ABCF->ab-angle b

Percentage Accurate: 18.3% → 35.3%

Time: 9.5s

Alternatives: 12

Speedup: 7.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.3% 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: 35.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)\\ \mathbf{if}\;C \leq 5.4 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(-2 \cdot \left(\left(\left(\mathsf{hypot}\left(B, A - C\right) - C\right) - A\right) \cdot F\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(A - \mathsf{fma}\left(-1, A, 0.5 \cdot \frac{{B}^{2}}{C}\right)\right) \cdot \left(\left(F + F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma (* C 4.0) A (* (- B) B))))
   (if (<= C 5.4e-49)
     (/
      (sqrt
       (*
        (fma (* C A) -4.0 (* B B))
        (* -2.0 (* (- (- (hypot B (- A C)) C) A) F))))
      t_0)
     (/
      (sqrt
       (*
        (- A (fma -1.0 A (* 0.5 (/ (pow B 2.0) C))))
        (* (+ F F) (fma -4.0 (* C A) (* B B)))))
      t_0))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma((C * 4.0), A, (-B * B));
	double tmp;
	if (C <= 5.4e-49) {
		tmp = sqrt((fma((C * A), -4.0, (B * B)) * (-2.0 * (((hypot(B, (A - C)) - C) - A) * F)))) / t_0;
	} else {
		tmp = sqrt(((A - fma(-1.0, A, (0.5 * (pow(B, 2.0) / C)))) * ((F + F) * fma(-4.0, (C * A), (B * B))))) / t_0;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(Float64(C * 4.0), A, Float64(Float64(-B) * B))
	tmp = 0.0
	if (C <= 5.4e-49)
		tmp = Float64(sqrt(Float64(fma(Float64(C * A), -4.0, Float64(B * B)) * Float64(-2.0 * Float64(Float64(Float64(hypot(B, Float64(A - C)) - C) - A) * F)))) / t_0);
	else
		tmp = Float64(sqrt(Float64(Float64(A - fma(-1.0, A, Float64(0.5 * Float64((B ^ 2.0) / C)))) * Float64(Float64(F + F) * fma(-4.0, Float64(C * A), Float64(B * B))))) / t_0);
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(C * 4.0), $MachinePrecision] * A + N[((-B) * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, 5.4e-49], N[(N[Sqrt[N[(N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $MachinePrecision] * N[(-2.0 * N[(N[(N[(N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision] - A), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(N[(A - N[(-1.0 * A + N[(0.5 * N[(N[Power[B, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(F + F), $MachinePrecision] * N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if C < 5.3999999999999999e-49

   1. Initial program 28.3%

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

   3. Applied rewrites 36.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 36.7

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

   5. Applied rewrites 36.7%

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

      1. lift--.f64 N/A

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

      2. sub-negate2 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l- N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate2 N/A

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

      9. lower--.f64 36.9

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

   7. Applied rewrites 36.9%

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

   9. Applied rewrites 36.9%

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

   if 5.3999999999999999e-49 < C

   1. Initial program 8.2%

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

   3. Applied rewrites 9.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 9.7

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

   5. Applied rewrites 9.7%

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

      1. lift--.f64 N/A

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

      2. sub-negate2 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l- N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate2 N/A

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

      9. lower--.f64 13.4

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

   7. Applied rewrites 13.4%

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

   8. Taylor expanded in C around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 33.7

         \[\leadsto \frac{\sqrt{\left(A - \mathsf{fma}\left(-1, A, 0.5 \cdot \frac{{B}^{2}}{C}\right)\right) \cdot \left(\left(F + F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

   10. Applied rewrites 33.7%

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

Alternative 2: 33.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)\\ t_1 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{-\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_1}\\ t_3 := B \cdot B - \left(4 \cdot C\right) \cdot A\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot A\right) \cdot \left(\left(F + F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{t\_0}\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-191}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot \left(1 + \frac{C}{A}\right) - \sqrt{\mathsf{fma}\left(C - A, C - A, B \cdot B\right)}\right) \cdot \left(\left(2 \cdot t\_3\right) \cdot F\right)}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma (* C 4.0) A (* (- B) B)))
        (t_1 (- (pow B 2.0) (* (* 4.0 A) C)))
        (t_2
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_1 F))
             (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
          t_1))
        (t_3 (- (* B B) (* (* 4.0 C) A))))
   (if (<= t_2 (- INFINITY))
     (/ (sqrt (* (* 2.0 A) (* (+ F F) (fma -4.0 (* C A) (* B B))))) t_0)
     (if (<= t_2 -5e-191)
       (/
        (-
         (sqrt
          (*
           (- (* A (+ 1.0 (/ C A))) (sqrt (fma (- C A) (- C A) (* B B))))
           (* (* 2.0 t_3) F))))
        t_3)
       (/ (sqrt (* -8.0 (* A (* C (* F (- A (* -1.0 A))))))) t_0)))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma((C * 4.0), A, (-B * B));
	double t_1 = pow(B, 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, 2.0)))))) / t_1;
	double t_3 = (B * B) - ((4.0 * C) * A);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = sqrt(((2.0 * A) * ((F + F) * fma(-4.0, (C * A), (B * B))))) / t_0;
	} else if (t_2 <= -5e-191) {
		tmp = -sqrt((((A * (1.0 + (C / A))) - sqrt(fma((C - A), (C - A), (B * B)))) * ((2.0 * t_3) * F))) / t_3;
	} else {
		tmp = sqrt((-8.0 * (A * (C * (F * (A - (-1.0 * A))))))) / t_0;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(Float64(C * 4.0), A, Float64(Float64(-B) * B))
	t_1 = Float64((B ^ 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 ^ 2.0))))))) / t_1)
	t_3 = Float64(Float64(B * B) - Float64(Float64(4.0 * C) * A))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(Float64(2.0 * A) * Float64(Float64(F + F) * fma(-4.0, Float64(C * A), Float64(B * B))))) / t_0);
	elseif (t_2 <= -5e-191)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(A * Float64(1.0 + Float64(C / A))) - sqrt(fma(Float64(C - A), Float64(C - A), Float64(B * B)))) * Float64(Float64(2.0 * t_3) * F)))) / t_3);
	else
		tmp = Float64(sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A - Float64(-1.0 * A))))))) / t_0);
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(C * 4.0), $MachinePrecision] * A + N[((-B) * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B, 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, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(B * B), $MachinePrecision] - N[(N[(4.0 * C), $MachinePrecision] * A), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[Sqrt[N[(N[(2.0 * A), $MachinePrecision] * N[(N[(F + F), $MachinePrecision] * N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[t$95$2, -5e-191], N[((-N[Sqrt[N[(N[(N[(A * N[(1.0 + N[(C / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(N[(C - A), $MachinePrecision] * N[(C - A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * t$95$3), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision], N[(N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A - N[(-1.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $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))) < -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 17.8%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 17.8

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

   5. Applied rewrites 17.8%

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

      1. lift--.f64 N/A

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

      2. sub-negate2 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l- N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate2 N/A

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

      9. lower--.f64 20.1

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

   7. Applied rewrites 20.1%

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

   8. Taylor expanded in A around -inf

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

      1. lower-*.f64 30.9

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

   10. Applied rewrites 30.9%

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

   if -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))) < -5.0000000000000001e-191

   1. Initial program 97.2%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-square-pow2 N/A

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

      4. pow2 N/A

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

      5. flip-+ N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 67.3%

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

   5. Applied rewrites 67.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

   7. Applied rewrites 97.3%

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

   8. Taylor expanded in A around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 96.0

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

   10. Applied rewrites 96.0%

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

   if -5.0000000000000001e-191 < (/.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 6.7%

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

   3. Applied rewrites 9.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 9.9

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

   5. Applied rewrites 9.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate2 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l- N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate2 N/A

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

      9. lower--.f64 12.0

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

   7. Applied rewrites 12.0%

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

   8. Taylor expanded in C around inf

      \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \color{blue}{\left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \color{blue}{\left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \color{blue}{\left(F \cdot \left(A - -1 \cdot A\right)\right)}\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \color{blue}{\left(A - -1 \cdot A\right)}\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - \color{blue}{-1 \cdot A}\right)\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      6. lower-*.f64 21.6

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot \color{blue}{A}\right)\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

   10. Applied rewrites 21.6%

       \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 33.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)\\ t_1 := B \cdot B - \left(4 \cdot C\right) \cdot A\\ t_2 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{-\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_2}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot A\right) \cdot \left(\left(F + F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{t\_0}\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-191}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(C - A, C - A, B \cdot B\right)}\right) \cdot \left(\left(2 \cdot t\_1\right) \cdot F\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma (* C 4.0) A (* (- B) B)))
        (t_1 (- (* B B) (* (* 4.0 C) A)))
        (t_2 (- (pow B 2.0) (* (* 4.0 A) C)))
        (t_3
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_2 F))
             (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
          t_2)))
   (if (<= t_3 (- INFINITY))
     (/ (sqrt (* (* 2.0 A) (* (+ F F) (fma -4.0 (* C A) (* B B))))) t_0)
     (if (<= t_3 -5e-191)
       (/
        (-
         (sqrt
          (*
           (- (+ C A) (sqrt (fma (- C A) (- C A) (* B B))))
           (* (* 2.0 t_1) F))))
        t_1)
       (/ (sqrt (* -8.0 (* A (* C (* F (- A (* -1.0 A))))))) t_0)))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma((C * 4.0), A, (-B * B));
	double t_1 = (B * B) - ((4.0 * C) * A);
	double t_2 = pow(B, 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, 2.0)))))) / t_2;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = sqrt(((2.0 * A) * ((F + F) * fma(-4.0, (C * A), (B * B))))) / t_0;
	} else if (t_3 <= -5e-191) {
		tmp = -sqrt((((C + A) - sqrt(fma((C - A), (C - A), (B * B)))) * ((2.0 * t_1) * F))) / t_1;
	} else {
		tmp = sqrt((-8.0 * (A * (C * (F * (A - (-1.0 * A))))))) / t_0;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(Float64(C * 4.0), A, Float64(Float64(-B) * B))
	t_1 = Float64(Float64(B * B) - Float64(Float64(4.0 * C) * A))
	t_2 = Float64((B ^ 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 ^ 2.0))))))) / t_2)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(Float64(2.0 * A) * Float64(Float64(F + F) * fma(-4.0, Float64(C * A), Float64(B * B))))) / t_0);
	elseif (t_3 <= -5e-191)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(C + A) - sqrt(fma(Float64(C - A), Float64(C - A), Float64(B * B)))) * Float64(Float64(2.0 * t_1) * F)))) / t_1);
	else
		tmp = Float64(sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A - Float64(-1.0 * A))))))) / t_0);
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(C * 4.0), $MachinePrecision] * A + N[((-B) * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - N[(N[(4.0 * C), $MachinePrecision] * A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B, 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, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[Sqrt[N[(N[(2.0 * A), $MachinePrecision] * N[(N[(F + F), $MachinePrecision] * N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[t$95$3, -5e-191], N[((-N[Sqrt[N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(N[(C - A), $MachinePrecision] * N[(C - A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[(N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A - N[(-1.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $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))) < -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 17.8%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 17.8

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

   5. Applied rewrites 17.8%

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

      1. lift--.f64 N/A

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

      2. sub-negate2 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l- N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate2 N/A

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

      9. lower--.f64 20.1

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

   7. Applied rewrites 20.1%

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

   8. Taylor expanded in A around -inf

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

      1. lower-*.f64 30.9

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

   10. Applied rewrites 30.9%

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

   if -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))) < -5.0000000000000001e-191

   1. Initial program 97.2%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-square-pow2 N/A

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

      4. pow2 N/A

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

      5. flip-+ N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 67.3%

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

   5. Applied rewrites 67.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

   7. Applied rewrites 97.3%

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

   if -5.0000000000000001e-191 < (/.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 6.7%

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

   3. Applied rewrites 9.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 9.9

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

   5. Applied rewrites 9.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate2 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l- N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate2 N/A

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

      9. lower--.f64 12.0

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

   7. Applied rewrites 12.0%

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

   8. Taylor expanded in C around inf

      \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \color{blue}{\left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \color{blue}{\left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \color{blue}{\left(F \cdot \left(A - -1 \cdot A\right)\right)}\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \color{blue}{\left(A - -1 \cdot A\right)}\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - \color{blue}{-1 \cdot A}\right)\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      6. lower-*.f64 21.6

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot \color{blue}{A}\right)\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

   10. Applied rewrites 21.6%

       \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 33.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)\\ t_1 := B \cdot B - \left(4 \cdot C\right) \cdot A\\ t_2 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{-\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_2}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot A\right) \cdot \left(\left(F + F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{t\_0}\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-191}:\\ \;\;\;\;\frac{-\sqrt{\left(A - \sqrt{\mathsf{fma}\left(C - A, C - A, B \cdot B\right)}\right) \cdot \left(\left(2 \cdot t\_1\right) \cdot F\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma (* C 4.0) A (* (- B) B)))
        (t_1 (- (* B B) (* (* 4.0 C) A)))
        (t_2 (- (pow B 2.0) (* (* 4.0 A) C)))
        (t_3
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_2 F))
             (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
          t_2)))
   (if (<= t_3 (- INFINITY))
     (/ (sqrt (* (* 2.0 A) (* (+ F F) (fma -4.0 (* C A) (* B B))))) t_0)
     (if (<= t_3 -5e-191)
       (/
        (-
         (sqrt
          (* (- A (sqrt (fma (- C A) (- C A) (* B B)))) (* (* 2.0 t_1) F))))
        t_1)
       (/ (sqrt (* -8.0 (* A (* C (* F (- A (* -1.0 A))))))) t_0)))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma((C * 4.0), A, (-B * B));
	double t_1 = (B * B) - ((4.0 * C) * A);
	double t_2 = pow(B, 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, 2.0)))))) / t_2;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = sqrt(((2.0 * A) * ((F + F) * fma(-4.0, (C * A), (B * B))))) / t_0;
	} else if (t_3 <= -5e-191) {
		tmp = -sqrt(((A - sqrt(fma((C - A), (C - A), (B * B)))) * ((2.0 * t_1) * F))) / t_1;
	} else {
		tmp = sqrt((-8.0 * (A * (C * (F * (A - (-1.0 * A))))))) / t_0;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(Float64(C * 4.0), A, Float64(Float64(-B) * B))
	t_1 = Float64(Float64(B * B) - Float64(Float64(4.0 * C) * A))
	t_2 = Float64((B ^ 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 ^ 2.0))))))) / t_2)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(Float64(2.0 * A) * Float64(Float64(F + F) * fma(-4.0, Float64(C * A), Float64(B * B))))) / t_0);
	elseif (t_3 <= -5e-191)
		tmp = Float64(Float64(-sqrt(Float64(Float64(A - sqrt(fma(Float64(C - A), Float64(C - A), Float64(B * B)))) * Float64(Float64(2.0 * t_1) * F)))) / t_1);
	else
		tmp = Float64(sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A - Float64(-1.0 * A))))))) / t_0);
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(C * 4.0), $MachinePrecision] * A + N[((-B) * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - N[(N[(4.0 * C), $MachinePrecision] * A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B, 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, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[Sqrt[N[(N[(2.0 * A), $MachinePrecision] * N[(N[(F + F), $MachinePrecision] * N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[t$95$3, -5e-191], N[((-N[Sqrt[N[(N[(A - N[Sqrt[N[(N[(C - A), $MachinePrecision] * N[(C - A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[(N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A - N[(-1.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $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))) < -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 17.8%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 17.8

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

   5. Applied rewrites 17.8%

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

      1. lift--.f64 N/A

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

      2. sub-negate2 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l- N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate2 N/A

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

      9. lower--.f64 20.1

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

   7. Applied rewrites 20.1%

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

   8. Taylor expanded in A around -inf

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

      1. lower-*.f64 30.9

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

   10. Applied rewrites 30.9%

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

   if -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))) < -5.0000000000000001e-191

   1. Initial program 97.2%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-square-pow2 N/A

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

      4. pow2 N/A

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

      5. flip-+ N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 67.3%

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

   5. Applied rewrites 67.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

   7. Applied rewrites 97.3%

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

   8. Taylor expanded in A around inf

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

   10. Applied rewrites 94.5%

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

   if -5.0000000000000001e-191 < (/.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 6.7%

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

   3. Applied rewrites 9.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 9.9

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

   5. Applied rewrites 9.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate2 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l- N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate2 N/A

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

      9. lower--.f64 12.0

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

   7. Applied rewrites 12.0%

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

   8. Taylor expanded in C around inf

      \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \color{blue}{\left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \color{blue}{\left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \color{blue}{\left(F \cdot \left(A - -1 \cdot A\right)\right)}\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \color{blue}{\left(A - -1 \cdot A\right)}\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - \color{blue}{-1 \cdot A}\right)\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      6. lower-*.f64 21.6

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot \color{blue}{A}\right)\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

   10. Applied rewrites 21.6%

       \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 26.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \left(F + F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ t_1 := \mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)\\ t_2 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{-\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_2}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{+158}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot A\right) \cdot t\_0}}{t\_1}\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-191}:\\ \;\;\;\;\frac{\sqrt{\left(A - B\right) \cdot t\_0}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* (+ F F) (fma -4.0 (* C A) (* B B))))
        (t_1 (fma (* C 4.0) A (* (- B) B)))
        (t_2 (- (pow B 2.0) (* (* 4.0 A) C)))
        (t_3
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_2 F))
             (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
          t_2)))
   (if (<= t_3 -1e+158)
     (/ (sqrt (* (* 2.0 A) t_0)) t_1)
     (if (<= t_3 -5e-191)
       (/ (sqrt (* (- A B) t_0)) t_1)
       (/ (sqrt (* -8.0 (* A (* C (* F (- A (* -1.0 A))))))) t_1)))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = (F + F) * fma(-4.0, (C * A), (B * B));
	double t_1 = fma((C * 4.0), A, (-B * B));
	double t_2 = pow(B, 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, 2.0)))))) / t_2;
	double tmp;
	if (t_3 <= -1e+158) {
		tmp = sqrt(((2.0 * A) * t_0)) / t_1;
	} else if (t_3 <= -5e-191) {
		tmp = sqrt(((A - B) * t_0)) / t_1;
	} else {
		tmp = sqrt((-8.0 * (A * (C * (F * (A - (-1.0 * A))))))) / t_1;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64(Float64(F + F) * fma(-4.0, Float64(C * A), Float64(B * B)))
	t_1 = fma(Float64(C * 4.0), A, Float64(Float64(-B) * B))
	t_2 = Float64((B ^ 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 ^ 2.0))))))) / t_2)
	tmp = 0.0
	if (t_3 <= -1e+158)
		tmp = Float64(sqrt(Float64(Float64(2.0 * A) * t_0)) / t_1);
	elseif (t_3 <= -5e-191)
		tmp = Float64(sqrt(Float64(Float64(A - B) * t_0)) / t_1);
	else
		tmp = Float64(sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A - Float64(-1.0 * A))))))) / t_1);
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(F + F), $MachinePrecision] * N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(C * 4.0), $MachinePrecision] * A + N[((-B) * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B, 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, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, -1e+158], N[(N[Sqrt[N[(N[(2.0 * A), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, -5e-191], N[(N[Sqrt[N[(N[(A - B), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A - N[(-1.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $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))) < -9.99999999999999953e157

   1. Initial program 8.6%

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

   3. Applied rewrites 22.3%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 22.3

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

   5. Applied rewrites 22.3%

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

      1. lift--.f64 N/A

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

      2. sub-negate2 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l- N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate2 N/A

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

      9. lower--.f64 24.6

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

   7. Applied rewrites 24.6%

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

   8. Taylor expanded in A around -inf

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

      1. lower-*.f64 33.5

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

   10. Applied rewrites 33.5%

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

   if -9.99999999999999953e157 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.0000000000000001e-191

   1. Initial program 97.4%

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

   3. Applied rewrites 97.4%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 97.4

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

   5. Applied rewrites 97.4%

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

      1. lift--.f64 N/A

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

      2. sub-negate2 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l- N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate2 N/A

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

      9. lower--.f64 97.4

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

   7. Applied rewrites 97.4%

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

   8. Taylor expanded in B around inf

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

   10. Applied rewrites 37.7%

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

   if -5.0000000000000001e-191 < (/.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 6.7%

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

   3. Applied rewrites 9.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 9.9

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

   5. Applied rewrites 9.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate2 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l- N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate2 N/A

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

      9. lower--.f64 12.0

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

   7. Applied rewrites 12.0%

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

   8. Taylor expanded in C around inf

      \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \color{blue}{\left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \color{blue}{\left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \color{blue}{\left(F \cdot \left(A - -1 \cdot A\right)\right)}\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \color{blue}{\left(A - -1 \cdot A\right)}\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - \color{blue}{-1 \cdot A}\right)\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      6. lower-*.f64 21.6

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot \color{blue}{A}\right)\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

   10. Applied rewrites 21.6%

       \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 24.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)\\ t_1 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{-\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_1}\\ t_3 := \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}{t\_0}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+199}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-191}:\\ \;\;\;\;\frac{\sqrt{\left(A - B\right) \cdot \left(\left(F + F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma (* C 4.0) A (* (- B) B)))
        (t_1 (- (pow B 2.0) (* (* 4.0 A) C)))
        (t_2
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_1 F))
             (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
          t_1))
        (t_3 (/ (sqrt (* -8.0 (* A (* C (* F (- A (* -1.0 A))))))) t_0)))
   (if (<= t_2 -1e+199)
     t_3
     (if (<= t_2 -5e-191)
       (/ (sqrt (* (- A B) (* (+ F F) (fma -4.0 (* C A) (* B B))))) t_0)
       t_3))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma((C * 4.0), A, (-B * B));
	double t_1 = pow(B, 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, 2.0)))))) / t_1;
	double t_3 = sqrt((-8.0 * (A * (C * (F * (A - (-1.0 * A))))))) / t_0;
	double tmp;
	if (t_2 <= -1e+199) {
		tmp = t_3;
	} else if (t_2 <= -5e-191) {
		tmp = sqrt(((A - B) * ((F + F) * fma(-4.0, (C * A), (B * B))))) / t_0;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(Float64(C * 4.0), A, Float64(Float64(-B) * B))
	t_1 = Float64((B ^ 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 ^ 2.0))))))) / t_1)
	t_3 = Float64(sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A - Float64(-1.0 * A))))))) / t_0)
	tmp = 0.0
	if (t_2 <= -1e+199)
		tmp = t_3;
	elseif (t_2 <= -5e-191)
		tmp = Float64(sqrt(Float64(Float64(A - B) * Float64(Float64(F + F) * fma(-4.0, Float64(C * A), Float64(B * B))))) / t_0);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(C * 4.0), $MachinePrecision] * A + N[((-B) * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B, 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, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A - N[(-1.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+199], t$95$3, If[LessEqual[t$95$2, -5e-191], N[(N[Sqrt[N[(N[(A - B), $MachinePrecision] * N[(N[(F + F), $MachinePrecision] * N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.0000000000000001e199 or -5.0000000000000001e-191 < (/.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 6.5%

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

   3. Applied rewrites 12.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 12.2

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

   5. Applied rewrites 12.2%

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

      1. lift--.f64 N/A

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

      2. sub-negate2 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l- N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate2 N/A

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

      9. lower--.f64 14.4

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

   7. Applied rewrites 14.4%

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

   8. Taylor expanded in C around inf

      \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \color{blue}{\left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \color{blue}{\left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \color{blue}{\left(F \cdot \left(A - -1 \cdot A\right)\right)}\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \color{blue}{\left(A - -1 \cdot A\right)}\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - \color{blue}{-1 \cdot A}\right)\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      6. lower-*.f64 22.7

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot \color{blue}{A}\right)\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

   10. Applied rewrites 22.7%

       \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

   if -1.0000000000000001e199 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.0000000000000001e-191

   1. Initial program 97.3%

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

   3. Applied rewrites 97.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 97.2

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

   5. Applied rewrites 97.2%

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

      1. lift--.f64 N/A

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

      2. sub-negate2 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l- N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate2 N/A

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

      9. lower--.f64 97.3

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

   7. Applied rewrites 97.3%

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

   8. Taylor expanded in B around inf

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

   10. Applied rewrites 37.0%

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

Alternative 7: 31.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)\\ \mathbf{if}\;C \leq 3.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(-2 \cdot \left(\left(\left(\mathsf{hypot}\left(B, A - C\right) - C\right) - A\right) \cdot F\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma (* C 4.0) A (* (- B) B))))
   (if (<= C 3.6e-48)
     (/
      (sqrt
       (*
        (fma (* C A) -4.0 (* B B))
        (* -2.0 (* (- (- (hypot B (- A C)) C) A) F))))
      t_0)
     (/ (sqrt (* -8.0 (* A (* C (* F (- A (* -1.0 A))))))) t_0))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma((C * 4.0), A, (-B * B));
	double tmp;
	if (C <= 3.6e-48) {
		tmp = sqrt((fma((C * A), -4.0, (B * B)) * (-2.0 * (((hypot(B, (A - C)) - C) - A) * F)))) / t_0;
	} else {
		tmp = sqrt((-8.0 * (A * (C * (F * (A - (-1.0 * A))))))) / t_0;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(Float64(C * 4.0), A, Float64(Float64(-B) * B))
	tmp = 0.0
	if (C <= 3.6e-48)
		tmp = Float64(sqrt(Float64(fma(Float64(C * A), -4.0, Float64(B * B)) * Float64(-2.0 * Float64(Float64(Float64(hypot(B, Float64(A - C)) - C) - A) * F)))) / t_0);
	else
		tmp = Float64(sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A - Float64(-1.0 * A))))))) / t_0);
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(C * 4.0), $MachinePrecision] * A + N[((-B) * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, 3.6e-48], N[(N[Sqrt[N[(N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $MachinePrecision] * N[(-2.0 * N[(N[(N[(N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision] - A), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A - N[(-1.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if C < 3.6000000000000002e-48

   1. Initial program 28.3%

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

   3. Applied rewrites 36.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 36.7

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

   5. Applied rewrites 36.7%

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

      1. lift--.f64 N/A

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

      2. sub-negate2 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l- N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate2 N/A

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

      9. lower--.f64 36.9

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

   7. Applied rewrites 36.9%

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

   9. Applied rewrites 36.9%

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

   if 3.6000000000000002e-48 < C

   1. Initial program 8.2%

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

   3. Applied rewrites 9.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 9.6

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

   5. Applied rewrites 9.6%

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

      1. lift--.f64 N/A

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

      2. sub-negate2 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l- N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate2 N/A

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

      9. lower--.f64 13.3

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

   7. Applied rewrites 13.3%

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

   8. Taylor expanded in C around inf

      \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \color{blue}{\left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \color{blue}{\left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \color{blue}{\left(F \cdot \left(A - -1 \cdot A\right)\right)}\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \color{blue}{\left(A - -1 \cdot A\right)}\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - \color{blue}{-1 \cdot A}\right)\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      6. lower-*.f64 25.8

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot \color{blue}{A}\right)\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

   10. Applied rewrites 25.8%

       \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 31.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \left(-B\right) \cdot B\\ \mathbf{if}\;C \leq 2.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}}{\mathsf{fma}\left(4 \cdot C, A, t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, t\_0\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* (- B) B)))
   (if (<= C 2.4e-48)
     (/
      (sqrt
       (*
        (* (- (+ C A) (hypot B (- A C))) (* F 2.0))
        (fma (* -4.0 C) A (* B B))))
      (fma (* 4.0 C) A t_0))
     (/
      (sqrt (* -8.0 (* A (* C (* F (- A (* -1.0 A)))))))
      (fma (* C 4.0) A t_0)))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = -B * B;
	double tmp;
	if (C <= 2.4e-48) {
		tmp = sqrt(((((C + A) - hypot(B, (A - C))) * (F * 2.0)) * fma((-4.0 * C), A, (B * B)))) / fma((4.0 * C), A, t_0);
	} else {
		tmp = sqrt((-8.0 * (A * (C * (F * (A - (-1.0 * A))))))) / fma((C * 4.0), A, t_0);
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64(Float64(-B) * B)
	tmp = 0.0
	if (C <= 2.4e-48)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(C + A) - hypot(B, Float64(A - C))) * Float64(F * 2.0)) * fma(Float64(-4.0 * C), A, Float64(B * B)))) / fma(Float64(4.0 * C), A, t_0));
	else
		tmp = Float64(sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A - Float64(-1.0 * A))))))) / fma(Float64(C * 4.0), A, t_0));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[((-B) * B), $MachinePrecision]}, If[LessEqual[C, 2.4e-48], N[(N[Sqrt[N[(N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * C), $MachinePrecision] * A + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A - N[(-1.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(C * 4.0), $MachinePrecision] * A + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if C < 2.4e-48

   1. Initial program 28.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate2 N/A

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

      5. distribute-lft-neg-out N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 36.9%

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

   5. Applied rewrites 36.8%

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

   if 2.4e-48 < C

   1. Initial program 8.2%

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

   3. Applied rewrites 9.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 9.6

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

   5. Applied rewrites 9.6%

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

      1. lift--.f64 N/A

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

      2. sub-negate2 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l- N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate2 N/A

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

      9. lower--.f64 13.3

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

   7. Applied rewrites 13.3%

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

   8. Taylor expanded in C around inf

      \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \color{blue}{\left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \color{blue}{\left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \color{blue}{\left(F \cdot \left(A - -1 \cdot A\right)\right)}\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \color{blue}{\left(A - -1 \cdot A\right)}\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - \color{blue}{-1 \cdot A}\right)\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      6. lower-*.f64 25.8

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot \color{blue}{A}\right)\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

   10. Applied rewrites 25.8%

       \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 31.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)\\ \mathbf{if}\;C \leq 3.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{\left(A - \left(\mathsf{hypot}\left(A - C, B\right) - C\right)\right) \cdot \left(\left(F + F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma (* C 4.0) A (* (- B) B))))
   (if (<= C 3.6e-48)
     (/
      (sqrt
       (*
        (- A (- (hypot (- A C) B) C))
        (* (+ F F) (fma -4.0 (* C A) (* B B)))))
      t_0)
     (/ (sqrt (* -8.0 (* A (* C (* F (- A (* -1.0 A))))))) t_0))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma((C * 4.0), A, (-B * B));
	double tmp;
	if (C <= 3.6e-48) {
		tmp = sqrt(((A - (hypot((A - C), B) - C)) * ((F + F) * fma(-4.0, (C * A), (B * B))))) / t_0;
	} else {
		tmp = sqrt((-8.0 * (A * (C * (F * (A - (-1.0 * A))))))) / t_0;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(Float64(C * 4.0), A, Float64(Float64(-B) * B))
	tmp = 0.0
	if (C <= 3.6e-48)
		tmp = Float64(sqrt(Float64(Float64(A - Float64(hypot(Float64(A - C), B) - C)) * Float64(Float64(F + F) * fma(-4.0, Float64(C * A), Float64(B * B))))) / t_0);
	else
		tmp = Float64(sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A - Float64(-1.0 * A))))))) / t_0);
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(C * 4.0), $MachinePrecision] * A + N[((-B) * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, 3.6e-48], N[(N[Sqrt[N[(N[(A - N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision] * N[(N[(F + F), $MachinePrecision] * N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A - N[(-1.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if C < 3.6000000000000002e-48

   1. Initial program 28.3%

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

   3. Applied rewrites 36.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 36.7

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

   5. Applied rewrites 36.7%

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

      1. lift--.f64 N/A

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

      2. sub-negate2 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l- N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate2 N/A

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

      9. lower--.f64 36.9

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

   7. Applied rewrites 36.9%

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

   if 3.6000000000000002e-48 < C

   1. Initial program 8.2%

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

   3. Applied rewrites 9.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 9.6

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

   5. Applied rewrites 9.6%

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

      1. lift--.f64 N/A

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

      2. sub-negate2 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l- N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate2 N/A

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

      9. lower--.f64 13.3

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

   7. Applied rewrites 13.3%

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

   8. Taylor expanded in C around inf

      \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \color{blue}{\left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \color{blue}{\left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \color{blue}{\left(F \cdot \left(A - -1 \cdot A\right)\right)}\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \color{blue}{\left(A - -1 \cdot A\right)}\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - \color{blue}{-1 \cdot A}\right)\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      6. lower-*.f64 25.8

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot \color{blue}{A}\right)\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

   10. Applied rewrites 25.8%

       \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 31.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)\\ \mathbf{if}\;C \leq 3.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - \mathsf{hypot}\left(A - C, B\right)\right) + C\right) \cdot \left(\left(F + F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma (* C 4.0) A (* (- B) B))))
   (if (<= C 3.6e-48)
     (/
      (sqrt
       (*
        (+ (- A (hypot (- A C) B)) C)
        (* (+ F F) (fma -4.0 (* C A) (* B B)))))
      t_0)
     (/ (sqrt (* -8.0 (* A (* C (* F (- A (* -1.0 A))))))) t_0))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma((C * 4.0), A, (-B * B));
	double tmp;
	if (C <= 3.6e-48) {
		tmp = sqrt((((A - hypot((A - C), B)) + C) * ((F + F) * fma(-4.0, (C * A), (B * B))))) / t_0;
	} else {
		tmp = sqrt((-8.0 * (A * (C * (F * (A - (-1.0 * A))))))) / t_0;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(Float64(C * 4.0), A, Float64(Float64(-B) * B))
	tmp = 0.0
	if (C <= 3.6e-48)
		tmp = Float64(sqrt(Float64(Float64(Float64(A - hypot(Float64(A - C), B)) + C) * Float64(Float64(F + F) * fma(-4.0, Float64(C * A), Float64(B * B))))) / t_0);
	else
		tmp = Float64(sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A - Float64(-1.0 * A))))))) / t_0);
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(C * 4.0), $MachinePrecision] * A + N[((-B) * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, 3.6e-48], N[(N[Sqrt[N[(N[(N[(A - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision] + C), $MachinePrecision] * N[(N[(F + F), $MachinePrecision] * N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A - N[(-1.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if C < 3.6000000000000002e-48

   1. Initial program 28.3%

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

   3. Applied rewrites 36.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 36.7

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

   5. Applied rewrites 36.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 36.7

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

   7. Applied rewrites 36.7%

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

   if 3.6000000000000002e-48 < C

   1. Initial program 8.2%

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

   3. Applied rewrites 9.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 9.6

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

   5. Applied rewrites 9.6%

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

      1. lift--.f64 N/A

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

      2. sub-negate2 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l- N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate2 N/A

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

      9. lower--.f64 13.3

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

   7. Applied rewrites 13.3%

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

   8. Taylor expanded in C around inf

      \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \color{blue}{\left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \color{blue}{\left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \color{blue}{\left(F \cdot \left(A - -1 \cdot A\right)\right)}\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \color{blue}{\left(A - -1 \cdot A\right)}\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - \color{blue}{-1 \cdot A}\right)\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

      6. lower-*.f64 25.8

         \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot \color{blue}{A}\right)\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

   10. Applied rewrites 25.8%

       \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 22.7% accurate, 7.2× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (/
  (sqrt (* -8.0 (* A (* C (* F (- A (* -1.0 A)))))))
  (fma (* C 4.0) A (* (- B) B))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	return sqrt((-8.0 * (A * (C * (F * (A - (-1.0 * A))))))) / fma((C * 4.0), A, (-B * B));
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	return Float64(sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A - Float64(-1.0 * A))))))) / fma(Float64(C * 4.0), A, Float64(Float64(-B) * B)))
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := N[(N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A - N[(-1.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(C * 4.0), $MachinePrecision] * A + N[((-B) * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 23.2%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 23.2

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

5. Applied rewrites 23.2%

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

   1. lift--.f64 N/A

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

   2. sub-negate2 N/A

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

   3. lift-+.f64 N/A

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

   4. associate--l- N/A

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

   5. lift--.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift--.f64 N/A

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

   8. sub-negate2 N/A

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

   9. lower--.f64 25.1

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

7. Applied rewrites 25.1%

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

8. Taylor expanded in C around inf

   \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\sqrt{-8 \cdot \color{blue}{\left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \color{blue}{\left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \color{blue}{\left(F \cdot \left(A - -1 \cdot A\right)\right)}\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \color{blue}{\left(A - -1 \cdot A\right)}\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

   5. lower--.f64 N/A

      \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - \color{blue}{-1 \cdot A}\right)\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

   6. lower-*.f64 22.7

      \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot \color{blue}{A}\right)\right)\right)\right)}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]

10. Applied rewrites 22.7%

    \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)} \]
11. Add Preprocessing

Alternative 12: 0.0% accurate, 11.7× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ -1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{-2}\right) \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F) :precision binary64 (* -1.0 (* (sqrt (/ F B)) (sqrt -2.0))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	return -1.0 * (sqrt((F / B)) * sqrt(-2.0));
}

Fortran

NOTE: A, B, 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, 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
    code = (-1.0d0) * (sqrt((f / b)) * sqrt((-2.0d0)))
end function

Java

assert A < B && B < C && C < F;
public static double code(double A, double B, double C, double F) {
	return -1.0 * (Math.sqrt((F / B)) * Math.sqrt(-2.0));
}

Python

[A, B, C, F] = sort([A, B, C, F])
def code(A, B, C, F):
	return -1.0 * (math.sqrt((F / B)) * math.sqrt(-2.0))

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	return Float64(-1.0 * Float64(sqrt(Float64(F / B)) * sqrt(-2.0)))
end

MATLAB

A, B, C, F = num2cell(sort([A, B, C, F])){:}
function tmp = code(A, B, C, F)
	tmp = -1.0 * (sqrt((F / B)) * sqrt(-2.0));
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := N[(-1.0 * N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * N[Sqrt[-2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. sub-negate2 N/A

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

   5. distribute-lft-neg-out N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-*.f64 N/A

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

3. Applied rewrites 25.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 25.2

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

   6. lift-hypot.f64 N/A

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

   7. pow2 N/A

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

   8. lift-pow.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-pow.f64 N/A

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

   11. pow2 N/A

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

   12. lower-hypot.f64 25.2

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 25.2

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

   16. lift-fma.f64 N/A

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

5. Applied rewrites 25.2%

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

7. Applied rewrites 20.2%

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

8. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{-2}\right)} \]
9. 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 0.0

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

10. Applied rewrites 0.0%

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

Reproduce

herbie shell --seed 2025111 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :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))))