ABCF->ab-angle a

Percentage Accurate: 18.9% → 55.0%

Time: 10.4s

Alternatives: 18

Speedup: 11.7×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 55.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)\\ t_1 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{-\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_1}\\ t_3 := \left(-B\_m\right) \cdot B\_m\\ t_4 := \mathsf{fma}\left(4 \cdot C, A, t\_3\right)\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{-208}:\\ \;\;\;\;\sqrt{F \cdot 2} \cdot \frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)} \cdot \sqrt{\left(\mathsf{hypot}\left(B\_m, A - C\right) + A\right) + C}}{t\_4}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-1, \frac{{B\_m}^{2}}{A}, 4 \cdot C\right) \cdot F} \cdot \sqrt{t\_0}}{\mathsf{fma}\left(C \cdot 4, A, t\_3\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C} \cdot \frac{\sqrt{\left(t\_0 \cdot 2\right) \cdot F}}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\sqrt{\frac{F}{B\_m}} \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* A C) -4.0 (* B_m B_m)))
        (t_1 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_2
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_1 F))
             (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0)))))))
          t_1))
        (t_3 (* (- B_m) B_m))
        (t_4 (fma (* 4.0 C) A t_3)))
   (if (<= t_2 -4e-208)
     (*
      (sqrt (* F 2.0))
      (/
       (*
        (sqrt (fma -4.0 (* C A) (* B_m B_m)))
        (sqrt (+ (+ (hypot B_m (- A C)) A) C)))
       t_4))
     (if (<= t_2 0.0)
       (/
        (* (sqrt (* (fma -1.0 (/ (pow B_m 2.0) A) (* 4.0 C)) F)) (sqrt t_0))
        (fma (* C 4.0) A t_3))
       (if (<= t_2 INFINITY)
         (*
          (sqrt (+ (+ (hypot (- A C) B_m) A) C))
          (/ (sqrt (* (* t_0 2.0) F)) t_4))
         (* -1.0 (* (sqrt (/ F B_m)) (sqrt 2.0))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((A * C), -4.0, (B_m * B_m));
	double t_1 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_2 = -sqrt(((2.0 * (t_1 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_1;
	double t_3 = -B_m * B_m;
	double t_4 = fma((4.0 * C), A, t_3);
	double tmp;
	if (t_2 <= -4e-208) {
		tmp = sqrt((F * 2.0)) * ((sqrt(fma(-4.0, (C * A), (B_m * B_m))) * sqrt(((hypot(B_m, (A - C)) + A) + C))) / t_4);
	} else if (t_2 <= 0.0) {
		tmp = (sqrt((fma(-1.0, (pow(B_m, 2.0) / A), (4.0 * C)) * F)) * sqrt(t_0)) / fma((C * 4.0), A, t_3);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((hypot((A - C), B_m) + A) + C)) * (sqrt(((t_0 * 2.0) * F)) / t_4);
	} else {
		tmp = -1.0 * (sqrt((F / B_m)) * sqrt(2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(A * C), -4.0, Float64(B_m * B_m))
	t_1 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_2 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))))))) / t_1)
	t_3 = Float64(Float64(-B_m) * B_m)
	t_4 = fma(Float64(4.0 * C), A, t_3)
	tmp = 0.0
	if (t_2 <= -4e-208)
		tmp = Float64(sqrt(Float64(F * 2.0)) * Float64(Float64(sqrt(fma(-4.0, Float64(C * A), Float64(B_m * B_m))) * sqrt(Float64(Float64(hypot(B_m, Float64(A - C)) + A) + C))) / t_4));
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(fma(-1.0, Float64((B_m ^ 2.0) / A), Float64(4.0 * C)) * F)) * sqrt(t_0)) / fma(Float64(C * 4.0), A, t_3));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)) * Float64(sqrt(Float64(Float64(t_0 * 2.0) * F)) / t_4));
	else
		tmp = Float64(-1.0 * Float64(sqrt(Float64(F / B_m)) * sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[((-B$95$m) * B$95$m), $MachinePrecision]}, Block[{t$95$4 = N[(N[(4.0 * C), $MachinePrecision] * A + t$95$3), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-208], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(N[Sqrt[N[(N[(-1.0 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision] + N[(4.0 * C), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(N[(C * 4.0), $MachinePrecision] * A + t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(N[(t$95$0 * 2.0), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.0000000000000004e-208

   1. Initial program 43.0%

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

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

   3. Applied rewrites 66.8%

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

   5. Applied rewrites 81.2%

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

   7. Applied rewrites 81.2%

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

   if -4.0000000000000004e-208 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 4.5%

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

   3. Applied rewrites 10.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. sqrt-prod N/A

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

      6. pow1/2 N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 17.6%

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

   6. Taylor expanded in A around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 44.5

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

   8. Applied rewrites 44.5%

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

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

   1. Initial program 40.9%

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

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

   3. Applied rewrites 83.4%

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

   5. Applied rewrites 83.2%

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

   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. 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

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

   3. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 1.2

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 1.2

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

   5. Applied rewrites 1.2%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 31.8

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

   8. Applied rewrites 31.8%

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

Alternative 2: 50.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)\\ t_1 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{-\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_1}\\ t_3 := \left(-B\_m\right) \cdot B\_m\\ t_4 := \mathsf{fma}\left(C \cdot 4, A, t\_3\right)\\ t_5 := \frac{\sqrt{\mathsf{fma}\left(-1, \frac{{B\_m}^{2}}{A}, 4 \cdot C\right) \cdot F} \cdot \sqrt{t\_0}}{t\_4}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-208}:\\ \;\;\;\;\frac{\sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)} \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(B\_m, A - C\right) + A\right) + C\right)}}{t\_4}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C} \cdot \frac{\sqrt{\left(t\_0 \cdot 2\right) \cdot F}}{\mathsf{fma}\left(4 \cdot C, A, t\_3\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\sqrt{\frac{F}{B\_m}} \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* A C) -4.0 (* B_m B_m)))
        (t_1 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_2
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_1 F))
             (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0)))))))
          t_1))
        (t_3 (* (- B_m) B_m))
        (t_4 (fma (* C 4.0) A t_3))
        (t_5
         (/
          (* (sqrt (* (fma -1.0 (/ (pow B_m 2.0) A) (* 4.0 C)) F)) (sqrt t_0))
          t_4)))
   (if (<= t_2 (- INFINITY))
     t_5
     (if (<= t_2 -4e-208)
       (/
        (*
         (sqrt (* F (fma -4.0 (* C A) (* B_m B_m))))
         (sqrt (* 2.0 (+ (+ (hypot B_m (- A C)) A) C))))
        t_4)
       (if (<= t_2 0.0)
         t_5
         (if (<= t_2 INFINITY)
           (*
            (sqrt (+ (+ (hypot (- A C) B_m) A) C))
            (/ (sqrt (* (* t_0 2.0) F)) (fma (* 4.0 C) A t_3)))
           (* -1.0 (* (sqrt (/ F B_m)) (sqrt 2.0)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((A * C), -4.0, (B_m * B_m));
	double t_1 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_2 = -sqrt(((2.0 * (t_1 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_1;
	double t_3 = -B_m * B_m;
	double t_4 = fma((C * 4.0), A, t_3);
	double t_5 = (sqrt((fma(-1.0, (pow(B_m, 2.0) / A), (4.0 * C)) * F)) * sqrt(t_0)) / t_4;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_5;
	} else if (t_2 <= -4e-208) {
		tmp = (sqrt((F * fma(-4.0, (C * A), (B_m * B_m)))) * sqrt((2.0 * ((hypot(B_m, (A - C)) + A) + C)))) / t_4;
	} else if (t_2 <= 0.0) {
		tmp = t_5;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((hypot((A - C), B_m) + A) + C)) * (sqrt(((t_0 * 2.0) * F)) / fma((4.0 * C), A, t_3));
	} else {
		tmp = -1.0 * (sqrt((F / B_m)) * sqrt(2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(A * C), -4.0, Float64(B_m * B_m))
	t_1 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_2 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))))))) / t_1)
	t_3 = Float64(Float64(-B_m) * B_m)
	t_4 = fma(Float64(C * 4.0), A, t_3)
	t_5 = Float64(Float64(sqrt(Float64(fma(-1.0, Float64((B_m ^ 2.0) / A), Float64(4.0 * C)) * F)) * sqrt(t_0)) / t_4)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_5;
	elseif (t_2 <= -4e-208)
		tmp = Float64(Float64(sqrt(Float64(F * fma(-4.0, Float64(C * A), Float64(B_m * B_m)))) * sqrt(Float64(2.0 * Float64(Float64(hypot(B_m, Float64(A - C)) + A) + C)))) / t_4);
	elseif (t_2 <= 0.0)
		tmp = t_5;
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)) * Float64(sqrt(Float64(Float64(t_0 * 2.0) * F)) / fma(Float64(4.0 * C), A, t_3)));
	else
		tmp = Float64(-1.0 * Float64(sqrt(Float64(F / B_m)) * sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[((-B$95$m) * B$95$m), $MachinePrecision]}, Block[{t$95$4 = N[(N[(C * 4.0), $MachinePrecision] * A + t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[Sqrt[N[(N[(-1.0 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision] + N[(4.0 * C), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$5, If[LessEqual[t$95$2, -4e-208], N[(N[(N[Sqrt[N[(F * N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$2, 0.0], t$95$5, If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(N[(t$95$0 * 2.0), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * C), $MachinePrecision] * A + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0 or -4.0000000000000004e-208 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.8%

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

   3. Applied rewrites 16.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. sqrt-prod N/A

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

      6. pow1/2 N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 33.3%

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

   6. Taylor expanded in A around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 46.4

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

   8. Applied rewrites 46.4%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.0000000000000004e-208

   1. Initial program 97.8%

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

   3. Applied rewrites 97.8%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 97.8

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

   5. Applied rewrites 97.8%

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

   7. Applied rewrites 98.0%

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

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

   1. Initial program 40.9%

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

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

   3. Applied rewrites 83.4%

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

   5. Applied rewrites 83.2%

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

   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. 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

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

   3. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 1.2

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 1.2

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

   5. Applied rewrites 1.2%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 31.8

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

   8. Applied rewrites 31.8%

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

Alternative 3: 49.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_2 := \mathsf{fma}\left(C \cdot 4, A, \left(-B\_m\right) \cdot B\_m\right)\\ t_3 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{-\sqrt{\left(2 \cdot \left(t\_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot C\right) \cdot F} \cdot \sqrt{t\_0}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq -4 \cdot 10^{-208}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C\right) \cdot \left(\left(F + F\right) \cdot t\_1\right)}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-164}:\\ \;\;\;\;\frac{\sqrt{4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(-2 \cdot C - 2 \cdot C\right)\right)\right)\right)}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{-\sqrt{C + C}}{t\_0} \cdot \left(\sqrt{2} \cdot \sqrt{F \cdot t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\sqrt{\frac{F}{B\_m}} \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* A C) -4.0 (* B_m B_m)))
        (t_1 (fma -4.0 (* C A) (* B_m B_m)))
        (t_2 (fma (* C 4.0) A (* (- B_m) B_m)))
        (t_3 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_4
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_3 F))
             (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0)))))))
          t_3)))
   (if (<= t_4 (- INFINITY))
     (/ (* (sqrt (* (* 4.0 C) F)) (sqrt t_0)) t_2)
     (if (<= t_4 -4e-208)
       (/ (sqrt (* (+ (+ (hypot (- A C) B_m) A) C) (* (+ F F) t_1))) t_2)
       (if (<= t_4 2e-164)
         (/ (sqrt (* 4.0 (* A (* C (* F (- (* -2.0 C) (* 2.0 C))))))) t_2)
         (if (<= t_4 INFINITY)
           (* (/ (- (sqrt (+ C C))) t_0) (* (sqrt 2.0) (sqrt (* F t_1))))
           (* -1.0 (* (sqrt (/ F B_m)) (sqrt 2.0)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((A * C), -4.0, (B_m * B_m));
	double t_1 = fma(-4.0, (C * A), (B_m * B_m));
	double t_2 = fma((C * 4.0), A, (-B_m * B_m));
	double t_3 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_4 = -sqrt(((2.0 * (t_3 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_3;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = (sqrt(((4.0 * C) * F)) * sqrt(t_0)) / t_2;
	} else if (t_4 <= -4e-208) {
		tmp = sqrt((((hypot((A - C), B_m) + A) + C) * ((F + F) * t_1))) / t_2;
	} else if (t_4 <= 2e-164) {
		tmp = sqrt((4.0 * (A * (C * (F * ((-2.0 * C) - (2.0 * C))))))) / t_2;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (-sqrt((C + C)) / t_0) * (sqrt(2.0) * sqrt((F * t_1)));
	} else {
		tmp = -1.0 * (sqrt((F / B_m)) * sqrt(2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(A * C), -4.0, Float64(B_m * B_m))
	t_1 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_2 = fma(Float64(C * 4.0), A, Float64(Float64(-B_m) * B_m))
	t_3 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_4 = Float64(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_3)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(Float64(4.0 * C) * F)) * sqrt(t_0)) / t_2);
	elseif (t_4 <= -4e-208)
		tmp = Float64(sqrt(Float64(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C) * Float64(Float64(F + F) * t_1))) / t_2);
	elseif (t_4 <= 2e-164)
		tmp = Float64(sqrt(Float64(4.0 * Float64(A * Float64(C * Float64(F * Float64(Float64(-2.0 * C) - Float64(2.0 * C))))))) / t_2);
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(Float64(-sqrt(Float64(C + C))) / t_0) * Float64(sqrt(2.0) * sqrt(Float64(F * t_1))));
	else
		tmp = Float64(-1.0 * Float64(sqrt(Float64(F / B_m)) * sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(C * 4.0), $MachinePrecision] * A + N[((-B$95$m) * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[Sqrt[N[(N[(4.0 * C), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, -4e-208], N[(N[Sqrt[N[(N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision] * N[(N[(F + F), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, 2e-164], N[(N[Sqrt[N[(4.0 * N[(A * N[(C * N[(F * N[(N[(-2.0 * C), $MachinePrecision] - N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[((-N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.2%

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

   3. Applied rewrites 19.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. sqrt-prod N/A

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

      6. pow1/2 N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 44.5%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 47.3

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

   8. Applied rewrites 47.3%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.0000000000000004e-208

   1. Initial program 97.8%

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

   3. Applied rewrites 97.8%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 97.8

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

   5. Applied rewrites 97.8%

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

   if -4.0000000000000004e-208 < (/.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))) < 1.99999999999999992e-164

   1. Initial program 5.2%

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

   3. Applied rewrites 11.6%

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

   5. Applied rewrites 4.1%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 36.9

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

   8. Applied rewrites 36.9%

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

   if 1.99999999999999992e-164 < (/.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 40.3%

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

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

   3. Applied rewrites 83.3%

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

   5. Applied rewrites 83.1%

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

   7. Applied rewrites 83.0%

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

   8. Taylor expanded in A around -inf

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

   10. Applied rewrites 83.0%

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

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

   1. Initial program 0.0%

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

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

   3. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 1.2

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 1.2

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

   5. Applied rewrites 1.2%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 31.8

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

   8. Applied rewrites 31.8%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 4: 49.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_2 := \mathsf{fma}\left(C \cdot 4, A, \left(-B\_m\right) \cdot B\_m\right)\\ t_3 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{-\sqrt{\left(2 \cdot \left(t\_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot C\right) \cdot F} \cdot \sqrt{t\_0}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq -4 \cdot 10^{-208}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(B\_m, A - C\right) + C\right) + A\right) \cdot \left(\left(F + F\right) \cdot t\_1\right)}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-164}:\\ \;\;\;\;\frac{\sqrt{4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(-2 \cdot C - 2 \cdot C\right)\right)\right)\right)}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{-\sqrt{C + C}}{t\_0} \cdot \left(\sqrt{2} \cdot \sqrt{F \cdot t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\sqrt{\frac{F}{B\_m}} \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* A C) -4.0 (* B_m B_m)))
        (t_1 (fma -4.0 (* C A) (* B_m B_m)))
        (t_2 (fma (* C 4.0) A (* (- B_m) B_m)))
        (t_3 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_4
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_3 F))
             (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0)))))))
          t_3)))
   (if (<= t_4 (- INFINITY))
     (/ (* (sqrt (* (* 4.0 C) F)) (sqrt t_0)) t_2)
     (if (<= t_4 -4e-208)
       (/ (sqrt (* (+ (+ (hypot B_m (- A C)) C) A) (* (+ F F) t_1))) t_2)
       (if (<= t_4 2e-164)
         (/ (sqrt (* 4.0 (* A (* C (* F (- (* -2.0 C) (* 2.0 C))))))) t_2)
         (if (<= t_4 INFINITY)
           (* (/ (- (sqrt (+ C C))) t_0) (* (sqrt 2.0) (sqrt (* F t_1))))
           (* -1.0 (* (sqrt (/ F B_m)) (sqrt 2.0)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((A * C), -4.0, (B_m * B_m));
	double t_1 = fma(-4.0, (C * A), (B_m * B_m));
	double t_2 = fma((C * 4.0), A, (-B_m * B_m));
	double t_3 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_4 = -sqrt(((2.0 * (t_3 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_3;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = (sqrt(((4.0 * C) * F)) * sqrt(t_0)) / t_2;
	} else if (t_4 <= -4e-208) {
		tmp = sqrt((((hypot(B_m, (A - C)) + C) + A) * ((F + F) * t_1))) / t_2;
	} else if (t_4 <= 2e-164) {
		tmp = sqrt((4.0 * (A * (C * (F * ((-2.0 * C) - (2.0 * C))))))) / t_2;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (-sqrt((C + C)) / t_0) * (sqrt(2.0) * sqrt((F * t_1)));
	} else {
		tmp = -1.0 * (sqrt((F / B_m)) * sqrt(2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(A * C), -4.0, Float64(B_m * B_m))
	t_1 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_2 = fma(Float64(C * 4.0), A, Float64(Float64(-B_m) * B_m))
	t_3 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_4 = Float64(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_3)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(Float64(4.0 * C) * F)) * sqrt(t_0)) / t_2);
	elseif (t_4 <= -4e-208)
		tmp = Float64(sqrt(Float64(Float64(Float64(hypot(B_m, Float64(A - C)) + C) + A) * Float64(Float64(F + F) * t_1))) / t_2);
	elseif (t_4 <= 2e-164)
		tmp = Float64(sqrt(Float64(4.0 * Float64(A * Float64(C * Float64(F * Float64(Float64(-2.0 * C) - Float64(2.0 * C))))))) / t_2);
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(Float64(-sqrt(Float64(C + C))) / t_0) * Float64(sqrt(2.0) * sqrt(Float64(F * t_1))));
	else
		tmp = Float64(-1.0 * Float64(sqrt(Float64(F / B_m)) * sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(C * 4.0), $MachinePrecision] * A + N[((-B$95$m) * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[Sqrt[N[(N[(4.0 * C), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, -4e-208], N[(N[Sqrt[N[(N[(N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + C), $MachinePrecision] + A), $MachinePrecision] * N[(N[(F + F), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, 2e-164], N[(N[Sqrt[N[(4.0 * N[(A * N[(C * N[(F * N[(N[(-2.0 * C), $MachinePrecision] - N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[((-N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.2%

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

   3. Applied rewrites 19.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. sqrt-prod N/A

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

      6. pow1/2 N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 44.5%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 47.3

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

   8. Applied rewrites 47.3%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.0000000000000004e-208

   1. Initial program 97.8%

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

   3. Applied rewrites 97.8%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 97.8

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

   5. Applied rewrites 97.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 97.8

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

      8. lift-hypot.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 97.8

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

   7. Applied rewrites 97.8%

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

   if -4.0000000000000004e-208 < (/.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))) < 1.99999999999999992e-164

   1. Initial program 5.2%

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

   3. Applied rewrites 11.6%

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

   5. Applied rewrites 4.1%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 36.9

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

   8. Applied rewrites 36.9%

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

   if 1.99999999999999992e-164 < (/.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 40.3%

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

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

   3. Applied rewrites 83.3%

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

   5. Applied rewrites 83.1%

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

   7. Applied rewrites 83.0%

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

   8. Taylor expanded in A around -inf

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

   10. Applied rewrites 83.0%

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

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

   1. Initial program 0.0%

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

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

   3. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 1.2

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 1.2

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

   5. Applied rewrites 1.2%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 31.8

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

   8. Applied rewrites 31.8%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 5: 55.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)\\ t_1 := \sqrt{t\_0}\\ t_2 := \sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C}\\ t_3 := \left(-B\_m\right) \cdot B\_m\\ t_4 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_5 := \frac{-\sqrt{\left(2 \cdot \left(t\_4 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_4}\\ \mathbf{if}\;t\_5 \leq -4 \cdot 10^{-208}:\\ \;\;\;\;\sqrt{F \cdot 2} \cdot \left(t\_1 \cdot \frac{-t\_2}{t\_0}\right)\\ \mathbf{elif}\;t\_5 \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-1, \frac{{B\_m}^{2}}{A}, 4 \cdot C\right) \cdot F} \cdot t\_1}{\mathsf{fma}\left(C \cdot 4, A, t\_3\right)}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;t\_2 \cdot \frac{\sqrt{\left(t\_0 \cdot 2\right) \cdot F}}{\mathsf{fma}\left(4 \cdot C, A, t\_3\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\sqrt{\frac{F}{B\_m}} \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* A C) -4.0 (* B_m B_m)))
        (t_1 (sqrt t_0))
        (t_2 (sqrt (+ (+ (hypot (- A C) B_m) A) C)))
        (t_3 (* (- B_m) B_m))
        (t_4 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_5
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_4 F))
             (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0)))))))
          t_4)))
   (if (<= t_5 -4e-208)
     (* (sqrt (* F 2.0)) (* t_1 (/ (- t_2) t_0)))
     (if (<= t_5 0.0)
       (/
        (* (sqrt (* (fma -1.0 (/ (pow B_m 2.0) A) (* 4.0 C)) F)) t_1)
        (fma (* C 4.0) A t_3))
       (if (<= t_5 INFINITY)
         (* t_2 (/ (sqrt (* (* t_0 2.0) F)) (fma (* 4.0 C) A t_3)))
         (* -1.0 (* (sqrt (/ F B_m)) (sqrt 2.0))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((A * C), -4.0, (B_m * B_m));
	double t_1 = sqrt(t_0);
	double t_2 = sqrt(((hypot((A - C), B_m) + A) + C));
	double t_3 = -B_m * B_m;
	double t_4 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_5 = -sqrt(((2.0 * (t_4 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_4;
	double tmp;
	if (t_5 <= -4e-208) {
		tmp = sqrt((F * 2.0)) * (t_1 * (-t_2 / t_0));
	} else if (t_5 <= 0.0) {
		tmp = (sqrt((fma(-1.0, (pow(B_m, 2.0) / A), (4.0 * C)) * F)) * t_1) / fma((C * 4.0), A, t_3);
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = t_2 * (sqrt(((t_0 * 2.0) * F)) / fma((4.0 * C), A, t_3));
	} else {
		tmp = -1.0 * (sqrt((F / B_m)) * sqrt(2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(A * C), -4.0, Float64(B_m * B_m))
	t_1 = sqrt(t_0)
	t_2 = sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C))
	t_3 = Float64(Float64(-B_m) * B_m)
	t_4 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_5 = Float64(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_4)
	tmp = 0.0
	if (t_5 <= -4e-208)
		tmp = Float64(sqrt(Float64(F * 2.0)) * Float64(t_1 * Float64(Float64(-t_2) / t_0)));
	elseif (t_5 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(fma(-1.0, Float64((B_m ^ 2.0) / A), Float64(4.0 * C)) * F)) * t_1) / fma(Float64(C * 4.0), A, t_3));
	elseif (t_5 <= Inf)
		tmp = Float64(t_2 * Float64(sqrt(Float64(Float64(t_0 * 2.0) * F)) / fma(Float64(4.0 * C), A, t_3)));
	else
		tmp = Float64(-1.0 * Float64(sqrt(Float64(F / B_m)) * sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[((-B$95$m) * B$95$m), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$4 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$4), $MachinePrecision]}, If[LessEqual[t$95$5, -4e-208], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[((-t$95$2) / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.0], N[(N[(N[Sqrt[N[(N[(-1.0 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision] + N[(4.0 * C), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[(C * 4.0), $MachinePrecision] * A + t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(t$95$2 * N[(N[Sqrt[N[(N[(t$95$0 * 2.0), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * C), $MachinePrecision] * A + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.0000000000000004e-208

   1. Initial program 43.0%

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

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

   3. Applied rewrites 66.8%

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

   5. Applied rewrites 81.2%

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

   if -4.0000000000000004e-208 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 4.5%

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

   3. Applied rewrites 10.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. sqrt-prod N/A

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

      6. pow1/2 N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 17.6%

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

   6. Taylor expanded in A around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 44.5

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

   8. Applied rewrites 44.5%

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

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

   1. Initial program 40.9%

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

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

   3. Applied rewrites 83.4%

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

   5. Applied rewrites 83.2%

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

   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. 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

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

   3. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 1.2

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 1.2

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

   5. Applied rewrites 1.2%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 31.8

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

   8. Applied rewrites 31.8%

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

Alternative 6: 54.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)\\ t_1 := \sqrt{t\_0}\\ t_2 := \left(-B\_m\right) \cdot B\_m\\ t_3 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{-\sqrt{\left(2 \cdot \left(t\_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -4 \cdot 10^{-208}:\\ \;\;\;\;\sqrt{F \cdot 2} \cdot \left(t\_1 \cdot \frac{-\sqrt{\left(\mathsf{hypot}\left(B\_m, A - C\right) + C\right) + A}}{t\_0}\right)\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-1, \frac{{B\_m}^{2}}{A}, 4 \cdot C\right) \cdot F} \cdot t\_1}{\mathsf{fma}\left(C \cdot 4, A, t\_2\right)}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C} \cdot \frac{\sqrt{\left(t\_0 \cdot 2\right) \cdot F}}{\mathsf{fma}\left(4 \cdot C, A, t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\sqrt{\frac{F}{B\_m}} \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* A C) -4.0 (* B_m B_m)))
        (t_1 (sqrt t_0))
        (t_2 (* (- B_m) B_m))
        (t_3 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_4
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_3 F))
             (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0)))))))
          t_3)))
   (if (<= t_4 -4e-208)
     (*
      (sqrt (* F 2.0))
      (* t_1 (/ (- (sqrt (+ (+ (hypot B_m (- A C)) C) A))) t_0)))
     (if (<= t_4 0.0)
       (/
        (* (sqrt (* (fma -1.0 (/ (pow B_m 2.0) A) (* 4.0 C)) F)) t_1)
        (fma (* C 4.0) A t_2))
       (if (<= t_4 INFINITY)
         (*
          (sqrt (+ (+ (hypot (- A C) B_m) A) C))
          (/ (sqrt (* (* t_0 2.0) F)) (fma (* 4.0 C) A t_2)))
         (* -1.0 (* (sqrt (/ F B_m)) (sqrt 2.0))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((A * C), -4.0, (B_m * B_m));
	double t_1 = sqrt(t_0);
	double t_2 = -B_m * B_m;
	double t_3 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_4 = -sqrt(((2.0 * (t_3 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_3;
	double tmp;
	if (t_4 <= -4e-208) {
		tmp = sqrt((F * 2.0)) * (t_1 * (-sqrt(((hypot(B_m, (A - C)) + C) + A)) / t_0));
	} else if (t_4 <= 0.0) {
		tmp = (sqrt((fma(-1.0, (pow(B_m, 2.0) / A), (4.0 * C)) * F)) * t_1) / fma((C * 4.0), A, t_2);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt(((hypot((A - C), B_m) + A) + C)) * (sqrt(((t_0 * 2.0) * F)) / fma((4.0 * C), A, t_2));
	} else {
		tmp = -1.0 * (sqrt((F / B_m)) * sqrt(2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(A * C), -4.0, Float64(B_m * B_m))
	t_1 = sqrt(t_0)
	t_2 = Float64(Float64(-B_m) * B_m)
	t_3 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_4 = Float64(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_3)
	tmp = 0.0
	if (t_4 <= -4e-208)
		tmp = Float64(sqrt(Float64(F * 2.0)) * Float64(t_1 * Float64(Float64(-sqrt(Float64(Float64(hypot(B_m, Float64(A - C)) + C) + A))) / t_0)));
	elseif (t_4 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(fma(-1.0, Float64((B_m ^ 2.0) / A), Float64(4.0 * C)) * F)) * t_1) / fma(Float64(C * 4.0), A, t_2));
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)) * Float64(sqrt(Float64(Float64(t_0 * 2.0) * F)) / fma(Float64(4.0 * C), A, t_2)));
	else
		tmp = Float64(-1.0 * Float64(sqrt(Float64(F / B_m)) * sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[((-B$95$m) * B$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, -4e-208], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[((-N[Sqrt[N[(N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + C), $MachinePrecision] + A), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(N[(N[Sqrt[N[(N[(-1.0 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision] + N[(4.0 * C), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[(C * 4.0), $MachinePrecision] * A + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(N[(t$95$0 * 2.0), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * C), $MachinePrecision] * A + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.0000000000000004e-208

   1. Initial program 43.0%

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

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

   3. Applied rewrites 66.8%

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

   5. Applied rewrites 81.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 79.1

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

      8. lift-hypot.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 79.1

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

   7. Applied rewrites 79.1%

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

   if -4.0000000000000004e-208 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 4.5%

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

   3. Applied rewrites 10.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. sqrt-prod N/A

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

      6. pow1/2 N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 17.6%

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

   6. Taylor expanded in A around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 44.5

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

   8. Applied rewrites 44.5%

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

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

   1. Initial program 40.9%

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

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

   3. Applied rewrites 83.4%

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

   5. Applied rewrites 83.2%

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

   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. 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

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

   3. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 1.2

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 1.2

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

   5. Applied rewrites 1.2%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 31.8

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

   8. Applied rewrites 31.8%

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

Alternative 7: 54.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)\\ t_1 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{-\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_1}\\ t_3 := \left(-B\_m\right) \cdot B\_m\\ t_4 := \mathsf{fma}\left(4 \cdot C, A, t\_3\right)\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{-208}:\\ \;\;\;\;\sqrt{2 \cdot \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)} \cdot \left(\sqrt{F} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(B\_m, A - C\right) + A\right) + C}}{t\_4}\right)\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-1, \frac{{B\_m}^{2}}{A}, 4 \cdot C\right) \cdot F} \cdot \sqrt{t\_0}}{\mathsf{fma}\left(C \cdot 4, A, t\_3\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C} \cdot \frac{\sqrt{\left(t\_0 \cdot 2\right) \cdot F}}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\sqrt{\frac{F}{B\_m}} \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* A C) -4.0 (* B_m B_m)))
        (t_1 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_2
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_1 F))
             (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0)))))))
          t_1))
        (t_3 (* (- B_m) B_m))
        (t_4 (fma (* 4.0 C) A t_3)))
   (if (<= t_2 -4e-208)
     (*
      (sqrt (* 2.0 (fma -4.0 (* C A) (* B_m B_m))))
      (* (sqrt F) (/ (sqrt (+ (+ (hypot B_m (- A C)) A) C)) t_4)))
     (if (<= t_2 0.0)
       (/
        (* (sqrt (* (fma -1.0 (/ (pow B_m 2.0) A) (* 4.0 C)) F)) (sqrt t_0))
        (fma (* C 4.0) A t_3))
       (if (<= t_2 INFINITY)
         (*
          (sqrt (+ (+ (hypot (- A C) B_m) A) C))
          (/ (sqrt (* (* t_0 2.0) F)) t_4))
         (* -1.0 (* (sqrt (/ F B_m)) (sqrt 2.0))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((A * C), -4.0, (B_m * B_m));
	double t_1 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_2 = -sqrt(((2.0 * (t_1 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_1;
	double t_3 = -B_m * B_m;
	double t_4 = fma((4.0 * C), A, t_3);
	double tmp;
	if (t_2 <= -4e-208) {
		tmp = sqrt((2.0 * fma(-4.0, (C * A), (B_m * B_m)))) * (sqrt(F) * (sqrt(((hypot(B_m, (A - C)) + A) + C)) / t_4));
	} else if (t_2 <= 0.0) {
		tmp = (sqrt((fma(-1.0, (pow(B_m, 2.0) / A), (4.0 * C)) * F)) * sqrt(t_0)) / fma((C * 4.0), A, t_3);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((hypot((A - C), B_m) + A) + C)) * (sqrt(((t_0 * 2.0) * F)) / t_4);
	} else {
		tmp = -1.0 * (sqrt((F / B_m)) * sqrt(2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(A * C), -4.0, Float64(B_m * B_m))
	t_1 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_2 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))))))) / t_1)
	t_3 = Float64(Float64(-B_m) * B_m)
	t_4 = fma(Float64(4.0 * C), A, t_3)
	tmp = 0.0
	if (t_2 <= -4e-208)
		tmp = Float64(sqrt(Float64(2.0 * fma(-4.0, Float64(C * A), Float64(B_m * B_m)))) * Float64(sqrt(F) * Float64(sqrt(Float64(Float64(hypot(B_m, Float64(A - C)) + A) + C)) / t_4)));
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(fma(-1.0, Float64((B_m ^ 2.0) / A), Float64(4.0 * C)) * F)) * sqrt(t_0)) / fma(Float64(C * 4.0), A, t_3));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)) * Float64(sqrt(Float64(Float64(t_0 * 2.0) * F)) / t_4));
	else
		tmp = Float64(-1.0 * Float64(sqrt(Float64(F / B_m)) * sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[((-B$95$m) * B$95$m), $MachinePrecision]}, Block[{t$95$4 = N[(N[(4.0 * C), $MachinePrecision] * A + t$95$3), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-208], N[(N[Sqrt[N[(2.0 * N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[(N[Sqrt[N[(N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(N[Sqrt[N[(N[(-1.0 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision] + N[(4.0 * C), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(N[(C * 4.0), $MachinePrecision] * A + t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(N[(t$95$0 * 2.0), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.0000000000000004e-208

   1. Initial program 43.0%

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

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

   3. Applied rewrites 66.8%

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

   5. Applied rewrites 66.7%

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

   7. Applied rewrites 78.8%

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

   if -4.0000000000000004e-208 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 4.5%

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

   3. Applied rewrites 10.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. sqrt-prod N/A

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

      6. pow1/2 N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 17.6%

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

   6. Taylor expanded in A around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 44.5

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

   8. Applied rewrites 44.5%

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

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

   1. Initial program 40.9%

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

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

   3. Applied rewrites 83.4%

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

   5. Applied rewrites 83.2%

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

   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. 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

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

   3. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 1.2

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 1.2

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

   5. Applied rewrites 1.2%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 31.8

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

   8. Applied rewrites 31.8%

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

Alternative 8: 41.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(C \cdot 4, A, \left(-B\_m\right) \cdot B\_m\right)\\ t_2 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{-\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_2}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+100}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot C\right) \cdot F} \cdot \sqrt{t\_0}}{t\_1}\\ \mathbf{elif}\;t\_3 \leq -4 \cdot 10^{-208}:\\ \;\;\;\;\frac{-\sqrt{B\_m}}{t\_0} \cdot \left(\sqrt{2} \cdot \sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(-2 \cdot C - 2 \cdot C\right)\right)\right)\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\sqrt{\frac{F}{B\_m}} \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* A C) -4.0 (* B_m B_m)))
        (t_1 (fma (* C 4.0) A (* (- B_m) B_m)))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_2 F))
             (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0)))))))
          t_2)))
   (if (<= t_3 -2e+100)
     (/ (* (sqrt (* (* 4.0 C) F)) (sqrt t_0)) t_1)
     (if (<= t_3 -4e-208)
       (*
        (/ (- (sqrt B_m)) t_0)
        (* (sqrt 2.0) (sqrt (* F (fma -4.0 (* C A) (* B_m B_m))))))
       (if (<= t_3 INFINITY)
         (/ (sqrt (* 4.0 (* A (* C (* F (- (* -2.0 C) (* 2.0 C))))))) t_1)
         (* -1.0 (* (sqrt (/ F B_m)) (sqrt 2.0))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((A * C), -4.0, (B_m * B_m));
	double t_1 = fma((C * 4.0), A, (-B_m * B_m));
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_3 = -sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_2;
	double tmp;
	if (t_3 <= -2e+100) {
		tmp = (sqrt(((4.0 * C) * F)) * sqrt(t_0)) / t_1;
	} else if (t_3 <= -4e-208) {
		tmp = (-sqrt(B_m) / t_0) * (sqrt(2.0) * sqrt((F * fma(-4.0, (C * A), (B_m * B_m)))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((4.0 * (A * (C * (F * ((-2.0 * C) - (2.0 * C))))))) / t_1;
	} else {
		tmp = -1.0 * (sqrt((F / B_m)) * sqrt(2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(A * C), -4.0, Float64(B_m * B_m))
	t_1 = fma(Float64(C * 4.0), A, Float64(Float64(-B_m) * B_m))
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(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_2)
	tmp = 0.0
	if (t_3 <= -2e+100)
		tmp = Float64(Float64(sqrt(Float64(Float64(4.0 * C) * F)) * sqrt(t_0)) / t_1);
	elseif (t_3 <= -4e-208)
		tmp = Float64(Float64(Float64(-sqrt(B_m)) / t_0) * Float64(sqrt(2.0) * sqrt(Float64(F * fma(-4.0, Float64(C * A), Float64(B_m * B_m))))));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(4.0 * Float64(A * Float64(C * Float64(F * Float64(Float64(-2.0 * C) - Float64(2.0 * C))))))) / t_1);
	else
		tmp = Float64(-1.0 * Float64(sqrt(Float64(F / B_m)) * sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(C * 4.0), $MachinePrecision] * A + N[((-B$95$m) * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+100], N[(N[(N[Sqrt[N[(N[(4.0 * C), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, -4e-208], N[(N[((-N[Sqrt[B$95$m], $MachinePrecision]) / t$95$0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(4.0 * N[(A * N[(C * N[(F * N[(N[(-2.0 * C), $MachinePrecision] - N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], N[(-1.0 * N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -2.00000000000000003e100

   1. Initial program 16.7%

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

   3. Applied rewrites 30.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. sqrt-prod N/A

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

      6. pow1/2 N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 49.3

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

   8. Applied rewrites 49.3%

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

   if -2.00000000000000003e100 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.0000000000000004e-208

   1. Initial program 97.8%

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

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

   3. Applied rewrites 98.1%

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

   5. Applied rewrites 98.0%

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

   7. Applied rewrites 97.8%

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

   8. Taylor expanded in B around inf

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

   10. Applied rewrites 64.5%

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

   if -4.0000000000000004e-208 < (/.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 19.2%

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

   3. Applied rewrites 31.5%

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 40.8

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

   8. Applied rewrites 40.8%

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

   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. 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

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

   3. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 1.2

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 1.2

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

   5. Applied rewrites 1.2%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 31.8

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

   8. Applied rewrites 31.8%

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

Alternative 9: 41.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)}\\ t_1 := \mathsf{fma}\left(C \cdot 4, A, \left(-B\_m\right) \cdot B\_m\right)\\ t_2 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{-\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_2}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+100}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot C\right) \cdot F} \cdot t\_0}{t\_1}\\ \mathbf{elif}\;t\_3 \leq -4 \cdot 10^{-208}:\\ \;\;\;\;\frac{\sqrt{\left(B\_m \cdot 2\right) \cdot F} \cdot t\_0}{t\_1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(-2 \cdot C - 2 \cdot C\right)\right)\right)\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\sqrt{\frac{F}{B\_m}} \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* A C) -4.0 (* B_m B_m))))
        (t_1 (fma (* C 4.0) A (* (- B_m) B_m)))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_2 F))
             (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0)))))))
          t_2)))
   (if (<= t_3 -2e+100)
     (/ (* (sqrt (* (* 4.0 C) F)) t_0) t_1)
     (if (<= t_3 -4e-208)
       (/ (* (sqrt (* (* B_m 2.0) F)) t_0) t_1)
       (if (<= t_3 INFINITY)
         (/ (sqrt (* 4.0 (* A (* C (* F (- (* -2.0 C) (* 2.0 C))))))) t_1)
         (* -1.0 (* (sqrt (/ F B_m)) (sqrt 2.0))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = sqrt(fma((A * C), -4.0, (B_m * B_m)));
	double t_1 = fma((C * 4.0), A, (-B_m * B_m));
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_3 = -sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_2;
	double tmp;
	if (t_3 <= -2e+100) {
		tmp = (sqrt(((4.0 * C) * F)) * t_0) / t_1;
	} else if (t_3 <= -4e-208) {
		tmp = (sqrt(((B_m * 2.0) * F)) * t_0) / t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((4.0 * (A * (C * (F * ((-2.0 * C) - (2.0 * C))))))) / t_1;
	} else {
		tmp = -1.0 * (sqrt((F / B_m)) * sqrt(2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = sqrt(fma(Float64(A * C), -4.0, Float64(B_m * B_m)))
	t_1 = fma(Float64(C * 4.0), A, Float64(Float64(-B_m) * B_m))
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(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_2)
	tmp = 0.0
	if (t_3 <= -2e+100)
		tmp = Float64(Float64(sqrt(Float64(Float64(4.0 * C) * F)) * t_0) / t_1);
	elseif (t_3 <= -4e-208)
		tmp = Float64(Float64(sqrt(Float64(Float64(B_m * 2.0) * F)) * t_0) / t_1);
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(4.0 * Float64(A * Float64(C * Float64(F * Float64(Float64(-2.0 * C) - Float64(2.0 * C))))))) / t_1);
	else
		tmp = Float64(-1.0 * Float64(sqrt(Float64(F / B_m)) * sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(C * 4.0), $MachinePrecision] * A + N[((-B$95$m) * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+100], N[(N[(N[Sqrt[N[(N[(4.0 * C), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, -4e-208], N[(N[(N[Sqrt[N[(N[(B$95$m * 2.0), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(4.0 * N[(A * N[(C * N[(F * N[(N[(-2.0 * C), $MachinePrecision] - N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], N[(-1.0 * N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -2.00000000000000003e100

   1. Initial program 16.7%

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

   3. Applied rewrites 30.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. sqrt-prod N/A

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

      6. pow1/2 N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 49.3

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

   8. Applied rewrites 49.3%

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

   if -2.00000000000000003e100 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.0000000000000004e-208

   1. Initial program 97.8%

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

   3. Applied rewrites 97.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. sqrt-prod N/A

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

      6. pow1/2 N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in B around inf

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

   8. Applied rewrites 64.6%

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

   if -4.0000000000000004e-208 < (/.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 19.2%

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

   3. Applied rewrites 31.5%

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 40.8

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

   8. Applied rewrites 40.8%

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

   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. 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

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

   3. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 1.2

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 1.2

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

   5. Applied rewrites 1.2%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 31.8

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

   8. Applied rewrites 31.8%

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

Alternative 10: 54.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{-\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_1}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{-208}:\\ \;\;\;\;\sqrt{F \cdot 2} \cdot \frac{\sqrt{t\_0} \cdot \sqrt{\left(\mathsf{hypot}\left(B\_m, A - C\right) + A\right) + C}}{\mathsf{fma}\left(4 \cdot C, A, \left(-B\_m\right) \cdot B\_m\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{-\sqrt{\left(C + -0.5 \cdot \frac{{B\_m}^{2}}{A}\right) + C}}{\mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)} \cdot \left(\sqrt{2} \cdot \sqrt{F \cdot t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\sqrt{\frac{F}{B\_m}} \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_2
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_1 F))
             (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0)))))))
          t_1)))
   (if (<= t_2 -4e-208)
     (*
      (sqrt (* F 2.0))
      (/
       (* (sqrt t_0) (sqrt (+ (+ (hypot B_m (- A C)) A) C)))
       (fma (* 4.0 C) A (* (- B_m) B_m))))
     (if (<= t_2 INFINITY)
       (*
        (/
         (- (sqrt (+ (+ C (* -0.5 (/ (pow B_m 2.0) A))) C)))
         (fma (* A C) -4.0 (* B_m B_m)))
        (* (sqrt 2.0) (sqrt (* F t_0))))
       (* -1.0 (* (sqrt (/ F B_m)) (sqrt 2.0)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_2 = -sqrt(((2.0 * (t_1 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_1;
	double tmp;
	if (t_2 <= -4e-208) {
		tmp = sqrt((F * 2.0)) * ((sqrt(t_0) * sqrt(((hypot(B_m, (A - C)) + A) + C))) / fma((4.0 * C), A, (-B_m * B_m)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (-sqrt(((C + (-0.5 * (pow(B_m, 2.0) / A))) + C)) / fma((A * C), -4.0, (B_m * B_m))) * (sqrt(2.0) * sqrt((F * t_0)));
	} else {
		tmp = -1.0 * (sqrt((F / B_m)) * sqrt(2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_2 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))))))) / t_1)
	tmp = 0.0
	if (t_2 <= -4e-208)
		tmp = Float64(sqrt(Float64(F * 2.0)) * Float64(Float64(sqrt(t_0) * sqrt(Float64(Float64(hypot(B_m, Float64(A - C)) + A) + C))) / fma(Float64(4.0 * C), A, Float64(Float64(-B_m) * B_m))));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(Float64(-sqrt(Float64(Float64(C + Float64(-0.5 * Float64((B_m ^ 2.0) / A))) + C))) / fma(Float64(A * C), -4.0, Float64(B_m * B_m))) * Float64(sqrt(2.0) * sqrt(Float64(F * t_0))));
	else
		tmp = Float64(-1.0 * Float64(sqrt(Float64(F / B_m)) * sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-208], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Sqrt[N[(N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(4.0 * C), $MachinePrecision] * A + N[((-B$95$m) * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[((-N[Sqrt[N[(N[(C + N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision]) / N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 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))) < -4.0000000000000004e-208

   1. Initial program 43.0%

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

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

   3. Applied rewrites 66.8%

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

   5. Applied rewrites 81.2%

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

   7. Applied rewrites 81.2%

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

   if -4.0000000000000004e-208 < (/.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 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. 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

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

   3. Applied rewrites 40.5%

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

   5. Applied rewrites 40.4%

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

   7. Applied rewrites 40.3%

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

   8. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 59.7

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

   10. Applied rewrites 59.7%

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

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

   1. Initial program 0.0%

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

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

   3. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 1.2

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 1.2

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

   5. Applied rewrites 1.2%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 31.8

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

   8. Applied rewrites 31.8%

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

Alternative 11: 45.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-64}:\\ \;\;\;\;\frac{-\sqrt{C + C}}{\mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)} \cdot \left(\sqrt{2} \cdot \sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C\right) \cdot 2\right) \cdot F} \cdot B\_m}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\_m\right) \cdot B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\sqrt{\frac{F}{B\_m}} \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-64)
   (*
    (/ (- (sqrt (+ C C))) (fma (* A C) -4.0 (* B_m B_m)))
    (* (sqrt 2.0) (sqrt (* F (fma -4.0 (* C A) (* B_m B_m))))))
   (if (<= (pow B_m 2.0) 5e+305)
     (/
      (* (sqrt (* (* (+ (+ (hypot (- A C) B_m) A) C) 2.0) F)) B_m)
      (fma (* C 4.0) A (* (- B_m) B_m)))
     (* -1.0 (* (sqrt (/ F B_m)) (sqrt 2.0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-64) {
		tmp = (-sqrt((C + C)) / fma((A * C), -4.0, (B_m * B_m))) * (sqrt(2.0) * sqrt((F * fma(-4.0, (C * A), (B_m * B_m)))));
	} else if (pow(B_m, 2.0) <= 5e+305) {
		tmp = (sqrt(((((hypot((A - C), B_m) + A) + C) * 2.0) * F)) * B_m) / fma((C * 4.0), A, (-B_m * B_m));
	} else {
		tmp = -1.0 * (sqrt((F / B_m)) * sqrt(2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-64)
		tmp = Float64(Float64(Float64(-sqrt(Float64(C + C))) / fma(Float64(A * C), -4.0, Float64(B_m * B_m))) * Float64(sqrt(2.0) * sqrt(Float64(F * fma(-4.0, Float64(C * A), Float64(B_m * B_m))))));
	elseif ((B_m ^ 2.0) <= 5e+305)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C) * 2.0) * F)) * B_m) / fma(Float64(C * 4.0), A, Float64(Float64(-B_m) * B_m)));
	else
		tmp = Float64(-1.0 * Float64(sqrt(Float64(F / B_m)) * sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-64], N[(N[((-N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision]) / N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+305], N[(N[(N[Sqrt[N[(N[(N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * B$95$m), $MachinePrecision] / N[(N[(C * 4.0), $MachinePrecision] * A + N[((-B$95$m) * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999965e-65

   1. Initial program 22.2%

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

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

   3. Applied rewrites 38.9%

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

   5. Applied rewrites 38.9%

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

   7. Applied rewrites 38.8%

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

   8. Taylor expanded in A around -inf

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

   10. Applied rewrites 44.7%

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

   if 9.99999999999999965e-65 < (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000009e305

   1. Initial program 30.7%

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

   3. Applied rewrites 37.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. sqrt-prod N/A

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

      6. pow1/2 N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 47.8%

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

   6. Taylor expanded in A around 0

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

   8. Applied rewrites 43.9%

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

   if 5.00000000000000009e305 < (pow.f64 B #s(literal 2 binary64))

   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. 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

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

   3. Applied rewrites 0.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 1.5

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 1.5

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

   5. Applied rewrites 1.5%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 49.9

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

   8. Applied rewrites 49.9%

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

Alternative 12: 43.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{-\sqrt{C + C}}{t\_0} \cdot \left(\sqrt{2} \cdot \sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{\left(\sqrt{B\_m \cdot F} \cdot \sqrt{2}\right) \cdot \sqrt{t\_0}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\_m\right) \cdot B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\sqrt{\frac{F}{B\_m}} \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* A C) -4.0 (* B_m B_m))))
   (if (<= (pow B_m 2.0) 5e+15)
     (*
      (/ (- (sqrt (+ C C))) t_0)
      (* (sqrt 2.0) (sqrt (* F (fma -4.0 (* C A) (* B_m B_m))))))
     (if (<= (pow B_m 2.0) 5e+305)
       (/
        (* (* (sqrt (* B_m F)) (sqrt 2.0)) (sqrt t_0))
        (fma (* C 4.0) A (* (- B_m) B_m)))
       (* -1.0 (* (sqrt (/ F B_m)) (sqrt 2.0)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((A * C), -4.0, (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 5e+15) {
		tmp = (-sqrt((C + C)) / t_0) * (sqrt(2.0) * sqrt((F * fma(-4.0, (C * A), (B_m * B_m)))));
	} else if (pow(B_m, 2.0) <= 5e+305) {
		tmp = ((sqrt((B_m * F)) * sqrt(2.0)) * sqrt(t_0)) / fma((C * 4.0), A, (-B_m * B_m));
	} else {
		tmp = -1.0 * (sqrt((F / B_m)) * sqrt(2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(A * C), -4.0, Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+15)
		tmp = Float64(Float64(Float64(-sqrt(Float64(C + C))) / t_0) * Float64(sqrt(2.0) * sqrt(Float64(F * fma(-4.0, Float64(C * A), Float64(B_m * B_m))))));
	elseif ((B_m ^ 2.0) <= 5e+305)
		tmp = Float64(Float64(Float64(sqrt(Float64(B_m * F)) * sqrt(2.0)) * sqrt(t_0)) / fma(Float64(C * 4.0), A, Float64(Float64(-B_m) * B_m)));
	else
		tmp = Float64(-1.0 * Float64(sqrt(Float64(F / B_m)) * sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+15], N[(N[((-N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+305], N[(N[(N[(N[Sqrt[N[(B$95$m * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(N[(C * 4.0), $MachinePrecision] * A + N[((-B$95$m) * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5e15

   1. Initial program 23.9%

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

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

   3. Applied rewrites 40.5%

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

   5. Applied rewrites 40.5%

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

   7. Applied rewrites 40.4%

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

   8. Taylor expanded in A around -inf

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

   10. Applied rewrites 43.8%

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

   if 5e15 < (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000009e305

   1. Initial program 29.4%

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

   3. Applied rewrites 35.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. sqrt-prod N/A

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

      6. pow1/2 N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 49.6%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 35.6

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

   8. Applied rewrites 35.6%

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

   if 5.00000000000000009e305 < (pow.f64 B #s(literal 2 binary64))

   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. 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

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

   3. Applied rewrites 0.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 1.5

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 1.5

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

   5. Applied rewrites 1.5%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 49.9

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

   8. Applied rewrites 49.9%

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

Alternative 13: 48.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 8.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{-\sqrt{C + C}}{t\_0} \cdot \left(\sqrt{2} \cdot \sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}\right)\\ \mathbf{elif}\;B\_m \leq 1.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C}}{t\_0} \cdot \left(\sqrt{2} \cdot \left(B\_m \cdot \sqrt{F}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\sqrt{\frac{F}{B\_m}} \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* A C) -4.0 (* B_m B_m))))
   (if (<= B_m 8.5e-31)
     (*
      (/ (- (sqrt (+ C C))) t_0)
      (* (sqrt 2.0) (sqrt (* F (fma -4.0 (* C A) (* B_m B_m))))))
     (if (<= B_m 1.5e+153)
       (*
        (/ (- (sqrt (+ (+ (hypot (- A C) B_m) A) C))) t_0)
        (* (sqrt 2.0) (* B_m (sqrt F))))
       (* -1.0 (* (sqrt (/ F B_m)) (sqrt 2.0)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((A * C), -4.0, (B_m * B_m));
	double tmp;
	if (B_m <= 8.5e-31) {
		tmp = (-sqrt((C + C)) / t_0) * (sqrt(2.0) * sqrt((F * fma(-4.0, (C * A), (B_m * B_m)))));
	} else if (B_m <= 1.5e+153) {
		tmp = (-sqrt(((hypot((A - C), B_m) + A) + C)) / t_0) * (sqrt(2.0) * (B_m * sqrt(F)));
	} else {
		tmp = -1.0 * (sqrt((F / B_m)) * sqrt(2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(A * C), -4.0, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 8.5e-31)
		tmp = Float64(Float64(Float64(-sqrt(Float64(C + C))) / t_0) * Float64(sqrt(2.0) * sqrt(Float64(F * fma(-4.0, Float64(C * A), Float64(B_m * B_m))))));
	elseif (B_m <= 1.5e+153)
		tmp = Float64(Float64(Float64(-sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C))) / t_0) * Float64(sqrt(2.0) * Float64(B_m * sqrt(F))));
	else
		tmp = Float64(-1.0 * Float64(sqrt(Float64(F / B_m)) * sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 8.5e-31], N[(N[((-N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.5e+153], N[(N[((-N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(B$95$m * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 8.5000000000000007e-31

   1. Initial program 22.3%

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

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

   3. Applied rewrites 38.9%

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

   5. Applied rewrites 38.9%

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

   7. Applied rewrites 38.8%

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

   8. Taylor expanded in A around -inf

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

   10. Applied rewrites 44.6%

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

   if 8.5000000000000007e-31 < B < 1.50000000000000009e153

   1. Initial program 30.7%

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

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

   3. Applied rewrites 49.3%

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

   5. Applied rewrites 49.2%

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

   7. Applied rewrites 49.2%

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

   8. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 53.4

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

   10. Applied rewrites 53.4%

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

   if 1.50000000000000009e153 < 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. 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

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

   3. Applied rewrites 0.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 1.5

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 1.5

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

   5. Applied rewrites 1.5%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 49.8

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

   8. Applied rewrites 49.8%

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

Alternative 14: 48.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 8.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{-\sqrt{C + C}}{t\_0} \cdot \left(\sqrt{2} \cdot \sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}\right)\\ \mathbf{elif}\;B\_m \leq 1.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C}}{t\_0} \cdot \left(\left(B\_m \cdot \sqrt{2}\right) \cdot \sqrt{F}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\sqrt{\frac{F}{B\_m}} \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* A C) -4.0 (* B_m B_m))))
   (if (<= B_m 8.5e-31)
     (*
      (/ (- (sqrt (+ C C))) t_0)
      (* (sqrt 2.0) (sqrt (* F (fma -4.0 (* C A) (* B_m B_m))))))
     (if (<= B_m 1.5e+153)
       (*
        (/ (- (sqrt (+ (+ (hypot (- A C) B_m) A) C))) t_0)
        (* (* B_m (sqrt 2.0)) (sqrt F)))
       (* -1.0 (* (sqrt (/ F B_m)) (sqrt 2.0)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((A * C), -4.0, (B_m * B_m));
	double tmp;
	if (B_m <= 8.5e-31) {
		tmp = (-sqrt((C + C)) / t_0) * (sqrt(2.0) * sqrt((F * fma(-4.0, (C * A), (B_m * B_m)))));
	} else if (B_m <= 1.5e+153) {
		tmp = (-sqrt(((hypot((A - C), B_m) + A) + C)) / t_0) * ((B_m * sqrt(2.0)) * sqrt(F));
	} else {
		tmp = -1.0 * (sqrt((F / B_m)) * sqrt(2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(A * C), -4.0, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 8.5e-31)
		tmp = Float64(Float64(Float64(-sqrt(Float64(C + C))) / t_0) * Float64(sqrt(2.0) * sqrt(Float64(F * fma(-4.0, Float64(C * A), Float64(B_m * B_m))))));
	elseif (B_m <= 1.5e+153)
		tmp = Float64(Float64(Float64(-sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C))) / t_0) * Float64(Float64(B_m * sqrt(2.0)) * sqrt(F)));
	else
		tmp = Float64(-1.0 * Float64(sqrt(Float64(F / B_m)) * sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 8.5e-31], N[(N[((-N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.5e+153], N[(N[((-N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision] * N[(N[(B$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 8.5000000000000007e-31

   1. Initial program 22.3%

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

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

   3. Applied rewrites 38.9%

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

   5. Applied rewrites 38.9%

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

   7. Applied rewrites 38.8%

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

   8. Taylor expanded in A around -inf

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

   10. Applied rewrites 44.6%

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

   if 8.5000000000000007e-31 < B < 1.50000000000000009e153

   1. Initial program 30.7%

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

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

   3. Applied rewrites 49.3%

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

   5. Applied rewrites 49.2%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \sqrt{\left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot 2\right) \cdot F}} \]

   7. Applied rewrites 49.2%

      \[\leadsto \frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)} \]

   8. Taylor expanded in A around 0

      \[\leadsto \frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \color{blue}{\left(\left(B \cdot \sqrt{2}\right) \cdot \sqrt{F}\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \left(\left(B \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{F}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \left(\left(B \cdot \sqrt{2}\right) \cdot \sqrt{\color{blue}{F}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \left(\left(B \cdot \sqrt{2}\right) \cdot \sqrt{F}\right) \]

      4. lower-sqrt.f64 53.4

         \[\leadsto \frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \left(\left(B \cdot \sqrt{2}\right) \cdot \sqrt{F}\right) \]

   10. Applied rewrites 53.4%

       \[\leadsto \frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \color{blue}{\left(\left(B \cdot \sqrt{2}\right) \cdot \sqrt{F}\right)} \]

   if 1.50000000000000009e153 < 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. 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

         \[\leadsto \frac{\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}^{\frac{1}{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied rewrites 0.2%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \color{blue}{\left(-4 \cdot \left(C \cdot A\right) + B \cdot B\right)}} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(2 \cdot F\right) \cdot \left(-4 \cdot \left(C \cdot A\right)\right) + \left(2 \cdot F\right) \cdot \left(B \cdot B\right)}} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \left(-4 \cdot \color{blue}{\left(C \cdot A\right)}\right) + \left(2 \cdot F\right) \cdot \left(B \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \color{blue}{\left(\left(-4 \cdot C\right) \cdot A\right)} + \left(2 \cdot F\right) \cdot \left(B \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(2 \cdot F\right) \cdot \left(-4 \cdot C\right)\right) \cdot A} + \left(2 \cdot F\right) \cdot \left(B \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\left(2 \cdot F\right) \cdot \left(-4 \cdot C\right), A, \left(2 \cdot F\right) \cdot \left(B \cdot B\right)\right)}} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(2 \cdot F\right) \cdot \left(-4 \cdot C\right)}, A, \left(2 \cdot F\right) \cdot \left(B \cdot B\right)\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(2 \cdot F\right)} \cdot \left(-4 \cdot C\right), A, \left(2 \cdot F\right) \cdot \left(B \cdot B\right)\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(F \cdot 2\right)} \cdot \left(-4 \cdot C\right), A, \left(2 \cdot F\right) \cdot \left(B \cdot B\right)\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(F \cdot 2\right)} \cdot \left(-4 \cdot C\right), A, \left(2 \cdot F\right) \cdot \left(B \cdot B\right)\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      12. *-commutative N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \color{blue}{\left(C \cdot -4\right)}, A, \left(2 \cdot F\right) \cdot \left(B \cdot B\right)\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \color{blue}{\left(C \cdot -4\right)}, A, \left(2 \cdot F\right) \cdot \left(B \cdot B\right)\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \left(2 \cdot F\right) \cdot \color{blue}{\left(B \cdot B\right)}\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      15. associate-*r* N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \color{blue}{\left(\left(2 \cdot F\right) \cdot B\right) \cdot B}\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \color{blue}{\left(\left(2 \cdot F\right) \cdot B\right) \cdot B}\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      17. lower-*.f64 1.5

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \color{blue}{\left(\left(2 \cdot F\right) \cdot B\right)} \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \left(\color{blue}{\left(2 \cdot F\right)} \cdot B\right) \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      19. *-commutative N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \left(\color{blue}{\left(F \cdot 2\right)} \cdot B\right) \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      20. lower-*.f64 1.5

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \left(\color{blue}{\left(F \cdot 2\right)} \cdot B\right) \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 1.5%

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \left(\left(F \cdot 2\right) \cdot B\right) \cdot B\right)}} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \color{blue}{\sqrt{2}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{\color{blue}{2}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) \]

      5. lower-sqrt.f64 49.8

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) \]

   8. Applied rewrites 49.8%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 38.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot 4, A, \left(-B\_m\right) \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 7 \cdot 10^{-279}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot C\right) \cdot F} \cdot \sqrt{\mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)}}{t\_0}\\ \mathbf{elif}\;B\_m \leq 7.7 \cdot 10^{-31}:\\ \;\;\;\;\frac{\sqrt{4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(-2 \cdot C - 2 \cdot C\right)\right)\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\sqrt{\frac{F}{B\_m}} \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* C 4.0) A (* (- B_m) B_m))))
   (if (<= B_m 7e-279)
     (/ (* (sqrt (* (* 4.0 C) F)) (sqrt (fma (* A C) -4.0 (* B_m B_m)))) t_0)
     (if (<= B_m 7.7e-31)
       (/ (sqrt (* 4.0 (* A (* C (* F (- (* -2.0 C) (* 2.0 C))))))) t_0)
       (* -1.0 (* (sqrt (/ F B_m)) (sqrt 2.0)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((C * 4.0), A, (-B_m * B_m));
	double tmp;
	if (B_m <= 7e-279) {
		tmp = (sqrt(((4.0 * C) * F)) * sqrt(fma((A * C), -4.0, (B_m * B_m)))) / t_0;
	} else if (B_m <= 7.7e-31) {
		tmp = sqrt((4.0 * (A * (C * (F * ((-2.0 * C) - (2.0 * C))))))) / t_0;
	} else {
		tmp = -1.0 * (sqrt((F / B_m)) * sqrt(2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(C * 4.0), A, Float64(Float64(-B_m) * B_m))
	tmp = 0.0
	if (B_m <= 7e-279)
		tmp = Float64(Float64(sqrt(Float64(Float64(4.0 * C) * F)) * sqrt(fma(Float64(A * C), -4.0, Float64(B_m * B_m)))) / t_0);
	elseif (B_m <= 7.7e-31)
		tmp = Float64(sqrt(Float64(4.0 * Float64(A * Float64(C * Float64(F * Float64(Float64(-2.0 * C) - Float64(2.0 * C))))))) / t_0);
	else
		tmp = Float64(-1.0 * Float64(sqrt(Float64(F / B_m)) * sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(C * 4.0), $MachinePrecision] * A + N[((-B$95$m) * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 7e-279], N[(N[(N[Sqrt[N[(N[(4.0 * C), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 7.7e-31], N[(N[Sqrt[N[(4.0 * N[(A * N[(C * N[(F * N[(N[(-2.0 * C), $MachinePrecision] - N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(-1.0 * N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 7.00000000000000019e-279

   1. Initial program 18.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied rewrites 29.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)}} \]
   4. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \color{blue}{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(2 \cdot F\right)\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      5. sqrt-prod N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(2 \cdot F\right)} \cdot \sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      6. pow1/2 N/A

         \[\leadsto \frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(2 \cdot F\right)} \cdot \color{blue}{{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(2 \cdot F\right)} \cdot {\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

   5. Applied rewrites 17.4%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot 2\right) \cdot F} \cdot \sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

   6. Taylor expanded in A around -inf

      \[\leadsto \frac{\sqrt{\color{blue}{\left(4 \cdot C\right)} \cdot F} \cdot \sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 28.5

         \[\leadsto \frac{\sqrt{\left(4 \cdot \color{blue}{C}\right) \cdot F} \cdot \sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

   8. Applied rewrites 28.5%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(4 \cdot C\right)} \cdot F} \cdot \sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

   if 7.00000000000000019e-279 < B < 7.70000000000000012e-31

   1. Initial program 22.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied rewrites 33.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)}} \]

   5. Applied rewrites 4.9%

      \[\leadsto \frac{\sqrt{\color{blue}{\frac{\left({\left(A + C\right)}^{2} - {\left(\mathsf{hypot}\left(A - C, B\right)\right)}^{2}\right) \cdot \left(\left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot 2\right) \cdot F\right)}{\left(A + C\right) - \mathsf{hypot}\left(A - C, B\right)}}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

   6. Taylor expanded in A around -inf

      \[\leadsto \frac{\sqrt{\color{blue}{4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(-2 \cdot C - 2 \cdot C\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{4 \cdot \color{blue}{\left(A \cdot \left(C \cdot \left(F \cdot \left(-2 \cdot C - 2 \cdot C\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{4 \cdot \left(A \cdot \color{blue}{\left(C \cdot \left(F \cdot \left(-2 \cdot C - 2 \cdot C\right)\right)\right)}\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{4 \cdot \left(A \cdot \left(C \cdot \color{blue}{\left(F \cdot \left(-2 \cdot C - 2 \cdot C\right)\right)}\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \color{blue}{\left(-2 \cdot C - 2 \cdot C\right)}\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sqrt{4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(-2 \cdot C - \color{blue}{2 \cdot C}\right)\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(-2 \cdot C - \color{blue}{2} \cdot C\right)\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      7. lower-*.f64 34.9

         \[\leadsto \frac{\sqrt{4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(-2 \cdot C - 2 \cdot \color{blue}{C}\right)\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

   8. Applied rewrites 34.9%

      \[\leadsto \frac{\sqrt{\color{blue}{4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(-2 \cdot C - 2 \cdot C\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

   if 7.70000000000000012e-31 < B

   1. Initial program 16.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. 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

         \[\leadsto \frac{\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}^{\frac{1}{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied rewrites 26.0%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \color{blue}{\left(-4 \cdot \left(C \cdot A\right) + B \cdot B\right)}} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(2 \cdot F\right) \cdot \left(-4 \cdot \left(C \cdot A\right)\right) + \left(2 \cdot F\right) \cdot \left(B \cdot B\right)}} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \left(-4 \cdot \color{blue}{\left(C \cdot A\right)}\right) + \left(2 \cdot F\right) \cdot \left(B \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \color{blue}{\left(\left(-4 \cdot C\right) \cdot A\right)} + \left(2 \cdot F\right) \cdot \left(B \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(2 \cdot F\right) \cdot \left(-4 \cdot C\right)\right) \cdot A} + \left(2 \cdot F\right) \cdot \left(B \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\left(2 \cdot F\right) \cdot \left(-4 \cdot C\right), A, \left(2 \cdot F\right) \cdot \left(B \cdot B\right)\right)}} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(2 \cdot F\right) \cdot \left(-4 \cdot C\right)}, A, \left(2 \cdot F\right) \cdot \left(B \cdot B\right)\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(2 \cdot F\right)} \cdot \left(-4 \cdot C\right), A, \left(2 \cdot F\right) \cdot \left(B \cdot B\right)\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(F \cdot 2\right)} \cdot \left(-4 \cdot C\right), A, \left(2 \cdot F\right) \cdot \left(B \cdot B\right)\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(F \cdot 2\right)} \cdot \left(-4 \cdot C\right), A, \left(2 \cdot F\right) \cdot \left(B \cdot B\right)\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      12. *-commutative N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \color{blue}{\left(C \cdot -4\right)}, A, \left(2 \cdot F\right) \cdot \left(B \cdot B\right)\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \color{blue}{\left(C \cdot -4\right)}, A, \left(2 \cdot F\right) \cdot \left(B \cdot B\right)\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \left(2 \cdot F\right) \cdot \color{blue}{\left(B \cdot B\right)}\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      15. associate-*r* N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \color{blue}{\left(\left(2 \cdot F\right) \cdot B\right) \cdot B}\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \color{blue}{\left(\left(2 \cdot F\right) \cdot B\right) \cdot B}\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      17. lower-*.f64 25.5

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \color{blue}{\left(\left(2 \cdot F\right) \cdot B\right)} \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \left(\color{blue}{\left(2 \cdot F\right)} \cdot B\right) \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      19. *-commutative N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \left(\color{blue}{\left(F \cdot 2\right)} \cdot B\right) \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      20. lower-*.f64 25.5

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \left(\color{blue}{\left(F \cdot 2\right)} \cdot B\right) \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 25.5%

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \left(\left(F \cdot 2\right) \cdot B\right) \cdot B\right)}} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \color{blue}{\sqrt{2}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{\color{blue}{2}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) \]

      5. lower-sqrt.f64 42.4

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) \]

   8. Applied rewrites 42.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 39.0% accurate, 6.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 7.7 \cdot 10^{-31}:\\ \;\;\;\;\frac{\sqrt{4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(-2 \cdot C - 2 \cdot C\right)\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\_m\right) \cdot B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\sqrt{\frac{F}{B\_m}} \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 7.7e-31)
   (/
    (sqrt (* 4.0 (* A (* C (* F (- (* -2.0 C) (* 2.0 C)))))))
    (fma (* C 4.0) A (* (- B_m) B_m)))
   (* -1.0 (* (sqrt (/ F B_m)) (sqrt 2.0)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 7.7e-31) {
		tmp = sqrt((4.0 * (A * (C * (F * ((-2.0 * C) - (2.0 * C))))))) / fma((C * 4.0), A, (-B_m * B_m));
	} else {
		tmp = -1.0 * (sqrt((F / B_m)) * sqrt(2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 7.7e-31)
		tmp = Float64(sqrt(Float64(4.0 * Float64(A * Float64(C * Float64(F * Float64(Float64(-2.0 * C) - Float64(2.0 * C))))))) / fma(Float64(C * 4.0), A, Float64(Float64(-B_m) * B_m)));
	else
		tmp = Float64(-1.0 * Float64(sqrt(Float64(F / B_m)) * sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 7.7e-31], N[(N[Sqrt[N[(4.0 * N[(A * N[(C * N[(F * N[(N[(-2.0 * C), $MachinePrecision] - N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(C * 4.0), $MachinePrecision] * A + N[((-B$95$m) * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 7.70000000000000012e-31

   1. Initial program 22.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied rewrites 32.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)}} \]

   5. Applied rewrites 4.5%

      \[\leadsto \frac{\sqrt{\color{blue}{\frac{\left({\left(A + C\right)}^{2} - {\left(\mathsf{hypot}\left(A - C, B\right)\right)}^{2}\right) \cdot \left(\left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot 2\right) \cdot F\right)}{\left(A + C\right) - \mathsf{hypot}\left(A - C, B\right)}}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

   6. Taylor expanded in A around -inf

      \[\leadsto \frac{\sqrt{\color{blue}{4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(-2 \cdot C - 2 \cdot C\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{4 \cdot \color{blue}{\left(A \cdot \left(C \cdot \left(F \cdot \left(-2 \cdot C - 2 \cdot C\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{4 \cdot \left(A \cdot \color{blue}{\left(C \cdot \left(F \cdot \left(-2 \cdot C - 2 \cdot C\right)\right)\right)}\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{4 \cdot \left(A \cdot \left(C \cdot \color{blue}{\left(F \cdot \left(-2 \cdot C - 2 \cdot C\right)\right)}\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \color{blue}{\left(-2 \cdot C - 2 \cdot C\right)}\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sqrt{4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(-2 \cdot C - \color{blue}{2 \cdot C}\right)\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(-2 \cdot C - \color{blue}{2} \cdot C\right)\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      7. lower-*.f64 34.9

         \[\leadsto \frac{\sqrt{4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(-2 \cdot C - 2 \cdot \color{blue}{C}\right)\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

   8. Applied rewrites 34.9%

      \[\leadsto \frac{\sqrt{\color{blue}{4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(-2 \cdot C - 2 \cdot C\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

   if 7.70000000000000012e-31 < B

   1. Initial program 16.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. 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

         \[\leadsto \frac{\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}^{\frac{1}{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied rewrites 26.0%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \color{blue}{\left(-4 \cdot \left(C \cdot A\right) + B \cdot B\right)}} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(2 \cdot F\right) \cdot \left(-4 \cdot \left(C \cdot A\right)\right) + \left(2 \cdot F\right) \cdot \left(B \cdot B\right)}} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \left(-4 \cdot \color{blue}{\left(C \cdot A\right)}\right) + \left(2 \cdot F\right) \cdot \left(B \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \color{blue}{\left(\left(-4 \cdot C\right) \cdot A\right)} + \left(2 \cdot F\right) \cdot \left(B \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(2 \cdot F\right) \cdot \left(-4 \cdot C\right)\right) \cdot A} + \left(2 \cdot F\right) \cdot \left(B \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\left(2 \cdot F\right) \cdot \left(-4 \cdot C\right), A, \left(2 \cdot F\right) \cdot \left(B \cdot B\right)\right)}} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(2 \cdot F\right) \cdot \left(-4 \cdot C\right)}, A, \left(2 \cdot F\right) \cdot \left(B \cdot B\right)\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(2 \cdot F\right)} \cdot \left(-4 \cdot C\right), A, \left(2 \cdot F\right) \cdot \left(B \cdot B\right)\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(F \cdot 2\right)} \cdot \left(-4 \cdot C\right), A, \left(2 \cdot F\right) \cdot \left(B \cdot B\right)\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(F \cdot 2\right)} \cdot \left(-4 \cdot C\right), A, \left(2 \cdot F\right) \cdot \left(B \cdot B\right)\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      12. *-commutative N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \color{blue}{\left(C \cdot -4\right)}, A, \left(2 \cdot F\right) \cdot \left(B \cdot B\right)\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \color{blue}{\left(C \cdot -4\right)}, A, \left(2 \cdot F\right) \cdot \left(B \cdot B\right)\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \left(2 \cdot F\right) \cdot \color{blue}{\left(B \cdot B\right)}\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      15. associate-*r* N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \color{blue}{\left(\left(2 \cdot F\right) \cdot B\right) \cdot B}\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \color{blue}{\left(\left(2 \cdot F\right) \cdot B\right) \cdot B}\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      17. lower-*.f64 25.5

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \color{blue}{\left(\left(2 \cdot F\right) \cdot B\right)} \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \left(\color{blue}{\left(2 \cdot F\right)} \cdot B\right) \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      19. *-commutative N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \left(\color{blue}{\left(F \cdot 2\right)} \cdot B\right) \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      20. lower-*.f64 25.5

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \left(\color{blue}{\left(F \cdot 2\right)} \cdot B\right) \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 25.5%

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \left(\left(F \cdot 2\right) \cdot B\right) \cdot B\right)}} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \color{blue}{\sqrt{2}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{\color{blue}{2}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) \]

      5. lower-sqrt.f64 42.4

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) \]

   8. Applied rewrites 42.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 27.2% accurate, 11.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -1 \cdot \left(\sqrt{\frac{F}{B\_m}} \cdot \sqrt{2}\right) \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (* -1.0 (* (sqrt (/ F B_m)) (sqrt 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 -1.0 * (sqrt((F / B_m)) * sqrt(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 = (-1.0d0) * (sqrt((f / b_m)) * sqrt(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 -1.0 * (Math.sqrt((F / B_m)) * Math.sqrt(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 -1.0 * (math.sqrt((F / B_m)) * math.sqrt(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(-1.0 * Float64(sqrt(Float64(F / B_m)) * sqrt(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 = -1.0 * (sqrt((F / B_m)) * sqrt(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[(-1.0 * N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.9%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. 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

      \[\leadsto \frac{\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}^{\frac{1}{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

3. Applied rewrites 31.8%

   \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\sqrt{\color{blue}{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. lift-fma.f64 N/A

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \color{blue}{\left(-4 \cdot \left(C \cdot A\right) + B \cdot B\right)}} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. distribute-lft-in N/A

      \[\leadsto \frac{\sqrt{\color{blue}{\left(2 \cdot F\right) \cdot \left(-4 \cdot \left(C \cdot A\right)\right) + \left(2 \cdot F\right) \cdot \left(B \cdot B\right)}} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \left(-4 \cdot \color{blue}{\left(C \cdot A\right)}\right) + \left(2 \cdot F\right) \cdot \left(B \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. associate-*r* N/A

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \color{blue}{\left(\left(-4 \cdot C\right) \cdot A\right)} + \left(2 \cdot F\right) \cdot \left(B \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. associate-*r* N/A

      \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(2 \cdot F\right) \cdot \left(-4 \cdot C\right)\right) \cdot A} + \left(2 \cdot F\right) \cdot \left(B \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\left(2 \cdot F\right) \cdot \left(-4 \cdot C\right), A, \left(2 \cdot F\right) \cdot \left(B \cdot B\right)\right)}} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(2 \cdot F\right) \cdot \left(-4 \cdot C\right)}, A, \left(2 \cdot F\right) \cdot \left(B \cdot B\right)\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(2 \cdot F\right)} \cdot \left(-4 \cdot C\right), A, \left(2 \cdot F\right) \cdot \left(B \cdot B\right)\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   10. *-commutative N/A

       \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(F \cdot 2\right)} \cdot \left(-4 \cdot C\right), A, \left(2 \cdot F\right) \cdot \left(B \cdot B\right)\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(F \cdot 2\right)} \cdot \left(-4 \cdot C\right), A, \left(2 \cdot F\right) \cdot \left(B \cdot B\right)\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   12. *-commutative N/A

       \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \color{blue}{\left(C \cdot -4\right)}, A, \left(2 \cdot F\right) \cdot \left(B \cdot B\right)\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \color{blue}{\left(C \cdot -4\right)}, A, \left(2 \cdot F\right) \cdot \left(B \cdot B\right)\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \left(2 \cdot F\right) \cdot \color{blue}{\left(B \cdot B\right)}\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   15. associate-*r* N/A

       \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \color{blue}{\left(\left(2 \cdot F\right) \cdot B\right) \cdot B}\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \color{blue}{\left(\left(2 \cdot F\right) \cdot B\right) \cdot B}\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   17. lower-*.f64 29.6

       \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \color{blue}{\left(\left(2 \cdot F\right) \cdot B\right)} \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   18. lift-*.f64 N/A

       \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \left(\color{blue}{\left(2 \cdot F\right)} \cdot B\right) \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   19. *-commutative N/A

       \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \left(\color{blue}{\left(F \cdot 2\right)} \cdot B\right) \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   20. lower-*.f64 29.6

       \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \left(\color{blue}{\left(F \cdot 2\right)} \cdot B\right) \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

5. Applied rewrites 29.6%

   \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\left(F \cdot 2\right) \cdot \left(C \cdot -4\right), A, \left(\left(F \cdot 2\right) \cdot B\right) \cdot B\right)}} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

6. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \color{blue}{\sqrt{2}}\right) \]

   3. lower-sqrt.f64 N/A

      \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{\color{blue}{2}}\right) \]

   4. lower-/.f64 N/A

      \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) \]

   5. lower-sqrt.f64 27.2

      \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) \]

8. Applied rewrites 27.2%

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
9. Add Preprocessing

Alternative 18: 5.3% accurate, 11.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -1 \cdot \left(\sqrt{\frac{1}{B\_m}} \cdot \sqrt{2}\right) \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (* -1.0 (* (sqrt (/ 1.0 B_m)) (sqrt 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 -1.0 * (sqrt((1.0 / B_m)) * sqrt(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 = (-1.0d0) * (sqrt((1.0d0 / b_m)) * sqrt(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 -1.0 * (Math.sqrt((1.0 / B_m)) * Math.sqrt(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 -1.0 * (math.sqrt((1.0 / B_m)) * math.sqrt(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(-1.0 * Float64(sqrt(Float64(1.0 / B_m)) * sqrt(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 = -1.0 * (sqrt((1.0 / B_m)) * sqrt(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[(-1.0 * N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.9%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

3. Applied rewrites 25.7%

   \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)}} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\color{blue}{\left(2 \cdot F\right)} \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

   2. count-2-rev N/A

      \[\leadsto \frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\color{blue}{\left(F + F\right)} \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

   3. lower-+.f64 25.7

      \[\leadsto \frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\color{blue}{\left(F + F\right)} \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

5. Applied rewrites 25.7%

   \[\leadsto \frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\color{blue}{\left(F + F\right)} \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

7. Applied rewrites 2.8%

   \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(\left(\mathsf{hypot}\left(B, A - C\right) + A\right) + C\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot 2}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

8. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{1}{B}} \cdot \sqrt{2}\right)} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{1}{B}} \cdot \sqrt{2}\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto -1 \cdot \left(\sqrt{\frac{1}{B}} \cdot \color{blue}{\sqrt{2}}\right) \]

   3. lower-sqrt.f64 N/A

      \[\leadsto -1 \cdot \left(\sqrt{\frac{1}{B}} \cdot \sqrt{\color{blue}{2}}\right) \]

   4. lower-/.f64 N/A

      \[\leadsto -1 \cdot \left(\sqrt{\frac{1}{B}} \cdot \sqrt{2}\right) \]

   5. lower-sqrt.f64 5.3

      \[\leadsto -1 \cdot \left(\sqrt{\frac{1}{B}} \cdot \sqrt{2}\right) \]

10. Applied rewrites 5.3%

    \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{1}{B}} \cdot \sqrt{2}\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025111 
(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))))