ABCF->ab-angle b

Percentage Accurate: 19.5% → 48.8%

Time: 16.9s

Alternatives: 14

Speedup: 12.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.5% 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: 48.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 9.2 \cdot 10^{-41}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot 2\right) \cdot \left(2 \cdot \left(A \cdot F\right)\right)}}{-\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;B\_m \leq 4.8 \cdot 10^{+46}:\\ \;\;\;\;-\sqrt{\frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\_m\right)}{t\_0} \cdot \left(F \cdot 2\right)}\\ \mathbf{elif}\;B\_m \leq 9 \cdot 10^{+226}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - \mathsf{hypot}\left(B\_m, A\right)\right) \cdot F\right) \cdot 2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{A + C}{B\_m} - 1}{B\_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m))))
   (if (<= B_m 9.2e-41)
     (/
      (sqrt (* (* t_0 2.0) (* 2.0 (* A F))))
      (- (fma B_m B_m (* -4.0 (* A C)))))
     (if (<= B_m 4.8e+46)
       (- (sqrt (* (/ (- (+ C A) (hypot (- A C) B_m)) t_0) (* F 2.0))))
       (if (<= B_m 9e+226)
         (/ (sqrt (* (* (- A (hypot B_m A)) F) 2.0)) (- B_m))
         (* (sqrt (* F (/ (- (/ (+ A C) B_m) 1.0) B_m))) (- (sqrt 2.0))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double tmp;
	if (B_m <= 9.2e-41) {
		tmp = sqrt(((t_0 * 2.0) * (2.0 * (A * F)))) / -fma(B_m, B_m, (-4.0 * (A * C)));
	} else if (B_m <= 4.8e+46) {
		tmp = -sqrt(((((C + A) - hypot((A - C), B_m)) / t_0) * (F * 2.0)));
	} else if (B_m <= 9e+226) {
		tmp = sqrt((((A - hypot(B_m, A)) * F) * 2.0)) / -B_m;
	} else {
		tmp = sqrt((F * ((((A + C) / B_m) - 1.0) / B_m))) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 9.2e-41)
		tmp = Float64(sqrt(Float64(Float64(t_0 * 2.0) * Float64(2.0 * Float64(A * F)))) / Float64(-fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))));
	elseif (B_m <= 4.8e+46)
		tmp = Float64(-sqrt(Float64(Float64(Float64(Float64(C + A) - hypot(Float64(A - C), B_m)) / t_0) * Float64(F * 2.0))));
	elseif (B_m <= 9e+226)
		tmp = Float64(sqrt(Float64(Float64(Float64(A - hypot(B_m, A)) * F) * 2.0)) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(A + C) / B_m) - 1.0) / B_m))) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 9.2e-41], N[(N[Sqrt[N[(N[(t$95$0 * 2.0), $MachinePrecision] * N[(2.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 4.8e+46], (-N[Sqrt[N[(N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[B$95$m, 9e+226], N[(N[Sqrt[N[(N[(N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(F * N[(N[(N[(N[(A + C), $MachinePrecision] / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 9.20000000000000041e-41

   1. Initial program 18.8%

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

   3. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 23.6

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

   5. Applied rewrites 23.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 23.8%

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

   8. Taylor expanded in B around 0

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

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

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

      2. unpow2 N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 23.8

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

   10. Applied rewrites 23.8%

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

   11. Taylor expanded in A around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 24.8

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

   13. Applied rewrites 24.8%

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

   if 9.20000000000000041e-41 < B < 4.80000000000000017e46

   1. Initial program 41.1%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 54.6%

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

   7. Applied rewrites 54.8%

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

   if 4.80000000000000017e46 < B < 8.99999999999999978e226

   1. Initial program 13.2%

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

   3. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 3.7

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

   5. Applied rewrites 3.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 3.7%

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

   8. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 47.6

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

   10. Applied rewrites 47.6%

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

   12. Applied rewrites 47.7%

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

   if 8.99999999999999978e226 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 4.8%

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

   6. Taylor expanded in B around inf

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

   8. Applied rewrites 57.9%

      \[\leadsto -\sqrt{F \cdot \frac{\frac{A + C}{B} - 1}{B}} \cdot \sqrt{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 31.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9.2 \cdot 10^{-41}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2\right) \cdot \left(2 \cdot \left(A \cdot F\right)\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{+46}:\\ \;\;\;\;-\sqrt{\frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(F \cdot 2\right)}\\ \mathbf{elif}\;B \leq 9 \cdot 10^{+226}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot F\right) \cdot 2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{A + C}{B} - 1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 42.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ t_1 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_1}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+56}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot F}}{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-210}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{F \cdot \left(-B\_m\right)}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot t\_0\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{-\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{A + C}{B\_m} - 1}{B\_m}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0)))
        (t_1 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* t_1 F))
            (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_1))))
   (if (<= t_2 -5e+56)
     (/ (sqrt (* (* 4.0 A) F)) (- (sqrt (fma -4.0 (* C A) (* B_m B_m)))))
     (if (<= t_2 -1e-210)
       (* (/ (sqrt 2.0) (- B_m)) (sqrt (* F (- B_m))))
       (if (<= t_2 0.0)
         (* (sqrt (* -0.5 (/ F C))) t_0)
         (if (<= t_2 INFINITY)
           (/
            (sqrt (* -16.0 (* (* A A) (* C F))))
            (- (fma B_m B_m (* -4.0 (* A C)))))
           (* (sqrt (* F (/ (- (/ (+ A C) B_m) 1.0) B_m))) t_0)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0);
	double t_1 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_2 = sqrt(((2.0 * (t_1 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_1;
	double tmp;
	if (t_2 <= -5e+56) {
		tmp = sqrt(((4.0 * A) * F)) / -sqrt(fma(-4.0, (C * A), (B_m * B_m)));
	} else if (t_2 <= -1e-210) {
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * -B_m));
	} else if (t_2 <= 0.0) {
		tmp = sqrt((-0.5 * (F / C))) * t_0;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((-16.0 * ((A * A) * (C * F)))) / -fma(B_m, B_m, (-4.0 * (A * C)));
	} else {
		tmp = sqrt((F * ((((A + C) / B_m) - 1.0) / B_m))) * t_0;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	t_1 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_1))
	tmp = 0.0
	if (t_2 <= -5e+56)
		tmp = Float64(sqrt(Float64(Float64(4.0 * A) * F)) / Float64(-sqrt(fma(-4.0, Float64(C * A), Float64(B_m * B_m)))));
	elseif (t_2 <= -1e-210)
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(F * Float64(-B_m))));
	elseif (t_2 <= 0.0)
		tmp = Float64(sqrt(Float64(-0.5 * Float64(F / C))) * t_0);
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(Float64(A * A) * Float64(C * F)))) / Float64(-fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(A + C) / B_m) - 1.0) / B_m))) * t_0);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+56], N[(N[Sqrt[N[(N[(4.0 * A), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$2, -1e-210], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(F * (-B$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(-0.5 * N[(F / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(-16.0 * N[(N[(A * A), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * N[(N[(N[(N[(A + C), $MachinePrecision] / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.00000000000000024e56

   1. Initial program 26.6%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 63.1%

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

   7. Applied rewrites 61.6%

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

   8. Taylor expanded in A around -inf

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

   10. Applied rewrites 38.3%

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

   if -5.00000000000000024e56 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-210

   1. Initial program 99.7%

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

   3. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 21.0

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

   5. Applied rewrites 21.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 21.0%

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

   8. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 48.1

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

   10. Applied rewrites 48.1%

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

   11. Taylor expanded in A around 0

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

   13. Applied rewrites 44.1%

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

   if -1e-210 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 0.0

   1. Initial program 3.3%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 5.0%

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

   6. Taylor expanded in A around -inf

      \[\leadsto -\sqrt{\frac{-1}{2} \cdot \frac{F}{C}} \cdot \sqrt{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 21.8%

      \[\leadsto -\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \sqrt{2} \]

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

   1. Initial program 44.8%

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

   3. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 50.0

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

   5. Applied rewrites 50.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 46.7%

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

   8. Taylor expanded in B around 0

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

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

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

      2. unpow2 N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 46.7

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

   10. Applied rewrites 46.7%

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

   11. Taylor expanded in A around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 32.0

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

   13. Applied rewrites 32.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 5.0%

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

   6. Taylor expanded in B around inf

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

   8. Applied rewrites 18.5%

      \[\leadsto -\sqrt{F \cdot \frac{\frac{A + C}{B} - 1}{B}} \cdot \sqrt{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 27.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -5 \cdot 10^{+56}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot F}}{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-210}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(-B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 0:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{A + C}{B} - 1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 42.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ t_1 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_1}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+56}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot F}}{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-210}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{F \cdot \left(-B\_m\right)}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot t\_0\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{-\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{-B\_m}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0)))
        (t_1 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* t_1 F))
            (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_1))))
   (if (<= t_2 -5e+56)
     (/ (sqrt (* (* 4.0 A) F)) (- (sqrt (fma -4.0 (* C A) (* B_m B_m)))))
     (if (<= t_2 -1e-210)
       (* (/ (sqrt 2.0) (- B_m)) (sqrt (* F (- B_m))))
       (if (<= t_2 0.0)
         (* (sqrt (* -0.5 (/ F C))) t_0)
         (if (<= t_2 INFINITY)
           (/
            (sqrt (* -16.0 (* (* A A) (* C F))))
            (- (fma B_m B_m (* -4.0 (* A C)))))
           (* (sqrt (/ F (- B_m))) t_0)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0);
	double t_1 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_2 = sqrt(((2.0 * (t_1 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_1;
	double tmp;
	if (t_2 <= -5e+56) {
		tmp = sqrt(((4.0 * A) * F)) / -sqrt(fma(-4.0, (C * A), (B_m * B_m)));
	} else if (t_2 <= -1e-210) {
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * -B_m));
	} else if (t_2 <= 0.0) {
		tmp = sqrt((-0.5 * (F / C))) * t_0;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((-16.0 * ((A * A) * (C * F)))) / -fma(B_m, B_m, (-4.0 * (A * C)));
	} else {
		tmp = sqrt((F / -B_m)) * t_0;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	t_1 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_1))
	tmp = 0.0
	if (t_2 <= -5e+56)
		tmp = Float64(sqrt(Float64(Float64(4.0 * A) * F)) / Float64(-sqrt(fma(-4.0, Float64(C * A), Float64(B_m * B_m)))));
	elseif (t_2 <= -1e-210)
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(F * Float64(-B_m))));
	elseif (t_2 <= 0.0)
		tmp = Float64(sqrt(Float64(-0.5 * Float64(F / C))) * t_0);
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(Float64(A * A) * Float64(C * F)))) / Float64(-fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))));
	else
		tmp = Float64(sqrt(Float64(F / Float64(-B_m))) * t_0);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+56], N[(N[Sqrt[N[(N[(4.0 * A), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$2, -1e-210], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(F * (-B$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(-0.5 * N[(F / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(-16.0 * N[(N[(A * A), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F / (-B$95$m)), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.00000000000000024e56

   1. Initial program 26.6%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 63.1%

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

   7. Applied rewrites 61.6%

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

   8. Taylor expanded in A around -inf

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

   10. Applied rewrites 38.3%

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

   if -5.00000000000000024e56 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-210

   1. Initial program 99.7%

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

   3. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 21.0

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

   5. Applied rewrites 21.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 21.0%

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

   8. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 48.1

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

   10. Applied rewrites 48.1%

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

   11. Taylor expanded in A around 0

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

   13. Applied rewrites 44.1%

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

   if -1e-210 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 0.0

   1. Initial program 3.3%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 5.0%

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

   6. Taylor expanded in A around -inf

      \[\leadsto -\sqrt{\frac{-1}{2} \cdot \frac{F}{C}} \cdot \sqrt{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 21.8%

      \[\leadsto -\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \sqrt{2} \]

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

   1. Initial program 44.8%

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

   3. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 50.0

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

   5. Applied rewrites 50.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 46.7%

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

   8. Taylor expanded in B around 0

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

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

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

      2. unpow2 N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 46.7

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

   10. Applied rewrites 46.7%

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

   11. Taylor expanded in A around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 32.0

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

   13. Applied rewrites 32.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 5.0%

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

   6. Taylor expanded in B around inf

      \[\leadsto -\sqrt{-1 \cdot \frac{F}{B}} \cdot \sqrt{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 17.3%

      \[\leadsto -\sqrt{-1 \cdot \frac{F}{B}} \cdot \sqrt{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 26.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -5 \cdot 10^{+56}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot F}}{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-210}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(-B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 0:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{-B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 43.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_0}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+56}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot F}}{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}}\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-210}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{F \cdot \left(-B\_m\right)}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}{-\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{A + C}{B\_m} - 1}{B\_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_1
         (/
          (sqrt
           (*
            (* 2.0 (* t_0 F))
            (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_0))))
   (if (<= t_1 -5e+56)
     (/ (sqrt (* (* 4.0 A) F)) (- (sqrt (fma -4.0 (* C A) (* B_m B_m)))))
     (if (<= t_1 -1e-210)
       (* (/ (sqrt 2.0) (- B_m)) (sqrt (* F (- B_m))))
       (if (<= t_1 INFINITY)
         (/
          (sqrt (* -8.0 (* A (* C (* F (+ A A))))))
          (- (fma B_m B_m (* -4.0 (* A C)))))
         (* (sqrt (* F (/ (- (/ (+ A C) B_m) 1.0) B_m))) (- (sqrt 2.0))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_0;
	double tmp;
	if (t_1 <= -5e+56) {
		tmp = sqrt(((4.0 * A) * F)) / -sqrt(fma(-4.0, (C * A), (B_m * B_m)));
	} else if (t_1 <= -1e-210) {
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * -B_m));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((-8.0 * (A * (C * (F * (A + A)))))) / -fma(B_m, B_m, (-4.0 * (A * C)));
	} else {
		tmp = sqrt((F * ((((A + C) / B_m) - 1.0) / B_m))) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_0))
	tmp = 0.0
	if (t_1 <= -5e+56)
		tmp = Float64(sqrt(Float64(Float64(4.0 * A) * F)) / Float64(-sqrt(fma(-4.0, Float64(C * A), Float64(B_m * B_m)))));
	elseif (t_1 <= -1e-210)
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(F * Float64(-B_m))));
	elseif (t_1 <= Inf)
		tmp = Float64(sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A + A)))))) / Float64(-fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(A + C) / B_m) - 1.0) / B_m))) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+56], N[(N[Sqrt[N[(N[(4.0 * A), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$1, -1e-210], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(F * (-B$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * N[(N[(N[(N[(A + C), $MachinePrecision] / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.00000000000000024e56

   1. Initial program 26.6%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 63.1%

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

   7. Applied rewrites 61.6%

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

   8. Taylor expanded in A around -inf

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

   10. Applied rewrites 38.3%

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

   if -5.00000000000000024e56 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-210

   1. Initial program 99.7%

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

   3. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 21.0

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

   5. Applied rewrites 21.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 21.0%

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

   8. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 48.1

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

   10. Applied rewrites 48.1%

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

   11. Taylor expanded in A around 0

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

   13. Applied rewrites 44.1%

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

   if -1e-210 < (/.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 20.8%

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

   3. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 40.8

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

   5. Applied rewrites 40.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 39.3%

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

   8. Taylor expanded in B around 0

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

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

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

      2. unpow2 N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 39.3

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

   10. Applied rewrites 39.3%

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

   11. Taylor expanded in C around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower--.f64 N/A

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

       6. lower-*.f64 34.6

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

   13. Applied rewrites 34.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 5.0%

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

   6. Taylor expanded in B around inf

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

   8. Applied rewrites 18.5%

      \[\leadsto -\sqrt{F \cdot \frac{\frac{A + C}{B} - 1}{B}} \cdot \sqrt{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 29.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -5 \cdot 10^{+56}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot F}}{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-210}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(-B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{A + C}{B} - 1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 48.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 10^{-40}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right) \cdot 2\right) \cdot \left(2 \cdot \left(A \cdot F\right)\right)}}{-\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;B\_m \leq 4.8 \cdot 10^{+46}:\\ \;\;\;\;-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(A - C, B\_m\right)\right) \cdot \frac{F \cdot 2}{\mathsf{fma}\left(C \cdot -4, A, B\_m \cdot B\_m\right)}}\\ \mathbf{elif}\;B\_m \leq 9 \cdot 10^{+226}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - \mathsf{hypot}\left(B\_m, A\right)\right) \cdot F\right) \cdot 2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{A + C}{B\_m} - 1}{B\_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1e-40)
   (/
    (sqrt (* (* (fma -4.0 (* C A) (* B_m B_m)) 2.0) (* 2.0 (* A F))))
    (- (fma B_m B_m (* -4.0 (* A C)))))
   (if (<= B_m 4.8e+46)
     (-
      (sqrt
       (*
        (- (+ A C) (hypot (- A C) B_m))
        (/ (* F 2.0) (fma (* C -4.0) A (* B_m B_m))))))
     (if (<= B_m 9e+226)
       (/ (sqrt (* (* (- A (hypot B_m A)) F) 2.0)) (- B_m))
       (* (sqrt (* F (/ (- (/ (+ A C) B_m) 1.0) B_m))) (- (sqrt 2.0)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1e-40) {
		tmp = sqrt(((fma(-4.0, (C * A), (B_m * B_m)) * 2.0) * (2.0 * (A * F)))) / -fma(B_m, B_m, (-4.0 * (A * C)));
	} else if (B_m <= 4.8e+46) {
		tmp = -sqrt((((A + C) - hypot((A - C), B_m)) * ((F * 2.0) / fma((C * -4.0), A, (B_m * B_m)))));
	} else if (B_m <= 9e+226) {
		tmp = sqrt((((A - hypot(B_m, A)) * F) * 2.0)) / -B_m;
	} else {
		tmp = sqrt((F * ((((A + C) / B_m) - 1.0) / B_m))) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1e-40)
		tmp = Float64(sqrt(Float64(Float64(fma(-4.0, Float64(C * A), Float64(B_m * B_m)) * 2.0) * Float64(2.0 * Float64(A * F)))) / Float64(-fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))));
	elseif (B_m <= 4.8e+46)
		tmp = Float64(-sqrt(Float64(Float64(Float64(A + C) - hypot(Float64(A - C), B_m)) * Float64(Float64(F * 2.0) / fma(Float64(C * -4.0), A, Float64(B_m * B_m))))));
	elseif (B_m <= 9e+226)
		tmp = Float64(sqrt(Float64(Float64(Float64(A - hypot(B_m, A)) * F) * 2.0)) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(A + C) / B_m) - 1.0) / B_m))) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1e-40], N[(N[Sqrt[N[(N[(N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(2.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 4.8e+46], (-N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(F * 2.0), $MachinePrecision] / N[(N[(C * -4.0), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[B$95$m, 9e+226], N[(N[Sqrt[N[(N[(N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(F * N[(N[(N[(N[(A + C), $MachinePrecision] / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if B < 9.9999999999999993e-41

   1. Initial program 18.8%

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

   3. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 23.6

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

   5. Applied rewrites 23.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 23.8%

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

   8. Taylor expanded in B around 0

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

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

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

      2. unpow2 N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 23.8

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

   10. Applied rewrites 23.8%

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

   11. Taylor expanded in A around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 24.8

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

   13. Applied rewrites 24.8%

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

   if 9.9999999999999993e-41 < B < 4.80000000000000017e46

   1. Initial program 41.1%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 54.6%

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

   7. Applied rewrites 47.3%

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

   9. Applied rewrites 48.3%

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

   if 4.80000000000000017e46 < B < 8.99999999999999978e226

   1. Initial program 13.2%

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

   3. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 3.7

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

   5. Applied rewrites 3.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 3.7%

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

   8. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 47.6

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

   10. Applied rewrites 47.6%

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

   12. Applied rewrites 47.7%

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

   if 8.99999999999999978e226 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 4.8%

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

   6. Taylor expanded in B around inf

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

   8. Applied rewrites 57.9%

      \[\leadsto -\sqrt{F \cdot \frac{\frac{A + C}{B} - 1}{B}} \cdot \sqrt{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 10^{-40}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2\right) \cdot \left(2 \cdot \left(A \cdot F\right)\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{+46}:\\ \;\;\;\;-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(A - C, B\right)\right) \cdot \frac{F \cdot 2}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}}\\ \mathbf{elif}\;B \leq 9 \cdot 10^{+226}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot F\right) \cdot 2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{A + C}{B} - 1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 46.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 8.8 \cdot 10^{+73}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right) \cdot 2\right) \cdot \left(F \cdot \mathsf{fma}\left(-0.5 \cdot B\_m, \frac{B\_m}{C}, A + A\right)\right)}}{-\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;B\_m \leq 9 \cdot 10^{+226}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - \mathsf{hypot}\left(B\_m, A\right)\right) \cdot F\right) \cdot 2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{A + C}{B\_m} - 1}{B\_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 8.8e+73)
   (/
    (sqrt
     (*
      (* (fma -4.0 (* C A) (* B_m B_m)) 2.0)
      (* F (fma (* -0.5 B_m) (/ B_m C) (+ A A)))))
    (- (fma B_m B_m (* -4.0 (* A C)))))
   (if (<= B_m 9e+226)
     (/ (sqrt (* (* (- A (hypot B_m A)) F) 2.0)) (- B_m))
     (* (sqrt (* F (/ (- (/ (+ A C) B_m) 1.0) B_m))) (- (sqrt 2.0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 8.8e+73) {
		tmp = sqrt(((fma(-4.0, (C * A), (B_m * B_m)) * 2.0) * (F * fma((-0.5 * B_m), (B_m / C), (A + A))))) / -fma(B_m, B_m, (-4.0 * (A * C)));
	} else if (B_m <= 9e+226) {
		tmp = sqrt((((A - hypot(B_m, A)) * F) * 2.0)) / -B_m;
	} else {
		tmp = sqrt((F * ((((A + C) / B_m) - 1.0) / B_m))) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 8.8e+73)
		tmp = Float64(sqrt(Float64(Float64(fma(-4.0, Float64(C * A), Float64(B_m * B_m)) * 2.0) * Float64(F * fma(Float64(-0.5 * B_m), Float64(B_m / C), Float64(A + A))))) / Float64(-fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))));
	elseif (B_m <= 9e+226)
		tmp = Float64(sqrt(Float64(Float64(Float64(A - hypot(B_m, A)) * F) * 2.0)) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(A + C) / B_m) - 1.0) / B_m))) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 8.8e+73], N[(N[Sqrt[N[(N[(N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(F * N[(N[(-0.5 * B$95$m), $MachinePrecision] * N[(B$95$m / C), $MachinePrecision] + N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 9e+226], N[(N[Sqrt[N[(N[(N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(F * N[(N[(N[(N[(A + C), $MachinePrecision] / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 8.8e73

   1. Initial program 20.7%

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

   3. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 22.3

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

   5. Applied rewrites 22.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 22.4%

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

   8. Taylor expanded in B around 0

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

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

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

      2. unpow2 N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 22.4

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

   10. Applied rewrites 22.4%

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

   12. Applied rewrites 22.4%

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

   if 8.8e73 < B < 8.99999999999999978e226

   1. Initial program 10.8%

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

   3. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 3.8

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

   5. Applied rewrites 3.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 3.7%

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

   8. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 47.3

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

   10. Applied rewrites 47.3%

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

   12. Applied rewrites 47.4%

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

   if 8.99999999999999978e226 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 4.8%

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

   6. Taylor expanded in B around inf

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

   8. Applied rewrites 57.9%

      \[\leadsto -\sqrt{F \cdot \frac{\frac{A + C}{B} - 1}{B}} \cdot \sqrt{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8.8 \cdot 10^{+73}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2\right) \cdot \left(F \cdot \mathsf{fma}\left(-0.5 \cdot B, \frac{B}{C}, A + A\right)\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;B \leq 9 \cdot 10^{+226}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot F\right) \cdot 2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{A + C}{B} - 1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.1% accurate, 4.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.05 \cdot 10^{+74}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right) \cdot 2\right) \cdot \left(F \cdot \mathsf{fma}\left(-0.5 \cdot B\_m, \frac{B\_m}{C}, A + A\right)\right)}}{-\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;B\_m \leq 2.7 \cdot 10^{+226}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{F \cdot \left(A - B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{A + C}{B\_m} - 1}{B\_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.05e+74)
   (/
    (sqrt
     (*
      (* (fma -4.0 (* C A) (* B_m B_m)) 2.0)
      (* F (fma (* -0.5 B_m) (/ B_m C) (+ A A)))))
    (- (fma B_m B_m (* -4.0 (* A C)))))
   (if (<= B_m 2.7e+226)
     (* (/ (sqrt 2.0) (- B_m)) (sqrt (* F (- A B_m))))
     (* (sqrt (* F (/ (- (/ (+ A C) B_m) 1.0) B_m))) (- (sqrt 2.0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.05e+74) {
		tmp = sqrt(((fma(-4.0, (C * A), (B_m * B_m)) * 2.0) * (F * fma((-0.5 * B_m), (B_m / C), (A + A))))) / -fma(B_m, B_m, (-4.0 * (A * C)));
	} else if (B_m <= 2.7e+226) {
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (A - B_m)));
	} else {
		tmp = sqrt((F * ((((A + C) / B_m) - 1.0) / B_m))) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.05e+74)
		tmp = Float64(sqrt(Float64(Float64(fma(-4.0, Float64(C * A), Float64(B_m * B_m)) * 2.0) * Float64(F * fma(Float64(-0.5 * B_m), Float64(B_m / C), Float64(A + A))))) / Float64(-fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))));
	elseif (B_m <= 2.7e+226)
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(F * Float64(A - B_m))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(A + C) / B_m) - 1.0) / B_m))) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.05e+74], N[(N[Sqrt[N[(N[(N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(F * N[(N[(-0.5 * B$95$m), $MachinePrecision] * N[(B$95$m / C), $MachinePrecision] + N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 2.7e+226], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(F * N[(A - B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * N[(N[(N[(N[(A + C), $MachinePrecision] / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.0499999999999999e74

   1. Initial program 20.7%

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

   3. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 22.3

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

   5. Applied rewrites 22.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 22.4%

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

   8. Taylor expanded in B around 0

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

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

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

      2. unpow2 N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 22.4

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

   10. Applied rewrites 22.4%

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

   12. Applied rewrites 22.4%

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

   if 1.0499999999999999e74 < B < 2.7000000000000003e226

   1. Initial program 10.8%

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

   3. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 3.8

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

   5. Applied rewrites 3.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 3.7%

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

   8. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 47.3

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

   10. Applied rewrites 47.3%

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

   11. Taylor expanded in A around 0

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

   13. Applied rewrites 36.4%

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

   if 2.7000000000000003e226 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 4.8%

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

   6. Taylor expanded in B around inf

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

   8. Applied rewrites 57.9%

      \[\leadsto -\sqrt{F \cdot \frac{\frac{A + C}{B} - 1}{B}} \cdot \sqrt{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 26.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.05 \cdot 10^{+74}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2\right) \cdot \left(F \cdot \mathsf{fma}\left(-0.5 \cdot B, \frac{B}{C}, A + A\right)\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;B \leq 2.7 \cdot 10^{+226}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{A + C}{B} - 1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.6% accurate, 6.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 5 \cdot 10^{+64}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right) \cdot 2\right) \cdot \left(2 \cdot \left(A \cdot F\right)\right)}}{-\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;B\_m \leq 2.7 \cdot 10^{+226}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{F \cdot \left(A - B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{A + C}{B\_m} - 1}{B\_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 5e+64)
   (/
    (sqrt (* (* (fma -4.0 (* C A) (* B_m B_m)) 2.0) (* 2.0 (* A F))))
    (- (fma B_m B_m (* -4.0 (* A C)))))
   (if (<= B_m 2.7e+226)
     (* (/ (sqrt 2.0) (- B_m)) (sqrt (* F (- A B_m))))
     (* (sqrt (* F (/ (- (/ (+ A C) B_m) 1.0) B_m))) (- (sqrt 2.0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 5e+64) {
		tmp = sqrt(((fma(-4.0, (C * A), (B_m * B_m)) * 2.0) * (2.0 * (A * F)))) / -fma(B_m, B_m, (-4.0 * (A * C)));
	} else if (B_m <= 2.7e+226) {
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (A - B_m)));
	} else {
		tmp = sqrt((F * ((((A + C) / B_m) - 1.0) / B_m))) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 5e+64)
		tmp = Float64(sqrt(Float64(Float64(fma(-4.0, Float64(C * A), Float64(B_m * B_m)) * 2.0) * Float64(2.0 * Float64(A * F)))) / Float64(-fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))));
	elseif (B_m <= 2.7e+226)
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(F * Float64(A - B_m))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(A + C) / B_m) - 1.0) / B_m))) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 5e+64], N[(N[Sqrt[N[(N[(N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(2.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 2.7e+226], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(F * N[(A - B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * N[(N[(N[(N[(A + C), $MachinePrecision] / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 5e64

   1. Initial program 20.8%

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

   3. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 22.3

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

   5. Applied rewrites 22.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 22.5%

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

   8. Taylor expanded in B around 0

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

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

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

      2. unpow2 N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 22.5

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

   10. Applied rewrites 22.5%

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

   11. Taylor expanded in A around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 23.1

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

   13. Applied rewrites 23.1%

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

   if 5e64 < B < 2.7000000000000003e226

   1. Initial program 10.5%

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

   3. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 3.7

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

   5. Applied rewrites 3.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 3.7%

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

   8. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 46.0

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

   10. Applied rewrites 46.0%

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

   11. Taylor expanded in A around 0

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

   13. Applied rewrites 35.2%

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

   if 2.7000000000000003e226 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 4.8%

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

   6. Taylor expanded in B around inf

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

   8. Applied rewrites 57.9%

      \[\leadsto -\sqrt{F \cdot \frac{\frac{A + C}{B} - 1}{B}} \cdot \sqrt{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 27.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5 \cdot 10^{+64}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2\right) \cdot \left(2 \cdot \left(A \cdot F\right)\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;B \leq 2.7 \cdot 10^{+226}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{A + C}{B} - 1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 40.3% accurate, 8.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;B\_m \leq 1.02 \cdot 10^{+74}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot t\_0\\ \mathbf{elif}\;B\_m \leq 2.7 \cdot 10^{+226}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{F \cdot \left(A - B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{-B\_m}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))))
   (if (<= B_m 1.02e+74)
     (* (sqrt (* -0.5 (/ F C))) t_0)
     (if (<= B_m 2.7e+226)
       (* (/ (sqrt 2.0) (- B_m)) (sqrt (* F (- A B_m))))
       (* (sqrt (/ F (- B_m))) t_0)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0);
	double tmp;
	if (B_m <= 1.02e+74) {
		tmp = sqrt((-0.5 * (F / C))) * t_0;
	} else if (B_m <= 2.7e+226) {
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (A - B_m)));
	} else {
		tmp = sqrt((F / -B_m)) * t_0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -sqrt(2.0d0)
    if (b_m <= 1.02d+74) then
        tmp = sqrt(((-0.5d0) * (f / c))) * t_0
    else if (b_m <= 2.7d+226) then
        tmp = (sqrt(2.0d0) / -b_m) * sqrt((f * (a - b_m)))
    else
        tmp = sqrt((f / -b_m)) * t_0
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = -Math.sqrt(2.0);
	double tmp;
	if (B_m <= 1.02e+74) {
		tmp = Math.sqrt((-0.5 * (F / C))) * t_0;
	} else if (B_m <= 2.7e+226) {
		tmp = (Math.sqrt(2.0) / -B_m) * Math.sqrt((F * (A - B_m)));
	} else {
		tmp = Math.sqrt((F / -B_m)) * t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = -math.sqrt(2.0)
	tmp = 0
	if B_m <= 1.02e+74:
		tmp = math.sqrt((-0.5 * (F / C))) * t_0
	elif B_m <= 2.7e+226:
		tmp = (math.sqrt(2.0) / -B_m) * math.sqrt((F * (A - B_m)))
	else:
		tmp = math.sqrt((F / -B_m)) * t_0
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (B_m <= 1.02e+74)
		tmp = Float64(sqrt(Float64(-0.5 * Float64(F / C))) * t_0);
	elseif (B_m <= 2.7e+226)
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(F * Float64(A - B_m))));
	else
		tmp = Float64(sqrt(Float64(F / Float64(-B_m))) * t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = -sqrt(2.0);
	tmp = 0.0;
	if (B_m <= 1.02e+74)
		tmp = sqrt((-0.5 * (F / C))) * t_0;
	elseif (B_m <= 2.7e+226)
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (A - B_m)));
	else
		tmp = sqrt((F / -B_m)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[B$95$m, 1.02e+74], N[(N[Sqrt[N[(-0.5 * N[(F / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 2.7e+226], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(F * N[(A - B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / (-B$95$m)), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.02000000000000005e74

   1. Initial program 20.7%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 23.5%

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

   6. Taylor expanded in A around -inf

      \[\leadsto -\sqrt{\frac{-1}{2} \cdot \frac{F}{C}} \cdot \sqrt{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 14.5%

      \[\leadsto -\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \sqrt{2} \]

   if 1.02000000000000005e74 < B < 2.7000000000000003e226

   1. Initial program 10.8%

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

   3. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 3.8

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

   5. Applied rewrites 3.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 3.7%

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

   8. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 47.3

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

   10. Applied rewrites 47.3%

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

   11. Taylor expanded in A around 0

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

   13. Applied rewrites 36.4%

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

   if 2.7000000000000003e226 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 4.8%

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

   6. Taylor expanded in B around inf

      \[\leadsto -\sqrt{-1 \cdot \frac{F}{B}} \cdot \sqrt{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.2%

      \[\leadsto -\sqrt{-1 \cdot \frac{F}{B}} \cdot \sqrt{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 20.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.02 \cdot 10^{+74}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 2.7 \cdot 10^{+226}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{-B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 38.5% accurate, 9.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;B\_m \leq 2.9 \cdot 10^{+131}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{-B\_m}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))))
   (if (<= B_m 2.9e+131)
     (* (sqrt (* -0.5 (/ F C))) t_0)
     (* (sqrt (/ F (- B_m))) t_0))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0);
	double tmp;
	if (B_m <= 2.9e+131) {
		tmp = sqrt((-0.5 * (F / C))) * t_0;
	} else {
		tmp = sqrt((F / -B_m)) * t_0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -sqrt(2.0d0)
    if (b_m <= 2.9d+131) then
        tmp = sqrt(((-0.5d0) * (f / c))) * t_0
    else
        tmp = sqrt((f / -b_m)) * t_0
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = -Math.sqrt(2.0);
	double tmp;
	if (B_m <= 2.9e+131) {
		tmp = Math.sqrt((-0.5 * (F / C))) * t_0;
	} else {
		tmp = Math.sqrt((F / -B_m)) * t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = -math.sqrt(2.0)
	tmp = 0
	if B_m <= 2.9e+131:
		tmp = math.sqrt((-0.5 * (F / C))) * t_0
	else:
		tmp = math.sqrt((F / -B_m)) * t_0
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (B_m <= 2.9e+131)
		tmp = Float64(sqrt(Float64(-0.5 * Float64(F / C))) * t_0);
	else
		tmp = Float64(sqrt(Float64(F / Float64(-B_m))) * t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = -sqrt(2.0);
	tmp = 0.0;
	if (B_m <= 2.9e+131)
		tmp = sqrt((-0.5 * (F / C))) * t_0;
	else
		tmp = sqrt((F / -B_m)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[B$95$m, 2.9e+131], N[(N[Sqrt[N[(-0.5 * N[(F / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[N[(F / (-B$95$m)), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 2.9000000000000001e131

   1. Initial program 20.5%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 24.9%

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

   6. Taylor expanded in A around -inf

      \[\leadsto -\sqrt{\frac{-1}{2} \cdot \frac{F}{C}} \cdot \sqrt{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 13.7%

      \[\leadsto -\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \sqrt{2} \]

   if 2.9000000000000001e131 < B

   1. Initial program 3.0%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 12.4%

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

   6. Taylor expanded in B around inf

      \[\leadsto -\sqrt{-1 \cdot \frac{F}{B}} \cdot \sqrt{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.4%

      \[\leadsto -\sqrt{-1 \cdot \frac{F}{B}} \cdot \sqrt{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 18.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.9 \cdot 10^{+131}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{-B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 2.0% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.0%

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

3. Taylor expanded in B around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. rem-square-sqrt N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-/.f64 2.1

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

5. Applied rewrites 2.1%

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

7. Applied rewrites 2.1%

   \[\leadsto -\sqrt{\frac{F}{B} \cdot 2} \cdot -1 \]
8. Step-by-step derivation

9. Applied rewrites 2.1%

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

11. Applied rewrites 2.1%

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

Alternative 12: 27.2% accurate, 12.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.0%

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

3. Taylor expanded in F around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

5. Applied rewrites 23.2%

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

6. Taylor expanded in B around inf

   \[\leadsto -\sqrt{-1 \cdot \frac{F}{B}} \cdot \sqrt{2} \]
7. Step-by-step derivation

8. Applied rewrites 11.6%

   \[\leadsto -\sqrt{-1 \cdot \frac{F}{B}} \cdot \sqrt{2} \]

9. Final simplification 11.6%

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

Alternative 13: 1.7% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.0%

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

3. Taylor expanded in B around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. rem-square-sqrt N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-/.f64 2.1

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

5. Applied rewrites 2.1%

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

7. Applied rewrites 2.1%

   \[\leadsto -\sqrt{\frac{F}{B} \cdot 2} \cdot -1 \]
8. Step-by-step derivation

9. Applied rewrites 2.1%

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

11. Applied rewrites 2.0%

    \[\leadsto \sqrt{\frac{\mathsf{fma}\left(F, B, B \cdot F\right)}{B \cdot B}} \]
12. Add Preprocessing

Alternative 14: 1.6% accurate, 18.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.0%

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

3. Taylor expanded in B around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. rem-square-sqrt N/A

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

   8. lower-*.f64 N/A

      \[\leadsto -\color{blue}{\left(\sqrt{2} \cdot -1\right)} \cdot \sqrt{\frac{F}{B}} \]

   9. lower-sqrt.f64 N/A

      \[\leadsto -\left(\color{blue}{\sqrt{2}} \cdot -1\right) \cdot \sqrt{\frac{F}{B}} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto -\left(\sqrt{2} \cdot -1\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

   11. lower-/.f64 2.1

       \[\leadsto -\left(\sqrt{2} \cdot -1\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 2.1%

   \[\leadsto \color{blue}{-\left(\sqrt{2} \cdot -1\right) \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

7. Applied rewrites 2.1%

   \[\leadsto -\sqrt{\frac{F}{B} \cdot 2} \cdot -1 \]
8. Step-by-step derivation

9. Applied rewrites 2.1%

   \[\leadsto \color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]
10. Step-by-step derivation

11. Applied rewrites 2.1%

    \[\leadsto \sqrt{F \cdot \frac{2}{B}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024351 
(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))))