ABCF->ab-angle a

Percentage Accurate: 19.2% → 64.1%

Time: 10.9s

Alternatives: 17

Speedup: 20.5×

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: 19.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 64.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{2 \cdot C}\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := {B\_m}^{2} - t\_1\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_2}\\ t_4 := B\_m \cdot B\_m - t\_1\\ t_5 := 2 \cdot \left(t\_4 \cdot F\right)\\ t_6 := \left(-B\_m\right) \cdot B\_m + t\_1\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot \left(\sqrt{t\_4} \cdot \left(-\sqrt{F}\right)\right)\right) \cdot t\_0}{t\_2}\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-209}:\\ \;\;\;\;\frac{\sqrt{t\_5 \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\_m\right)\right)}}{t\_6}\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt{t\_5 \cdot \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, 2 \cdot C\right)}}{t\_6}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_5} \cdot \left(-t\_0\right)}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right)\\ \end{array} \end{array} \]

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 (* 2.0 C)))
        (t_1 (* (* 4.0 A) C))
        (t_2 (- (pow B_m 2.0) t_1))
        (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 (- (* B_m B_m) t_1))
        (t_5 (* 2.0 (* t_4 F)))
        (t_6 (+ (* (- B_m) B_m) t_1)))
   (if (<= t_3 (- INFINITY))
     (/ (* (* (sqrt 2.0) (* (sqrt t_4) (- (sqrt F)))) t_0) t_2)
     (if (<= t_3 -2e-209)
       (/ (sqrt (* t_5 (+ (+ A C) (hypot (- A C) B_m)))) t_6)
       (if (<= t_3 4e-163)
         (/ (sqrt (* t_5 (fma -0.5 (/ (* B_m B_m) A) (* 2.0 C)))) t_6)
         (if (<= t_3 INFINITY)
           (/ (* (sqrt t_5) (- t_0)) t_4)
           (*
            (/ (sqrt 2.0) (- B_m))
            (* (sqrt F) (sqrt (+ C (hypot B_m C)))))))))))

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((2.0 * C));
	double t_1 = (4.0 * A) * C;
	double t_2 = pow(B_m, 2.0) - t_1;
	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 = (B_m * B_m) - t_1;
	double t_5 = 2.0 * (t_4 * F);
	double t_6 = (-B_m * B_m) + t_1;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = ((sqrt(2.0) * (sqrt(t_4) * -sqrt(F))) * t_0) / t_2;
	} else if (t_3 <= -2e-209) {
		tmp = sqrt((t_5 * ((A + C) + hypot((A - C), B_m)))) / t_6;
	} else if (t_3 <= 4e-163) {
		tmp = sqrt((t_5 * fma(-0.5, ((B_m * B_m) / A), (2.0 * C)))) / t_6;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt(t_5) * -t_0) / t_4;
	} else {
		tmp = (sqrt(2.0) / -B_m) * (sqrt(F) * sqrt((C + hypot(B_m, C))));
	}
	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(Float64(2.0 * C))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64((B_m ^ 2.0) - t_1)
	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(Float64(B_m * B_m) - t_1)
	t_5 = Float64(2.0 * Float64(t_4 * F))
	t_6 = Float64(Float64(Float64(-B_m) * B_m) + t_1)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(sqrt(2.0) * Float64(sqrt(t_4) * Float64(-sqrt(F)))) * t_0) / t_2);
	elseif (t_3 <= -2e-209)
		tmp = Float64(sqrt(Float64(t_5 * Float64(Float64(A + C) + hypot(Float64(A - C), B_m)))) / t_6);
	elseif (t_3 <= 4e-163)
		tmp = Float64(sqrt(Float64(t_5 * fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(2.0 * C)))) / t_6);
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(t_5) * Float64(-t_0)) / t_4);
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * Float64(sqrt(F) * sqrt(Float64(C + hypot(B_m, C)))));
	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[(2.0 * C), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $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 = N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[(2.0 * N[(t$95$4 * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[((-B$95$m) * B$95$m), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[t$95$4], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, -2e-209], N[(N[Sqrt[N[(t$95$5 * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$6), $MachinePrecision], If[LessEqual[t$95$3, 4e-163], N[(N[Sqrt[N[(t$95$5 * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$6), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[t$95$5], $MachinePrecision] * (-t$95$0)), $MachinePrecision] / t$95$4), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.0%

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

   4. Applied rewrites 35.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 35.5

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

   6. Applied rewrites 35.5%

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

   7. Taylor expanded in A around -inf

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 15.0

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

   9. Applied rewrites 15.0%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift--.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. sqrt-prod N/A

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

       8. lower-*.f64 N/A

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

       9. pow2 N/A

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

       10. lower-sqrt.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. lift--.f64 N/A

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

       16. lower-sqrt.f64 22.0

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

   11. Applied rewrites 22.0%

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

   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))) < -2.0000000000000001e-209

   1. Initial program 96.5%

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

   4. Applied rewrites 96.5%

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

   if -2.0000000000000001e-209 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 3.99999999999999969e-163

   1. Initial program 6.0%

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

   4. Applied rewrites 10.4%

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

   5. Taylor expanded in A around -inf

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 24.1

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

   7. Applied rewrites 24.1%

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

   9. Applied rewrites 27.8%

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

   if 3.99999999999999969e-163 < (/.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 21.8%

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

   4. Applied rewrites 89.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 88.8

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

   6. Applied rewrites 88.8%

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

   7. Taylor expanded in A around -inf

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 39.9

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

   9. Applied rewrites 39.9%

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

   11. Applied rewrites 39.9%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 18.1

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

   5. Applied rewrites 18.1%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

      3. lift-+.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

      4. lift-hypot.f64 N/A

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

      5. sqrt-prod N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-hypot.f64 N/A

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\right) \]

      14. lift-+.f64 32.4

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\right) \]

   7. Applied rewrites 32.4%

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \color{blue}{\sqrt{C + \mathsf{hypot}\left(B, C\right)}}\right)\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 40.9%

   \[\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:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot \left(\sqrt{B \cdot B - \left(4 \cdot A\right) \cdot C} \cdot \left(-\sqrt{F}\right)\right)\right) \cdot \sqrt{2 \cdot C}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -2 \cdot 10^{-209}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)}}{\left(-B\right) \cdot B + \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 4 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, 2 \cdot C\right)}}{\left(-B\right) \cdot B + \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{B \cdot B - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 55.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{2}}{-B\_m}\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := B\_m \cdot B\_m - t\_1\\ t_3 := 2 \cdot \left(t\_2 \cdot F\right)\\ t_4 := {B\_m}^{2} - t\_1\\ t_5 := \frac{\sqrt{\left(2 \cdot \left(t\_4 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_4}\\ t_6 := \frac{\sqrt{t\_3} \cdot \left(-\sqrt{2 \cdot C}\right)}{t\_2}\\ \mathbf{if}\;t\_5 \leq -1 \cdot 10^{+97}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_5 \leq -2 \cdot 10^{-209}:\\ \;\;\;\;t\_0 \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)}\\ \mathbf{elif}\;t\_5 \leq 4 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt{t\_3 \cdot \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, 2 \cdot C\right)}}{\left(-B\_m\right) \cdot B\_m + t\_1}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;t\_6\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\sqrt{B\_m}, \sqrt{F}, 0.5 \cdot \left(\sqrt{\frac{F}{B\_m}} \cdot C\right)\right)\\ \end{array} \end{array} \]

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 2.0) (- B_m)))
        (t_1 (* (* 4.0 A) C))
        (t_2 (- (* B_m B_m) t_1))
        (t_3 (* 2.0 (* t_2 F)))
        (t_4 (- (pow B_m 2.0) t_1))
        (t_5
         (/
          (sqrt
           (*
            (* 2.0 (* t_4 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_4)))
        (t_6 (/ (* (sqrt t_3) (- (sqrt (* 2.0 C)))) t_2)))
   (if (<= t_5 -1e+97)
     t_6
     (if (<= t_5 -2e-209)
       (* t_0 (sqrt (* F (+ C (sqrt (fma B_m B_m (* C C)))))))
       (if (<= t_5 4e-163)
         (/
          (sqrt (* t_3 (fma -0.5 (/ (* B_m B_m) A) (* 2.0 C))))
          (+ (* (- B_m) B_m) t_1))
         (if (<= t_5 INFINITY)
           t_6
           (*
            t_0
            (fma (sqrt B_m) (sqrt F) (* 0.5 (* (sqrt (/ F B_m)) C))))))))))

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(2.0) / -B_m;
	double t_1 = (4.0 * A) * C;
	double t_2 = (B_m * B_m) - t_1;
	double t_3 = 2.0 * (t_2 * F);
	double t_4 = pow(B_m, 2.0) - t_1;
	double t_5 = sqrt(((2.0 * (t_4 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_4;
	double t_6 = (sqrt(t_3) * -sqrt((2.0 * C))) / t_2;
	double tmp;
	if (t_5 <= -1e+97) {
		tmp = t_6;
	} else if (t_5 <= -2e-209) {
		tmp = t_0 * sqrt((F * (C + sqrt(fma(B_m, B_m, (C * C))))));
	} else if (t_5 <= 4e-163) {
		tmp = sqrt((t_3 * fma(-0.5, ((B_m * B_m) / A), (2.0 * C)))) / ((-B_m * B_m) + t_1);
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = t_6;
	} else {
		tmp = t_0 * fma(sqrt(B_m), sqrt(F), (0.5 * (sqrt((F / B_m)) * C)));
	}
	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(sqrt(2.0) / Float64(-B_m))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(Float64(B_m * B_m) - t_1)
	t_3 = Float64(2.0 * Float64(t_2 * F))
	t_4 = Float64((B_m ^ 2.0) - t_1)
	t_5 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_4 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_4))
	t_6 = Float64(Float64(sqrt(t_3) * Float64(-sqrt(Float64(2.0 * C)))) / t_2)
	tmp = 0.0
	if (t_5 <= -1e+97)
		tmp = t_6;
	elseif (t_5 <= -2e-209)
		tmp = Float64(t_0 * sqrt(Float64(F * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C)))))));
	elseif (t_5 <= 4e-163)
		tmp = Float64(sqrt(Float64(t_3 * fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(2.0 * C)))) / Float64(Float64(Float64(-B_m) * B_m) + t_1));
	elseif (t_5 <= Inf)
		tmp = t_6;
	else
		tmp = Float64(t_0 * fma(sqrt(B_m), sqrt(F), Float64(0.5 * Float64(sqrt(Float64(F / B_m)) * C))));
	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[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$4 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$4)), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[Sqrt[t$95$3], $MachinePrecision] * (-N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$5, -1e+97], t$95$6, If[LessEqual[t$95$5, -2e-209], N[(t$95$0 * N[Sqrt[N[(F * N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 4e-163], N[(N[Sqrt[N[(t$95$3 * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[((-B$95$m) * B$95$m), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, Infinity], t$95$6, N[(t$95$0 * N[(N[Sqrt[B$95$m], $MachinePrecision] * N[Sqrt[F], $MachinePrecision] + N[(0.5 * N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.0000000000000001e97 or 3.99999999999999969e-163 < (/.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 13.4%

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

   4. Applied rewrites 54.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 54.5

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

   6. Applied rewrites 54.5%

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

   7. Taylor expanded in A around -inf

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 23.3

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

   9. Applied rewrites 23.3%

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

   11. Applied rewrites 23.3%

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

   if -1.0000000000000001e97 < (/.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-209

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 39.0

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

   5. Applied rewrites 39.0%

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

      1. lift-hypot.f64 N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 39.0

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

   7. Applied rewrites 39.0%

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

   if -2.0000000000000001e-209 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 3.99999999999999969e-163

   1. Initial program 6.0%

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

   4. Applied rewrites 10.4%

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

   5. Taylor expanded in A around -inf

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 24.1

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

   7. Applied rewrites 24.1%

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

   9. Applied rewrites 27.8%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 18.1

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

   5. Applied rewrites 18.1%

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

   6. Taylor expanded in C around 0

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

      1. sqrt-prod N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 30.6

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \mathsf{fma}\left(\sqrt{B}, \sqrt{F}, 0.5 \cdot \left(\sqrt{\frac{F}{B}} \cdot C\right)\right)\right) \]

   8. Applied rewrites 30.6%

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \mathsf{fma}\left(\sqrt{B}, \color{blue}{\sqrt{F}}, 0.5 \cdot \left(\sqrt{\frac{F}{B}} \cdot C\right)\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 28.9%

   \[\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^{+97}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{B \cdot B - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -2 \cdot 10^{-209}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\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 4 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, 2 \cdot C\right)}}{\left(-B\right) \cdot B + \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{B \cdot B - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \mathsf{fma}\left(\sqrt{B}, \sqrt{F}, 0.5 \cdot \left(\sqrt{\frac{F}{B}} \cdot C\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{B\_m \cdot B\_m}{A}\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \sqrt{2} \cdot \left(\sqrt{B\_m \cdot B\_m - t\_1} \cdot \left(-\sqrt{F}\right)\right)\\ t_3 := {B\_m}^{2} - t\_1\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(t\_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{t\_2 \cdot \sqrt{2 \cdot C}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq -2 \cdot 10^{-209}:\\ \;\;\;\;\frac{t\_2 \cdot \sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\_m\right)}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(A \cdot \left(t\_0 - 4 \cdot C\right)\right) \cdot F\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(-0.5, t\_0, 2 \cdot C\right)}\right)}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right)\\ \end{array} \end{array} \]

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 (/ (* B_m B_m) A))
        (t_1 (* (* 4.0 A) C))
        (t_2 (* (sqrt 2.0) (* (sqrt (- (* B_m B_m) t_1)) (- (sqrt F)))))
        (t_3 (- (pow B_m 2.0) t_1))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* t_3 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_3))))
   (if (<= t_4 (- INFINITY))
     (/ (* t_2 (sqrt (* 2.0 C))) t_3)
     (if (<= t_4 -2e-209)
       (/ (* t_2 (sqrt (+ (+ A C) (hypot (- A C) B_m)))) t_3)
       (if (<= t_4 INFINITY)
         (/
          (*
           (sqrt (* 2.0 (* (* A (- t_0 (* 4.0 C))) F)))
           (- (sqrt (fma -0.5 t_0 (* 2.0 C)))))
          t_3)
         (*
          (/ (sqrt 2.0) (- B_m))
          (* (sqrt F) (sqrt (+ C (hypot B_m C))))))))))

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 = (B_m * B_m) / A;
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(2.0) * (sqrt(((B_m * B_m) - t_1)) * -sqrt(F));
	double t_3 = pow(B_m, 2.0) - t_1;
	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 tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = (t_2 * sqrt((2.0 * C))) / t_3;
	} else if (t_4 <= -2e-209) {
		tmp = (t_2 * sqrt(((A + C) + hypot((A - C), B_m)))) / t_3;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt((2.0 * ((A * (t_0 - (4.0 * C))) * F))) * -sqrt(fma(-0.5, t_0, (2.0 * C)))) / t_3;
	} else {
		tmp = (sqrt(2.0) / -B_m) * (sqrt(F) * sqrt((C + hypot(B_m, C))));
	}
	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(B_m * B_m) / A)
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(2.0) * Float64(sqrt(Float64(Float64(B_m * B_m) - t_1)) * Float64(-sqrt(F))))
	t_3 = Float64((B_m ^ 2.0) - t_1)
	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))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(t_2 * sqrt(Float64(2.0 * C))) / t_3);
	elseif (t_4 <= -2e-209)
		tmp = Float64(Float64(t_2 * sqrt(Float64(Float64(A + C) + hypot(Float64(A - C), B_m)))) / t_3);
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(Float64(A * Float64(t_0 - Float64(4.0 * C))) * F))) * Float64(-sqrt(fma(-0.5, t_0, Float64(2.0 * C))))) / t_3);
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * Float64(sqrt(F) * sqrt(Float64(C + hypot(B_m, C)))));
	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[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $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]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(t$95$2 * N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, -2e-209], N[(N[(t$95$2 * N[Sqrt[N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Sqrt[N[(2.0 * N[(N[(A * N[(t$95$0 - N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(-0.5 * t$95$0 + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.0%

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

   4. Applied rewrites 35.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 35.5

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

   6. Applied rewrites 35.5%

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

   7. Taylor expanded in A around -inf

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 15.0

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

   9. Applied rewrites 15.0%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift--.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. sqrt-prod N/A

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

       8. lower-*.f64 N/A

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

       9. pow2 N/A

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

       10. lower-sqrt.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. lift--.f64 N/A

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

       16. lower-sqrt.f64 22.0

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

   11. Applied rewrites 22.0%

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

   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))) < -2.0000000000000001e-209

   1. Initial program 96.5%

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

   4. Applied rewrites 97.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 97.6

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

   6. Applied rewrites 97.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sqrt-prod N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower--.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-sqrt.f64 99.3

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

   8. Applied rewrites 99.3%

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

   if -2.0000000000000001e-209 < (/.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 12.5%

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

   4. Applied rewrites 42.6%

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

   5. Taylor expanded in A around -inf

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 30.6

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

   7. Applied rewrites 30.6%

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

   8. Taylor expanded in A around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 30.6

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

   10. Applied rewrites 30.6%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 18.1

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

   5. Applied rewrites 18.1%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

      3. lift-+.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

      4. lift-hypot.f64 N/A

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

      5. sqrt-prod N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-hypot.f64 N/A

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\right) \]

      14. lift-+.f64 32.4

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\right) \]

   7. Applied rewrites 32.4%

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \color{blue}{\sqrt{C + \mathsf{hypot}\left(B, C\right)}}\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -\infty:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot \left(\sqrt{B \cdot B - \left(4 \cdot A\right) \cdot C} \cdot \left(-\sqrt{F}\right)\right)\right) \cdot \sqrt{2 \cdot C}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -2 \cdot 10^{-209}:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot \left(\sqrt{B \cdot B - \left(4 \cdot A\right) \cdot C} \cdot \left(-\sqrt{F}\right)\right)\right) \cdot \sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(A \cdot \left(\frac{B \cdot B}{A} - 4 \cdot C\right)\right) \cdot F\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, 2 \cdot C\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := B\_m \cdot B\_m - t\_0\\ t_2 := {B\_m}^{2} - t\_0\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_2}\\ t_4 := \frac{B\_m \cdot B\_m}{A}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot \left(\sqrt{t\_1} \cdot \left(-\sqrt{F}\right)\right)\right) \cdot \sqrt{2 \cdot C}}{t\_2}\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-209}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(t\_1 \cdot F\right)} \cdot \left(-\sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\_m\right)}\right)}{t\_2}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(A \cdot \left(t\_4 - 4 \cdot C\right)\right) \cdot F\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(-0.5, t\_4, 2 \cdot C\right)}\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right)\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C))
        (t_1 (- (* B_m B_m) t_0))
        (t_2 (- (pow B_m 2.0) t_0))
        (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 (/ (* B_m B_m) A)))
   (if (<= t_3 (- INFINITY))
     (/ (* (* (sqrt 2.0) (* (sqrt t_1) (- (sqrt F)))) (sqrt (* 2.0 C))) t_2)
     (if (<= t_3 -2e-209)
       (/
        (* (sqrt (* 2.0 (* t_1 F))) (- (sqrt (+ (+ A C) (hypot (- A C) B_m)))))
        t_2)
       (if (<= t_3 INFINITY)
         (/
          (*
           (sqrt (* 2.0 (* (* A (- t_4 (* 4.0 C))) F)))
           (- (sqrt (fma -0.5 t_4 (* 2.0 C)))))
          t_2)
         (*
          (/ (sqrt 2.0) (- B_m))
          (* (sqrt F) (sqrt (+ C (hypot B_m C))))))))))

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 = (4.0 * A) * C;
	double t_1 = (B_m * B_m) - t_0;
	double t_2 = pow(B_m, 2.0) - t_0;
	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 = (B_m * B_m) / A;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = ((sqrt(2.0) * (sqrt(t_1) * -sqrt(F))) * sqrt((2.0 * C))) / t_2;
	} else if (t_3 <= -2e-209) {
		tmp = (sqrt((2.0 * (t_1 * F))) * -sqrt(((A + C) + hypot((A - C), B_m)))) / t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt((2.0 * ((A * (t_4 - (4.0 * C))) * F))) * -sqrt(fma(-0.5, t_4, (2.0 * C)))) / t_2;
	} else {
		tmp = (sqrt(2.0) / -B_m) * (sqrt(F) * sqrt((C + hypot(B_m, C))));
	}
	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(4.0 * A) * C)
	t_1 = Float64(Float64(B_m * B_m) - t_0)
	t_2 = Float64((B_m ^ 2.0) - t_0)
	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(Float64(B_m * B_m) / A)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(sqrt(2.0) * Float64(sqrt(t_1) * Float64(-sqrt(F)))) * sqrt(Float64(2.0 * C))) / t_2);
	elseif (t_3 <= -2e-209)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(t_1 * F))) * Float64(-sqrt(Float64(Float64(A + C) + hypot(Float64(A - C), B_m))))) / t_2);
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(Float64(A * Float64(t_4 - Float64(4.0 * C))) * F))) * Float64(-sqrt(fma(-0.5, t_4, Float64(2.0 * C))))) / t_2);
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * Float64(sqrt(F) * sqrt(Float64(C + hypot(B_m, C)))));
	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), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $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 = N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[t$95$1], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, -2e-209], N[(N[(N[Sqrt[N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(2.0 * N[(N[(A * N[(t$95$4 - N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(-0.5 * t$95$4 + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.0%

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

   4. Applied rewrites 35.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 35.5

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

   6. Applied rewrites 35.5%

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

   7. Taylor expanded in A around -inf

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 15.0

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

   9. Applied rewrites 15.0%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift--.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. sqrt-prod N/A

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

       8. lower-*.f64 N/A

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

       9. pow2 N/A

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

       10. lower-sqrt.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. lift--.f64 N/A

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

       16. lower-sqrt.f64 22.0

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

   11. Applied rewrites 22.0%

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

   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))) < -2.0000000000000001e-209

   1. Initial program 96.5%

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

   4. Applied rewrites 97.6%

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

   if -2.0000000000000001e-209 < (/.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 12.5%

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

   4. Applied rewrites 42.6%

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

   5. Taylor expanded in A around -inf

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 30.6

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

   7. Applied rewrites 30.6%

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

   8. Taylor expanded in A around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 30.6

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

   10. Applied rewrites 30.6%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 18.1

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

   5. Applied rewrites 18.1%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

      3. lift-+.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

      4. lift-hypot.f64 N/A

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

      5. sqrt-prod N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-hypot.f64 N/A

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\right) \]

      14. lift-+.f64 32.4

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\right) \]

   7. Applied rewrites 32.4%

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \color{blue}{\sqrt{C + \mathsf{hypot}\left(B, C\right)}}\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -\infty:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot \left(\sqrt{B \cdot B - \left(4 \cdot A\right) \cdot C} \cdot \left(-\sqrt{F}\right)\right)\right) \cdot \sqrt{2 \cdot C}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -2 \cdot 10^{-209}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)} \cdot \left(-\sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(A \cdot \left(\frac{B \cdot B}{A} - 4 \cdot C\right)\right) \cdot F\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, 2 \cdot C\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{B\_m \cdot B\_m}{A}\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := B\_m \cdot B\_m - t\_1\\ t_3 := {B\_m}^{2} - t\_1\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(t\_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot \left(\sqrt{t\_2} \cdot \left(-\sqrt{F}\right)\right)\right) \cdot \sqrt{2 \cdot C}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq -2 \cdot 10^{-209}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\_m\right)\right)}}{\left(-B\_m\right) \cdot B\_m + t\_1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(A \cdot \left(t\_0 - 4 \cdot C\right)\right) \cdot F\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(-0.5, t\_0, 2 \cdot C\right)}\right)}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right)\\ \end{array} \end{array} \]

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 (/ (* B_m B_m) A))
        (t_1 (* (* 4.0 A) C))
        (t_2 (- (* B_m B_m) t_1))
        (t_3 (- (pow B_m 2.0) t_1))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* t_3 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_3))))
   (if (<= t_4 (- INFINITY))
     (/ (* (* (sqrt 2.0) (* (sqrt t_2) (- (sqrt F)))) (sqrt (* 2.0 C))) t_3)
     (if (<= t_4 -2e-209)
       (/
        (sqrt (* (* 2.0 (* t_2 F)) (+ (+ A C) (hypot (- A C) B_m))))
        (+ (* (- B_m) B_m) t_1))
       (if (<= t_4 INFINITY)
         (/
          (*
           (sqrt (* 2.0 (* (* A (- t_0 (* 4.0 C))) F)))
           (- (sqrt (fma -0.5 t_0 (* 2.0 C)))))
          t_3)
         (*
          (/ (sqrt 2.0) (- B_m))
          (* (sqrt F) (sqrt (+ C (hypot B_m C))))))))))

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 = (B_m * B_m) / A;
	double t_1 = (4.0 * A) * C;
	double t_2 = (B_m * B_m) - t_1;
	double t_3 = pow(B_m, 2.0) - t_1;
	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 tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = ((sqrt(2.0) * (sqrt(t_2) * -sqrt(F))) * sqrt((2.0 * C))) / t_3;
	} else if (t_4 <= -2e-209) {
		tmp = sqrt(((2.0 * (t_2 * F)) * ((A + C) + hypot((A - C), B_m)))) / ((-B_m * B_m) + t_1);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt((2.0 * ((A * (t_0 - (4.0 * C))) * F))) * -sqrt(fma(-0.5, t_0, (2.0 * C)))) / t_3;
	} else {
		tmp = (sqrt(2.0) / -B_m) * (sqrt(F) * sqrt((C + hypot(B_m, C))));
	}
	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(B_m * B_m) / A)
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(Float64(B_m * B_m) - t_1)
	t_3 = Float64((B_m ^ 2.0) - t_1)
	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))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(sqrt(2.0) * Float64(sqrt(t_2) * Float64(-sqrt(F)))) * sqrt(Float64(2.0 * C))) / t_3);
	elseif (t_4 <= -2e-209)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) + hypot(Float64(A - C), B_m)))) / Float64(Float64(Float64(-B_m) * B_m) + t_1));
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(Float64(A * Float64(t_0 - Float64(4.0 * C))) * F))) * Float64(-sqrt(fma(-0.5, t_0, Float64(2.0 * C))))) / t_3);
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * Float64(sqrt(F) * sqrt(Float64(C + hypot(B_m, C)))));
	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[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $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]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[t$95$2], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, -2e-209], N[(N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[((-B$95$m) * B$95$m), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Sqrt[N[(2.0 * N[(N[(A * N[(t$95$0 - N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(-0.5 * t$95$0 + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.0%

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

   4. Applied rewrites 35.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 35.5

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

   6. Applied rewrites 35.5%

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

   7. Taylor expanded in A around -inf

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 15.0

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

   9. Applied rewrites 15.0%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift--.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. sqrt-prod N/A

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

       8. lower-*.f64 N/A

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

       9. pow2 N/A

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

       10. lower-sqrt.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. lift--.f64 N/A

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

       16. lower-sqrt.f64 22.0

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

   11. Applied rewrites 22.0%

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

   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))) < -2.0000000000000001e-209

   1. Initial program 96.5%

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

   4. Applied rewrites 96.5%

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

   if -2.0000000000000001e-209 < (/.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 12.5%

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

   4. Applied rewrites 42.6%

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

   5. Taylor expanded in A around -inf

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 30.6

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

   7. Applied rewrites 30.6%

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

   8. Taylor expanded in A around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 30.6

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

   10. Applied rewrites 30.6%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 18.1

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

   5. Applied rewrites 18.1%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

      3. lift-+.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

      4. lift-hypot.f64 N/A

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

      5. sqrt-prod N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-hypot.f64 N/A

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\right) \]

      14. lift-+.f64 32.4

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\right) \]

   7. Applied rewrites 32.4%

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \color{blue}{\sqrt{C + \mathsf{hypot}\left(B, C\right)}}\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -\infty:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot \left(\sqrt{B \cdot B - \left(4 \cdot A\right) \cdot C} \cdot \left(-\sqrt{F}\right)\right)\right) \cdot \sqrt{2 \cdot C}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -2 \cdot 10^{-209}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)}}{\left(-B\right) \cdot B + \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(A \cdot \left(\frac{B \cdot B}{A} - 4 \cdot C\right)\right) \cdot F\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, 2 \cdot C\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := B\_m \cdot B\_m - t\_0\\ t_2 := 2 \cdot \left(t\_1 \cdot F\right)\\ t_3 := {B\_m}^{2} - t\_0\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(t\_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot \left(\sqrt{t\_1} \cdot \left(-\sqrt{F}\right)\right)\right) \cdot \sqrt{2 \cdot C}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq -2 \cdot 10^{-209}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\_m\right)\right)}}{\left(-B\_m\right) \cdot B\_m + t\_0}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_2} \cdot \left(-\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, 2 \cdot C\right)}\right)}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right)\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C))
        (t_1 (- (* B_m B_m) t_0))
        (t_2 (* 2.0 (* t_1 F)))
        (t_3 (- (pow B_m 2.0) t_0))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* t_3 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_3))))
   (if (<= t_4 (- INFINITY))
     (/ (* (* (sqrt 2.0) (* (sqrt t_1) (- (sqrt F)))) (sqrt (* 2.0 C))) t_3)
     (if (<= t_4 -2e-209)
       (/
        (sqrt (* t_2 (+ (+ A C) (hypot (- A C) B_m))))
        (+ (* (- B_m) B_m) t_0))
       (if (<= t_4 INFINITY)
         (/
          (* (sqrt t_2) (- (sqrt (fma -0.5 (/ (* B_m B_m) A) (* 2.0 C)))))
          t_3)
         (*
          (/ (sqrt 2.0) (- B_m))
          (* (sqrt F) (sqrt (+ C (hypot B_m C))))))))))

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 = (4.0 * A) * C;
	double t_1 = (B_m * B_m) - t_0;
	double t_2 = 2.0 * (t_1 * F);
	double t_3 = pow(B_m, 2.0) - t_0;
	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 tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = ((sqrt(2.0) * (sqrt(t_1) * -sqrt(F))) * sqrt((2.0 * C))) / t_3;
	} else if (t_4 <= -2e-209) {
		tmp = sqrt((t_2 * ((A + C) + hypot((A - C), B_m)))) / ((-B_m * B_m) + t_0);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt(t_2) * -sqrt(fma(-0.5, ((B_m * B_m) / A), (2.0 * C)))) / t_3;
	} else {
		tmp = (sqrt(2.0) / -B_m) * (sqrt(F) * sqrt((C + hypot(B_m, C))));
	}
	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(4.0 * A) * C)
	t_1 = Float64(Float64(B_m * B_m) - t_0)
	t_2 = Float64(2.0 * Float64(t_1 * F))
	t_3 = Float64((B_m ^ 2.0) - t_0)
	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))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(sqrt(2.0) * Float64(sqrt(t_1) * Float64(-sqrt(F)))) * sqrt(Float64(2.0 * C))) / t_3);
	elseif (t_4 <= -2e-209)
		tmp = Float64(sqrt(Float64(t_2 * Float64(Float64(A + C) + hypot(Float64(A - C), B_m)))) / Float64(Float64(Float64(-B_m) * B_m) + t_0));
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(t_2) * Float64(-sqrt(fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(2.0 * C))))) / t_3);
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * Float64(sqrt(F) * sqrt(Float64(C + hypot(B_m, C)))));
	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), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $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]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[t$95$1], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, -2e-209], N[(N[Sqrt[N[(t$95$2 * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[((-B$95$m) * B$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Sqrt[t$95$2], $MachinePrecision] * (-N[Sqrt[N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.0%

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

   4. Applied rewrites 35.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 35.5

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

   6. Applied rewrites 35.5%

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

   7. Taylor expanded in A around -inf

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 15.0

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

   9. Applied rewrites 15.0%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift--.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. sqrt-prod N/A

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

       8. lower-*.f64 N/A

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

       9. pow2 N/A

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

       10. lower-sqrt.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. lift--.f64 N/A

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

       16. lower-sqrt.f64 22.0

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

   11. Applied rewrites 22.0%

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

   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))) < -2.0000000000000001e-209

   1. Initial program 96.5%

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

   4. Applied rewrites 96.5%

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

   if -2.0000000000000001e-209 < (/.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 12.5%

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

   4. Applied rewrites 42.6%

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

   5. Taylor expanded in A around -inf

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 30.6

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

   7. Applied rewrites 30.6%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 18.1

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

   5. Applied rewrites 18.1%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

      3. lift-+.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

      4. lift-hypot.f64 N/A

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

      5. sqrt-prod N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-hypot.f64 N/A

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\right) \]

      14. lift-+.f64 32.4

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\right) \]

   7. Applied rewrites 32.4%

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \color{blue}{\sqrt{C + \mathsf{hypot}\left(B, C\right)}}\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -\infty:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot \left(\sqrt{B \cdot B - \left(4 \cdot A\right) \cdot C} \cdot \left(-\sqrt{F}\right)\right)\right) \cdot \sqrt{2 \cdot C}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -2 \cdot 10^{-209}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)}}{\left(-B\right) \cdot B + \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, 2 \cdot C\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := B\_m \cdot B\_m - t\_0\\ t_2 := 2 \cdot \left(t\_1 \cdot F\right)\\ \mathbf{if}\;B\_m \leq 1.85 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{t\_2} \cdot \left(-\sqrt{2 \cdot C}\right)}{t\_1}\\ \mathbf{elif}\;B\_m \leq 9.2 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\_m\right)\right)}}{\left(-B\_m\right) \cdot B\_m + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right)\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C))
        (t_1 (- (* B_m B_m) t_0))
        (t_2 (* 2.0 (* t_1 F))))
   (if (<= B_m 1.85e-60)
     (/ (* (sqrt t_2) (- (sqrt (* 2.0 C)))) t_1)
     (if (<= B_m 9.2e+51)
       (/
        (sqrt (* t_2 (+ (+ A C) (hypot (- A C) B_m))))
        (+ (* (- B_m) B_m) t_0))
       (* (/ (sqrt 2.0) (- B_m)) (* (sqrt F) (sqrt (+ C (hypot B_m C)))))))))

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 = (4.0 * A) * C;
	double t_1 = (B_m * B_m) - t_0;
	double t_2 = 2.0 * (t_1 * F);
	double tmp;
	if (B_m <= 1.85e-60) {
		tmp = (sqrt(t_2) * -sqrt((2.0 * C))) / t_1;
	} else if (B_m <= 9.2e+51) {
		tmp = sqrt((t_2 * ((A + C) + hypot((A - C), B_m)))) / ((-B_m * B_m) + t_0);
	} else {
		tmp = (sqrt(2.0) / -B_m) * (sqrt(F) * sqrt((C + hypot(B_m, C))));
	}
	return tmp;
}

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 t_0 = (4.0 * A) * C;
	double t_1 = (B_m * B_m) - t_0;
	double t_2 = 2.0 * (t_1 * F);
	double tmp;
	if (B_m <= 1.85e-60) {
		tmp = (Math.sqrt(t_2) * -Math.sqrt((2.0 * C))) / t_1;
	} else if (B_m <= 9.2e+51) {
		tmp = Math.sqrt((t_2 * ((A + C) + Math.hypot((A - C), B_m)))) / ((-B_m * B_m) + t_0);
	} else {
		tmp = (Math.sqrt(2.0) / -B_m) * (Math.sqrt(F) * Math.sqrt((C + Math.hypot(B_m, C))));
	}
	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):
	t_0 = (4.0 * A) * C
	t_1 = (B_m * B_m) - t_0
	t_2 = 2.0 * (t_1 * F)
	tmp = 0
	if B_m <= 1.85e-60:
		tmp = (math.sqrt(t_2) * -math.sqrt((2.0 * C))) / t_1
	elif B_m <= 9.2e+51:
		tmp = math.sqrt((t_2 * ((A + C) + math.hypot((A - C), B_m)))) / ((-B_m * B_m) + t_0)
	else:
		tmp = (math.sqrt(2.0) / -B_m) * (math.sqrt(F) * math.sqrt((C + math.hypot(B_m, C))))
	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(4.0 * A) * C)
	t_1 = Float64(Float64(B_m * B_m) - t_0)
	t_2 = Float64(2.0 * Float64(t_1 * F))
	tmp = 0.0
	if (B_m <= 1.85e-60)
		tmp = Float64(Float64(sqrt(t_2) * Float64(-sqrt(Float64(2.0 * C)))) / t_1);
	elseif (B_m <= 9.2e+51)
		tmp = Float64(sqrt(Float64(t_2 * Float64(Float64(A + C) + hypot(Float64(A - C), B_m)))) / Float64(Float64(Float64(-B_m) * B_m) + t_0));
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * Float64(sqrt(F) * sqrt(Float64(C + hypot(B_m, C)))));
	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)
	t_0 = (4.0 * A) * C;
	t_1 = (B_m * B_m) - t_0;
	t_2 = 2.0 * (t_1 * F);
	tmp = 0.0;
	if (B_m <= 1.85e-60)
		tmp = (sqrt(t_2) * -sqrt((2.0 * C))) / t_1;
	elseif (B_m <= 9.2e+51)
		tmp = sqrt((t_2 * ((A + C) + hypot((A - C), B_m)))) / ((-B_m * B_m) + t_0);
	else
		tmp = (sqrt(2.0) / -B_m) * (sqrt(F) * sqrt((C + hypot(B_m, C))));
	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_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.85e-60], N[(N[(N[Sqrt[t$95$2], $MachinePrecision] * (-N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[B$95$m, 9.2e+51], N[(N[Sqrt[N[(t$95$2 * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[((-B$95$m) * B$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.85000000000000012e-60

   1. Initial program 18.9%

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

   4. Applied rewrites 36.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 36.3

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

   6. Applied rewrites 36.3%

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

   7. Taylor expanded in A around -inf

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 14.7

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

   9. Applied rewrites 14.7%

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

   11. Applied rewrites 14.7%

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

   if 1.85000000000000012e-60 < B < 9.2000000000000002e51

   1. Initial program 43.8%

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

   4. Applied rewrites 53.5%

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

   if 9.2000000000000002e51 < B

   1. Initial program 15.1%

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

   3. Taylor expanded in A around 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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 51.6

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

   5. Applied rewrites 51.6%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

      3. lift-+.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

      4. lift-hypot.f64 N/A

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

      5. sqrt-prod N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-hypot.f64 N/A

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\right) \]

      14. lift-+.f64 76.1

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\right) \]

   7. Applied rewrites 76.1%

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \color{blue}{\sqrt{C + \mathsf{hypot}\left(B, C\right)}}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.85 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{B \cdot B - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq 9.2 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)}}{\left(-B\right) \cdot B + \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := B\_m \cdot B\_m - \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 2.2 \cdot 10^{-82}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(t\_0 \cdot F\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right)\\ \end{array} \end{array} \]

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 (- (* B_m B_m) (* (* 4.0 A) C))))
   (if (<= B_m 2.2e-82)
     (/ (* (sqrt (* 2.0 (* t_0 F))) (- (sqrt (* 2.0 C)))) t_0)
     (* (/ (sqrt 2.0) (- B_m)) (* (sqrt F) (sqrt (+ C (hypot B_m C))))))))

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 = (B_m * B_m) - ((4.0 * A) * C);
	double tmp;
	if (B_m <= 2.2e-82) {
		tmp = (sqrt((2.0 * (t_0 * F))) * -sqrt((2.0 * C))) / t_0;
	} else {
		tmp = (sqrt(2.0) / -B_m) * (sqrt(F) * sqrt((C + hypot(B_m, C))));
	}
	return tmp;
}

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 t_0 = (B_m * B_m) - ((4.0 * A) * C);
	double tmp;
	if (B_m <= 2.2e-82) {
		tmp = (Math.sqrt((2.0 * (t_0 * F))) * -Math.sqrt((2.0 * C))) / t_0;
	} else {
		tmp = (Math.sqrt(2.0) / -B_m) * (Math.sqrt(F) * Math.sqrt((C + Math.hypot(B_m, C))));
	}
	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):
	t_0 = (B_m * B_m) - ((4.0 * A) * C)
	tmp = 0
	if B_m <= 2.2e-82:
		tmp = (math.sqrt((2.0 * (t_0 * F))) * -math.sqrt((2.0 * C))) / t_0
	else:
		tmp = (math.sqrt(2.0) / -B_m) * (math.sqrt(F) * math.sqrt((C + math.hypot(B_m, C))))
	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(B_m * B_m) - Float64(Float64(4.0 * A) * C))
	tmp = 0.0
	if (B_m <= 2.2e-82)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(t_0 * F))) * Float64(-sqrt(Float64(2.0 * C)))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * Float64(sqrt(F) * sqrt(Float64(C + hypot(B_m, C)))));
	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)
	t_0 = (B_m * B_m) - ((4.0 * A) * C);
	tmp = 0.0;
	if (B_m <= 2.2e-82)
		tmp = (sqrt((2.0 * (t_0 * F))) * -sqrt((2.0 * C))) / t_0;
	else
		tmp = (sqrt(2.0) / -B_m) * (sqrt(F) * sqrt((C + hypot(B_m, C))));
	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_] := Block[{t$95$0 = N[(N[(B$95$m * B$95$m), $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 2.2e-82], N[(N[(N[Sqrt[N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 2.19999999999999986e-82

   1. Initial program 19.2%

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

   4. Applied rewrites 36.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 36.5

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

   6. Applied rewrites 36.5%

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

   7. Taylor expanded in A around -inf

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 15.0

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

   9. Applied rewrites 15.0%

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

   11. Applied rewrites 15.0%

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

   if 2.19999999999999986e-82 < B

   1. Initial program 22.0%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 47.2

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

   5. Applied rewrites 47.2%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

      3. lift-+.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

      4. lift-hypot.f64 N/A

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

      5. sqrt-prod N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-hypot.f64 N/A

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\right) \]

      14. lift-+.f64 65.3

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\right) \]

   7. Applied rewrites 65.3%

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \color{blue}{\sqrt{C + \mathsf{hypot}\left(B, C\right)}}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.2 \cdot 10^{-82}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{B \cdot B - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{2}}{-B\_m}\\ t_1 := B\_m \cdot B\_m - \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 10^{+42}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(t\_1 \cdot F\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{t\_1}\\ \mathbf{elif}\;B\_m \leq 2.2 \cdot 10^{+187}:\\ \;\;\;\;t\_0 \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\sqrt{B\_m}, \sqrt{F}, 0.5 \cdot \left(A \cdot \sqrt{\frac{F}{B\_m}}\right)\right)\\ \end{array} \end{array} \]

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 2.0) (- B_m))) (t_1 (- (* B_m B_m) (* (* 4.0 A) C))))
   (if (<= B_m 1e+42)
     (/ (* (sqrt (* 2.0 (* t_1 F))) (- (sqrt (* 2.0 C)))) t_1)
     (if (<= B_m 2.2e+187)
       (* t_0 (sqrt (* F (+ C (hypot B_m C)))))
       (* t_0 (fma (sqrt B_m) (sqrt F) (* 0.5 (* A (sqrt (/ F B_m))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = sqrt(2.0) / -B_m;
	double t_1 = (B_m * B_m) - ((4.0 * A) * C);
	double tmp;
	if (B_m <= 1e+42) {
		tmp = (sqrt((2.0 * (t_1 * F))) * -sqrt((2.0 * C))) / t_1;
	} else if (B_m <= 2.2e+187) {
		tmp = t_0 * sqrt((F * (C + hypot(B_m, C))));
	} else {
		tmp = t_0 * fma(sqrt(B_m), sqrt(F), (0.5 * (A * sqrt((F / B_m)))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(sqrt(2.0) / Float64(-B_m))
	t_1 = Float64(Float64(B_m * B_m) - Float64(Float64(4.0 * A) * C))
	tmp = 0.0
	if (B_m <= 1e+42)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(t_1 * F))) * Float64(-sqrt(Float64(2.0 * C)))) / t_1);
	elseif (B_m <= 2.2e+187)
		tmp = Float64(t_0 * sqrt(Float64(F * Float64(C + hypot(B_m, C)))));
	else
		tmp = Float64(t_0 * fma(sqrt(B_m), sqrt(F), Float64(0.5 * Float64(A * sqrt(Float64(F / B_m))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B$95$m * B$95$m), $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1e+42], N[(N[(N[Sqrt[N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[B$95$m, 2.2e+187], N[(t$95$0 * N[Sqrt[N[(F * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[Sqrt[B$95$m], $MachinePrecision] * N[Sqrt[F], $MachinePrecision] + N[(0.5 * N[(A * N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.00000000000000004e42

   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

   4. Applied rewrites 37.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 37.9

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

   6. Applied rewrites 37.9%

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

   7. Taylor expanded in A around -inf

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 14.0

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

   9. Applied rewrites 14.0%

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

   11. Applied rewrites 14.0%

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

   if 1.00000000000000004e42 < B < 2.1999999999999998e187

   1. Initial program 28.7%

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

   3. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 59.4

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

   5. Applied rewrites 59.4%

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

   if 2.1999999999999998e187 < B

   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 C around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 44.9

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(A, B\right)\right)}\right) \]

   5. Applied rewrites 44.9%

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

   6. Taylor expanded in A around 0

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

      1. sqrt-prod N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 96.4

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \mathsf{fma}\left(\sqrt{B}, \sqrt{F}, 0.5 \cdot \left(A \cdot \sqrt{\frac{F}{B}}\right)\right)\right) \]

   8. Applied rewrites 96.4%

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \mathsf{fma}\left(\sqrt{B}, \color{blue}{\sqrt{F}}, 0.5 \cdot \left(A \cdot \sqrt{\frac{F}{B}}\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 27.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 10^{+42}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{B \cdot B - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{+187}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \mathsf{fma}\left(\sqrt{B}, \sqrt{F}, 0.5 \cdot \left(A \cdot \sqrt{\frac{F}{B}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.6% accurate, 4.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{2}}{-B\_m}\\ \mathbf{if}\;B\_m \leq 4.7 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B\_m \leq 2.7 \cdot 10^{-26}:\\ \;\;\;\;t\_0 \cdot \sqrt{-0.5 \cdot \frac{\left(B\_m \cdot B\_m\right) \cdot F}{A}}\\ \mathbf{elif}\;B\_m \leq 8.2 \cdot 10^{+93}:\\ \;\;\;\;t\_0 \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\sqrt{B\_m}, \sqrt{F}, 0.5 \cdot \left(\sqrt{\frac{F}{B\_m}} \cdot C\right)\right)\\ \end{array} \end{array} \]

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 2.0) (- B_m))))
   (if (<= B_m 4.7e-114)
     (sqrt (/ (- F) A))
     (if (<= B_m 2.7e-26)
       (* t_0 (sqrt (* -0.5 (/ (* (* B_m B_m) F) A))))
       (if (<= B_m 8.2e+93)
         (* t_0 (sqrt (* F (+ C (sqrt (fma B_m B_m (* C C)))))))
         (* t_0 (fma (sqrt B_m) (sqrt F) (* 0.5 (* (sqrt (/ F B_m)) C)))))))))

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(2.0) / -B_m;
	double tmp;
	if (B_m <= 4.7e-114) {
		tmp = sqrt((-F / A));
	} else if (B_m <= 2.7e-26) {
		tmp = t_0 * sqrt((-0.5 * (((B_m * B_m) * F) / A)));
	} else if (B_m <= 8.2e+93) {
		tmp = t_0 * sqrt((F * (C + sqrt(fma(B_m, B_m, (C * C))))));
	} else {
		tmp = t_0 * fma(sqrt(B_m), sqrt(F), (0.5 * (sqrt((F / B_m)) * C)));
	}
	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(sqrt(2.0) / Float64(-B_m))
	tmp = 0.0
	if (B_m <= 4.7e-114)
		tmp = sqrt(Float64(Float64(-F) / A));
	elseif (B_m <= 2.7e-26)
		tmp = Float64(t_0 * sqrt(Float64(-0.5 * Float64(Float64(Float64(B_m * B_m) * F) / A))));
	elseif (B_m <= 8.2e+93)
		tmp = Float64(t_0 * sqrt(Float64(F * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C)))))));
	else
		tmp = Float64(t_0 * fma(sqrt(B_m), sqrt(F), Float64(0.5 * Float64(sqrt(Float64(F / B_m)) * C))));
	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[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]}, If[LessEqual[B$95$m, 4.7e-114], N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision], If[LessEqual[B$95$m, 2.7e-26], N[(t$95$0 * N[Sqrt[N[(-0.5 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * F), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 8.2e+93], N[(t$95$0 * N[Sqrt[N[(F * N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[Sqrt[B$95$m], $MachinePrecision] * N[Sqrt[F], $MachinePrecision] + N[(0.5 * N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 4.70000000000000006e-114

   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 F around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 14.9%

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

   6. Taylor expanded in A around -inf

      \[\leadsto \sqrt{-1 \cdot \frac{F}{A}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{-1 \cdot \frac{F}{A}} \]

      2. lower-/.f64 13.0

         \[\leadsto \sqrt{-1 \cdot \frac{F}{A}} \]

   8. Applied rewrites 13.0%

      \[\leadsto \sqrt{-1 \cdot \frac{F}{A}} \]

   if 4.70000000000000006e-114 < B < 2.69999999999999982e-26

   1. Initial program 10.8%

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

   3. Taylor expanded in C around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 10.6

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(A, B\right)\right)}\right) \]

   5. Applied rewrites 10.6%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\frac{-1}{2} \cdot \frac{\left(B \cdot B\right) \cdot F}{A}}\right) \]

      5. lift-*.f64 6.5

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{-0.5 \cdot \frac{\left(B \cdot B\right) \cdot F}{A}}\right) \]

   8. Applied rewrites 6.5%

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{-0.5 \cdot \frac{\left(B \cdot B\right) \cdot F}{A}}\right) \]

   if 2.69999999999999982e-26 < B < 8.2000000000000002e93

   1. Initial program 47.5%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 55.5

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

   5. Applied rewrites 55.5%

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

      1. lift-hypot.f64 N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 48.3

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

   7. Applied rewrites 48.3%

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

   if 8.2000000000000002e93 < B

   1. Initial program 7.4%

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

   3. Taylor expanded in A around 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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 49.3

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

   5. Applied rewrites 49.3%

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

   6. Taylor expanded in C around 0

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

      1. sqrt-prod N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 72.1

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \mathsf{fma}\left(\sqrt{B}, \sqrt{F}, 0.5 \cdot \left(\sqrt{\frac{F}{B}} \cdot C\right)\right)\right) \]

   8. Applied rewrites 72.1%

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \mathsf{fma}\left(\sqrt{B}, \color{blue}{\sqrt{F}}, 0.5 \cdot \left(\sqrt{\frac{F}{B}} \cdot C\right)\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 26.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.7 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 2.7 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{-0.5 \cdot \frac{\left(B \cdot B\right) \cdot F}{A}}\\ \mathbf{elif}\;B \leq 8.2 \cdot 10^{+93}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \mathsf{fma}\left(\sqrt{B}, \sqrt{F}, 0.5 \cdot \left(\sqrt{\frac{F}{B}} \cdot C\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.4% accurate, 4.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{2}}{-B\_m}\\ \mathbf{if}\;B\_m \leq 4.7 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B\_m \leq 2.7 \cdot 10^{-26}:\\ \;\;\;\;t\_0 \cdot \sqrt{-0.5 \cdot \frac{\left(B\_m \cdot B\_m\right) \cdot F}{A}}\\ \mathbf{elif}\;B\_m \leq 8.2 \cdot 10^{+93}:\\ \;\;\;\;t\_0 \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\sqrt{B\_m}, \sqrt{F}, 0.5 \cdot \left(A \cdot \sqrt{\frac{F}{B\_m}}\right)\right)\\ \end{array} \end{array} \]

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 2.0) (- B_m))))
   (if (<= B_m 4.7e-114)
     (sqrt (/ (- F) A))
     (if (<= B_m 2.7e-26)
       (* t_0 (sqrt (* -0.5 (/ (* (* B_m B_m) F) A))))
       (if (<= B_m 8.2e+93)
         (* t_0 (sqrt (* F (+ C (sqrt (fma B_m B_m (* C C)))))))
         (* t_0 (fma (sqrt B_m) (sqrt F) (* 0.5 (* A (sqrt (/ F B_m)))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = sqrt(2.0) / -B_m;
	double tmp;
	if (B_m <= 4.7e-114) {
		tmp = sqrt((-F / A));
	} else if (B_m <= 2.7e-26) {
		tmp = t_0 * sqrt((-0.5 * (((B_m * B_m) * F) / A)));
	} else if (B_m <= 8.2e+93) {
		tmp = t_0 * sqrt((F * (C + sqrt(fma(B_m, B_m, (C * C))))));
	} else {
		tmp = t_0 * fma(sqrt(B_m), sqrt(F), (0.5 * (A * sqrt((F / B_m)))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(sqrt(2.0) / Float64(-B_m))
	tmp = 0.0
	if (B_m <= 4.7e-114)
		tmp = sqrt(Float64(Float64(-F) / A));
	elseif (B_m <= 2.7e-26)
		tmp = Float64(t_0 * sqrt(Float64(-0.5 * Float64(Float64(Float64(B_m * B_m) * F) / A))));
	elseif (B_m <= 8.2e+93)
		tmp = Float64(t_0 * sqrt(Float64(F * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C)))))));
	else
		tmp = Float64(t_0 * fma(sqrt(B_m), sqrt(F), Float64(0.5 * Float64(A * sqrt(Float64(F / B_m))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]}, If[LessEqual[B$95$m, 4.7e-114], N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision], If[LessEqual[B$95$m, 2.7e-26], N[(t$95$0 * N[Sqrt[N[(-0.5 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * F), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 8.2e+93], N[(t$95$0 * N[Sqrt[N[(F * N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[Sqrt[B$95$m], $MachinePrecision] * N[Sqrt[F], $MachinePrecision] + N[(0.5 * N[(A * N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 4.70000000000000006e-114

   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 F around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 14.9%

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

   6. Taylor expanded in A around -inf

      \[\leadsto \sqrt{-1 \cdot \frac{F}{A}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{-1 \cdot \frac{F}{A}} \]

      2. lower-/.f64 13.0

         \[\leadsto \sqrt{-1 \cdot \frac{F}{A}} \]

   8. Applied rewrites 13.0%

      \[\leadsto \sqrt{-1 \cdot \frac{F}{A}} \]

   if 4.70000000000000006e-114 < B < 2.69999999999999982e-26

   1. Initial program 10.8%

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

   3. Taylor expanded in C around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 10.6

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(A, B\right)\right)}\right) \]

   5. Applied rewrites 10.6%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\frac{-1}{2} \cdot \frac{\left(B \cdot B\right) \cdot F}{A}}\right) \]

      5. lift-*.f64 6.5

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{-0.5 \cdot \frac{\left(B \cdot B\right) \cdot F}{A}}\right) \]

   8. Applied rewrites 6.5%

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{-0.5 \cdot \frac{\left(B \cdot B\right) \cdot F}{A}}\right) \]

   if 2.69999999999999982e-26 < B < 8.2000000000000002e93

   1. Initial program 47.5%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 55.5

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

   5. Applied rewrites 55.5%

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

      1. lift-hypot.f64 N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 48.3

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

   7. Applied rewrites 48.3%

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

   if 8.2000000000000002e93 < B

   1. Initial program 7.4%

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

   3. Taylor expanded in C around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

      8. unpow2 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{A \cdot A + {B}^{2}}\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{A \cdot A + B \cdot B}\right)}\right) \]

      10. lower-hypot.f64 47.3

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(A, B\right)\right)}\right) \]

   5. Applied rewrites 47.3%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(A, B\right)\right)}\right)} \]

   6. Taylor expanded in A around 0

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B \cdot F} + \color{blue}{\frac{1}{2} \cdot \left(A \cdot \sqrt{\frac{F}{B}}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. sqrt-prod N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B} \cdot \sqrt{F} + \frac{1}{2} \cdot \left(\color{blue}{A} \cdot \sqrt{\frac{F}{B}}\right)\right)\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \mathsf{fma}\left(\sqrt{B}, \sqrt{F}, \frac{1}{2} \cdot \left(A \cdot \sqrt{\frac{F}{B}}\right)\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \mathsf{fma}\left(\sqrt{B}, \sqrt{F}, \frac{1}{2} \cdot \left(A \cdot \sqrt{\frac{F}{B}}\right)\right)\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \mathsf{fma}\left(\sqrt{B}, \sqrt{F}, \frac{1}{2} \cdot \left(A \cdot \sqrt{\frac{F}{B}}\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \mathsf{fma}\left(\sqrt{B}, \sqrt{F}, \frac{1}{2} \cdot \left(A \cdot \sqrt{\frac{F}{B}}\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \mathsf{fma}\left(\sqrt{B}, \sqrt{F}, \frac{1}{2} \cdot \left(A \cdot \sqrt{\frac{F}{B}}\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \mathsf{fma}\left(\sqrt{B}, \sqrt{F}, \frac{1}{2} \cdot \left(A \cdot \sqrt{\frac{F}{B}}\right)\right)\right) \]

      8. lower-/.f64 72.3

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \mathsf{fma}\left(\sqrt{B}, \sqrt{F}, 0.5 \cdot \left(A \cdot \sqrt{\frac{F}{B}}\right)\right)\right) \]

   8. Applied rewrites 72.3%

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \mathsf{fma}\left(\sqrt{B}, \color{blue}{\sqrt{F}}, 0.5 \cdot \left(A \cdot \sqrt{\frac{F}{B}}\right)\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 26.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.7 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 2.7 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{-0.5 \cdot \frac{\left(B \cdot B\right) \cdot F}{A}}\\ \mathbf{elif}\;B \leq 8.2 \cdot 10^{+93}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \mathsf{fma}\left(\sqrt{B}, \sqrt{F}, 0.5 \cdot \left(A \cdot \sqrt{\frac{F}{B}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.8% accurate, 5.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := B\_m \cdot B\_m - \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 1.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(t\_0 \cdot F\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \mathsf{fma}\left(\sqrt{B\_m}, \sqrt{F}, 0.5 \cdot \left(\sqrt{\frac{F}{B\_m}} \cdot C\right)\right)\\ \end{array} \end{array} \]

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 (- (* B_m B_m) (* (* 4.0 A) C))))
   (if (<= B_m 1.5e+47)
     (/ (* (sqrt (* 2.0 (* t_0 F))) (- (sqrt (* 2.0 C)))) t_0)
     (*
      (/ (sqrt 2.0) (- B_m))
      (fma (sqrt B_m) (sqrt F) (* 0.5 (* (sqrt (/ F B_m)) C)))))))

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 = (B_m * B_m) - ((4.0 * A) * C);
	double tmp;
	if (B_m <= 1.5e+47) {
		tmp = (sqrt((2.0 * (t_0 * F))) * -sqrt((2.0 * C))) / t_0;
	} else {
		tmp = (sqrt(2.0) / -B_m) * fma(sqrt(B_m), sqrt(F), (0.5 * (sqrt((F / B_m)) * C)));
	}
	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(B_m * B_m) - Float64(Float64(4.0 * A) * C))
	tmp = 0.0
	if (B_m <= 1.5e+47)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(t_0 * F))) * Float64(-sqrt(Float64(2.0 * C)))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * fma(sqrt(B_m), sqrt(F), Float64(0.5 * Float64(sqrt(Float64(F / B_m)) * C))));
	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[(B$95$m * B$95$m), $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.5e+47], N[(N[(N[Sqrt[N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[(N[Sqrt[B$95$m], $MachinePrecision] * N[Sqrt[F], $MachinePrecision] + N[(0.5 * N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 1.5000000000000001e47

   1. Initial program 21.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 38.5%

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)}} \cdot \sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)}} \cdot \sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)}} \cdot \sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{-\sqrt{2 \cdot \left(\color{blue}{\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)} \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(B \cdot B - \color{blue}{\left(4 \cdot A\right)} \cdot C\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(B \cdot B - \color{blue}{\left(4 \cdot A\right) \cdot C}\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. sqrt-prod N/A

         \[\leadsto \frac{-\color{blue}{\left(\sqrt{2} \cdot \sqrt{\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F}\right)} \cdot \sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{-\color{blue}{\left(\sqrt{2} \cdot \sqrt{\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F}\right)} \cdot \sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{-\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F}\right) \cdot \sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{-\left(\sqrt{2} \cdot \color{blue}{\sqrt{\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F}}\right) \cdot \sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{-\left(\sqrt{2} \cdot \sqrt{\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right) \cdot F}\right) \cdot \sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{-\left(\sqrt{2} \cdot \sqrt{\left(B \cdot B - \color{blue}{\left(4 \cdot A\right) \cdot C}\right) \cdot F}\right) \cdot \sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{-\left(\sqrt{2} \cdot \sqrt{\left(B \cdot B - \color{blue}{\left(4 \cdot A\right)} \cdot C\right) \cdot F}\right) \cdot \sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      15. lift--.f64 N/A

          \[\leadsto \frac{-\left(\sqrt{2} \cdot \sqrt{\color{blue}{\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)} \cdot F}\right) \cdot \sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      16. lift-*.f64 38.5

          \[\leadsto \frac{-\left(\sqrt{2} \cdot \sqrt{\color{blue}{\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F}}\right) \cdot \sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Applied rewrites 38.5%

      \[\leadsto \frac{-\color{blue}{\left(\sqrt{2} \cdot \sqrt{\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F}\right)} \cdot \sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \frac{-\left(\sqrt{2} \cdot \sqrt{\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F}\right) \cdot \sqrt{\color{blue}{2 \cdot C}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   8. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{-\left(\sqrt{2} \cdot \sqrt{\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F}\right) \cdot \sqrt{2 \cdot C}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. pow2 N/A

         \[\leadsto \frac{-\left(\sqrt{2} \cdot \sqrt{\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F}\right) \cdot \sqrt{2 \cdot C}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lift-*.f64 14.4

         \[\leadsto \frac{-\left(\sqrt{2} \cdot \sqrt{\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F}\right) \cdot \sqrt{2 \cdot \color{blue}{C}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. Applied rewrites 14.4%

      \[\leadsto \frac{-\left(\sqrt{2} \cdot \sqrt{\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F}\right) \cdot \sqrt{\color{blue}{2 \cdot C}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   10. Step-by-step derivation

   11. Applied rewrites 14.3%

       \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)} \cdot \sqrt{2 \cdot C}}{B \cdot B - \left(4 \cdot A\right) \cdot C}} \]

   if 1.5000000000000001e47 < B

   1. Initial program 15.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 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. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F} \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right) \]

      8. unpow2 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{B \cdot B + {C}^{2}}\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{B \cdot B + C \cdot C}\right)}\right) \]

      10. lower-hypot.f64 51.6

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

   5. Applied rewrites 51.6%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)} \]

   6. Taylor expanded in C around 0

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B \cdot F} + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{F}{B}} \cdot C\right)}\right)\right) \]
   7. Step-by-step derivation

      1. sqrt-prod N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B} \cdot \sqrt{F} + \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{F}{B}}} \cdot C\right)\right)\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \mathsf{fma}\left(\sqrt{B}, \sqrt{F}, \frac{1}{2} \cdot \left(\sqrt{\frac{F}{B}} \cdot C\right)\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \mathsf{fma}\left(\sqrt{B}, \sqrt{F}, \frac{1}{2} \cdot \left(\sqrt{\frac{F}{B}} \cdot C\right)\right)\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \mathsf{fma}\left(\sqrt{B}, \sqrt{F}, \frac{1}{2} \cdot \left(\sqrt{\frac{F}{B}} \cdot C\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \mathsf{fma}\left(\sqrt{B}, \sqrt{F}, \frac{1}{2} \cdot \left(\sqrt{\frac{F}{B}} \cdot C\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \mathsf{fma}\left(\sqrt{B}, \sqrt{F}, \frac{1}{2} \cdot \left(\sqrt{\frac{F}{B}} \cdot C\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \mathsf{fma}\left(\sqrt{B}, \sqrt{F}, \frac{1}{2} \cdot \left(\sqrt{\frac{F}{B}} \cdot C\right)\right)\right) \]

      8. lower-/.f64 66.1

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \mathsf{fma}\left(\sqrt{B}, \sqrt{F}, 0.5 \cdot \left(\sqrt{\frac{F}{B}} \cdot C\right)\right)\right) \]

   8. Applied rewrites 66.1%

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \mathsf{fma}\left(\sqrt{B}, \color{blue}{\sqrt{F}}, 0.5 \cdot \left(\sqrt{\frac{F}{B}} \cdot C\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{B \cdot B - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \mathsf{fma}\left(\sqrt{B}, \sqrt{F}, 0.5 \cdot \left(\sqrt{\frac{F}{B}} \cdot C\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 44.1% accurate, 8.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{+26}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{F \cdot \left(C + B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

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 (<= F -2e-310)
   (sqrt (/ (- F) A))
   (if (<= F 1.8e+26)
     (* (/ (sqrt 2.0) (- B_m)) (sqrt (* F (+ C B_m))))
     (- (sqrt (* (/ F B_m) 2.0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -2e-310) {
		tmp = sqrt((-F / A));
	} else if (F <= 1.8e+26) {
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (C + B_m)));
	} else {
		tmp = -sqrt(((F / B_m) * 2.0));
	}
	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 (f <= (-2d-310)) then
        tmp = sqrt((-f / a))
    else if (f <= 1.8d+26) then
        tmp = (sqrt(2.0d0) / -b_m) * sqrt((f * (c + b_m)))
    else
        tmp = -sqrt(((f / b_m) * 2.0d0))
    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 (F <= -2e-310) {
		tmp = Math.sqrt((-F / A));
	} else if (F <= 1.8e+26) {
		tmp = (Math.sqrt(2.0) / -B_m) * Math.sqrt((F * (C + B_m)));
	} else {
		tmp = -Math.sqrt(((F / B_m) * 2.0));
	}
	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 F <= -2e-310:
		tmp = math.sqrt((-F / A))
	elif F <= 1.8e+26:
		tmp = (math.sqrt(2.0) / -B_m) * math.sqrt((F * (C + B_m)))
	else:
		tmp = -math.sqrt(((F / B_m) * 2.0))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -2e-310)
		tmp = sqrt(Float64(Float64(-F) / A));
	elseif (F <= 1.8e+26)
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(F * Float64(C + B_m))));
	else
		tmp = Float64(-sqrt(Float64(Float64(F / B_m) * 2.0)));
	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 (F <= -2e-310)
		tmp = sqrt((-F / A));
	elseif (F <= 1.8e+26)
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (C + B_m)));
	else
		tmp = -sqrt(((F / B_m) * 2.0));
	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[F, -2e-310], N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision], If[LessEqual[F, 1.8e+26], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(F * N[(C + B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if F < -1.999999999999994e-310

   1. Initial program 19.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \left(\sqrt{-2} \cdot \sqrt{-1}\right)} \]
   4. Step-by-step derivation

      1. sqrt-unprod N/A

         \[\leadsto \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{-2 \cdot -1} \]

      2. metadata-eval N/A

         \[\leadsto \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2} \]

      3. sqrt-unprod N/A

         \[\leadsto \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

   5. Applied rewrites 57.6%

      \[\leadsto \color{blue}{\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}} \]

   6. Taylor expanded in A around -inf

      \[\leadsto \sqrt{-1 \cdot \frac{F}{A}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{-1 \cdot \frac{F}{A}} \]

      2. lower-/.f64 34.2

         \[\leadsto \sqrt{-1 \cdot \frac{F}{A}} \]

   8. Applied rewrites 34.2%

      \[\leadsto \sqrt{-1 \cdot \frac{F}{A}} \]

   if -1.999999999999994e-310 < F < 1.80000000000000012e26

   1. Initial program 25.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F} \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right) \]

      8. unpow2 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{B \cdot B + {C}^{2}}\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{B \cdot B + C \cdot C}\right)}\right) \]

      10. lower-hypot.f64 25.4

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

   5. Applied rewrites 25.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)} \]

   6. Taylor expanded in B around inf

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + B\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 19.6%

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + B\right)}\right) \]

   if 1.80000000000000012e26 < F

   1. Initial program 11.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F}{B} \cdot 2} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F}{B} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F}{B} \cdot 2} \]

      5. lower-/.f64 23.6

         \[\leadsto -1 \cdot \sqrt{\frac{F}{B} \cdot 2} \]

   5. Applied rewrites 23.6%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{B} \cdot 2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 23.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{+26}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(C + B\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.7% accurate, 8.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{+26}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{F \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

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 (<= F -2e-310)
   (sqrt (/ (- F) A))
   (if (<= F 1.6e+26)
     (* (/ (sqrt 2.0) (- B_m)) (sqrt (* F B_m)))
     (- (sqrt (* (/ F B_m) 2.0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -2e-310) {
		tmp = sqrt((-F / A));
	} else if (F <= 1.6e+26) {
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * B_m));
	} else {
		tmp = -sqrt(((F / B_m) * 2.0));
	}
	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 (f <= (-2d-310)) then
        tmp = sqrt((-f / a))
    else if (f <= 1.6d+26) then
        tmp = (sqrt(2.0d0) / -b_m) * sqrt((f * b_m))
    else
        tmp = -sqrt(((f / b_m) * 2.0d0))
    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 (F <= -2e-310) {
		tmp = Math.sqrt((-F / A));
	} else if (F <= 1.6e+26) {
		tmp = (Math.sqrt(2.0) / -B_m) * Math.sqrt((F * B_m));
	} else {
		tmp = -Math.sqrt(((F / B_m) * 2.0));
	}
	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 F <= -2e-310:
		tmp = math.sqrt((-F / A))
	elif F <= 1.6e+26:
		tmp = (math.sqrt(2.0) / -B_m) * math.sqrt((F * B_m))
	else:
		tmp = -math.sqrt(((F / B_m) * 2.0))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -2e-310)
		tmp = sqrt(Float64(Float64(-F) / A));
	elseif (F <= 1.6e+26)
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(F * B_m)));
	else
		tmp = Float64(-sqrt(Float64(Float64(F / B_m) * 2.0)));
	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 (F <= -2e-310)
		tmp = sqrt((-F / A));
	elseif (F <= 1.6e+26)
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * B_m));
	else
		tmp = -sqrt(((F / B_m) * 2.0));
	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[F, -2e-310], N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision], If[LessEqual[F, 1.6e+26], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(F * B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if F < -1.999999999999994e-310

   1. Initial program 19.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \left(\sqrt{-2} \cdot \sqrt{-1}\right)} \]
   4. Step-by-step derivation

      1. sqrt-unprod N/A

         \[\leadsto \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{-2 \cdot -1} \]

      2. metadata-eval N/A

         \[\leadsto \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2} \]

      3. sqrt-unprod N/A

         \[\leadsto \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

   5. Applied rewrites 57.6%

      \[\leadsto \color{blue}{\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}} \]

   6. Taylor expanded in A around -inf

      \[\leadsto \sqrt{-1 \cdot \frac{F}{A}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{-1 \cdot \frac{F}{A}} \]

      2. lower-/.f64 34.2

         \[\leadsto \sqrt{-1 \cdot \frac{F}{A}} \]

   8. Applied rewrites 34.2%

      \[\leadsto \sqrt{-1 \cdot \frac{F}{A}} \]

   if -1.999999999999994e-310 < F < 1.60000000000000014e26

   1. Initial program 25.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F} \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right) \]

      8. unpow2 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{B \cdot B + {C}^{2}}\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{B \cdot B + C \cdot C}\right)}\right) \]

      10. lower-hypot.f64 25.4

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right) \]

   5. Applied rewrites 25.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)} \]

   6. Taylor expanded in B around inf

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot B}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 19.9%

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot B}\right) \]

   if 1.60000000000000014e26 < F

   1. Initial program 11.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F}{B} \cdot 2} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F}{B} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F}{B} \cdot 2} \]

      5. lower-/.f64 23.6

         \[\leadsto -1 \cdot \sqrt{\frac{F}{B} \cdot 2} \]

   5. Applied rewrites 23.6%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{B} \cdot 2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 23.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{+26}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot B}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 38.7% accurate, 14.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 7.4 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 7.4e-9) (sqrt (/ (- F) A)) (- (sqrt (* (/ F B_m) 2.0)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 7.4e-9) {
		tmp = sqrt((-F / A));
	} else {
		tmp = -sqrt(((F / B_m) * 2.0));
	}
	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 (b_m <= 7.4d-9) then
        tmp = sqrt((-f / a))
    else
        tmp = -sqrt(((f / b_m) * 2.0d0))
    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 (B_m <= 7.4e-9) {
		tmp = Math.sqrt((-F / A));
	} else {
		tmp = -Math.sqrt(((F / B_m) * 2.0));
	}
	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 B_m <= 7.4e-9:
		tmp = math.sqrt((-F / A))
	else:
		tmp = -math.sqrt(((F / B_m) * 2.0))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 7.4e-9)
		tmp = sqrt(Float64(Float64(-F) / A));
	else
		tmp = Float64(-sqrt(Float64(Float64(F / B_m) * 2.0)));
	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 (B_m <= 7.4e-9)
		tmp = sqrt((-F / A));
	else
		tmp = -sqrt(((F / B_m) * 2.0));
	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[B$95$m, 7.4e-9], N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision], (-N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < 7.4e-9

   1. Initial program 19.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \left(\sqrt{-2} \cdot \sqrt{-1}\right)} \]
   4. Step-by-step derivation

      1. sqrt-unprod N/A

         \[\leadsto \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{-2 \cdot -1} \]

      2. metadata-eval N/A

         \[\leadsto \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2} \]

      3. sqrt-unprod N/A

         \[\leadsto \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

   5. Applied rewrites 13.6%

      \[\leadsto \color{blue}{\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}} \]

   6. Taylor expanded in A around -inf

      \[\leadsto \sqrt{-1 \cdot \frac{F}{A}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{-1 \cdot \frac{F}{A}} \]

      2. lower-/.f64 13.2

         \[\leadsto \sqrt{-1 \cdot \frac{F}{A}} \]

   8. Applied rewrites 13.2%

      \[\leadsto \sqrt{-1 \cdot \frac{F}{A}} \]

   if 7.4e-9 < B

   1. Initial program 21.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F}{B} \cdot 2} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F}{B} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F}{B} \cdot 2} \]

      5. lower-/.f64 48.2

         \[\leadsto -1 \cdot \sqrt{\frac{F}{B} \cdot 2} \]

   5. Applied rewrites 48.2%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{B} \cdot 2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7.4 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 1.5% accurate, 18.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \sqrt{-2 \cdot \frac{F}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (sqrt (* -2.0 (/ F B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return sqrt((-2.0 * (F / B_m)));
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt(((-2.0d0) * (f / b_m)))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt((-2.0 * (F / B_m)));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return math.sqrt((-2.0 * (F / B_m)))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return sqrt(Float64(-2.0 * Float64(F / B_m)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = sqrt((-2.0 * (F / B_m)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[Sqrt[N[(-2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 20.1%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in F around -inf

   \[\leadsto \color{blue}{\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \left(\sqrt{-2} \cdot \sqrt{-1}\right)} \]
4. Step-by-step derivation

   1. sqrt-unprod N/A

      \[\leadsto \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{-2 \cdot -1} \]

   2. metadata-eval N/A

      \[\leadsto \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2} \]

   3. sqrt-unprod N/A

      \[\leadsto \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

   5. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

5. Applied rewrites 10.7%

   \[\leadsto \color{blue}{\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}} \]

6. Taylor expanded in B around -inf

   \[\leadsto \sqrt{-2 \cdot \frac{F}{B}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \sqrt{-2 \cdot \frac{F}{B}} \]

   2. lift-/.f64 1.9

      \[\leadsto \sqrt{-2 \cdot \frac{F}{B}} \]

8. Applied rewrites 1.9%

   \[\leadsto \sqrt{-2 \cdot \frac{F}{B}} \]
9. Add Preprocessing

Alternative 17: 20.5% accurate, 20.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \sqrt{\frac{-F}{A}} \end{array} \]

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) A)))

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 / A));
}

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 / a))
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 / A));
}

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 / A))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return sqrt(Float64(Float64(-F) / A))
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 / A));
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) / A), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 20.1%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in F around -inf

   \[\leadsto \color{blue}{\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \left(\sqrt{-2} \cdot \sqrt{-1}\right)} \]
4. Step-by-step derivation

   1. sqrt-unprod N/A

      \[\leadsto \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{-2 \cdot -1} \]

   2. metadata-eval N/A

      \[\leadsto \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2} \]

   3. sqrt-unprod N/A

      \[\leadsto \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

   5. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

5. Applied rewrites 10.7%

   \[\leadsto \color{blue}{\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}} \]

6. Taylor expanded in A around -inf

   \[\leadsto \sqrt{-1 \cdot \frac{F}{A}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \sqrt{-1 \cdot \frac{F}{A}} \]

   2. lower-/.f64 10.5

      \[\leadsto \sqrt{-1 \cdot \frac{F}{A}} \]

8. Applied rewrites 10.5%

   \[\leadsto \sqrt{-1 \cdot \frac{F}{A}} \]

9. Final simplification 10.5%

   \[\leadsto \sqrt{\frac{-F}{A}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025085 
(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))))