ABCF->ab-angle a

Percentage Accurate: 18.6% → 60.2%

Time: 14.0s

Alternatives: 21

Speedup: 16.9×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.6% 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: 60.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C\\ t_1 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ t_2 := t\_1 \cdot 2\\ t_3 := -t\_1\\ t_4 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_5 := -t\_4\\ t_6 := \frac{\sqrt{\left(2 \cdot \left(t\_4 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_5}\\ \mathbf{if}\;t\_6 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\left(C + C\right) \cdot F} \cdot \sqrt{t\_2}}{t\_3}\\ \mathbf{elif}\;t\_6 \leq -1 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(F \cdot t\_0\right)}}{t\_5}\\ \mathbf{elif}\;t\_6 \leq 5 \cdot 10^{+171}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-1, \frac{\left(B\_m \cdot B\_m\right) \cdot F}{A}, \left(4 \cdot C\right) \cdot F\right) \cdot t\_1}}{t\_3}\\ \mathbf{elif}\;t\_6 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot t\_1} \cdot \sqrt{t\_0}}{t\_5}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{hypot}\left(C, B\_m\right) + C} \cdot \sqrt{F}\right) \cdot \frac{\frac{2}{\sqrt{2}}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (+ (+ (hypot (- A C) B_m) A) C))
        (t_1 (fma (* -4.0 A) C (* B_m B_m)))
        (t_2 (* t_1 2.0))
        (t_3 (- t_1))
        (t_4 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_5 (- t_4))
        (t_6
         (/
          (sqrt
           (*
            (* 2.0 (* t_4 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          t_5)))
   (if (<= t_6 (- INFINITY))
     (/ (* (sqrt (* (+ C C) F)) (sqrt t_2)) t_3)
     (if (<= t_6 -1e-186)
       (/ (sqrt (* t_2 (* F t_0))) t_5)
       (if (<= t_6 5e+171)
         (/
          (sqrt (* (fma -1.0 (/ (* (* B_m B_m) F) A) (* (* 4.0 C) F)) t_1))
          t_3)
         (if (<= t_6 INFINITY)
           (/ (* (sqrt (* (* 2.0 F) t_1)) (sqrt t_0)) t_5)
           (*
            (* (sqrt (+ (hypot C B_m) C)) (sqrt F))
            (/ (/ 2.0 (sqrt 2.0)) (- B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (hypot((A - C), B_m) + A) + C;
	double t_1 = fma((-4.0 * A), C, (B_m * B_m));
	double t_2 = t_1 * 2.0;
	double t_3 = -t_1;
	double t_4 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_5 = -t_4;
	double t_6 = sqrt(((2.0 * (t_4 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_5;
	double tmp;
	if (t_6 <= -((double) INFINITY)) {
		tmp = (sqrt(((C + C) * F)) * sqrt(t_2)) / t_3;
	} else if (t_6 <= -1e-186) {
		tmp = sqrt((t_2 * (F * t_0))) / t_5;
	} else if (t_6 <= 5e+171) {
		tmp = sqrt((fma(-1.0, (((B_m * B_m) * F) / A), ((4.0 * C) * F)) * t_1)) / t_3;
	} else if (t_6 <= ((double) INFINITY)) {
		tmp = (sqrt(((2.0 * F) * t_1)) * sqrt(t_0)) / t_5;
	} else {
		tmp = (sqrt((hypot(C, B_m) + C)) * sqrt(F)) * ((2.0 / sqrt(2.0)) / -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(Float64(hypot(Float64(A - C), B_m) + A) + C)
	t_1 = fma(Float64(-4.0 * A), C, Float64(B_m * B_m))
	t_2 = Float64(t_1 * 2.0)
	t_3 = Float64(-t_1)
	t_4 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_5 = Float64(-t_4)
	t_6 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_4 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / t_5)
	tmp = 0.0
	if (t_6 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(Float64(C + C) * F)) * sqrt(t_2)) / t_3);
	elseif (t_6 <= -1e-186)
		tmp = Float64(sqrt(Float64(t_2 * Float64(F * t_0))) / t_5);
	elseif (t_6 <= 5e+171)
		tmp = Float64(sqrt(Float64(fma(-1.0, Float64(Float64(Float64(B_m * B_m) * F) / A), Float64(Float64(4.0 * C) * F)) * t_1)) / t_3);
	elseif (t_6 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * F) * t_1)) * sqrt(t_0)) / t_5);
	else
		tmp = Float64(Float64(sqrt(Float64(hypot(C, B_m) + C)) * sqrt(F)) * Float64(Float64(2.0 / sqrt(2.0)) / Float64(-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[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * 2.0), $MachinePrecision]}, Block[{t$95$3 = (-t$95$1)}, Block[{t$95$4 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = (-t$95$4)}, Block[{t$95$6 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$4 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision]}, If[LessEqual[t$95$6, (-Infinity)], N[(N[(N[Sqrt[N[(N[(C + C), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$6, -1e-186], N[(N[Sqrt[N[(t$95$2 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision], If[LessEqual[t$95$6, 5e+171], N[(N[Sqrt[N[(N[(-1.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * F), $MachinePrecision] / A), $MachinePrecision] + N[(N[(4.0 * C), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$6, Infinity], N[(N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision], N[(N[(N[Sqrt[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 11.7%

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

   5. Taylor expanded in A around -inf

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

   7. Applied rewrites 15.7%

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

   9. Applied rewrites 15.7%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. sqrt-prod N/A

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

       8. pow1/2 N/A

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

       9. metadata-eval N/A

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

   11. Applied rewrites 24.0%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.9999999999999991e-187

   1. Initial program 97.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

   if -9.9999999999999991e-187 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 5.0000000000000004e171

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 23.6%

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

   5. Taylor expanded in A around -inf

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

   7. Applied rewrites 27.9%

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

   9. Applied rewrites 27.9%

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

   10. Taylor expanded in A around -inf

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

   12. Applied rewrites 36.3%

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

   if 5.0000000000000004e171 < (/.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 14.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

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

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 72.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0

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

   5. Applied rewrites 16.2%

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

   7. Applied rewrites 25.2%

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

   9. Applied rewrites 25.3%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\left(C + C\right) \cdot F} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right) \cdot 2}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right) \cdot 2\right) \cdot \left(F \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 5 \cdot 10^{+171}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-1, \frac{\left(B \cdot B\right) \cdot F}{A}, \left(4 \cdot C\right) \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot \sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{hypot}\left(C, B\right) + C} \cdot \sqrt{F}\right) \cdot \frac{\frac{2}{\sqrt{2}}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 60.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ t_1 := t\_0 \cdot 2\\ t_2 := -t\_0\\ t_3 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_4 := -t\_3\\ t_5 := \frac{\sqrt{\left(2 \cdot \left(t\_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_4}\\ \mathbf{if}\;t\_5 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\left(C + C\right) \cdot F} \cdot \sqrt{t\_1}}{t\_2}\\ \mathbf{elif}\;t\_5 \leq -1 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(F \cdot \left(\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C\right)\right)}}{t\_4}\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{+171}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-1, \frac{\left(B\_m \cdot B\_m\right) \cdot F}{A}, \left(4 \cdot C\right) \cdot F\right) \cdot t\_0}}{t\_2}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;\frac{\sqrt{C + C} \cdot \sqrt{\left(F \cdot 2\right) \cdot t\_0}}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{hypot}\left(C, B\_m\right) + C} \cdot \sqrt{F}\right) \cdot \frac{\frac{2}{\sqrt{2}}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 A) C (* B_m B_m)))
        (t_1 (* t_0 2.0))
        (t_2 (- t_0))
        (t_3 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_4 (- t_3))
        (t_5
         (/
          (sqrt
           (*
            (* 2.0 (* t_3 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          t_4)))
   (if (<= t_5 (- INFINITY))
     (/ (* (sqrt (* (+ C C) F)) (sqrt t_1)) t_2)
     (if (<= t_5 -1e-186)
       (/ (sqrt (* t_1 (* F (+ (+ (hypot (- A C) B_m) A) C)))) t_4)
       (if (<= t_5 5e+171)
         (/
          (sqrt (* (fma -1.0 (/ (* (* B_m B_m) F) A) (* (* 4.0 C) F)) t_0))
          t_2)
         (if (<= t_5 INFINITY)
           (/ (* (sqrt (+ C C)) (sqrt (* (* F 2.0) t_0))) t_4)
           (*
            (* (sqrt (+ (hypot C B_m) C)) (sqrt F))
            (/ (/ 2.0 (sqrt 2.0)) (- B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * A), C, (B_m * B_m));
	double t_1 = t_0 * 2.0;
	double t_2 = -t_0;
	double t_3 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_4 = -t_3;
	double t_5 = sqrt(((2.0 * (t_3 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_4;
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = (sqrt(((C + C) * F)) * sqrt(t_1)) / t_2;
	} else if (t_5 <= -1e-186) {
		tmp = sqrt((t_1 * (F * ((hypot((A - C), B_m) + A) + C)))) / t_4;
	} else if (t_5 <= 5e+171) {
		tmp = sqrt((fma(-1.0, (((B_m * B_m) * F) / A), ((4.0 * C) * F)) * t_0)) / t_2;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = (sqrt((C + C)) * sqrt(((F * 2.0) * t_0))) / t_4;
	} else {
		tmp = (sqrt((hypot(C, B_m) + C)) * sqrt(F)) * ((2.0 / sqrt(2.0)) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * A), C, Float64(B_m * B_m))
	t_1 = Float64(t_0 * 2.0)
	t_2 = Float64(-t_0)
	t_3 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_4 = Float64(-t_3)
	t_5 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_3 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / t_4)
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(Float64(C + C) * F)) * sqrt(t_1)) / t_2);
	elseif (t_5 <= -1e-186)
		tmp = Float64(sqrt(Float64(t_1 * Float64(F * Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)))) / t_4);
	elseif (t_5 <= 5e+171)
		tmp = Float64(sqrt(Float64(fma(-1.0, Float64(Float64(Float64(B_m * B_m) * F) / A), Float64(Float64(4.0 * C) * F)) * t_0)) / t_2);
	elseif (t_5 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(C + C)) * sqrt(Float64(Float64(F * 2.0) * t_0))) / t_4);
	else
		tmp = Float64(Float64(sqrt(Float64(hypot(C, B_m) + C)) * sqrt(F)) * Float64(Float64(2.0 / sqrt(2.0)) / Float64(-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[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * 2.0), $MachinePrecision]}, Block[{t$95$2 = (-t$95$0)}, Block[{t$95$3 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = (-t$95$3)}, Block[{t$95$5 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]}, If[LessEqual[t$95$5, (-Infinity)], N[(N[(N[Sqrt[N[(N[(C + C), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$5, -1e-186], N[(N[Sqrt[N[(t$95$1 * N[(F * N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$5, 5e+171], N[(N[Sqrt[N[(N[(-1.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * F), $MachinePrecision] / A), $MachinePrecision] + N[(N[(4.0 * C), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(N[(N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], N[(N[(N[Sqrt[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 11.7%

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

   5. Taylor expanded in A around -inf

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

   7. Applied rewrites 15.7%

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

   9. Applied rewrites 15.7%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. sqrt-prod N/A

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

       8. pow1/2 N/A

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

       9. metadata-eval N/A

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

   11. Applied rewrites 24.0%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.9999999999999991e-187

   1. Initial program 97.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

   if -9.9999999999999991e-187 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 5.0000000000000004e171

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 23.6%

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

   5. Taylor expanded in A around -inf

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

   7. Applied rewrites 27.9%

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

   9. Applied rewrites 27.9%

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

   10. Taylor expanded in A around -inf

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

   12. Applied rewrites 36.3%

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

   if 5.0000000000000004e171 < (/.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 14.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 35.6%

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

   5. Taylor expanded in A around -inf

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

   7. Applied rewrites 25.1%

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

   9. Applied rewrites 46.3%

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

   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

   5. Applied rewrites 16.2%

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

   7. Applied rewrites 25.2%

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

   9. Applied rewrites 25.3%

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

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\left(C + C\right) \cdot F} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right) \cdot 2}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right) \cdot 2\right) \cdot \left(F \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 5 \cdot 10^{+171}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-1, \frac{\left(B \cdot B\right) \cdot F}{A}, \left(4 \cdot C\right) \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{C + C} \cdot \sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{hypot}\left(C, B\right) + C} \cdot \sqrt{F}\right) \cdot \frac{\frac{2}{\sqrt{2}}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ t_1 := -t\_0\\ t_2 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := -t\_2\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\left(C + C\right) \cdot F} \cdot \sqrt{t\_0 \cdot 2}}{t\_1}\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{\left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)\right) \cdot t\_0} \cdot \frac{-\sqrt{2}}{\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+171}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-1, \frac{\left(B\_m \cdot B\_m\right) \cdot F}{A}, \left(4 \cdot C\right) \cdot F\right) \cdot t\_0}}{t\_1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{C + C} \cdot \sqrt{\left(F \cdot 2\right) \cdot t\_0}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{hypot}\left(C, B\_m\right) + C} \cdot \sqrt{F}\right) \cdot \frac{\frac{2}{\sqrt{2}}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 A) C (* B_m B_m)))
        (t_1 (- t_0))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3 (- t_2))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* t_2 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          t_3)))
   (if (<= t_4 (- INFINITY))
     (/ (* (sqrt (* (+ C C) F)) (sqrt (* t_0 2.0))) t_1)
     (if (<= t_4 -1e-186)
       (*
        (sqrt (* (* F (+ A (+ C (hypot B_m (- A C))))) t_0))
        (/ (- (sqrt 2.0)) (fma B_m B_m (* -4.0 (* A C)))))
       (if (<= t_4 5e+171)
         (/
          (sqrt (* (fma -1.0 (/ (* (* B_m B_m) F) A) (* (* 4.0 C) F)) t_0))
          t_1)
         (if (<= t_4 INFINITY)
           (/ (* (sqrt (+ C C)) (sqrt (* (* F 2.0) t_0))) t_3)
           (*
            (* (sqrt (+ (hypot C B_m) C)) (sqrt F))
            (/ (/ 2.0 (sqrt 2.0)) (- B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * A), C, (B_m * B_m));
	double t_1 = -t_0;
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_3 = -t_2;
	double t_4 = sqrt(((2.0 * (t_2 * 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(((C + C) * F)) * sqrt((t_0 * 2.0))) / t_1;
	} else if (t_4 <= -1e-186) {
		tmp = sqrt(((F * (A + (C + hypot(B_m, (A - C))))) * t_0)) * (-sqrt(2.0) / fma(B_m, B_m, (-4.0 * (A * C))));
	} else if (t_4 <= 5e+171) {
		tmp = sqrt((fma(-1.0, (((B_m * B_m) * F) / A), ((4.0 * C) * F)) * t_0)) / t_1;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt((C + C)) * sqrt(((F * 2.0) * t_0))) / t_3;
	} else {
		tmp = (sqrt((hypot(C, B_m) + C)) * sqrt(F)) * ((2.0 / sqrt(2.0)) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * A), C, Float64(B_m * B_m))
	t_1 = Float64(-t_0)
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(-t_2)
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / t_3)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(Float64(C + C) * F)) * sqrt(Float64(t_0 * 2.0))) / t_1);
	elseif (t_4 <= -1e-186)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(A + Float64(C + hypot(B_m, Float64(A - C))))) * t_0)) * Float64(Float64(-sqrt(2.0)) / fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))));
	elseif (t_4 <= 5e+171)
		tmp = Float64(sqrt(Float64(fma(-1.0, Float64(Float64(Float64(B_m * B_m) * F) / A), Float64(Float64(4.0 * C) * F)) * t_0)) / t_1);
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(C + C)) * sqrt(Float64(Float64(F * 2.0) * t_0))) / t_3);
	else
		tmp = Float64(Float64(sqrt(Float64(hypot(C, B_m) + C)) * sqrt(F)) * Float64(Float64(2.0 / sqrt(2.0)) / Float64(-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[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = (-t$95$2)}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[Sqrt[N[(N[(C + C), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$4, -1e-186], N[(N[Sqrt[N[(N[(F * N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 5e+171], N[(N[Sqrt[N[(N[(-1.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * F), $MachinePrecision] / A), $MachinePrecision] + N[(N[(4.0 * C), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(N[Sqrt[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 11.7%

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

   5. Taylor expanded in A around -inf

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

   7. Applied rewrites 15.7%

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

   9. Applied rewrites 15.7%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. sqrt-prod N/A

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

       8. pow1/2 N/A

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

       9. metadata-eval N/A

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

   11. Applied rewrites 24.0%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.9999999999999991e-187

   1. Initial program 97.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in F around 0

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

   7. Applied rewrites 99.2%

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

   if -9.9999999999999991e-187 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 5.0000000000000004e171

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 23.6%

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

   5. Taylor expanded in A around -inf

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

   7. Applied rewrites 27.9%

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

   9. Applied rewrites 27.9%

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

   10. Taylor expanded in A around -inf

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

   12. Applied rewrites 36.3%

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

   if 5.0000000000000004e171 < (/.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 14.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 35.6%

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

   5. Taylor expanded in A around -inf

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

   7. Applied rewrites 25.1%

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

   9. Applied rewrites 46.3%

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

   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

   5. Applied rewrites 16.2%

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

   7. Applied rewrites 25.2%

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

   9. Applied rewrites 25.3%

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

4. Final simplification 37.9%

   \[\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{\sqrt{\left(C + C\right) \cdot F} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right) \cdot 2}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{\left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot \frac{-\sqrt{2}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \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 5 \cdot 10^{+171}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-1, \frac{\left(B \cdot B\right) \cdot F}{A}, \left(4 \cdot C\right) \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{C + C} \cdot \sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{hypot}\left(C, B\right) + C} \cdot \sqrt{F}\right) \cdot \frac{\frac{2}{\sqrt{2}}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ t_1 := -t\_0\\ t_2 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := -t\_2\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+216}:\\ \;\;\;\;\frac{\sqrt{\left(C + C\right) \cdot F} \cdot \sqrt{t\_0 \cdot 2}}{t\_1}\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot t\_0\right)}}{t\_1}\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+171}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-1, \frac{\left(B\_m \cdot B\_m\right) \cdot F}{A}, \left(4 \cdot C\right) \cdot F\right) \cdot t\_0}}{t\_1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{C + C} \cdot \sqrt{\left(F \cdot 2\right) \cdot t\_0}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{hypot}\left(C, B\_m\right) + C} \cdot \sqrt{F}\right) \cdot \frac{\frac{2}{\sqrt{2}}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 A) C (* B_m B_m)))
        (t_1 (- t_0))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3 (- t_2))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* t_2 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          t_3)))
   (if (<= t_4 -5e+216)
     (/ (* (sqrt (* (+ C C) F)) (sqrt (* t_0 2.0))) t_1)
     (if (<= t_4 -1e-186)
       (/ (sqrt (* (+ (+ (hypot (- A C) B_m) A) C) (* (* 2.0 F) t_0))) t_1)
       (if (<= t_4 5e+171)
         (/
          (sqrt (* (fma -1.0 (/ (* (* B_m B_m) F) A) (* (* 4.0 C) F)) t_0))
          t_1)
         (if (<= t_4 INFINITY)
           (/ (* (sqrt (+ C C)) (sqrt (* (* F 2.0) t_0))) t_3)
           (*
            (* (sqrt (+ (hypot C B_m) C)) (sqrt F))
            (/ (/ 2.0 (sqrt 2.0)) (- B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * A), C, (B_m * B_m));
	double t_1 = -t_0;
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_3 = -t_2;
	double t_4 = sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_3;
	double tmp;
	if (t_4 <= -5e+216) {
		tmp = (sqrt(((C + C) * F)) * sqrt((t_0 * 2.0))) / t_1;
	} else if (t_4 <= -1e-186) {
		tmp = sqrt((((hypot((A - C), B_m) + A) + C) * ((2.0 * F) * t_0))) / t_1;
	} else if (t_4 <= 5e+171) {
		tmp = sqrt((fma(-1.0, (((B_m * B_m) * F) / A), ((4.0 * C) * F)) * t_0)) / t_1;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt((C + C)) * sqrt(((F * 2.0) * t_0))) / t_3;
	} else {
		tmp = (sqrt((hypot(C, B_m) + C)) * sqrt(F)) * ((2.0 / sqrt(2.0)) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * A), C, Float64(B_m * B_m))
	t_1 = Float64(-t_0)
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(-t_2)
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / t_3)
	tmp = 0.0
	if (t_4 <= -5e+216)
		tmp = Float64(Float64(sqrt(Float64(Float64(C + C) * F)) * sqrt(Float64(t_0 * 2.0))) / t_1);
	elseif (t_4 <= -1e-186)
		tmp = Float64(sqrt(Float64(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C) * Float64(Float64(2.0 * F) * t_0))) / t_1);
	elseif (t_4 <= 5e+171)
		tmp = Float64(sqrt(Float64(fma(-1.0, Float64(Float64(Float64(B_m * B_m) * F) / A), Float64(Float64(4.0 * C) * F)) * t_0)) / t_1);
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(C + C)) * sqrt(Float64(Float64(F * 2.0) * t_0))) / t_3);
	else
		tmp = Float64(Float64(sqrt(Float64(hypot(C, B_m) + C)) * sqrt(F)) * Float64(Float64(2.0 / sqrt(2.0)) / Float64(-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[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = (-t$95$2)}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, -5e+216], N[(N[(N[Sqrt[N[(N[(C + C), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$4, -1e-186], N[(N[Sqrt[N[(N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$4, 5e+171], N[(N[Sqrt[N[(N[(-1.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * F), $MachinePrecision] / A), $MachinePrecision] + N[(N[(4.0 * C), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(N[Sqrt[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.9999999999999998e216

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 18.7%

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

   5. Taylor expanded in A around -inf

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

   7. Applied rewrites 14.6%

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

   9. Applied rewrites 14.6%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. sqrt-prod N/A

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

       8. pow1/2 N/A

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

       9. metadata-eval N/A

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

   11. Applied rewrites 22.2%

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

   if -4.9999999999999998e216 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.9999999999999991e-187

   1. Initial program 97.0%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 97.0%

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

   if -9.9999999999999991e-187 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 5.0000000000000004e171

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 23.6%

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

   5. Taylor expanded in A around -inf

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

   7. Applied rewrites 27.9%

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

   9. Applied rewrites 27.9%

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

   10. Taylor expanded in A around -inf

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

   12. Applied rewrites 36.3%

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

   if 5.0000000000000004e171 < (/.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 14.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 35.6%

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

   5. Taylor expanded in A around -inf

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

   7. Applied rewrites 25.1%

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

   9. Applied rewrites 46.3%

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

   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

   5. Applied rewrites 16.2%

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

   7. Applied rewrites 25.2%

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

   9. Applied rewrites 25.3%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -5 \cdot 10^{+216}:\\ \;\;\;\;\frac{\sqrt{\left(C + C\right) \cdot F} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right) \cdot 2}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 5 \cdot 10^{+171}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-1, \frac{\left(B \cdot B\right) \cdot F}{A}, \left(4 \cdot C\right) \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{C + C} \cdot \sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{hypot}\left(C, B\right) + C} \cdot \sqrt{F}\right) \cdot \frac{\frac{2}{\sqrt{2}}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ t_1 := -t\_0\\ t_2 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_2}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+216}:\\ \;\;\;\;\frac{\sqrt{\left(C + C\right) \cdot F} \cdot \sqrt{t\_0 \cdot 2}}{t\_1}\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot t\_0\right)}}{t\_1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-1, \frac{\left(B\_m \cdot B\_m\right) \cdot F}{A}, \left(4 \cdot C\right) \cdot F\right) \cdot t\_0}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{hypot}\left(C, B\_m\right) + C} \cdot \sqrt{F}\right) \cdot \frac{\frac{2}{\sqrt{2}}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 A) C (* B_m B_m)))
        (t_1 (- t_0))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* t_2 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_2))))
   (if (<= t_3 -5e+216)
     (/ (* (sqrt (* (+ C C) F)) (sqrt (* t_0 2.0))) t_1)
     (if (<= t_3 -1e-186)
       (/ (sqrt (* (+ (+ (hypot (- A C) B_m) A) C) (* (* 2.0 F) t_0))) t_1)
       (if (<= t_3 INFINITY)
         (/
          (sqrt (* (fma -1.0 (/ (* (* B_m B_m) F) A) (* (* 4.0 C) F)) t_0))
          t_1)
         (*
          (* (sqrt (+ (hypot C B_m) C)) (sqrt F))
          (/ (/ 2.0 (sqrt 2.0)) (- B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * A), C, (B_m * B_m));
	double t_1 = -t_0;
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_3 = sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_2;
	double tmp;
	if (t_3 <= -5e+216) {
		tmp = (sqrt(((C + C) * F)) * sqrt((t_0 * 2.0))) / t_1;
	} else if (t_3 <= -1e-186) {
		tmp = sqrt((((hypot((A - C), B_m) + A) + C) * ((2.0 * F) * t_0))) / t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((fma(-1.0, (((B_m * B_m) * F) / A), ((4.0 * C) * F)) * t_0)) / t_1;
	} else {
		tmp = (sqrt((hypot(C, B_m) + C)) * sqrt(F)) * ((2.0 / sqrt(2.0)) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * A), C, Float64(B_m * B_m))
	t_1 = Float64(-t_0)
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_2))
	tmp = 0.0
	if (t_3 <= -5e+216)
		tmp = Float64(Float64(sqrt(Float64(Float64(C + C) * F)) * sqrt(Float64(t_0 * 2.0))) / t_1);
	elseif (t_3 <= -1e-186)
		tmp = Float64(sqrt(Float64(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C) * Float64(Float64(2.0 * F) * t_0))) / t_1);
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(fma(-1.0, Float64(Float64(Float64(B_m * B_m) * F) / A), Float64(Float64(4.0 * C) * F)) * t_0)) / t_1);
	else
		tmp = Float64(Float64(sqrt(Float64(hypot(C, B_m) + C)) * sqrt(F)) * Float64(Float64(2.0 / sqrt(2.0)) / Float64(-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[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$2)), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+216], N[(N[(N[Sqrt[N[(N[(C + C), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, -1e-186], N[(N[Sqrt[N[(N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(-1.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * F), $MachinePrecision] / A), $MachinePrecision] + N[(N[(4.0 * C), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $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))) < -4.9999999999999998e216

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 18.7%

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

   5. Taylor expanded in A around -inf

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

   7. Applied rewrites 14.6%

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

   9. Applied rewrites 14.6%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. sqrt-prod N/A

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

       8. pow1/2 N/A

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

       9. metadata-eval N/A

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

   11. Applied rewrites 22.2%

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

   if -4.9999999999999998e216 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.9999999999999991e-187

   1. Initial program 97.0%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 97.0%

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

   if -9.9999999999999991e-187 < (/.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 18.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 26.8%

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

   5. Taylor expanded in A around -inf

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

   7. Applied rewrites 27.2%

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

   9. Applied rewrites 27.2%

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

   10. Taylor expanded in A around -inf

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

   12. Applied rewrites 33.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0

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

   5. Applied rewrites 16.2%

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

   7. Applied rewrites 25.2%

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

   9. Applied rewrites 25.3%

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

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -5 \cdot 10^{+216}:\\ \;\;\;\;\frac{\sqrt{\left(C + C\right) \cdot F} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right) \cdot 2}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-1, \frac{\left(B \cdot B\right) \cdot F}{A}, \left(4 \cdot C\right) \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{hypot}\left(C, B\right) + C} \cdot \sqrt{F}\right) \cdot \frac{\frac{2}{\sqrt{2}}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ t_1 := -t\_0\\ t_2 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_2}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+216}:\\ \;\;\;\;\frac{\sqrt{\left(C + C\right) \cdot F} \cdot \sqrt{t\_0 \cdot 2}}{t\_1}\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{\frac{\left(\left(\mathsf{hypot}\left(A - C, B\_m\right) + C\right) + A\right) \cdot F}{t\_0}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-1, \frac{\left(B\_m \cdot B\_m\right) \cdot F}{A}, \left(4 \cdot C\right) \cdot F\right) \cdot t\_0}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{hypot}\left(C, B\_m\right) + C} \cdot \sqrt{F}\right) \cdot \frac{\frac{2}{\sqrt{2}}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 A) C (* B_m B_m)))
        (t_1 (- t_0))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* t_2 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_2))))
   (if (<= t_3 -5e+216)
     (/ (* (sqrt (* (+ C C) F)) (sqrt (* t_0 2.0))) t_1)
     (if (<= t_3 -1e-186)
       (* (sqrt (/ (* (+ (+ (hypot (- A C) B_m) C) A) F) t_0)) (- (sqrt 2.0)))
       (if (<= t_3 INFINITY)
         (/
          (sqrt (* (fma -1.0 (/ (* (* B_m B_m) F) A) (* (* 4.0 C) F)) t_0))
          t_1)
         (*
          (* (sqrt (+ (hypot C B_m) C)) (sqrt F))
          (/ (/ 2.0 (sqrt 2.0)) (- B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * A), C, (B_m * B_m));
	double t_1 = -t_0;
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_3 = sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_2;
	double tmp;
	if (t_3 <= -5e+216) {
		tmp = (sqrt(((C + C) * F)) * sqrt((t_0 * 2.0))) / t_1;
	} else if (t_3 <= -1e-186) {
		tmp = sqrt(((((hypot((A - C), B_m) + C) + A) * F) / t_0)) * -sqrt(2.0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((fma(-1.0, (((B_m * B_m) * F) / A), ((4.0 * C) * F)) * t_0)) / t_1;
	} else {
		tmp = (sqrt((hypot(C, B_m) + C)) * sqrt(F)) * ((2.0 / sqrt(2.0)) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * A), C, Float64(B_m * B_m))
	t_1 = Float64(-t_0)
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_2))
	tmp = 0.0
	if (t_3 <= -5e+216)
		tmp = Float64(Float64(sqrt(Float64(Float64(C + C) * F)) * sqrt(Float64(t_0 * 2.0))) / t_1);
	elseif (t_3 <= -1e-186)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(hypot(Float64(A - C), B_m) + C) + A) * F) / t_0)) * Float64(-sqrt(2.0)));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(fma(-1.0, Float64(Float64(Float64(B_m * B_m) * F) / A), Float64(Float64(4.0 * C) * F)) * t_0)) / t_1);
	else
		tmp = Float64(Float64(sqrt(Float64(hypot(C, B_m) + C)) * sqrt(F)) * Float64(Float64(2.0 / sqrt(2.0)) / Float64(-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[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$2)), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+216], N[(N[(N[Sqrt[N[(N[(C + C), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, -1e-186], N[(N[Sqrt[N[(N[(N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] + A), $MachinePrecision] * F), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(-1.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * F), $MachinePrecision] / A), $MachinePrecision] + N[(N[(4.0 * C), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $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))) < -4.9999999999999998e216

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 18.7%

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

   5. Taylor expanded in A around -inf

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

   7. Applied rewrites 14.6%

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

   9. Applied rewrites 14.6%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. sqrt-prod N/A

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

       8. pow1/2 N/A

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

       9. metadata-eval N/A

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

   11. Applied rewrites 22.2%

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

   if -4.9999999999999998e216 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.9999999999999991e-187

   1. Initial program 97.0%

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

   3. Taylor expanded in F around 0

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

   5. Applied rewrites 92.5%

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

   if -9.9999999999999991e-187 < (/.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 18.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 26.8%

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

   5. Taylor expanded in A around -inf

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

   7. Applied rewrites 27.2%

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

   9. Applied rewrites 27.2%

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

   10. Taylor expanded in A around -inf

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

   12. Applied rewrites 33.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0

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

   5. Applied rewrites 16.2%

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

   7. Applied rewrites 25.2%

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

   9. Applied rewrites 25.3%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -5 \cdot 10^{+216}:\\ \;\;\;\;\frac{\sqrt{\left(C + C\right) \cdot F} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right) \cdot 2}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{\frac{\left(\left(\mathsf{hypot}\left(A - C, B\right) + C\right) + A\right) \cdot F}{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-1, \frac{\left(B \cdot B\right) \cdot F}{A}, \left(4 \cdot C\right) \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{hypot}\left(C, B\right) + C} \cdot \sqrt{F}\right) \cdot \frac{\frac{2}{\sqrt{2}}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 1.1 \cdot 10^{-12}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-1, \frac{\left(B\_m \cdot B\_m\right) \cdot F}{A}, \left(4 \cdot C\right) \cdot F\right) \cdot t\_0}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{hypot}\left(C, B\_m\right) + C} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 A) C (* B_m B_m))))
   (if (<= B_m 1.1e-12)
     (/
      (sqrt (* (fma -1.0 (/ (* (* B_m B_m) F) A) (* (* 4.0 C) F)) t_0))
      (- t_0))
     (* (* (sqrt (+ (hypot C B_m) C)) (sqrt F)) (/ (sqrt 2.0) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * A), C, (B_m * B_m));
	double tmp;
	if (B_m <= 1.1e-12) {
		tmp = sqrt((fma(-1.0, (((B_m * B_m) * F) / A), ((4.0 * C) * F)) * t_0)) / -t_0;
	} else {
		tmp = (sqrt((hypot(C, B_m) + C)) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * A), C, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 1.1e-12)
		tmp = Float64(sqrt(Float64(fma(-1.0, Float64(Float64(Float64(B_m * B_m) * F) / A), Float64(Float64(4.0 * C) * F)) * t_0)) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(Float64(hypot(C, B_m) + C)) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-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[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.1e-12], N[(N[Sqrt[N[(N[(-1.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * F), $MachinePrecision] / A), $MachinePrecision] + N[(N[(4.0 * C), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 1.09999999999999996e-12

   1. Initial program 19.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 24.1%

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

   5. Taylor expanded in A around -inf

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

   7. Applied rewrites 15.9%

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

   9. Applied rewrites 15.9%

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

   10. Taylor expanded in A around -inf

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

   12. Applied rewrites 17.4%

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

   if 1.09999999999999996e-12 < B

   1. Initial program 11.7%

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

   5. Applied rewrites 35.7%

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

   7. Applied rewrites 51.9%

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

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.1 \cdot 10^{-12}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-1, \frac{\left(B \cdot B\right) \cdot F}{A}, \left(4 \cdot C\right) \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{hypot}\left(C, B\right) + C} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 1.1 \cdot 10^{-12}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-1, \frac{\left(B\_m \cdot B\_m\right) \cdot F}{A}, \left(4 \cdot C\right) \cdot F\right) \cdot t\_0}}{-t\_0}\\ \mathbf{elif}\;B\_m \leq 9.2 \cdot 10^{+241}:\\ \;\;\;\;\sqrt{\mathsf{hypot}\left(C, B\_m\right) + C} \cdot \frac{\sqrt{F \cdot 2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B\_m + C \cdot \left(1 + 0.5 \cdot \frac{C}{B\_m}\right)} \cdot \left(-\sqrt{F}\right)\right) \cdot \frac{\sqrt{2}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 A) C (* B_m B_m))))
   (if (<= B_m 1.1e-12)
     (/
      (sqrt (* (fma -1.0 (/ (* (* B_m B_m) F) A) (* (* 4.0 C) F)) t_0))
      (- t_0))
     (if (<= B_m 9.2e+241)
       (* (sqrt (+ (hypot C B_m) C)) (/ (sqrt (* F 2.0)) (- B_m)))
       (*
        (* (sqrt (+ B_m (* C (+ 1.0 (* 0.5 (/ C B_m)))))) (- (sqrt F)))
        (/ (sqrt 2.0) B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * A), C, (B_m * B_m));
	double tmp;
	if (B_m <= 1.1e-12) {
		tmp = sqrt((fma(-1.0, (((B_m * B_m) * F) / A), ((4.0 * C) * F)) * t_0)) / -t_0;
	} else if (B_m <= 9.2e+241) {
		tmp = sqrt((hypot(C, B_m) + C)) * (sqrt((F * 2.0)) / -B_m);
	} else {
		tmp = (sqrt((B_m + (C * (1.0 + (0.5 * (C / B_m)))))) * -sqrt(F)) * (sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * A), C, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 1.1e-12)
		tmp = Float64(sqrt(Float64(fma(-1.0, Float64(Float64(Float64(B_m * B_m) * F) / A), Float64(Float64(4.0 * C) * F)) * t_0)) / Float64(-t_0));
	elseif (B_m <= 9.2e+241)
		tmp = Float64(sqrt(Float64(hypot(C, B_m) + C)) * Float64(sqrt(Float64(F * 2.0)) / Float64(-B_m)));
	else
		tmp = Float64(Float64(sqrt(Float64(B_m + Float64(C * Float64(1.0 + Float64(0.5 * Float64(C / B_m)))))) * Float64(-sqrt(F))) * Float64(sqrt(2.0) / 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[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.1e-12], N[(N[Sqrt[N[(N[(-1.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * F), $MachinePrecision] / A), $MachinePrecision] + N[(N[(4.0 * C), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[B$95$m, 9.2e+241], N[(N[Sqrt[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(B$95$m + N[(C * N[(1.0 + N[(0.5 * N[(C / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.09999999999999996e-12

   1. Initial program 19.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 24.1%

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

   5. Taylor expanded in A around -inf

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

   7. Applied rewrites 15.9%

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

   9. Applied rewrites 15.9%

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

   10. Taylor expanded in A around -inf

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

   12. Applied rewrites 17.4%

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

   if 1.09999999999999996e-12 < B < 9.1999999999999998e241

   1. Initial program 14.7%

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

   5. Applied rewrites 35.1%

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

   7. Applied rewrites 43.8%

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

   9. Applied rewrites 42.4%

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

   if 9.1999999999999998e241 < 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 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

   5. Applied rewrites 37.8%

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

   7. Applied rewrites 83.0%

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

   8. Taylor expanded in C around 0

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

   10. Applied rewrites 77.2%

       \[\leadsto -\left(\sqrt{B + C \cdot \left(1 + 0.5 \cdot \frac{C}{B}\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 26.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.1 \cdot 10^{-12}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-1, \frac{\left(B \cdot B\right) \cdot F}{A}, \left(4 \cdot C\right) \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 9.2 \cdot 10^{+241}:\\ \;\;\;\;\sqrt{\mathsf{hypot}\left(C, B\right) + C} \cdot \frac{\sqrt{F \cdot 2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B + C \cdot \left(1 + 0.5 \cdot \frac{C}{B}\right)} \cdot \left(-\sqrt{F}\right)\right) \cdot \frac{\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 1.1 \cdot 10^{-12}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-1, \frac{\left(B\_m \cdot B\_m\right) \cdot F}{A}, \left(4 \cdot C\right) \cdot F\right) \cdot t\_0}}{-t\_0}\\ \mathbf{elif}\;B\_m \leq 9.5 \cdot 10^{+186}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot F\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B\_m + C} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 A) C (* B_m B_m))))
   (if (<= B_m 1.1e-12)
     (/
      (sqrt (* (fma -1.0 (/ (* (* B_m B_m) F) A) (* (* 4.0 C) F)) t_0))
      (- t_0))
     (if (<= B_m 9.5e+186)
       (/ (sqrt (* 2.0 (* (+ (hypot C B_m) C) F))) (- B_m))
       (* (* (sqrt (+ B_m C)) (sqrt F)) (/ (sqrt 2.0) (- B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * A), C, (B_m * B_m));
	double tmp;
	if (B_m <= 1.1e-12) {
		tmp = sqrt((fma(-1.0, (((B_m * B_m) * F) / A), ((4.0 * C) * F)) * t_0)) / -t_0;
	} else if (B_m <= 9.5e+186) {
		tmp = sqrt((2.0 * ((hypot(C, B_m) + C) * F))) / -B_m;
	} else {
		tmp = (sqrt((B_m + C)) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * A), C, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 1.1e-12)
		tmp = Float64(sqrt(Float64(fma(-1.0, Float64(Float64(Float64(B_m * B_m) * F) / A), Float64(Float64(4.0 * C) * F)) * t_0)) / Float64(-t_0));
	elseif (B_m <= 9.5e+186)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(hypot(C, B_m) + C) * F))) / Float64(-B_m));
	else
		tmp = Float64(Float64(sqrt(Float64(B_m + C)) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-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[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.1e-12], N[(N[Sqrt[N[(N[(-1.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * F), $MachinePrecision] / A), $MachinePrecision] + N[(N[(4.0 * C), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[B$95$m, 9.5e+186], N[(N[Sqrt[N[(2.0 * N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.09999999999999996e-12

   1. Initial program 19.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 24.1%

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

   5. Taylor expanded in A around -inf

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

   7. Applied rewrites 15.9%

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

   9. Applied rewrites 15.9%

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

   10. Taylor expanded in A around -inf

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

   12. Applied rewrites 17.4%

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

   if 1.09999999999999996e-12 < B < 9.49999999999999999e186

   1. Initial program 19.8%

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

   3. Taylor expanded in A around 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

   5. Applied rewrites 30.6%

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

   7. Applied rewrites 33.2%

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

   9. Applied rewrites 30.6%

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

   if 9.49999999999999999e186 < 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 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

   5. Applied rewrites 43.0%

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

   7. Applied rewrites 78.6%

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

   8. Taylor expanded in B around inf

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

   10. Applied rewrites 73.4%

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

4. Final simplification 26.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.1 \cdot 10^{-12}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-1, \frac{\left(B \cdot B\right) \cdot F}{A}, \left(4 \cdot C\right) \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{+186}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B + C} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.8% accurate, 4.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 3.5 \cdot 10^{+60}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-1, \frac{\left(B\_m \cdot B\_m\right) \cdot F}{A}, \left(4 \cdot C\right) \cdot F\right) \cdot t\_0}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B\_m + C} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 A) C (* B_m B_m))))
   (if (<= B_m 3.5e+60)
     (/
      (sqrt (* (fma -1.0 (/ (* (* B_m B_m) F) A) (* (* 4.0 C) F)) t_0))
      (- t_0))
     (* (* (sqrt (+ B_m C)) (sqrt F)) (/ (sqrt 2.0) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * A), C, (B_m * B_m));
	double tmp;
	if (B_m <= 3.5e+60) {
		tmp = sqrt((fma(-1.0, (((B_m * B_m) * F) / A), ((4.0 * C) * F)) * t_0)) / -t_0;
	} else {
		tmp = (sqrt((B_m + C)) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * A), C, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 3.5e+60)
		tmp = Float64(sqrt(Float64(fma(-1.0, Float64(Float64(Float64(B_m * B_m) * F) / A), Float64(Float64(4.0 * C) * F)) * t_0)) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(Float64(B_m + C)) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-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[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 3.5e+60], N[(N[Sqrt[N[(N[(-1.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * F), $MachinePrecision] / A), $MachinePrecision] + N[(N[(4.0 * C), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 3.5000000000000002e60

   1. Initial program 20.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 25.1%

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

   5. Taylor expanded in A around -inf

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

   7. Applied rewrites 15.9%

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

   9. Applied rewrites 15.9%

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

   10. Taylor expanded in A around -inf

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

   12. Applied rewrites 17.4%

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

   if 3.5000000000000002e60 < B

   1. Initial program 5.9%

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

   3. Taylor expanded in A around 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

   5. Applied rewrites 39.8%

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

   7. Applied rewrites 62.1%

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

   8. Taylor expanded in B around inf

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

   10. Applied rewrites 55.4%

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

4. Final simplification 25.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.5 \cdot 10^{+60}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-1, \frac{\left(B \cdot B\right) \cdot F}{A}, \left(4 \cdot C\right) \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B + C} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.0% accurate, 5.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 3.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(C + C\right) \cdot F\right) \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)}}{-A \cdot \mathsf{fma}\left(-4, C, \frac{B\_m \cdot B\_m}{A}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B\_m + C} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 3.2e+47)
   (/
    (sqrt (* (* (* (+ C C) F) 2.0) (fma (* -4.0 A) C (* B_m B_m))))
    (- (* A (fma -4.0 C (/ (* B_m B_m) A)))))
   (* (* (sqrt (+ B_m C)) (sqrt F)) (/ (sqrt 2.0) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 3.2e+47) {
		tmp = sqrt(((((C + C) * F) * 2.0) * fma((-4.0 * A), C, (B_m * B_m)))) / -(A * fma(-4.0, C, ((B_m * B_m) / A)));
	} else {
		tmp = (sqrt((B_m + C)) * sqrt(F)) * (sqrt(2.0) / -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)
	tmp = 0.0
	if (B_m <= 3.2e+47)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(C + C) * F) * 2.0) * fma(Float64(-4.0 * A), C, Float64(B_m * B_m)))) / Float64(-Float64(A * fma(-4.0, C, Float64(Float64(B_m * B_m) / A)))));
	else
		tmp = Float64(Float64(sqrt(Float64(B_m + C)) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-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_] := If[LessEqual[B$95$m, 3.2e+47], N[(N[Sqrt[N[(N[(N[(N[(C + C), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(A * N[(-4.0 * C + N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 3.2e47

   1. Initial program 20.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 25.5%

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

   5. Taylor expanded in A around -inf

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

   7. Applied rewrites 16.2%

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

   9. Applied rewrites 16.2%

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

   10. Taylor expanded in A around inf

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

   12. Applied rewrites 16.2%

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

   if 3.2e47 < B

   1. Initial program 5.8%

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

   3. Taylor expanded in A around 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

   5. Applied rewrites 37.3%

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

   7. Applied rewrites 58.0%

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

   8. Taylor expanded in B around inf

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

   10. Applied rewrites 51.7%

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

4. Final simplification 24.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(C + C\right) \cdot F\right) \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}}{-A \cdot \mathsf{fma}\left(-4, C, \frac{B \cdot B}{A}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B + C} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.2% accurate, 6.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 3.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{\sqrt{\left(\left(4 \cdot C\right) \cdot F\right) \cdot t\_0}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B\_m + C} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 A) C (* B_m B_m))))
   (if (<= B_m 3.2e+47)
     (/ (sqrt (* (* (* 4.0 C) F) t_0)) (- t_0))
     (* (* (sqrt (+ B_m C)) (sqrt F)) (/ (sqrt 2.0) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * A), C, (B_m * B_m));
	double tmp;
	if (B_m <= 3.2e+47) {
		tmp = sqrt((((4.0 * C) * F) * t_0)) / -t_0;
	} else {
		tmp = (sqrt((B_m + C)) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * A), C, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 3.2e+47)
		tmp = Float64(sqrt(Float64(Float64(Float64(4.0 * C) * F) * t_0)) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(Float64(B_m + C)) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-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[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 3.2e+47], N[(N[Sqrt[N[(N[(N[(4.0 * C), $MachinePrecision] * F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 3.2e47

   1. Initial program 20.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 25.5%

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

   5. Taylor expanded in A around -inf

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

   7. Applied rewrites 16.2%

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

   9. Applied rewrites 16.2%

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

   10. Taylor expanded in A around -inf

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

   12. Applied rewrites 16.2%

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

   if 3.2e47 < B

   1. Initial program 5.8%

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

   3. Taylor expanded in A around 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

   5. Applied rewrites 37.3%

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

   7. Applied rewrites 58.0%

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

   8. Taylor expanded in B around inf

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

   10. Applied rewrites 51.7%

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

4. Final simplification 24.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{\sqrt{\left(\left(4 \cdot C\right) \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B + C} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 49.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.4 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(C + C\right) \cdot F\right) \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B\_m + C} \cdot \sqrt{F}\right) \cdot \frac{\frac{2}{\sqrt{2}}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 2.4e-89)
   (/
    (sqrt (* (* (* (+ C C) F) 2.0) (fma (* -4.0 A) C (* B_m B_m))))
    (* (* 4.0 A) C))
   (* (* (sqrt (+ B_m C)) (sqrt F)) (/ (/ 2.0 (sqrt 2.0)) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2.4e-89) {
		tmp = sqrt(((((C + C) * F) * 2.0) * fma((-4.0 * A), C, (B_m * B_m)))) / ((4.0 * A) * C);
	} else {
		tmp = (sqrt((B_m + C)) * sqrt(F)) * ((2.0 / sqrt(2.0)) / -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)
	tmp = 0.0
	if (B_m <= 2.4e-89)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(C + C) * F) * 2.0) * fma(Float64(-4.0 * A), C, Float64(B_m * B_m)))) / Float64(Float64(4.0 * A) * C));
	else
		tmp = Float64(Float64(sqrt(Float64(B_m + C)) * sqrt(F)) * Float64(Float64(2.0 / sqrt(2.0)) / Float64(-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_] := If[LessEqual[B$95$m, 2.4e-89], N[(N[Sqrt[N[(N[(N[(N[(C + C), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2.40000000000000016e-89

   1. Initial program 18.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 23.0%

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

   5. Taylor expanded in A around -inf

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

   7. Applied rewrites 15.3%

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

   9. Applied rewrites 15.3%

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

   10. Taylor expanded in A around inf

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

   12. Applied rewrites 15.2%

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

   if 2.40000000000000016e-89 < B

   1. Initial program 15.8%

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

   3. Taylor expanded in A around 0

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

   5. Applied rewrites 33.8%

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

   7. Applied rewrites 47.5%

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

   9. Applied rewrites 47.6%

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

   10. Taylor expanded in B around inf

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

   12. Applied rewrites 41.9%

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

4. Final simplification 24.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.4 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(C + C\right) \cdot F\right) \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B + C} \cdot \sqrt{F}\right) \cdot \frac{\frac{2}{\sqrt{2}}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 49.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.4 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(C + C\right) \cdot F\right) \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B\_m + C} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 2.4e-89)
   (/
    (sqrt (* (* (* (+ C C) F) 2.0) (fma (* -4.0 A) C (* B_m B_m))))
    (* (* 4.0 A) C))
   (* (* (sqrt (+ B_m C)) (sqrt F)) (/ (sqrt 2.0) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2.4e-89) {
		tmp = sqrt(((((C + C) * F) * 2.0) * fma((-4.0 * A), C, (B_m * B_m)))) / ((4.0 * A) * C);
	} else {
		tmp = (sqrt((B_m + C)) * sqrt(F)) * (sqrt(2.0) / -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)
	tmp = 0.0
	if (B_m <= 2.4e-89)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(C + C) * F) * 2.0) * fma(Float64(-4.0 * A), C, Float64(B_m * B_m)))) / Float64(Float64(4.0 * A) * C));
	else
		tmp = Float64(Float64(sqrt(Float64(B_m + C)) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-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_] := If[LessEqual[B$95$m, 2.4e-89], N[(N[Sqrt[N[(N[(N[(N[(C + C), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2.40000000000000016e-89

   1. Initial program 18.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 23.0%

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

   5. Taylor expanded in A around -inf

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

   7. Applied rewrites 15.3%

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

   9. Applied rewrites 15.3%

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

   10. Taylor expanded in A around inf

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

   12. Applied rewrites 15.2%

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

   if 2.40000000000000016e-89 < B

   1. Initial program 15.8%

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

   3. Taylor expanded in A around 0

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

   5. Applied rewrites 33.8%

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

   7. Applied rewrites 47.5%

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

   8. Taylor expanded in B around inf

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

   10. Applied rewrites 41.8%

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

4. Final simplification 24.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.4 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(C + C\right) \cdot F\right) \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B + C} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 44.5% accurate, 7.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.8 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B\_m + C} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.8e-89)
   (/ (sqrt (* -16.0 (* A (* (* C C) F)))) (- (fma (* -4.0 A) C (* B_m B_m))))
   (* (* (sqrt (+ B_m C)) (sqrt F)) (/ (sqrt 2.0) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.8e-89) {
		tmp = sqrt((-16.0 * (A * ((C * C) * F)))) / -fma((-4.0 * A), C, (B_m * B_m));
	} else {
		tmp = (sqrt((B_m + C)) * sqrt(F)) * (sqrt(2.0) / -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)
	tmp = 0.0
	if (B_m <= 1.8e-89)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(A * Float64(Float64(C * C) * F)))) / Float64(-fma(Float64(-4.0 * A), C, Float64(B_m * B_m))));
	else
		tmp = Float64(Float64(sqrt(Float64(B_m + C)) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-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_] := If[LessEqual[B$95$m, 1.8e-89], N[(N[Sqrt[N[(-16.0 * N[(A * N[(N[(C * C), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.80000000000000003e-89

   1. Initial program 18.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 23.0%

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

   5. Taylor expanded in A around -inf

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

   7. Applied rewrites 15.3%

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

   9. Applied rewrites 15.3%

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

   10. Taylor expanded in A around -inf

       \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 11.2%

       \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

   if 1.80000000000000003e-89 < B

   1. Initial program 15.8%

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

   3. Taylor expanded in A around 0

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

   5. Applied rewrites 33.8%

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

   7. Applied rewrites 47.5%

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

   8. Taylor expanded in B around inf

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

   10. Applied rewrites 41.8%

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

4. Final simplification 21.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.8 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B + C} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 37.4% accurate, 7.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{F}\\ \mathbf{if}\;C \leq 4.2 \cdot 10^{+177}:\\ \;\;\;\;t\_0 \cdot \sqrt{\frac{2}{B\_m}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + C} \cdot t\_0\right) \cdot \frac{\sqrt{2}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt F))))
   (if (<= C 4.2e+177)
     (* t_0 (sqrt (/ 2.0 B_m)))
     (* (* (sqrt (+ C C)) t_0) (/ (sqrt 2.0) B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(F);
	double tmp;
	if (C <= 4.2e+177) {
		tmp = t_0 * sqrt((2.0 / B_m));
	} else {
		tmp = (sqrt((C + C)) * t_0) * (sqrt(2.0) / B_m);
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = -sqrt(f)
    if (c <= 4.2d+177) then
        tmp = t_0 * sqrt((2.0d0 / b_m))
    else
        tmp = (sqrt((c + c)) * t_0) * (sqrt(2.0d0) / b_m)
    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 t_0 = -Math.sqrt(F);
	double tmp;
	if (C <= 4.2e+177) {
		tmp = t_0 * Math.sqrt((2.0 / B_m));
	} else {
		tmp = (Math.sqrt((C + C)) * t_0) * (Math.sqrt(2.0) / B_m);
	}
	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 = -math.sqrt(F)
	tmp = 0
	if C <= 4.2e+177:
		tmp = t_0 * math.sqrt((2.0 / B_m))
	else:
		tmp = (math.sqrt((C + C)) * t_0) * (math.sqrt(2.0) / 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(F))
	tmp = 0.0
	if (C <= 4.2e+177)
		tmp = Float64(t_0 * sqrt(Float64(2.0 / B_m)));
	else
		tmp = Float64(Float64(sqrt(Float64(C + C)) * t_0) * Float64(sqrt(2.0) / B_m));
	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 = -sqrt(F);
	tmp = 0.0;
	if (C <= 4.2e+177)
		tmp = t_0 * sqrt((2.0 / B_m));
	else
		tmp = (sqrt((C + C)) * t_0) * (sqrt(2.0) / B_m);
	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[Sqrt[F], $MachinePrecision])}, If[LessEqual[C, 4.2e+177], N[(t$95$0 * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if C < 4.20000000000000026e177

   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

   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

   5. Applied rewrites 12.0%

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

   7. Applied rewrites 12.1%

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

   9. Applied rewrites 15.6%

      \[\leadsto -\sqrt{F} \cdot \sqrt{\frac{2}{B}} \]

   if 4.20000000000000026e177 < C

   1. Initial program 1.4%

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

   5. Applied rewrites 11.2%

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

   7. Applied rewrites 23.5%

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

   8. Taylor expanded in B around 0

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

   10. Applied rewrites 19.7%

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

4. Final simplification 16.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 4.2 \cdot 10^{+177}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + C} \cdot \left(-\sqrt{F}\right)\right) \cdot \frac{\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 36.2% accurate, 8.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \left(\sqrt{B\_m + C} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (* (* (sqrt (+ B_m C)) (sqrt F)) (/ (sqrt 2.0) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return (sqrt((B_m + C)) * sqrt(F)) * (sqrt(2.0) / -B_m);
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return (Math.sqrt((B_m + C)) * Math.sqrt(F)) * (Math.sqrt(2.0) / -B_m);
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return (math.sqrt((B_m + C)) * math.sqrt(F)) * (math.sqrt(2.0) / -B_m)

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(Float64(sqrt(Float64(B_m + C)) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-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((B_m + C)) * sqrt(F)) * (sqrt(2.0) / -B_m);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.4%

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

5. Applied rewrites 13.7%

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

7. Applied rewrites 18.3%

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

8. Taylor expanded in B around inf

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

10. Applied rewrites 14.4%

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

11. Final simplification 14.4%

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

Alternative 18: 36.0% accurate, 10.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 4.4 \cdot 10^{+182}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{1}{B\_m}\right) \cdot \sqrt{C \cdot F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 4.4e+182)
   (* (- (sqrt F)) (sqrt (/ 2.0 B_m)))
   (* (* -2.0 (/ 1.0 B_m)) (sqrt (* C F)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 4.4e+182) {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	} else {
		tmp = (-2.0 * (1.0 / B_m)) * sqrt((C * F));
	}
	return tmp;
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= 4.4d+182) then
        tmp = -sqrt(f) * sqrt((2.0d0 / b_m))
    else
        tmp = ((-2.0d0) * (1.0d0 / b_m)) * sqrt((c * f))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 4.4e+182) {
		tmp = -Math.sqrt(F) * Math.sqrt((2.0 / B_m));
	} else {
		tmp = (-2.0 * (1.0 / B_m)) * Math.sqrt((C * F));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if C <= 4.4e+182:
		tmp = -math.sqrt(F) * math.sqrt((2.0 / B_m))
	else:
		tmp = (-2.0 * (1.0 / B_m)) * math.sqrt((C * F))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 4.4e+182)
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / B_m)));
	else
		tmp = Float64(Float64(-2.0 * Float64(1.0 / B_m)) * sqrt(Float64(C * F)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= 4.4e+182)
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	else
		tmp = (-2.0 * (1.0 / B_m)) * sqrt((C * F));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 4.4e+182], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(1.0 / B$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 4.39999999999999993e182

   1. Initial program 19.1%

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

   3. Taylor expanded in B around inf

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

   5. Applied rewrites 12.0%

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

   7. Applied rewrites 12.1%

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

   9. Applied rewrites 15.6%

      \[\leadsto -\sqrt{F} \cdot \sqrt{\frac{2}{B}} \]

   if 4.39999999999999993e182 < C

   1. Initial program 1.5%

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

   5. Applied rewrites 11.5%

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

   7. Applied rewrites 23.9%

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

   9. Applied rewrites 24.0%

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

   10. Taylor expanded in B around 0

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

   12. Applied rewrites 11.6%

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

4. Final simplification 15.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 4.4 \cdot 10^{+182}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{1}{B}\right) \cdot \sqrt{C \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 29.0% accurate, 11.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 8 \cdot 10^{+140}:\\ \;\;\;\;-\sqrt{\frac{F}{B\_m} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{1}{B\_m}\right) \cdot \sqrt{C \cdot F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 8e+140)
   (- (sqrt (* (/ F B_m) 2.0)))
   (* (* -2.0 (/ 1.0 B_m)) (sqrt (* C F)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 8e+140) {
		tmp = -sqrt(((F / B_m) * 2.0));
	} else {
		tmp = (-2.0 * (1.0 / B_m)) * sqrt((C * F));
	}
	return tmp;
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= 8d+140) then
        tmp = -sqrt(((f / b_m) * 2.0d0))
    else
        tmp = ((-2.0d0) * (1.0d0 / b_m)) * sqrt((c * f))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 8e+140) {
		tmp = -Math.sqrt(((F / B_m) * 2.0));
	} else {
		tmp = (-2.0 * (1.0 / B_m)) * Math.sqrt((C * F));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if C <= 8e+140:
		tmp = -math.sqrt(((F / B_m) * 2.0))
	else:
		tmp = (-2.0 * (1.0 / B_m)) * math.sqrt((C * F))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 8e+140)
		tmp = Float64(-sqrt(Float64(Float64(F / B_m) * 2.0)));
	else
		tmp = Float64(Float64(-2.0 * Float64(1.0 / B_m)) * sqrt(Float64(C * F)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= 8e+140)
		tmp = -sqrt(((F / B_m) * 2.0));
	else
		tmp = (-2.0 * (1.0 / B_m)) * sqrt((C * F));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 8e+140], (-N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), N[(N[(-2.0 * N[(1.0 / B$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 8.00000000000000047e140

   1. Initial program 20.0%

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

   3. Taylor expanded in B around inf

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

   5. Applied rewrites 12.5%

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

   7. Applied rewrites 12.6%

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B} \cdot 2}} \]

   if 8.00000000000000047e140 < C

   1. Initial program 1.4%

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

   5. Applied rewrites 13.8%

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

   7. Applied rewrites 22.6%

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

   9. Applied rewrites 22.6%

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

   10. Taylor expanded in B around 0

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

   12. Applied rewrites 8.7%

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

Alternative 20: 27.8% accurate, 16.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -\sqrt{\frac{F}{B\_m} \cdot 2} \end{array} \]

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 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) {
	return -sqrt(((F / B_m) * 2.0));
}

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 / b_m) * 2.0d0))
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 / B_m) * 2.0));
}

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 / B_m) * 2.0))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-sqrt(Float64(Float64(F / B_m) * 2.0)))
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 / B_m) * 2.0));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := (-N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 17.4%

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

3. Taylor expanded in B around inf

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

5. Applied rewrites 11.6%

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

7. Applied rewrites 11.7%

   \[\leadsto \color{blue}{-\sqrt{\frac{F}{B} \cdot 2}} \]
8. Add Preprocessing

Alternative 21: 27.8% accurate, 16.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -\sqrt{F \cdot \frac{2}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- (sqrt (* F (/ 2.0 B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -sqrt((F * (2.0 / B_m)));
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -Math.sqrt((F * (2.0 / B_m)));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -math.sqrt((F * (2.0 / B_m)))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-sqrt(Float64(F * Float64(2.0 / B_m))))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -sqrt((F * (2.0 / B_m)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := (-N[Sqrt[N[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 17.4%

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

3. Taylor expanded in B around inf

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

5. Applied rewrites 11.6%

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

7. Applied rewrites 11.7%

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

9. Applied rewrites 11.6%

   \[\leadsto -\sqrt{F \cdot \frac{2}{B}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025018 
(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))))