[\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 -2 \cdot 10^{+255}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot 2} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot -4, 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^{-216}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right) \cdot 2, F \cdot \mathsf{hypot}\left(A - C, B\right), \left(C + A\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right)\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{\left(\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)} \cdot \sqrt{F \cdot 2}\right) \cdot \sqrt{\mathsf{fma}\left(C \cdot -4, 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 \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot \sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\sqrt{2}\right) \cdot \frac{\sqrt{\mathsf{hypot}\left(C, B\right) + C}}{B}\right) \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C\\ t_1 := \sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C \cdot 2\right)}\\ t_2 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ t_3 := \left(2 \cdot F\right) \cdot t\_2\\ t_4 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_5 := -t\_4\\ t_6 := \frac{\sqrt{\left(2 \cdot \left(t\_4 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_5}\\ t_7 := \mathsf{fma}\left(C \cdot -4, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;t\_6 \leq -2 \cdot 10^{+255}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right) \cdot 2} \cdot \sqrt{F}\right) \cdot \frac{t\_1}{-t\_7}\\ \mathbf{elif}\;t\_6 \leq -2 \cdot 10^{-216}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot t\_3}}{-t\_2}\\ \mathbf{elif}\;t\_6 \leq 0:\\ \;\;\;\;\frac{\left(t\_1 \cdot \sqrt{F \cdot 2}\right) \cdot \sqrt{t\_7}}{t\_5}\\ \mathbf{elif}\;t\_6 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_3} \cdot \sqrt{t\_0}}{t\_5}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\sqrt{2}\right) \cdot \frac{\sqrt{\mathsf{hypot}\left(C, B\_m\right) + C}}{B\_m}\right) \cdot \sqrt{F}\\ \end{array} \end{array} \]

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 (+ (+ (hypot (- A C) B_m) A) C))
        (t_1 (sqrt (fma -0.5 (/ (* B_m B_m) A) (* C 2.0))))
        (t_2 (fma (* -4.0 A) C (* B_m B_m)))
        (t_3 (* (* 2.0 F) t_2))
        (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))
        (t_7 (fma (* C -4.0) A (* B_m B_m))))
   (if (<= t_6 -2e+255)
     (*
      (* (sqrt (* (fma (* -4.0 C) A (* B_m B_m)) 2.0)) (sqrt F))
      (/ t_1 (- t_7)))
     (if (<= t_6 -2e-216)
       (/ (sqrt (* t_0 t_3)) (- t_2))
       (if (<= t_6 0.0)
         (/ (* (* t_1 (sqrt (* F 2.0))) (sqrt t_7)) t_5)
         (if (<= t_6 INFINITY)
           (/ (* (sqrt t_3) (sqrt t_0)) t_5)
           (*
            (* (- (sqrt 2.0)) (/ (sqrt (+ (hypot C B_m) C)) 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 = (hypot((A - C), B_m) + A) + C;
	double t_1 = sqrt(fma(-0.5, ((B_m * B_m) / A), (C * 2.0)));
	double t_2 = fma((-4.0 * A), C, (B_m * B_m));
	double t_3 = (2.0 * F) * t_2;
	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 t_7 = fma((C * -4.0), A, (B_m * B_m));
	double tmp;
	if (t_6 <= -2e+255) {
		tmp = (sqrt((fma((-4.0 * C), A, (B_m * B_m)) * 2.0)) * sqrt(F)) * (t_1 / -t_7);
	} else if (t_6 <= -2e-216) {
		tmp = sqrt((t_0 * t_3)) / -t_2;
	} else if (t_6 <= 0.0) {
		tmp = ((t_1 * sqrt((F * 2.0))) * sqrt(t_7)) / t_5;
	} else if (t_6 <= ((double) INFINITY)) {
		tmp = (sqrt(t_3) * sqrt(t_0)) / t_5;
	} else {
		tmp = (-sqrt(2.0) * (sqrt((hypot(C, B_m) + C)) / 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 = Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)
	t_1 = sqrt(fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(C * 2.0)))
	t_2 = fma(Float64(-4.0 * A), C, Float64(B_m * B_m))
	t_3 = Float64(Float64(2.0 * F) * t_2)
	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)
	t_7 = fma(Float64(C * -4.0), A, Float64(B_m * B_m))
	tmp = 0.0
	if (t_6 <= -2e+255)
		tmp = Float64(Float64(sqrt(Float64(fma(Float64(-4.0 * C), A, Float64(B_m * B_m)) * 2.0)) * sqrt(F)) * Float64(t_1 / Float64(-t_7)));
	elseif (t_6 <= -2e-216)
		tmp = Float64(sqrt(Float64(t_0 * t_3)) / Float64(-t_2));
	elseif (t_6 <= 0.0)
		tmp = Float64(Float64(Float64(t_1 * sqrt(Float64(F * 2.0))) * sqrt(t_7)) / t_5);
	elseif (t_6 <= Inf)
		tmp = Float64(Float64(sqrt(t_3) * sqrt(t_0)) / t_5);
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) * Float64(sqrt(Float64(hypot(C, B_m) + C)) / 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[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * F), $MachinePrecision] * t$95$2), $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]}, Block[{t$95$7 = N[(N[(C * -4.0), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, -2e+255], N[(N[(N[Sqrt[N[(N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / (-t$95$7)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, -2e-216], N[(N[Sqrt[N[(t$95$0 * t$95$3), $MachinePrecision]], $MachinePrecision] / (-t$95$2)), $MachinePrecision], If[LessEqual[t$95$6, 0.0], N[(N[(N[(t$95$1 * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$7], $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision], If[LessEqual[t$95$6, Infinity], N[(N[(N[Sqrt[t$95$3], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) * N[(N[Sqrt[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $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))) < -1.99999999999999998e255

   1. Initial program 5.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 18.5

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

   5. Applied rewrites 18.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 25.7%

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

   9. Applied rewrites 25.7%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lift-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. metadata-eval N/A

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

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

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

       13. lift-*.f64 N/A

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

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

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

       15. lift-*.f64 N/A

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

       16. lower--.f64 N/A

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

       17. lift-*.f64 N/A

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

       18. pow2 N/A

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

       19. lift-pow.f64 N/A

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

   11. Applied rewrites 40.2%

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

   if -1.99999999999999998e255 < (/.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))) < -2.0000000000000001e-216

   1. Initial program 98.2%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 98.2%

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

   if -2.0000000000000001e-216 < (/.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 3.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 30.2

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

   5. Applied rewrites 30.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 24.8%

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

   9. Applied rewrites 29.7%

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

   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 42.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 86.3%

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

   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. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 21.5

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

   5. Applied rewrites 21.5%

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

   7. Applied rewrites 30.0%

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

   9. Applied rewrites 30.1%

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

4. Final simplification 47.6%

   \[\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 -2 \cdot 10^{+255}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot 2} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot -4, 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^{-216}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, 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{\left(\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)} \cdot \sqrt{F \cdot 2}\right) \cdot \sqrt{\mathsf{fma}\left(C \cdot -4, 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 \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot \sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\sqrt{2}\right) \cdot \frac{\sqrt{\mathsf{hypot}\left(C, B\right) + C}}{B}\right) \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C \cdot 2\right)}\\ t_1 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ t_2 := \left(2 \cdot F\right) \cdot t\_1\\ t_3 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_4 := -t\_3\\ t_5 := \frac{\sqrt{\left(2 \cdot \left(t\_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_4}\\ t_6 := \mathsf{fma}\left(C \cdot -4, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;t\_5 \leq -2 \cdot 10^{+255}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right) \cdot 2} \cdot \sqrt{F}\right) \cdot \frac{t\_0}{-t\_6}\\ \mathbf{elif}\;t\_5 \leq -2 \cdot 10^{-216}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C\right) \cdot t\_2}}{-t\_1}\\ \mathbf{elif}\;t\_5 \leq 0:\\ \;\;\;\;\frac{\left(t\_0 \cdot \sqrt{F \cdot 2}\right) \cdot \sqrt{t\_6}}{t\_4}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_2} \cdot \sqrt{2 \cdot C}}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\sqrt{2}\right) \cdot \frac{\sqrt{\mathsf{hypot}\left(C, B\_m\right) + C}}{B\_m}\right) \cdot \sqrt{F}\\ \end{array} \end{array} \]

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 (sqrt (fma -0.5 (/ (* B_m B_m) A) (* C 2.0))))
        (t_1 (fma (* -4.0 A) C (* B_m B_m)))
        (t_2 (* (* 2.0 F) t_1))
        (t_3 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_4 (- t_3))
        (t_5
         (/
          (sqrt
           (*
            (* 2.0 (* t_3 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          t_4))
        (t_6 (fma (* C -4.0) A (* B_m B_m))))
   (if (<= t_5 -2e+255)
     (*
      (* (sqrt (* (fma (* -4.0 C) A (* B_m B_m)) 2.0)) (sqrt F))
      (/ t_0 (- t_6)))
     (if (<= t_5 -2e-216)
       (/ (sqrt (* (+ (+ (hypot (- A C) B_m) A) C) t_2)) (- t_1))
       (if (<= t_5 0.0)
         (/ (* (* t_0 (sqrt (* F 2.0))) (sqrt t_6)) t_4)
         (if (<= t_5 INFINITY)
           (/ (* (sqrt t_2) (sqrt (* 2.0 C))) t_4)
           (*
            (* (- (sqrt 2.0)) (/ (sqrt (+ (hypot C B_m) C)) 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 = sqrt(fma(-0.5, ((B_m * B_m) / A), (C * 2.0)));
	double t_1 = fma((-4.0 * A), C, (B_m * B_m));
	double t_2 = (2.0 * F) * t_1;
	double t_3 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_4 = -t_3;
	double t_5 = sqrt(((2.0 * (t_3 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_4;
	double t_6 = fma((C * -4.0), A, (B_m * B_m));
	double tmp;
	if (t_5 <= -2e+255) {
		tmp = (sqrt((fma((-4.0 * C), A, (B_m * B_m)) * 2.0)) * sqrt(F)) * (t_0 / -t_6);
	} else if (t_5 <= -2e-216) {
		tmp = sqrt((((hypot((A - C), B_m) + A) + C) * t_2)) / -t_1;
	} else if (t_5 <= 0.0) {
		tmp = ((t_0 * sqrt((F * 2.0))) * sqrt(t_6)) / t_4;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = (sqrt(t_2) * sqrt((2.0 * C))) / t_4;
	} else {
		tmp = (-sqrt(2.0) * (sqrt((hypot(C, B_m) + C)) / 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 = sqrt(fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(C * 2.0)))
	t_1 = fma(Float64(-4.0 * A), C, Float64(B_m * B_m))
	t_2 = Float64(Float64(2.0 * F) * t_1)
	t_3 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_4 = Float64(-t_3)
	t_5 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_3 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / t_4)
	t_6 = fma(Float64(C * -4.0), A, Float64(B_m * B_m))
	tmp = 0.0
	if (t_5 <= -2e+255)
		tmp = Float64(Float64(sqrt(Float64(fma(Float64(-4.0 * C), A, Float64(B_m * B_m)) * 2.0)) * sqrt(F)) * Float64(t_0 / Float64(-t_6)));
	elseif (t_5 <= -2e-216)
		tmp = Float64(sqrt(Float64(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C) * t_2)) / Float64(-t_1));
	elseif (t_5 <= 0.0)
		tmp = Float64(Float64(Float64(t_0 * sqrt(Float64(F * 2.0))) * sqrt(t_6)) / t_4);
	elseif (t_5 <= Inf)
		tmp = Float64(Float64(sqrt(t_2) * sqrt(Float64(2.0 * C))) / t_4);
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) * Float64(sqrt(Float64(hypot(C, B_m) + C)) / 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[Sqrt[N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * F), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = (-t$95$3)}, Block[{t$95$5 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$3 * 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 = N[(N[(C * -4.0), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -2e+255], N[(N[(N[Sqrt[N[(N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / (-t$95$6)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, -2e-216], N[(N[Sqrt[N[(N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[t$95$5, 0.0], N[(N[(N[(t$95$0 * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$6], $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(N[(N[Sqrt[t$95$2], $MachinePrecision] * N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) * N[(N[Sqrt[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $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))) < -1.99999999999999998e255

   1. Initial program 5.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 18.5

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

   5. Applied rewrites 18.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 25.7%

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

   9. Applied rewrites 25.7%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lift-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. metadata-eval N/A

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

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

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

       13. lift-*.f64 N/A

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

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

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

       15. lift-*.f64 N/A

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

       16. lower--.f64 N/A

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

       17. lift-*.f64 N/A

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

       18. pow2 N/A

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

       19. lift-pow.f64 N/A

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

   11. Applied rewrites 40.2%

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

   if -1.99999999999999998e255 < (/.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))) < -2.0000000000000001e-216

   1. Initial program 98.2%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 98.2%

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

   if -2.0000000000000001e-216 < (/.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 3.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 30.2

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

   5. Applied rewrites 30.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 24.8%

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

   9. Applied rewrites 29.7%

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

   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 42.9%

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

   3. Taylor expanded in A around -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 26.1

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

   5. Applied rewrites 26.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 30.7%

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

   8. Taylor expanded in A around inf

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

   10. Applied rewrites 30.7%

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

   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. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 21.5

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

   5. Applied rewrites 21.5%

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

   7. Applied rewrites 30.0%

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

   9. Applied rewrites 30.1%

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

4. Final simplification 42.8%

   \[\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 -2 \cdot 10^{+255}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot 2} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot -4, 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^{-216}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, 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{\left(\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)} \cdot \sqrt{F \cdot 2}\right) \cdot \sqrt{\mathsf{fma}\left(C \cdot -4, 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 \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot \sqrt{2 \cdot C}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\sqrt{2}\right) \cdot \frac{\sqrt{\mathsf{hypot}\left(C, B\right) + C}}{B}\right) \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ t_1 := \left(2 \cdot F\right) \cdot t\_0\\ t_2 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := -t\_2\\ t_4 := \mathsf{fma}\left(C \cdot -4, A, B\_m \cdot B\_m\right)\\ t_5 := \frac{\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_3}\\ t_6 := \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C \cdot 2\right)}}{-t\_4}\\ \mathbf{if}\;t\_5 \leq -2 \cdot 10^{+255}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right) \cdot 2} \cdot \sqrt{F}\right) \cdot t\_6\\ \mathbf{elif}\;t\_5 \leq -2 \cdot 10^{-216}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C\right) \cdot t\_1}}{-t\_0}\\ \mathbf{elif}\;t\_5 \leq 0:\\ \;\;\;\;\sqrt{F \cdot 2} \cdot \left(\sqrt{t\_4} \cdot t\_6\right)\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_1} \cdot \sqrt{2 \cdot C}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\sqrt{2}\right) \cdot \frac{\sqrt{\mathsf{hypot}\left(C, B\_m\right) + C}}{B\_m}\right) \cdot \sqrt{F}\\ \end{array} \end{array} \]

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 A) C (* B_m B_m)))
        (t_1 (* (* 2.0 F) t_0))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3 (- t_2))
        (t_4 (fma (* C -4.0) A (* B_m B_m)))
        (t_5
         (/
          (sqrt
           (*
            (* 2.0 (* t_2 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          t_3))
        (t_6 (/ (sqrt (fma -0.5 (/ (* B_m B_m) A) (* C 2.0))) (- t_4))))
   (if (<= t_5 -2e+255)
     (* (* (sqrt (* (fma (* -4.0 C) A (* B_m B_m)) 2.0)) (sqrt F)) t_6)
     (if (<= t_5 -2e-216)
       (/ (sqrt (* (+ (+ (hypot (- A C) B_m) A) C) t_1)) (- t_0))
       (if (<= t_5 0.0)
         (* (sqrt (* F 2.0)) (* (sqrt t_4) t_6))
         (if (<= t_5 INFINITY)
           (/ (* (sqrt t_1) (sqrt (* 2.0 C))) t_3)
           (*
            (* (- (sqrt 2.0)) (/ (sqrt (+ (hypot C B_m) C)) 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 * A), C, (B_m * B_m));
	double t_1 = (2.0 * F) * t_0;
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_3 = -t_2;
	double t_4 = fma((C * -4.0), A, (B_m * B_m));
	double t_5 = sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_3;
	double t_6 = sqrt(fma(-0.5, ((B_m * B_m) / A), (C * 2.0))) / -t_4;
	double tmp;
	if (t_5 <= -2e+255) {
		tmp = (sqrt((fma((-4.0 * C), A, (B_m * B_m)) * 2.0)) * sqrt(F)) * t_6;
	} else if (t_5 <= -2e-216) {
		tmp = sqrt((((hypot((A - C), B_m) + A) + C) * t_1)) / -t_0;
	} else if (t_5 <= 0.0) {
		tmp = sqrt((F * 2.0)) * (sqrt(t_4) * t_6);
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = (sqrt(t_1) * sqrt((2.0 * C))) / t_3;
	} else {
		tmp = (-sqrt(2.0) * (sqrt((hypot(C, B_m) + C)) / 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(-4.0 * A), C, Float64(B_m * B_m))
	t_1 = Float64(Float64(2.0 * F) * t_0)
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(-t_2)
	t_4 = fma(Float64(C * -4.0), A, Float64(B_m * B_m))
	t_5 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / t_3)
	t_6 = Float64(sqrt(fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(C * 2.0))) / Float64(-t_4))
	tmp = 0.0
	if (t_5 <= -2e+255)
		tmp = Float64(Float64(sqrt(Float64(fma(Float64(-4.0 * C), A, Float64(B_m * B_m)) * 2.0)) * sqrt(F)) * t_6);
	elseif (t_5 <= -2e-216)
		tmp = Float64(sqrt(Float64(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C) * t_1)) / Float64(-t_0));
	elseif (t_5 <= 0.0)
		tmp = Float64(sqrt(Float64(F * 2.0)) * Float64(sqrt(t_4) * t_6));
	elseif (t_5 <= Inf)
		tmp = Float64(Float64(sqrt(t_1) * sqrt(Float64(2.0 * C))) / t_3);
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) * Float64(sqrt(Float64(hypot(C, B_m) + C)) / 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[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = (-t$95$2)}, Block[{t$95$4 = N[(N[(C * -4.0), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$2 * 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$3), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$4)), $MachinePrecision]}, If[LessEqual[t$95$5, -2e+255], N[(N[(N[Sqrt[N[(N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * t$95$6), $MachinePrecision], If[LessEqual[t$95$5, -2e-216], N[(N[Sqrt[N[(N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$5, 0.0], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[t$95$4], $MachinePrecision] * t$95$6), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) * N[(N[Sqrt[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $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))) < -1.99999999999999998e255

   1. Initial program 5.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 18.5

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

   5. Applied rewrites 18.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 25.7%

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

   9. Applied rewrites 25.7%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lift-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. metadata-eval N/A

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

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

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

       13. lift-*.f64 N/A

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

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

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

       15. lift-*.f64 N/A

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

       16. lower--.f64 N/A

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

       17. lift-*.f64 N/A

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

       18. pow2 N/A

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

       19. lift-pow.f64 N/A

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

   11. Applied rewrites 40.2%

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

   if -1.99999999999999998e255 < (/.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))) < -2.0000000000000001e-216

   1. Initial program 98.2%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 98.2%

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

   if -2.0000000000000001e-216 < (/.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 3.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 30.2

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

   5. Applied rewrites 30.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 24.8%

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

   9. Applied rewrites 29.6%

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

   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 42.9%

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

   3. Taylor expanded in A around -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 26.1

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

   5. Applied rewrites 26.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 30.7%

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

   8. Taylor expanded in A around inf

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

   10. Applied rewrites 30.7%

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

   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. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 21.5

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

   5. Applied rewrites 21.5%

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

   7. Applied rewrites 30.0%

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

   9. Applied rewrites 30.1%

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

4. Final simplification 42.8%

   \[\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 -2 \cdot 10^{+255}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot 2} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot -4, 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^{-216}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, 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:\\ \;\;\;\;\sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\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{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot \sqrt{2 \cdot C}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\sqrt{2}\right) \cdot \frac{\sqrt{\mathsf{hypot}\left(C, B\right) + C}}{B}\right) \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(C \cdot -4, A, B\_m \cdot B\_m\right)\\ t_2 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_2}\\ t_4 := -t\_1\\ t_5 := \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C \cdot 2\right)}}{t\_4}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+255}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right) \cdot 2} \cdot \sqrt{F}\right) \cdot t\_5\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-216}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot t\_0\right)}}{-t\_0}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{F \cdot 2} \cdot \left(\sqrt{t\_1} \cdot t\_5\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot t\_1} \cdot \frac{\sqrt{C} \cdot \sqrt{2}}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\sqrt{2}\right) \cdot \frac{\sqrt{\mathsf{hypot}\left(C, B\_m\right) + C}}{B\_m}\right) \cdot \sqrt{F}\\ \end{array} \end{array} \]

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 A) C (* B_m B_m)))
        (t_1 (fma (* C -4.0) A (* B_m B_m)))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* t_2 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_2)))
        (t_4 (- t_1))
        (t_5 (/ (sqrt (fma -0.5 (/ (* B_m B_m) A) (* C 2.0))) t_4)))
   (if (<= t_3 -2e+255)
     (* (* (sqrt (* (fma (* -4.0 C) A (* B_m B_m)) 2.0)) (sqrt F)) t_5)
     (if (<= t_3 -2e-216)
       (/ (sqrt (* (+ (+ (hypot (- A C) B_m) A) C) (* (* 2.0 F) t_0))) (- t_0))
       (if (<= t_3 0.0)
         (* (sqrt (* F 2.0)) (* (sqrt t_1) t_5))
         (if (<= t_3 INFINITY)
           (* (sqrt (* (* F 2.0) t_1)) (/ (* (sqrt C) (sqrt 2.0)) t_4))
           (*
            (* (- (sqrt 2.0)) (/ (sqrt (+ (hypot C B_m) C)) 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 * A), C, (B_m * B_m));
	double t_1 = fma((C * -4.0), A, (B_m * B_m));
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_3 = sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_2;
	double t_4 = -t_1;
	double t_5 = sqrt(fma(-0.5, ((B_m * B_m) / A), (C * 2.0))) / t_4;
	double tmp;
	if (t_3 <= -2e+255) {
		tmp = (sqrt((fma((-4.0 * C), A, (B_m * B_m)) * 2.0)) * sqrt(F)) * t_5;
	} else if (t_3 <= -2e-216) {
		tmp = sqrt((((hypot((A - C), B_m) + A) + C) * ((2.0 * F) * t_0))) / -t_0;
	} else if (t_3 <= 0.0) {
		tmp = sqrt((F * 2.0)) * (sqrt(t_1) * t_5);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((F * 2.0) * t_1)) * ((sqrt(C) * sqrt(2.0)) / t_4);
	} else {
		tmp = (-sqrt(2.0) * (sqrt((hypot(C, B_m) + C)) / 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(-4.0 * A), C, Float64(B_m * B_m))
	t_1 = fma(Float64(C * -4.0), A, Float64(B_m * B_m))
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_2))
	t_4 = Float64(-t_1)
	t_5 = Float64(sqrt(fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(C * 2.0))) / t_4)
	tmp = 0.0
	if (t_3 <= -2e+255)
		tmp = Float64(Float64(sqrt(Float64(fma(Float64(-4.0 * C), A, Float64(B_m * B_m)) * 2.0)) * sqrt(F)) * t_5);
	elseif (t_3 <= -2e-216)
		tmp = Float64(sqrt(Float64(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C) * Float64(Float64(2.0 * F) * t_0))) / Float64(-t_0));
	elseif (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(F * 2.0)) * Float64(sqrt(t_1) * t_5));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(F * 2.0) * t_1)) * Float64(Float64(sqrt(C) * sqrt(2.0)) / t_4));
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) * Float64(sqrt(Float64(hypot(C, B_m) + C)) / 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[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(C * -4.0), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$2 * 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$2)), $MachinePrecision]}, Block[{t$95$4 = (-t$95$1)}, Block[{t$95$5 = N[(N[Sqrt[N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+255], N[(N[(N[Sqrt[N[(N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision], If[LessEqual[t$95$3, -2e-216], N[(N[Sqrt[N[(N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[t$95$1], $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[C], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) * N[(N[Sqrt[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $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))) < -1.99999999999999998e255

   1. Initial program 5.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 18.5

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

   5. Applied rewrites 18.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 25.7%

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

   9. Applied rewrites 25.7%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lift-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. metadata-eval N/A

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

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

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

       13. lift-*.f64 N/A

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

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

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

       15. lift-*.f64 N/A

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

       16. lower--.f64 N/A

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

       17. lift-*.f64 N/A

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

       18. pow2 N/A

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

       19. lift-pow.f64 N/A

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

   11. Applied rewrites 40.2%

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

   if -1.99999999999999998e255 < (/.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))) < -2.0000000000000001e-216

   1. Initial program 98.2%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 98.2%

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

   if -2.0000000000000001e-216 < (/.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 3.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 30.2

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

   5. Applied rewrites 30.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 24.8%

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

   9. Applied rewrites 29.6%

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

   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 42.9%

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

   3. Taylor expanded in A around -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 26.1

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

   5. Applied rewrites 26.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 30.7%

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

   9. Applied rewrites 30.8%

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

   10. Taylor expanded in A around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-sqrt.f64 30.9

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

   12. Applied rewrites 30.9%

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

   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. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 21.5

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

   5. Applied rewrites 21.5%

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

   7. Applied rewrites 30.0%

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

   9. Applied rewrites 30.1%

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

4. Final simplification 42.8%

   \[\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 -2 \cdot 10^{+255}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot 2} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot -4, 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^{-216}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, 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:\\ \;\;\;\;\sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\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:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \frac{\sqrt{C} \cdot \sqrt{2}}{-\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\sqrt{2}\right) \cdot \frac{\sqrt{\mathsf{hypot}\left(C, B\right) + C}}{B}\right) \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot -4, A, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{hypot}\left(C, B\_m\right) + C\\ t_2 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_2}\\ t_4 := -t\_0\\ t_5 := \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C \cdot 2\right)}}{t\_4}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{+139}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right) \cdot 2} \cdot \sqrt{F}\right) \cdot t\_5\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-216}:\\ \;\;\;\;\frac{\sqrt{\left(t\_1 \cdot F\right) \cdot 2}}{-B\_m}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{F \cdot 2} \cdot \left(\sqrt{t\_0} \cdot t\_5\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot t\_0} \cdot \frac{\sqrt{C} \cdot \sqrt{2}}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\sqrt{2}\right) \cdot \frac{\sqrt{t\_1}}{B\_m}\right) \cdot \sqrt{F}\\ \end{array} \end{array} \]

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 -4.0) A (* B_m B_m)))
        (t_1 (+ (hypot C B_m) C))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* t_2 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_2)))
        (t_4 (- t_0))
        (t_5 (/ (sqrt (fma -0.5 (/ (* B_m B_m) A) (* C 2.0))) t_4)))
   (if (<= t_3 -1e+139)
     (* (* (sqrt (* (fma (* -4.0 C) A (* B_m B_m)) 2.0)) (sqrt F)) t_5)
     (if (<= t_3 -2e-216)
       (/ (sqrt (* (* t_1 F) 2.0)) (- B_m))
       (if (<= t_3 0.0)
         (* (sqrt (* F 2.0)) (* (sqrt t_0) t_5))
         (if (<= t_3 INFINITY)
           (* (sqrt (* (* F 2.0) t_0)) (/ (* (sqrt C) (sqrt 2.0)) t_4))
           (* (* (- (sqrt 2.0)) (/ (sqrt t_1) 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 * -4.0), A, (B_m * B_m));
	double t_1 = hypot(C, B_m) + C;
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_3 = sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_2;
	double t_4 = -t_0;
	double t_5 = sqrt(fma(-0.5, ((B_m * B_m) / A), (C * 2.0))) / t_4;
	double tmp;
	if (t_3 <= -1e+139) {
		tmp = (sqrt((fma((-4.0 * C), A, (B_m * B_m)) * 2.0)) * sqrt(F)) * t_5;
	} else if (t_3 <= -2e-216) {
		tmp = sqrt(((t_1 * F) * 2.0)) / -B_m;
	} else if (t_3 <= 0.0) {
		tmp = sqrt((F * 2.0)) * (sqrt(t_0) * t_5);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((F * 2.0) * t_0)) * ((sqrt(C) * sqrt(2.0)) / t_4);
	} else {
		tmp = (-sqrt(2.0) * (sqrt(t_1) / 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 * -4.0), A, Float64(B_m * B_m))
	t_1 = Float64(hypot(C, B_m) + C)
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_2))
	t_4 = Float64(-t_0)
	t_5 = Float64(sqrt(fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(C * 2.0))) / t_4)
	tmp = 0.0
	if (t_3 <= -1e+139)
		tmp = Float64(Float64(sqrt(Float64(fma(Float64(-4.0 * C), A, Float64(B_m * B_m)) * 2.0)) * sqrt(F)) * t_5);
	elseif (t_3 <= -2e-216)
		tmp = Float64(sqrt(Float64(Float64(t_1 * F) * 2.0)) / Float64(-B_m));
	elseif (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(F * 2.0)) * Float64(sqrt(t_0) * t_5));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(F * 2.0) * t_0)) * Float64(Float64(sqrt(C) * sqrt(2.0)) / t_4));
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) * Float64(sqrt(t_1) / 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 * -4.0), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$2 * 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$2)), $MachinePrecision]}, Block[{t$95$4 = (-t$95$0)}, Block[{t$95$5 = N[(N[Sqrt[N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]}, If[LessEqual[t$95$3, -1e+139], N[(N[(N[Sqrt[N[(N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision], If[LessEqual[t$95$3, -2e-216], N[(N[Sqrt[N[(N[(t$95$1 * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[C], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) * N[(N[Sqrt[t$95$1], $MachinePrecision] / B$95$m), $MachinePrecision]), $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))) < -1.00000000000000003e139

   1. Initial program 19.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 24.5

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

   5. Applied rewrites 24.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. distribute-rgt-neg-in N/A

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

   7. Applied rewrites 30.5%

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

   9. Applied rewrites 30.6%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lift-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. metadata-eval N/A

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

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

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

       13. lift-*.f64 N/A

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

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

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

       15. lift-*.f64 N/A

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

       16. lower--.f64 N/A

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

       17. lift-*.f64 N/A

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

       18. pow2 N/A

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

       19. lift-pow.f64 N/A

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

   11. Applied rewrites 42.9%

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

   if -1.00000000000000003e139 < (/.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))) < -2.0000000000000001e-216

   1. Initial program 97.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 34.6

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

   5. Applied rewrites 34.6%

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

   7. Applied rewrites 34.6%

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

   if -2.0000000000000001e-216 < (/.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 3.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 30.2

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

   5. Applied rewrites 30.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left({\color{blue}{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}}^{\frac{1}{2}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 24.8%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. Applied rewrites 29.6%

      \[\leadsto \color{blue}{\sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right)} \]

   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 42.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -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 26.1

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 26.1%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left({\color{blue}{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}}^{\frac{1}{2}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 30.7%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. Applied rewrites 30.8%

      \[\leadsto \color{blue}{\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)} \]

   10. Taylor expanded in A around -inf

       \[\leadsto \sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\color{blue}{\sqrt{C} \cdot \sqrt{2}}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\color{blue}{\sqrt{C} \cdot \sqrt{2}}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\color{blue}{\sqrt{C}} \cdot \sqrt{2}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       3. lower-sqrt.f64 30.9

          \[\leadsto \sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\sqrt{C} \cdot \color{blue}{\sqrt{2}}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

   12. Applied rewrites 30.9%

       \[\leadsto \sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\color{blue}{\sqrt{C} \cdot \sqrt{2}}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

   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. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. *-commutative N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \cdot \frac{\sqrt{2}}{B}} \]

      4. lower-*.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \cdot \frac{\sqrt{2}}{B}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \cdot \frac{\sqrt{2}}{B} \]

      6. *-commutative N/A

         \[\leadsto -\sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \cdot \frac{\sqrt{2}}{B} \]

      7. lower-*.f64 N/A

         \[\leadsto -\sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \cdot \frac{\sqrt{2}}{B} \]

      8. +-commutative N/A

         \[\leadsto -\sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      9. lower-+.f64 N/A

         \[\leadsto -\sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      10. +-commutative N/A

          \[\leadsto -\sqrt{\left(\sqrt{\color{blue}{{C}^{2} + {B}^{2}}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      11. unpow2 N/A

          \[\leadsto -\sqrt{\left(\sqrt{\color{blue}{C \cdot C} + {B}^{2}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      12. unpow2 N/A

          \[\leadsto -\sqrt{\left(\sqrt{C \cdot C + \color{blue}{B \cdot B}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      13. lower-hypot.f64 N/A

          \[\leadsto -\sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      14. lower-/.f64 N/A

          \[\leadsto -\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \color{blue}{\frac{\sqrt{2}}{B}} \]

      15. lower-sqrt.f64 21.5

          \[\leadsto -\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \frac{\color{blue}{\sqrt{2}}}{B} \]

   5. Applied rewrites 21.5%

      \[\leadsto \color{blue}{-\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B}} \]
   6. Step-by-step derivation

   7. Applied rewrites 30.0%

      \[\leadsto -\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \sqrt{F} \]
   8. Step-by-step derivation

   9. Applied rewrites 30.1%

      \[\leadsto -\left(\sqrt{2} \cdot \frac{\sqrt{\mathsf{hypot}\left(C, B\right) + C}}{B}\right) \cdot \sqrt{F} \]
3. Recombined 5 regimes into one program.

4. Final simplification 33.0%

   \[\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 -1 \cdot 10^{+139}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot 2} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot -4, 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^{-216}:\\ \;\;\;\;\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 0:\\ \;\;\;\;\sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\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:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \frac{\sqrt{C} \cdot \sqrt{2}}{-\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\sqrt{2}\right) \cdot \frac{\sqrt{\mathsf{hypot}\left(C, B\right) + C}}{B}\right) \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot -4, A, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{hypot}\left(C, B\_m\right) + C\\ t_2 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_2}\\ t_4 := -t\_0\\ t_5 := \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C \cdot 2\right)}}{t\_4}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{+139}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right) \cdot 2} \cdot \sqrt{F}\right) \cdot t\_5\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-216}:\\ \;\;\;\;\frac{\sqrt{\left(t\_1 \cdot F\right) \cdot 2}}{-B\_m}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{F \cdot 2} \cdot \left(\sqrt{t\_0} \cdot t\_5\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot t\_0} \cdot \frac{\sqrt{C} \cdot \sqrt{2}}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t\_1}}{-B\_m} \cdot \sqrt{F}\\ \end{array} \end{array} \]

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 -4.0) A (* B_m B_m)))
        (t_1 (+ (hypot C B_m) C))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* t_2 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_2)))
        (t_4 (- t_0))
        (t_5 (/ (sqrt (fma -0.5 (/ (* B_m B_m) A) (* C 2.0))) t_4)))
   (if (<= t_3 -1e+139)
     (* (* (sqrt (* (fma (* -4.0 C) A (* B_m B_m)) 2.0)) (sqrt F)) t_5)
     (if (<= t_3 -2e-216)
       (/ (sqrt (* (* t_1 F) 2.0)) (- B_m))
       (if (<= t_3 0.0)
         (* (sqrt (* F 2.0)) (* (sqrt t_0) t_5))
         (if (<= t_3 INFINITY)
           (* (sqrt (* (* F 2.0) t_0)) (/ (* (sqrt C) (sqrt 2.0)) t_4))
           (* (/ (sqrt (* 2.0 t_1)) (- 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 * -4.0), A, (B_m * B_m));
	double t_1 = hypot(C, B_m) + C;
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_3 = sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_2;
	double t_4 = -t_0;
	double t_5 = sqrt(fma(-0.5, ((B_m * B_m) / A), (C * 2.0))) / t_4;
	double tmp;
	if (t_3 <= -1e+139) {
		tmp = (sqrt((fma((-4.0 * C), A, (B_m * B_m)) * 2.0)) * sqrt(F)) * t_5;
	} else if (t_3 <= -2e-216) {
		tmp = sqrt(((t_1 * F) * 2.0)) / -B_m;
	} else if (t_3 <= 0.0) {
		tmp = sqrt((F * 2.0)) * (sqrt(t_0) * t_5);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((F * 2.0) * t_0)) * ((sqrt(C) * sqrt(2.0)) / t_4);
	} else {
		tmp = (sqrt((2.0 * t_1)) / -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 * -4.0), A, Float64(B_m * B_m))
	t_1 = Float64(hypot(C, B_m) + C)
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_2))
	t_4 = Float64(-t_0)
	t_5 = Float64(sqrt(fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(C * 2.0))) / t_4)
	tmp = 0.0
	if (t_3 <= -1e+139)
		tmp = Float64(Float64(sqrt(Float64(fma(Float64(-4.0 * C), A, Float64(B_m * B_m)) * 2.0)) * sqrt(F)) * t_5);
	elseif (t_3 <= -2e-216)
		tmp = Float64(sqrt(Float64(Float64(t_1 * F) * 2.0)) / Float64(-B_m));
	elseif (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(F * 2.0)) * Float64(sqrt(t_0) * t_5));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(F * 2.0) * t_0)) * Float64(Float64(sqrt(C) * sqrt(2.0)) / t_4));
	else
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_1)) / 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 * -4.0), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$2 * 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$2)), $MachinePrecision]}, Block[{t$95$4 = (-t$95$0)}, Block[{t$95$5 = N[(N[Sqrt[N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]}, If[LessEqual[t$95$3, -1e+139], N[(N[(N[Sqrt[N[(N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision], If[LessEqual[t$95$3, -2e-216], N[(N[Sqrt[N[(N[(t$95$1 * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[C], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(2.0 * t$95$1), $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))) < -1.00000000000000003e139

   1. Initial program 19.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 24.5

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 24.5%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left({\color{blue}{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}}^{\frac{1}{2}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 30.5%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. Applied rewrites 30.6%

      \[\leadsto \color{blue}{\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(F \cdot 2\right)} \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       4. associate-*l* N/A

          \[\leadsto \sqrt{\color{blue}{F \cdot \left(2 \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)\right)}} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)\right) \cdot F}} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       6. lift-fma.f64 N/A

          \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(\left(C \cdot -4\right) \cdot A + B \cdot B\right)}\right) \cdot F} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       7. +-commutative N/A

          \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(B \cdot B + \left(C \cdot -4\right) \cdot A\right)}\right) \cdot F} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(2 \cdot \left(B \cdot B + \color{blue}{\left(C \cdot -4\right)} \cdot A\right)\right) \cdot F} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       9. associate-*l* N/A

          \[\leadsto \sqrt{\left(2 \cdot \left(B \cdot B + \color{blue}{C \cdot \left(-4 \cdot A\right)}\right)\right) \cdot F} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       10. *-commutative N/A

           \[\leadsto \sqrt{\left(2 \cdot \left(B \cdot B + \color{blue}{\left(-4 \cdot A\right) \cdot C}\right)\right) \cdot F} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       11. metadata-eval N/A

           \[\leadsto \sqrt{\left(2 \cdot \left(B \cdot B + \left(\color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot A\right) \cdot C\right)\right) \cdot F} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       12. distribute-lft-neg-in N/A

           \[\leadsto \sqrt{\left(2 \cdot \left(B \cdot B + \color{blue}{\left(\mathsf{neg}\left(4 \cdot A\right)\right)} \cdot C\right)\right) \cdot F} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       13. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(2 \cdot \left(B \cdot B + \left(\mathsf{neg}\left(\color{blue}{4 \cdot A}\right)\right) \cdot C\right)\right) \cdot F} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       14. fp-cancel-sub-sign-inv N/A

           \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)}\right) \cdot F} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       15. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(2 \cdot \left(B \cdot B - \color{blue}{\left(4 \cdot A\right) \cdot C}\right)\right) \cdot F} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       16. lower--.f64 N/A

           \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)}\right) \cdot F} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       17. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(2 \cdot \left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right)\right) \cdot F} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       18. pow2 N/A

           \[\leadsto \sqrt{\left(2 \cdot \left(\color{blue}{{B}^{2}} - \left(4 \cdot A\right) \cdot C\right)\right) \cdot F} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       19. lift-pow.f64 N/A

           \[\leadsto \sqrt{\left(2 \cdot \left(\color{blue}{{B}^{2}} - \left(4 \cdot A\right) \cdot C\right)\right) \cdot F} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

   11. Applied rewrites 42.9%

       \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot 2} \cdot \sqrt{F}\right)} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

   if -1.00000000000000003e139 < (/.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))) < -2.0000000000000001e-216

   1. Initial program 97.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. *-commutative N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \cdot \frac{\sqrt{2}}{B}} \]

      4. lower-*.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \cdot \frac{\sqrt{2}}{B}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \cdot \frac{\sqrt{2}}{B} \]

      6. *-commutative N/A

         \[\leadsto -\sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \cdot \frac{\sqrt{2}}{B} \]

      7. lower-*.f64 N/A

         \[\leadsto -\sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \cdot \frac{\sqrt{2}}{B} \]

      8. +-commutative N/A

         \[\leadsto -\sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      9. lower-+.f64 N/A

         \[\leadsto -\sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      10. +-commutative N/A

          \[\leadsto -\sqrt{\left(\sqrt{\color{blue}{{C}^{2} + {B}^{2}}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      11. unpow2 N/A

          \[\leadsto -\sqrt{\left(\sqrt{\color{blue}{C \cdot C} + {B}^{2}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      12. unpow2 N/A

          \[\leadsto -\sqrt{\left(\sqrt{C \cdot C + \color{blue}{B \cdot B}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      13. lower-hypot.f64 N/A

          \[\leadsto -\sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      14. lower-/.f64 N/A

          \[\leadsto -\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \color{blue}{\frac{\sqrt{2}}{B}} \]

      15. lower-sqrt.f64 34.6

          \[\leadsto -\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \frac{\color{blue}{\sqrt{2}}}{B} \]

   5. Applied rewrites 34.6%

      \[\leadsto \color{blue}{-\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B}} \]
   6. Step-by-step derivation

   7. Applied rewrites 34.6%

      \[\leadsto \frac{\sqrt{\left(\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F\right) \cdot 2}}{\color{blue}{-B}} \]

   if -2.0000000000000001e-216 < (/.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 3.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 30.2

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 30.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left({\color{blue}{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}}^{\frac{1}{2}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 24.8%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. Applied rewrites 29.6%

      \[\leadsto \color{blue}{\sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right)} \]

   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 42.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -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 26.1

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 26.1%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left({\color{blue}{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}}^{\frac{1}{2}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 30.7%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. Applied rewrites 30.8%

      \[\leadsto \color{blue}{\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)} \]

   10. Taylor expanded in A around -inf

       \[\leadsto \sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\color{blue}{\sqrt{C} \cdot \sqrt{2}}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\color{blue}{\sqrt{C} \cdot \sqrt{2}}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\color{blue}{\sqrt{C}} \cdot \sqrt{2}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       3. lower-sqrt.f64 30.9

          \[\leadsto \sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\sqrt{C} \cdot \color{blue}{\sqrt{2}}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

   12. Applied rewrites 30.9%

       \[\leadsto \sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\color{blue}{\sqrt{C} \cdot \sqrt{2}}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

   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. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. *-commutative N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \cdot \frac{\sqrt{2}}{B}} \]

      4. lower-*.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \cdot \frac{\sqrt{2}}{B}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \cdot \frac{\sqrt{2}}{B} \]

      6. *-commutative N/A

         \[\leadsto -\sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \cdot \frac{\sqrt{2}}{B} \]

      7. lower-*.f64 N/A

         \[\leadsto -\sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \cdot \frac{\sqrt{2}}{B} \]

      8. +-commutative N/A

         \[\leadsto -\sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      9. lower-+.f64 N/A

         \[\leadsto -\sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      10. +-commutative N/A

          \[\leadsto -\sqrt{\left(\sqrt{\color{blue}{{C}^{2} + {B}^{2}}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      11. unpow2 N/A

          \[\leadsto -\sqrt{\left(\sqrt{\color{blue}{C \cdot C} + {B}^{2}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      12. unpow2 N/A

          \[\leadsto -\sqrt{\left(\sqrt{C \cdot C + \color{blue}{B \cdot B}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      13. lower-hypot.f64 N/A

          \[\leadsto -\sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      14. lower-/.f64 N/A

          \[\leadsto -\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \color{blue}{\frac{\sqrt{2}}{B}} \]

      15. lower-sqrt.f64 21.5

          \[\leadsto -\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \frac{\color{blue}{\sqrt{2}}}{B} \]

   5. Applied rewrites 21.5%

      \[\leadsto \color{blue}{-\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B}} \]
   6. Step-by-step derivation

   7. Applied rewrites 30.0%

      \[\leadsto -\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \sqrt{F} \]
   8. Step-by-step derivation

   9. Applied rewrites 30.1%

      \[\leadsto \frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(C, B\right) + C\right)}}{-B} \cdot \color{blue}{\sqrt{F}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 33.0%

   \[\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 -1 \cdot 10^{+139}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot 2} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot -4, 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^{-216}:\\ \;\;\;\;\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 0:\\ \;\;\;\;\sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\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:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \frac{\sqrt{C} \cdot \sqrt{2}}{-\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(C, B\right) + C\right)}}{-B} \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot -4, A, B\_m \cdot B\_m\right)\\ t_1 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_1}\\ t_3 := -t\_0\\ t_4 := \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C \cdot 2\right)}}{t\_3}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+139}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right) \cdot 2} \cdot \sqrt{F}\right) \cdot t\_4\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-216}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot F\right) \cdot 2}}{-B\_m}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{F \cdot 2} \cdot \left(\sqrt{t\_0} \cdot t\_4\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot t\_0} \cdot \frac{\sqrt{C} \cdot \sqrt{2}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-\sqrt{2}}{B\_m} \cdot \sqrt{B\_m + C}\right) \cdot \sqrt{F}\\ \end{array} \end{array} \]

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 -4.0) A (* B_m B_m)))
        (t_1 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* t_1 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_1)))
        (t_3 (- t_0))
        (t_4 (/ (sqrt (fma -0.5 (/ (* B_m B_m) A) (* C 2.0))) t_3)))
   (if (<= t_2 -1e+139)
     (* (* (sqrt (* (fma (* -4.0 C) A (* B_m B_m)) 2.0)) (sqrt F)) t_4)
     (if (<= t_2 -2e-216)
       (/ (sqrt (* (* (+ (hypot C B_m) C) F) 2.0)) (- B_m))
       (if (<= t_2 0.0)
         (* (sqrt (* F 2.0)) (* (sqrt t_0) t_4))
         (if (<= t_2 INFINITY)
           (* (sqrt (* (* F 2.0) t_0)) (/ (* (sqrt C) (sqrt 2.0)) t_3))
           (* (* (/ (- (sqrt 2.0)) B_m) (sqrt (+ B_m C))) (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 * -4.0), A, (B_m * B_m));
	double t_1 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_2 = sqrt(((2.0 * (t_1 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_1;
	double t_3 = -t_0;
	double t_4 = sqrt(fma(-0.5, ((B_m * B_m) / A), (C * 2.0))) / t_3;
	double tmp;
	if (t_2 <= -1e+139) {
		tmp = (sqrt((fma((-4.0 * C), A, (B_m * B_m)) * 2.0)) * sqrt(F)) * t_4;
	} else if (t_2 <= -2e-216) {
		tmp = sqrt((((hypot(C, B_m) + C) * F) * 2.0)) / -B_m;
	} else if (t_2 <= 0.0) {
		tmp = sqrt((F * 2.0)) * (sqrt(t_0) * t_4);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((F * 2.0) * t_0)) * ((sqrt(C) * sqrt(2.0)) / t_3);
	} else {
		tmp = ((-sqrt(2.0) / B_m) * sqrt((B_m + C))) * 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 * -4.0), A, Float64(B_m * B_m))
	t_1 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_1))
	t_3 = Float64(-t_0)
	t_4 = Float64(sqrt(fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(C * 2.0))) / t_3)
	tmp = 0.0
	if (t_2 <= -1e+139)
		tmp = Float64(Float64(sqrt(Float64(fma(Float64(-4.0 * C), A, Float64(B_m * B_m)) * 2.0)) * sqrt(F)) * t_4);
	elseif (t_2 <= -2e-216)
		tmp = Float64(sqrt(Float64(Float64(Float64(hypot(C, B_m) + C) * F) * 2.0)) / Float64(-B_m));
	elseif (t_2 <= 0.0)
		tmp = Float64(sqrt(Float64(F * 2.0)) * Float64(sqrt(t_0) * t_4));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(F * 2.0) * t_0)) * Float64(Float64(sqrt(C) * sqrt(2.0)) / t_3));
	else
		tmp = Float64(Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(B_m + C))) * 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 * -4.0), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$1 * 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$1)), $MachinePrecision]}, Block[{t$95$3 = (-t$95$0)}, Block[{t$95$4 = N[(N[Sqrt[N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+139], N[(N[(N[Sqrt[N[(N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision], If[LessEqual[t$95$2, -2e-216], 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$2, 0.0], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[C], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision]), $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))) < -1.00000000000000003e139

   1. Initial program 19.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 24.5

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 24.5%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left({\color{blue}{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}}^{\frac{1}{2}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 30.5%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. Applied rewrites 30.6%

      \[\leadsto \color{blue}{\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(F \cdot 2\right)} \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       4. associate-*l* N/A

          \[\leadsto \sqrt{\color{blue}{F \cdot \left(2 \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)\right)}} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)\right) \cdot F}} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       6. lift-fma.f64 N/A

          \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(\left(C \cdot -4\right) \cdot A + B \cdot B\right)}\right) \cdot F} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       7. +-commutative N/A

          \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(B \cdot B + \left(C \cdot -4\right) \cdot A\right)}\right) \cdot F} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(2 \cdot \left(B \cdot B + \color{blue}{\left(C \cdot -4\right)} \cdot A\right)\right) \cdot F} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       9. associate-*l* N/A

          \[\leadsto \sqrt{\left(2 \cdot \left(B \cdot B + \color{blue}{C \cdot \left(-4 \cdot A\right)}\right)\right) \cdot F} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       10. *-commutative N/A

           \[\leadsto \sqrt{\left(2 \cdot \left(B \cdot B + \color{blue}{\left(-4 \cdot A\right) \cdot C}\right)\right) \cdot F} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       11. metadata-eval N/A

           \[\leadsto \sqrt{\left(2 \cdot \left(B \cdot B + \left(\color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot A\right) \cdot C\right)\right) \cdot F} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       12. distribute-lft-neg-in N/A

           \[\leadsto \sqrt{\left(2 \cdot \left(B \cdot B + \color{blue}{\left(\mathsf{neg}\left(4 \cdot A\right)\right)} \cdot C\right)\right) \cdot F} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       13. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(2 \cdot \left(B \cdot B + \left(\mathsf{neg}\left(\color{blue}{4 \cdot A}\right)\right) \cdot C\right)\right) \cdot F} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       14. fp-cancel-sub-sign-inv N/A

           \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)}\right) \cdot F} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       15. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(2 \cdot \left(B \cdot B - \color{blue}{\left(4 \cdot A\right) \cdot C}\right)\right) \cdot F} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       16. lower--.f64 N/A

           \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)}\right) \cdot F} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       17. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(2 \cdot \left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right)\right) \cdot F} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       18. pow2 N/A

           \[\leadsto \sqrt{\left(2 \cdot \left(\color{blue}{{B}^{2}} - \left(4 \cdot A\right) \cdot C\right)\right) \cdot F} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       19. lift-pow.f64 N/A

           \[\leadsto \sqrt{\left(2 \cdot \left(\color{blue}{{B}^{2}} - \left(4 \cdot A\right) \cdot C\right)\right) \cdot F} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

   11. Applied rewrites 42.9%

       \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot 2} \cdot \sqrt{F}\right)} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

   if -1.00000000000000003e139 < (/.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))) < -2.0000000000000001e-216

   1. Initial program 97.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. *-commutative N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \cdot \frac{\sqrt{2}}{B}} \]

      4. lower-*.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \cdot \frac{\sqrt{2}}{B}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \cdot \frac{\sqrt{2}}{B} \]

      6. *-commutative N/A

         \[\leadsto -\sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \cdot \frac{\sqrt{2}}{B} \]

      7. lower-*.f64 N/A

         \[\leadsto -\sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \cdot \frac{\sqrt{2}}{B} \]

      8. +-commutative N/A

         \[\leadsto -\sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      9. lower-+.f64 N/A

         \[\leadsto -\sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      10. +-commutative N/A

          \[\leadsto -\sqrt{\left(\sqrt{\color{blue}{{C}^{2} + {B}^{2}}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      11. unpow2 N/A

          \[\leadsto -\sqrt{\left(\sqrt{\color{blue}{C \cdot C} + {B}^{2}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      12. unpow2 N/A

          \[\leadsto -\sqrt{\left(\sqrt{C \cdot C + \color{blue}{B \cdot B}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      13. lower-hypot.f64 N/A

          \[\leadsto -\sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      14. lower-/.f64 N/A

          \[\leadsto -\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \color{blue}{\frac{\sqrt{2}}{B}} \]

      15. lower-sqrt.f64 34.6

          \[\leadsto -\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \frac{\color{blue}{\sqrt{2}}}{B} \]

   5. Applied rewrites 34.6%

      \[\leadsto \color{blue}{-\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B}} \]
   6. Step-by-step derivation

   7. Applied rewrites 34.6%

      \[\leadsto \frac{\sqrt{\left(\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F\right) \cdot 2}}{\color{blue}{-B}} \]

   if -2.0000000000000001e-216 < (/.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 3.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 30.2

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 30.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left({\color{blue}{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}}^{\frac{1}{2}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 24.8%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. Applied rewrites 29.6%

      \[\leadsto \color{blue}{\sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right)} \]

   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 42.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -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 26.1

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 26.1%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left({\color{blue}{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}}^{\frac{1}{2}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 30.7%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. Applied rewrites 30.8%

      \[\leadsto \color{blue}{\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)} \]

   10. Taylor expanded in A around -inf

       \[\leadsto \sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\color{blue}{\sqrt{C} \cdot \sqrt{2}}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\color{blue}{\sqrt{C} \cdot \sqrt{2}}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\color{blue}{\sqrt{C}} \cdot \sqrt{2}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       3. lower-sqrt.f64 30.9

          \[\leadsto \sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\sqrt{C} \cdot \color{blue}{\sqrt{2}}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

   12. Applied rewrites 30.9%

       \[\leadsto \sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\color{blue}{\sqrt{C} \cdot \sqrt{2}}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

   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. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. *-commutative N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \cdot \frac{\sqrt{2}}{B}} \]

      4. lower-*.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \cdot \frac{\sqrt{2}}{B}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \cdot \frac{\sqrt{2}}{B} \]

      6. *-commutative N/A

         \[\leadsto -\sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \cdot \frac{\sqrt{2}}{B} \]

      7. lower-*.f64 N/A

         \[\leadsto -\sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \cdot \frac{\sqrt{2}}{B} \]

      8. +-commutative N/A

         \[\leadsto -\sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      9. lower-+.f64 N/A

         \[\leadsto -\sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      10. +-commutative N/A

          \[\leadsto -\sqrt{\left(\sqrt{\color{blue}{{C}^{2} + {B}^{2}}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      11. unpow2 N/A

          \[\leadsto -\sqrt{\left(\sqrt{\color{blue}{C \cdot C} + {B}^{2}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      12. unpow2 N/A

          \[\leadsto -\sqrt{\left(\sqrt{C \cdot C + \color{blue}{B \cdot B}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      13. lower-hypot.f64 N/A

          \[\leadsto -\sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      14. lower-/.f64 N/A

          \[\leadsto -\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \color{blue}{\frac{\sqrt{2}}{B}} \]

      15. lower-sqrt.f64 21.5

          \[\leadsto -\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \frac{\color{blue}{\sqrt{2}}}{B} \]

   5. Applied rewrites 21.5%

      \[\leadsto \color{blue}{-\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B}} \]
   6. Step-by-step derivation

   7. Applied rewrites 30.0%

      \[\leadsto -\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \sqrt{F} \]

   8. Taylor expanded in C around 0

      \[\leadsto -\left(\frac{\sqrt{2}}{B} \cdot \sqrt{B + C}\right) \cdot \sqrt{F} \]
   9. Step-by-step derivation

   10. Applied rewrites 27.6%

       \[\leadsto -\left(\frac{\sqrt{2}}{B} \cdot \sqrt{B + C}\right) \cdot \sqrt{F} \]
3. Recombined 5 regimes into one program.

4. Final simplification 31.9%

   \[\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 -1 \cdot 10^{+139}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot 2} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot -4, 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^{-216}:\\ \;\;\;\;\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 0:\\ \;\;\;\;\sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\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:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \frac{\sqrt{C} \cdot \sqrt{2}}{-\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-\sqrt{2}}{B} \cdot \sqrt{B + C}\right) \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.9% accurate, 4.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot -4, A, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ t_2 := \frac{B\_m \cdot B\_m}{A}\\ \mathbf{if}\;B\_m \leq 9 \cdot 10^{-225}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(t\_2, -0.5, C \cdot 2\right) \cdot \left(\left(2 \cdot F\right) \cdot t\_1\right)}}{-t\_1}\\ \mathbf{elif}\;B\_m \leq 23500000000000:\\ \;\;\;\;\sqrt{\left(F + F\right) \cdot t\_0} \cdot \frac{\sqrt{\mathsf{fma}\left(-0.5, t\_2, C \cdot 2\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-\sqrt{2}}{B\_m} \cdot \sqrt{B\_m + C}\right) \cdot \sqrt{F}\\ \end{array} \end{array} \]

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 -4.0) A (* B_m B_m)))
        (t_1 (fma (* -4.0 A) C (* B_m B_m)))
        (t_2 (/ (* B_m B_m) A)))
   (if (<= B_m 9e-225)
     (/ (sqrt (* (fma t_2 -0.5 (* C 2.0)) (* (* 2.0 F) t_1))) (- t_1))
     (if (<= B_m 23500000000000.0)
       (* (sqrt (* (+ F F) t_0)) (/ (sqrt (fma -0.5 t_2 (* C 2.0))) (- t_0)))
       (* (* (/ (- (sqrt 2.0)) B_m) (sqrt (+ B_m C))) (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 * -4.0), A, (B_m * B_m));
	double t_1 = fma((-4.0 * A), C, (B_m * B_m));
	double t_2 = (B_m * B_m) / A;
	double tmp;
	if (B_m <= 9e-225) {
		tmp = sqrt((fma(t_2, -0.5, (C * 2.0)) * ((2.0 * F) * t_1))) / -t_1;
	} else if (B_m <= 23500000000000.0) {
		tmp = sqrt(((F + F) * t_0)) * (sqrt(fma(-0.5, t_2, (C * 2.0))) / -t_0);
	} else {
		tmp = ((-sqrt(2.0) / B_m) * sqrt((B_m + C))) * 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 * -4.0), A, Float64(B_m * B_m))
	t_1 = fma(Float64(-4.0 * A), C, Float64(B_m * B_m))
	t_2 = Float64(Float64(B_m * B_m) / A)
	tmp = 0.0
	if (B_m <= 9e-225)
		tmp = Float64(sqrt(Float64(fma(t_2, -0.5, Float64(C * 2.0)) * Float64(Float64(2.0 * F) * t_1))) / Float64(-t_1));
	elseif (B_m <= 23500000000000.0)
		tmp = Float64(sqrt(Float64(Float64(F + F) * t_0)) * Float64(sqrt(fma(-0.5, t_2, Float64(C * 2.0))) / Float64(-t_0)));
	else
		tmp = Float64(Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(B_m + C))) * 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 * -4.0), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]}, If[LessEqual[B$95$m, 9e-225], N[(N[Sqrt[N[(N[(t$95$2 * -0.5 + N[(C * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[B$95$m, 23500000000000.0], N[(N[Sqrt[N[(N[(F + F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(-0.5 * t$95$2 + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 8.9999999999999999e-225

   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 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 16.2

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 16.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 16.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}} \]

   if 8.9999999999999999e-225 < B < 2.35e13

   1. Initial program 32.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 20.2

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 20.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left({\color{blue}{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}}^{\frac{1}{2}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 19.8%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. Applied rewrites 19.9%

      \[\leadsto \color{blue}{\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(F \cdot 2\right)} \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto \sqrt{\color{blue}{\left(2 \cdot F\right)} \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       3. count-2-rev N/A

          \[\leadsto \sqrt{\color{blue}{\left(F + F\right)} \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

       4. lower-+.f64 19.9

          \[\leadsto \sqrt{\color{blue}{\left(F + F\right)} \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

   11. Applied rewrites 19.9%

       \[\leadsto \sqrt{\color{blue}{\left(F + F\right)} \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

   if 2.35e13 < B

   1. Initial program 14.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 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. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. *-commutative N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \cdot \frac{\sqrt{2}}{B}} \]

      4. lower-*.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \cdot \frac{\sqrt{2}}{B}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \cdot \frac{\sqrt{2}}{B} \]

      6. *-commutative N/A

         \[\leadsto -\sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \cdot \frac{\sqrt{2}}{B} \]

      7. lower-*.f64 N/A

         \[\leadsto -\sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \cdot \frac{\sqrt{2}}{B} \]

      8. +-commutative N/A

         \[\leadsto -\sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      9. lower-+.f64 N/A

         \[\leadsto -\sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      10. +-commutative N/A

          \[\leadsto -\sqrt{\left(\sqrt{\color{blue}{{C}^{2} + {B}^{2}}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      11. unpow2 N/A

          \[\leadsto -\sqrt{\left(\sqrt{\color{blue}{C \cdot C} + {B}^{2}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      12. unpow2 N/A

          \[\leadsto -\sqrt{\left(\sqrt{C \cdot C + \color{blue}{B \cdot B}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      13. lower-hypot.f64 N/A

          \[\leadsto -\sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      14. lower-/.f64 N/A

          \[\leadsto -\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \color{blue}{\frac{\sqrt{2}}{B}} \]

      15. lower-sqrt.f64 53.6

          \[\leadsto -\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \frac{\color{blue}{\sqrt{2}}}{B} \]

   5. Applied rewrites 53.6%

      \[\leadsto \color{blue}{-\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B}} \]
   6. Step-by-step derivation

   7. Applied rewrites 72.6%

      \[\leadsto -\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \sqrt{F} \]

   8. Taylor expanded in C around 0

      \[\leadsto -\left(\frac{\sqrt{2}}{B} \cdot \sqrt{B + C}\right) \cdot \sqrt{F} \]
   9. Step-by-step derivation

   10. Applied rewrites 69.7%

       \[\leadsto -\left(\frac{\sqrt{2}}{B} \cdot \sqrt{B + C}\right) \cdot \sqrt{F} \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9 \cdot 10^{-225}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 23500000000000:\\ \;\;\;\;\sqrt{\left(F + F\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-\sqrt{2}}{B} \cdot \sqrt{B + C}\right) \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 53.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot -4, A, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 9 \cdot 10^{-225}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\frac{B\_m \cdot B\_m}{A}, -0.5, C \cdot 2\right) \cdot \left(\left(2 \cdot F\right) \cdot t\_1\right)}}{-t\_1}\\ \mathbf{elif}\;B\_m \leq 21500000000000:\\ \;\;\;\;\left(-\sqrt{\left(F \cdot 2\right) \cdot t\_0}\right) \cdot \frac{\sqrt{2 \cdot C}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-\sqrt{2}}{B\_m} \cdot \sqrt{B\_m + C}\right) \cdot \sqrt{F}\\ \end{array} \end{array} \]

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 -4.0) A (* B_m B_m)))
        (t_1 (fma (* -4.0 A) C (* B_m B_m))))
   (if (<= B_m 9e-225)
     (/
      (sqrt (* (fma (/ (* B_m B_m) A) -0.5 (* C 2.0)) (* (* 2.0 F) t_1)))
      (- t_1))
     (if (<= B_m 21500000000000.0)
       (* (- (sqrt (* (* F 2.0) t_0))) (/ (sqrt (* 2.0 C)) t_0))
       (* (* (/ (- (sqrt 2.0)) B_m) (sqrt (+ B_m C))) (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 * -4.0), A, (B_m * B_m));
	double t_1 = fma((-4.0 * A), C, (B_m * B_m));
	double tmp;
	if (B_m <= 9e-225) {
		tmp = sqrt((fma(((B_m * B_m) / A), -0.5, (C * 2.0)) * ((2.0 * F) * t_1))) / -t_1;
	} else if (B_m <= 21500000000000.0) {
		tmp = -sqrt(((F * 2.0) * t_0)) * (sqrt((2.0 * C)) / t_0);
	} else {
		tmp = ((-sqrt(2.0) / B_m) * sqrt((B_m + C))) * 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 * -4.0), A, Float64(B_m * B_m))
	t_1 = fma(Float64(-4.0 * A), C, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 9e-225)
		tmp = Float64(sqrt(Float64(fma(Float64(Float64(B_m * B_m) / A), -0.5, Float64(C * 2.0)) * Float64(Float64(2.0 * F) * t_1))) / Float64(-t_1));
	elseif (B_m <= 21500000000000.0)
		tmp = Float64(Float64(-sqrt(Float64(Float64(F * 2.0) * t_0))) * Float64(sqrt(Float64(2.0 * C)) / t_0));
	else
		tmp = Float64(Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(B_m + C))) * 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 * -4.0), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 9e-225], N[(N[Sqrt[N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5 + N[(C * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[B$95$m, 21500000000000.0], N[((-N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]) * N[(N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 8.9999999999999999e-225

   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 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 16.2

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 16.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 16.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}} \]

   if 8.9999999999999999e-225 < B < 2.15e13

   1. Initial program 32.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 20.2

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 20.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left({\color{blue}{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}}^{\frac{1}{2}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 19.8%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. Applied rewrites 19.9%

      \[\leadsto \color{blue}{\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)} \]

   10. Taylor expanded in A around inf

       \[\leadsto \sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\sqrt{2 \cdot \color{blue}{C}}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 23.7%

       \[\leadsto \sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\sqrt{2 \cdot \color{blue}{C}}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

   if 2.15e13 < B

   1. Initial program 14.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 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. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. *-commutative N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \cdot \frac{\sqrt{2}}{B}} \]

      4. lower-*.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \cdot \frac{\sqrt{2}}{B}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \cdot \frac{\sqrt{2}}{B} \]

      6. *-commutative N/A

         \[\leadsto -\sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \cdot \frac{\sqrt{2}}{B} \]

      7. lower-*.f64 N/A

         \[\leadsto -\sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \cdot \frac{\sqrt{2}}{B} \]

      8. +-commutative N/A

         \[\leadsto -\sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      9. lower-+.f64 N/A

         \[\leadsto -\sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      10. +-commutative N/A

          \[\leadsto -\sqrt{\left(\sqrt{\color{blue}{{C}^{2} + {B}^{2}}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      11. unpow2 N/A

          \[\leadsto -\sqrt{\left(\sqrt{\color{blue}{C \cdot C} + {B}^{2}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      12. unpow2 N/A

          \[\leadsto -\sqrt{\left(\sqrt{C \cdot C + \color{blue}{B \cdot B}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      13. lower-hypot.f64 N/A

          \[\leadsto -\sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      14. lower-/.f64 N/A

          \[\leadsto -\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \color{blue}{\frac{\sqrt{2}}{B}} \]

      15. lower-sqrt.f64 53.6

          \[\leadsto -\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \frac{\color{blue}{\sqrt{2}}}{B} \]

   5. Applied rewrites 53.6%

      \[\leadsto \color{blue}{-\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B}} \]
   6. Step-by-step derivation

   7. Applied rewrites 72.6%

      \[\leadsto -\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \sqrt{F} \]

   8. Taylor expanded in C around 0

      \[\leadsto -\left(\frac{\sqrt{2}}{B} \cdot \sqrt{B + C}\right) \cdot \sqrt{F} \]
   9. Step-by-step derivation

   10. Applied rewrites 69.7%

       \[\leadsto -\left(\frac{\sqrt{2}}{B} \cdot \sqrt{B + C}\right) \cdot \sqrt{F} \]
3. Recombined 3 regimes into one program.

4. Final simplification 29.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9 \cdot 10^{-225}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 21500000000000:\\ \;\;\;\;\left(-\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \cdot \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-\sqrt{2}}{B} \cdot \sqrt{B + C}\right) \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 52.6% accurate, 5.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot -4, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 21500000000000:\\ \;\;\;\;\left(-\sqrt{\left(F \cdot 2\right) \cdot t\_0}\right) \cdot \frac{\sqrt{2 \cdot C}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-\sqrt{2}}{B\_m} \cdot \sqrt{B\_m + C}\right) \cdot \sqrt{F}\\ \end{array} \end{array} \]

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 -4.0) A (* B_m B_m))))
   (if (<= B_m 21500000000000.0)
     (* (- (sqrt (* (* F 2.0) t_0))) (/ (sqrt (* 2.0 C)) t_0))
     (* (* (/ (- (sqrt 2.0)) B_m) (sqrt (+ B_m C))) (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 * -4.0), A, (B_m * B_m));
	double tmp;
	if (B_m <= 21500000000000.0) {
		tmp = -sqrt(((F * 2.0) * t_0)) * (sqrt((2.0 * C)) / t_0);
	} else {
		tmp = ((-sqrt(2.0) / B_m) * sqrt((B_m + C))) * 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 * -4.0), A, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 21500000000000.0)
		tmp = Float64(Float64(-sqrt(Float64(Float64(F * 2.0) * t_0))) * Float64(sqrt(Float64(2.0 * C)) / t_0));
	else
		tmp = Float64(Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(B_m + C))) * 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 * -4.0), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 21500000000000.0], N[((-N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]) * N[(N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 2.15e13

   1. Initial program 23.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 17.1

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, \color{blue}{2 \cdot C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 17.1%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left({\color{blue}{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}}^{\frac{1}{2}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(\frac{B \cdot B}{A}, \frac{-1}{2}, 2 \cdot C\right)\right)}^{\frac{1}{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 16.4%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. Applied rewrites 16.4%

      \[\leadsto \color{blue}{\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)} \]

   10. Taylor expanded in A around inf

       \[\leadsto \sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\sqrt{2 \cdot \color{blue}{C}}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 16.8%

       \[\leadsto \sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)} \cdot \left(-\frac{\sqrt{2 \cdot \color{blue}{C}}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \]

   if 2.15e13 < B

   1. Initial program 14.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 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. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. *-commutative N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \cdot \frac{\sqrt{2}}{B}} \]

      4. lower-*.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \cdot \frac{\sqrt{2}}{B}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \cdot \frac{\sqrt{2}}{B} \]

      6. *-commutative N/A

         \[\leadsto -\sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \cdot \frac{\sqrt{2}}{B} \]

      7. lower-*.f64 N/A

         \[\leadsto -\sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \cdot \frac{\sqrt{2}}{B} \]

      8. +-commutative N/A

         \[\leadsto -\sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      9. lower-+.f64 N/A

         \[\leadsto -\sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      10. +-commutative N/A

          \[\leadsto -\sqrt{\left(\sqrt{\color{blue}{{C}^{2} + {B}^{2}}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      11. unpow2 N/A

          \[\leadsto -\sqrt{\left(\sqrt{\color{blue}{C \cdot C} + {B}^{2}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      12. unpow2 N/A

          \[\leadsto -\sqrt{\left(\sqrt{C \cdot C + \color{blue}{B \cdot B}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      13. lower-hypot.f64 N/A

          \[\leadsto -\sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      14. lower-/.f64 N/A

          \[\leadsto -\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \color{blue}{\frac{\sqrt{2}}{B}} \]

      15. lower-sqrt.f64 53.6

          \[\leadsto -\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \frac{\color{blue}{\sqrt{2}}}{B} \]

   5. Applied rewrites 53.6%

      \[\leadsto \color{blue}{-\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B}} \]
   6. Step-by-step derivation

   7. Applied rewrites 72.6%

      \[\leadsto -\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \sqrt{F} \]

   8. Taylor expanded in C around 0

      \[\leadsto -\left(\frac{\sqrt{2}}{B} \cdot \sqrt{B + C}\right) \cdot \sqrt{F} \]
   9. Step-by-step derivation

   10. Applied rewrites 69.7%

       \[\leadsto -\left(\frac{\sqrt{2}}{B} \cdot \sqrt{B + C}\right) \cdot \sqrt{F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 21500000000000:\\ \;\;\;\;\left(-\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right) \cdot \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-\sqrt{2}}{B} \cdot \sqrt{B + C}\right) \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 36.8% accurate, 8.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \left(\frac{-\sqrt{2}}{B\_m} \cdot \sqrt{B\_m + C}\right) \cdot \sqrt{F} \end{array} \]

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 (+ B_m C))) (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((B_m + C))) * 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((b_m + c))) * 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((B_m + C))) * 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((B_m + C))) * 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(Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(B_m + C))) * 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((B_m + C))) * 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[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.2%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
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. lower-neg.f64 N/A

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

   3. *-commutative N/A

      \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \cdot \frac{\sqrt{2}}{B}} \]

   4. lower-*.f64 N/A

      \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \cdot \frac{\sqrt{2}}{B}} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \cdot \frac{\sqrt{2}}{B} \]

   6. *-commutative N/A

      \[\leadsto -\sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \cdot \frac{\sqrt{2}}{B} \]

   7. lower-*.f64 N/A

      \[\leadsto -\sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \cdot \frac{\sqrt{2}}{B} \]

   8. +-commutative N/A

      \[\leadsto -\sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \cdot \frac{\sqrt{2}}{B} \]

   9. lower-+.f64 N/A

      \[\leadsto -\sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \cdot \frac{\sqrt{2}}{B} \]

   10. +-commutative N/A

       \[\leadsto -\sqrt{\left(\sqrt{\color{blue}{{C}^{2} + {B}^{2}}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

   11. unpow2 N/A

       \[\leadsto -\sqrt{\left(\sqrt{\color{blue}{C \cdot C} + {B}^{2}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

   12. unpow2 N/A

       \[\leadsto -\sqrt{\left(\sqrt{C \cdot C + \color{blue}{B \cdot B}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

   13. lower-hypot.f64 N/A

       \[\leadsto -\sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

   14. lower-/.f64 N/A

       \[\leadsto -\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \color{blue}{\frac{\sqrt{2}}{B}} \]

   15. lower-sqrt.f64 15.5

       \[\leadsto -\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \frac{\color{blue}{\sqrt{2}}}{B} \]

5. Applied rewrites 15.5%

   \[\leadsto \color{blue}{-\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B}} \]
6. Step-by-step derivation

7. Applied rewrites 19.7%

   \[\leadsto -\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \sqrt{F} \]

8. Taylor expanded in C around 0

   \[\leadsto -\left(\frac{\sqrt{2}}{B} \cdot \sqrt{B + C}\right) \cdot \sqrt{F} \]
9. Step-by-step derivation

10. Applied rewrites 16.7%

    \[\leadsto -\left(\frac{\sqrt{2}}{B} \cdot \sqrt{B + C}\right) \cdot \sqrt{F} \]

11. Final simplification 16.7%

    \[\leadsto \left(\frac{-\sqrt{2}}{B} \cdot \sqrt{B + C}\right) \cdot \sqrt{F} \]
12. Add Preprocessing

Alternative 17: 28.8% accurate, 12.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 2.1 \cdot 10^{+137}:\\ \;\;\;\;-\sqrt{\frac{F \cdot 2}{B\_m}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{2}{B\_m} \cdot \sqrt{C \cdot F}\\ \end{array} \end{array} \]

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 2.1e+137)
   (- (sqrt (/ (* F 2.0) B_m)))
   (- (* (/ 2.0 B_m) (sqrt (* C 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 <= 2.1e+137) {
		tmp = -sqrt(((F * 2.0) / B_m));
	} else {
		tmp = -((2.0 / B_m) * sqrt((C * 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 <= 2.1d+137) then
        tmp = -sqrt(((f * 2.0d0) / b_m))
    else
        tmp = -((2.0d0 / b_m) * sqrt((c * 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 <= 2.1e+137) {
		tmp = -Math.sqrt(((F * 2.0) / B_m));
	} else {
		tmp = -((2.0 / B_m) * Math.sqrt((C * 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 <= 2.1e+137:
		tmp = -math.sqrt(((F * 2.0) / B_m))
	else:
		tmp = -((2.0 / B_m) * math.sqrt((C * 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 <= 2.1e+137)
		tmp = Float64(-sqrt(Float64(Float64(F * 2.0) / B_m)));
	else
		tmp = Float64(-Float64(Float64(2.0 / B_m) * sqrt(Float64(C * 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 <= 2.1e+137)
		tmp = -sqrt(((F * 2.0) / B_m));
	else
		tmp = -((2.0 / B_m) * sqrt((C * 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, 2.1e+137], (-N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision]), (-N[(N[(2.0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if C < 2.0999999999999999e137

   1. Initial program 23.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in 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. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{\frac{F}{B}}} \cdot \sqrt{2} \]

      5. lower-/.f64 N/A

         \[\leadsto -\sqrt{\color{blue}{\frac{F}{B}}} \cdot \sqrt{2} \]

      6. lower-sqrt.f64 13.0

         \[\leadsto -\sqrt{\frac{F}{B}} \cdot \color{blue}{\sqrt{2}} \]

   5. Applied rewrites 13.0%

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 13.0%

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B} \cdot 2}} \]
   8. Step-by-step derivation

   9. Applied rewrites 13.1%

      \[\leadsto -\sqrt{\frac{F \cdot 2}{B}} \]

   if 2.0999999999999999e137 < C

   1. Initial program 6.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. *-commutative N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \cdot \frac{\sqrt{2}}{B}} \]

      4. lower-*.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \cdot \frac{\sqrt{2}}{B}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \cdot \frac{\sqrt{2}}{B} \]

      6. *-commutative N/A

         \[\leadsto -\sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \cdot \frac{\sqrt{2}}{B} \]

      7. lower-*.f64 N/A

         \[\leadsto -\sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \cdot \frac{\sqrt{2}}{B} \]

      8. +-commutative N/A

         \[\leadsto -\sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      9. lower-+.f64 N/A

         \[\leadsto -\sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      10. +-commutative N/A

          \[\leadsto -\sqrt{\left(\sqrt{\color{blue}{{C}^{2} + {B}^{2}}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      11. unpow2 N/A

          \[\leadsto -\sqrt{\left(\sqrt{\color{blue}{C \cdot C} + {B}^{2}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      12. unpow2 N/A

          \[\leadsto -\sqrt{\left(\sqrt{C \cdot C + \color{blue}{B \cdot B}} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      13. lower-hypot.f64 N/A

          \[\leadsto -\sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      14. lower-/.f64 N/A

          \[\leadsto -\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \color{blue}{\frac{\sqrt{2}}{B}} \]

      15. lower-sqrt.f64 9.7

          \[\leadsto -\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \frac{\color{blue}{\sqrt{2}}}{B} \]

   5. Applied rewrites 9.7%

      \[\leadsto \color{blue}{-\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \frac{\sqrt{2}}{B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto -\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F} \]
   7. Step-by-step derivation

   8. Applied rewrites 4.9%

      \[\leadsto -\frac{2}{B} \cdot \sqrt{C \cdot F} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 36.1% 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 21.2%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in 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. lower-neg.f64 N/A

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

   3. lower-*.f64 N/A

      \[\leadsto -\color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto -\color{blue}{\sqrt{\frac{F}{B}}} \cdot \sqrt{2} \]

   5. lower-/.f64 N/A

      \[\leadsto -\sqrt{\color{blue}{\frac{F}{B}}} \cdot \sqrt{2} \]

   6. lower-sqrt.f64 11.6

      \[\leadsto -\sqrt{\frac{F}{B}} \cdot \color{blue}{\sqrt{2}} \]

5. Applied rewrites 11.6%

   \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]
6. Step-by-step derivation

7. Applied rewrites 11.6%

   \[\leadsto \color{blue}{-\sqrt{\frac{F}{B} \cdot 2}} \]
8. Step-by-step derivation

9. Applied rewrites 17.0%

   \[\leadsto -\frac{\sqrt{F \cdot 2}}{\sqrt{B}} \]

10. Final simplification 17.0%

    \[\leadsto \frac{\sqrt{F \cdot 2}}{-\sqrt{B}} \]
11. Add Preprocessing

Alternative 19: 28.0% accurate, 16.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -\sqrt{\frac{F \cdot 2}{B\_m}} \end{array} \]

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) 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) / 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) / 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) / 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) / 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 * 2.0) / 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) / 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 * 2.0), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 21.2%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in 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. lower-neg.f64 N/A

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

   3. lower-*.f64 N/A

      \[\leadsto -\color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto -\color{blue}{\sqrt{\frac{F}{B}}} \cdot \sqrt{2} \]

   5. lower-/.f64 N/A

      \[\leadsto -\sqrt{\color{blue}{\frac{F}{B}}} \cdot \sqrt{2} \]

   6. lower-sqrt.f64 11.6

      \[\leadsto -\sqrt{\frac{F}{B}} \cdot \color{blue}{\sqrt{2}} \]

5. Applied rewrites 11.6%

   \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]
6. Step-by-step derivation

7. Applied rewrites 11.6%

   \[\leadsto \color{blue}{-\sqrt{\frac{F}{B} \cdot 2}} \]
8. Step-by-step derivation

9. Applied rewrites 11.7%

   \[\leadsto -\sqrt{\frac{F \cdot 2}{B}} \]
10. Add Preprocessing

Alternative 20: 28.0% accurate, 16.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -\sqrt{F \cdot \frac{2}{B\_m}} \end{array} \]

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 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 / 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 / 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 / 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 / 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 * Float64(2.0 / 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 / 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[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 21.2%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in 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. lower-neg.f64 N/A

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

   3. lower-*.f64 N/A

      \[\leadsto -\color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto -\color{blue}{\sqrt{\frac{F}{B}}} \cdot \sqrt{2} \]

   5. lower-/.f64 N/A

      \[\leadsto -\sqrt{\color{blue}{\frac{F}{B}}} \cdot \sqrt{2} \]

   6. lower-sqrt.f64 11.6

      \[\leadsto -\sqrt{\frac{F}{B}} \cdot \color{blue}{\sqrt{2}} \]

5. Applied rewrites 11.6%

   \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]
6. Step-by-step derivation

7. Applied rewrites 11.6%

   \[\leadsto \color{blue}{-\sqrt{\frac{F}{B} \cdot 2}} \]
8. Step-by-step derivation

9. Applied rewrites 11.7%

   \[\leadsto -\sqrt{F \cdot \frac{2}{B}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025017 
(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))))