ABCF->ab-angle a

Percentage Accurate: 18.6% → 59.1%

Time: 9.1s

Alternatives: 11

Speedup: 15.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 59.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{F \cdot 2}\\ \mathbf{if}\;A \leq -3.2 \cdot 10^{-35}:\\ \;\;\;\;t\_0 \cdot \left(-1 \cdot \sqrt{{A}^{-1} \cdot -0.5}\right)\\ \mathbf{elif}\;A \leq 4.4 \cdot 10^{-209}:\\ \;\;\;\;t\_0 \cdot \left(-1 \cdot \left({B\_m}^{-1} \cdot \sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right)\right)\\ \mathbf{elif}\;A \leq 1.95 \cdot 10^{-28}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(C \cdot -4, A, B\_m \cdot B\_m\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C}\right)}{{B\_m}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (sqrt (* F 2.0))))
   (if (<= A -3.2e-35)
     (* t_0 (* -1.0 (sqrt (* (pow A -1.0) -0.5))))
     (if (<= A 4.4e-209)
       (* t_0 (* -1.0 (* (pow B_m -1.0) (sqrt (+ C (hypot B_m C))))))
       (if (<= A 1.95e-28)
         (/
          (*
           (sqrt (* (* 2.0 F) (fma (* C -4.0) A (* B_m B_m))))
           (- (sqrt (+ (+ (hypot (- A C) B_m) A) C))))
          (- (pow B_m 2.0) (* (* 4.0 A) C)))
         (* 0.25 (sqrt (* (/ F A) -16.0))))))))

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((F * 2.0));
	double tmp;
	if (A <= -3.2e-35) {
		tmp = t_0 * (-1.0 * sqrt((pow(A, -1.0) * -0.5)));
	} else if (A <= 4.4e-209) {
		tmp = t_0 * (-1.0 * (pow(B_m, -1.0) * sqrt((C + hypot(B_m, C)))));
	} else if (A <= 1.95e-28) {
		tmp = (sqrt(((2.0 * F) * fma((C * -4.0), A, (B_m * B_m)))) * -sqrt(((hypot((A - C), B_m) + A) + C))) / (pow(B_m, 2.0) - ((4.0 * A) * C));
	} else {
		tmp = 0.25 * sqrt(((F / A) * -16.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = sqrt(Float64(F * 2.0))
	tmp = 0.0
	if (A <= -3.2e-35)
		tmp = Float64(t_0 * Float64(-1.0 * sqrt(Float64((A ^ -1.0) * -0.5))));
	elseif (A <= 4.4e-209)
		tmp = Float64(t_0 * Float64(-1.0 * Float64((B_m ^ -1.0) * sqrt(Float64(C + hypot(B_m, C))))));
	elseif (A <= 1.95e-28)
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * F) * fma(Float64(C * -4.0), A, Float64(B_m * B_m)))) * Float64(-sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)))) / Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C)));
	else
		tmp = Float64(0.25 * sqrt(Float64(Float64(F / A) * -16.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[A, -3.2e-35], N[(t$95$0 * N[(-1.0 * N[Sqrt[N[(N[Power[A, -1.0], $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[A, 4.4e-209], N[(t$95$0 * N[(-1.0 * N[(N[Power[B$95$m, -1.0], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[A, 1.95e-28], N[(N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(N[(C * -4.0), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[Sqrt[N[(N[(F / A), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if A < -3.1999999999999998e-35

   1. Initial program 7.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

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

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 12.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 17.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-neg.f64 N/A

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

   7. Applied rewrites 17.0%

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

   8. Taylor expanded in A around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 53.9

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(-1 \cdot \sqrt{{A}^{-1} \cdot -0.5}\right) \]

   10. Applied rewrites 53.9%

       \[\leadsto \sqrt{F \cdot 2} \cdot \color{blue}{\left(-1 \cdot \sqrt{{A}^{-1} \cdot -0.5}\right)} \]

   if -3.1999999999999998e-35 < A < 4.40000000000000019e-209

   1. Initial program 31.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

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

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 48.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 55.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-neg.f64 N/A

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

   7. Applied rewrites 55.6%

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

   8. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. inv-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. pow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 64.8

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

   10. Applied rewrites 64.8%

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

   if 4.40000000000000019e-209 < A < 1.94999999999999999e-28

   1. Initial program 34.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

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

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 61.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 61.6

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

   5. Applied rewrites 61.6%

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

   if 1.94999999999999999e-28 < A

   1. Initial program 12.9%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 17.7%

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

   5. Applied rewrites 17.7%

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

   6. Taylor expanded in C around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\sqrt{\frac{F}{A}} \cdot \sqrt{-16}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\left(\sqrt{\frac{F}{A}} \cdot \sqrt{-16}\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{4} \cdot \sqrt{\frac{F}{A} \cdot -16} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \sqrt{\frac{F}{A} \cdot -16} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \sqrt{\frac{F}{A} \cdot -16} \]

      5. lower-/.f64 63.5

         \[\leadsto 0.25 \cdot \sqrt{\frac{F}{A} \cdot -16} \]

   8. Applied rewrites 63.5%

      \[\leadsto \color{blue}{0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 57.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := \sqrt{t\_0}\\ t_2 := -t\_0\\ t_3 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{-\sqrt{\left(2 \cdot \left(t\_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_3}\\ t_5 := \sqrt{F \cdot 2}\\ \mathbf{if}\;t\_4 \leq -1 \cdot 10^{+107}:\\ \;\;\;\;t\_5 \cdot \frac{t\_1 \cdot \sqrt{C + C}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq -2 \cdot 10^{-203}:\\ \;\;\;\;t\_5 \cdot \frac{t\_1 \cdot \sqrt{\left(B\_m + A\right) + C}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;t\_5 \cdot \frac{t\_1 \cdot \sqrt{\left(C + -0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right) + C}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \mathbf{else}:\\ \;\;\;\;t\_5 \cdot \left(-1 \cdot \frac{1}{\sqrt{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (sqrt t_0))
        (t_2 (- t_0))
        (t_3 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_4
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_3 F))
             (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0)))))))
          t_3))
        (t_5 (sqrt (* F 2.0))))
   (if (<= t_4 -1e+107)
     (* t_5 (/ (* t_1 (sqrt (+ C C))) t_2))
     (if (<= t_4 -2e-203)
       (* t_5 (/ (* t_1 (sqrt (+ (+ B_m A) C))) t_2))
       (if (<= t_4 0.0)
         (* t_5 (/ (* t_1 (sqrt (+ (+ C (* -0.5 (/ (* B_m B_m) A))) C))) t_2))
         (if (<= t_4 INFINITY)
           (* 0.25 (sqrt (* (/ F A) -16.0)))
           (* t_5 (* -1.0 (/ 1.0 (sqrt B_m))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = sqrt(t_0);
	double t_2 = -t_0;
	double t_3 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_4 = -sqrt(((2.0 * (t_3 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_3;
	double t_5 = sqrt((F * 2.0));
	double tmp;
	if (t_4 <= -1e+107) {
		tmp = t_5 * ((t_1 * sqrt((C + C))) / t_2);
	} else if (t_4 <= -2e-203) {
		tmp = t_5 * ((t_1 * sqrt(((B_m + A) + C))) / t_2);
	} else if (t_4 <= 0.0) {
		tmp = t_5 * ((t_1 * sqrt(((C + (-0.5 * ((B_m * B_m) / A))) + C))) / t_2);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = 0.25 * sqrt(((F / A) * -16.0));
	} else {
		tmp = t_5 * (-1.0 * (1.0 / sqrt(B_m)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = sqrt(t_0)
	t_2 = Float64(-t_0)
	t_3 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_4 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_3 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))))))) / t_3)
	t_5 = sqrt(Float64(F * 2.0))
	tmp = 0.0
	if (t_4 <= -1e+107)
		tmp = Float64(t_5 * Float64(Float64(t_1 * sqrt(Float64(C + C))) / t_2));
	elseif (t_4 <= -2e-203)
		tmp = Float64(t_5 * Float64(Float64(t_1 * sqrt(Float64(Float64(B_m + A) + C))) / t_2));
	elseif (t_4 <= 0.0)
		tmp = Float64(t_5 * Float64(Float64(t_1 * sqrt(Float64(Float64(C + Float64(-0.5 * Float64(Float64(B_m * B_m) / A))) + C))) / t_2));
	elseif (t_4 <= Inf)
		tmp = Float64(0.25 * sqrt(Float64(Float64(F / A) * -16.0)));
	else
		tmp = Float64(t_5 * Float64(-1.0 * Float64(1.0 / sqrt(B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$2 = (-t$95$0)}, Block[{t$95$3 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, -1e+107], N[(t$95$5 * N[(N[(t$95$1 * N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -2e-203], N[(t$95$5 * N[(N[(t$95$1 * N[Sqrt[N[(N[(B$95$m + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(t$95$5 * N[(N[(t$95$1 * N[Sqrt[N[(N[(C + N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(0.25 * N[Sqrt[N[(N[(F / A), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$5 * N[(-1.0 * N[(1.0 / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.9999999999999997e106

   1. Initial program 15.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

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

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 48.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 71.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-neg.f64 N/A

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

   7. Applied rewrites 71.9%

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

   8. Taylor expanded in A around -inf

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

   10. Applied rewrites 65.5%

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

   if -9.9999999999999997e106 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -2.0000000000000001e-203

   1. Initial program 97.4%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

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

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 97.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-neg.f64 N/A

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

   7. Applied rewrites 98.0%

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

   8. Taylor expanded in B around inf

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

   10. Applied rewrites 65.6%

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

   if -2.0000000000000001e-203 < (/.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 5.0%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

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

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 13.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 25.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-neg.f64 N/A

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

   7. Applied rewrites 25.1%

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

   8. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 54.0

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

   10. Applied rewrites 54.0%

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

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

   1. Initial program 40.4%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 56.8%

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

   5. Applied rewrites 56.7%

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

   6. Taylor expanded in C around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\sqrt{\frac{F}{A}} \cdot \sqrt{-16}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\left(\sqrt{\frac{F}{A}} \cdot \sqrt{-16}\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{4} \cdot \sqrt{\frac{F}{A} \cdot -16} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \sqrt{\frac{F}{A} \cdot -16} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \sqrt{\frac{F}{A} \cdot -16} \]

      5. lower-/.f64 79.3

         \[\leadsto 0.25 \cdot \sqrt{\frac{F}{A} \cdot -16} \]

   8. Applied rewrites 79.3%

      \[\leadsto \color{blue}{0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}} \]

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

   1. Initial program 0.0%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

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

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 0.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-neg.f64 N/A

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

   7. Applied rewrites 0.3%

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

   8. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 46.9

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

   10. Applied rewrites 46.9%

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

Alternative 3: 59.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{F \cdot 2}\\ t_1 := \mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;A \leq -3.2 \cdot 10^{-35}:\\ \;\;\;\;t\_0 \cdot \left(-1 \cdot \sqrt{{A}^{-1} \cdot -0.5}\right)\\ \mathbf{elif}\;A \leq 4.4 \cdot 10^{-209}:\\ \;\;\;\;t\_0 \cdot \left(-1 \cdot \left({B\_m}^{-1} \cdot \sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right)\right)\\ \mathbf{elif}\;A \leq 1.95 \cdot 10^{-28}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C} \cdot \frac{\sqrt{\left(t\_1 \cdot 2\right) \cdot F}}{-t\_1}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (sqrt (* F 2.0))) (t_1 (fma (* A C) -4.0 (* B_m B_m))))
   (if (<= A -3.2e-35)
     (* t_0 (* -1.0 (sqrt (* (pow A -1.0) -0.5))))
     (if (<= A 4.4e-209)
       (* t_0 (* -1.0 (* (pow B_m -1.0) (sqrt (+ C (hypot B_m C))))))
       (if (<= A 1.95e-28)
         (*
          (sqrt (+ (+ (hypot (- A C) B_m) A) C))
          (/ (sqrt (* (* t_1 2.0) F)) (- t_1)))
         (* 0.25 (sqrt (* (/ F A) -16.0))))))))

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((F * 2.0));
	double t_1 = fma((A * C), -4.0, (B_m * B_m));
	double tmp;
	if (A <= -3.2e-35) {
		tmp = t_0 * (-1.0 * sqrt((pow(A, -1.0) * -0.5)));
	} else if (A <= 4.4e-209) {
		tmp = t_0 * (-1.0 * (pow(B_m, -1.0) * sqrt((C + hypot(B_m, C)))));
	} else if (A <= 1.95e-28) {
		tmp = sqrt(((hypot((A - C), B_m) + A) + C)) * (sqrt(((t_1 * 2.0) * F)) / -t_1);
	} else {
		tmp = 0.25 * sqrt(((F / A) * -16.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = sqrt(Float64(F * 2.0))
	t_1 = fma(Float64(A * C), -4.0, Float64(B_m * B_m))
	tmp = 0.0
	if (A <= -3.2e-35)
		tmp = Float64(t_0 * Float64(-1.0 * sqrt(Float64((A ^ -1.0) * -0.5))));
	elseif (A <= 4.4e-209)
		tmp = Float64(t_0 * Float64(-1.0 * Float64((B_m ^ -1.0) * sqrt(Float64(C + hypot(B_m, C))))));
	elseif (A <= 1.95e-28)
		tmp = Float64(sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)) * Float64(sqrt(Float64(Float64(t_1 * 2.0) * F)) / Float64(-t_1)));
	else
		tmp = Float64(0.25 * sqrt(Float64(Float64(F / A) * -16.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[A, -3.2e-35], N[(t$95$0 * N[(-1.0 * N[Sqrt[N[(N[Power[A, -1.0], $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[A, 4.4e-209], N[(t$95$0 * N[(-1.0 * N[(N[Power[B$95$m, -1.0], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[A, 1.95e-28], N[(N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(N[(t$95$1 * 2.0), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[Sqrt[N[(N[(F / A), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if A < -3.1999999999999998e-35

   1. Initial program 7.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

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

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 12.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 17.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-neg.f64 N/A

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

   7. Applied rewrites 17.0%

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

   8. Taylor expanded in A around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 53.9

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(-1 \cdot \sqrt{{A}^{-1} \cdot -0.5}\right) \]

   10. Applied rewrites 53.9%

       \[\leadsto \sqrt{F \cdot 2} \cdot \color{blue}{\left(-1 \cdot \sqrt{{A}^{-1} \cdot -0.5}\right)} \]

   if -3.1999999999999998e-35 < A < 4.40000000000000019e-209

   1. Initial program 31.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

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

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 48.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 55.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-neg.f64 N/A

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

   7. Applied rewrites 55.6%

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

   8. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. inv-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. pow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 64.8

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

   10. Applied rewrites 64.8%

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

   if 4.40000000000000019e-209 < A < 1.94999999999999999e-28

   1. Initial program 34.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

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

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 61.6%

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

   5. Applied rewrites 61.6%

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

   if 1.94999999999999999e-28 < A

   1. Initial program 12.9%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 17.7%

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

   5. Applied rewrites 17.7%

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

   6. Taylor expanded in C around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\sqrt{\frac{F}{A}} \cdot \sqrt{-16}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\left(\sqrt{\frac{F}{A}} \cdot \sqrt{-16}\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{4} \cdot \sqrt{\frac{F}{A} \cdot -16} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \sqrt{\frac{F}{A} \cdot -16} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \sqrt{\frac{F}{A} \cdot -16} \]

      5. lower-/.f64 63.5

         \[\leadsto 0.25 \cdot \sqrt{\frac{F}{A} \cdot -16} \]

   8. Applied rewrites 63.5%

      \[\leadsto \color{blue}{0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 57.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{F \cdot 2}\\ t_1 := \mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 9.5 \cdot 10^{-60}:\\ \;\;\;\;t\_0 \cdot \left(-1 \cdot \sqrt{{A}^{-1} \cdot -0.5}\right)\\ \mathbf{elif}\;B\_m \leq 1.62 \cdot 10^{+153}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{t\_1} \cdot \frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C}}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(-1 \cdot \left({B\_m}^{-1} \cdot \sqrt{A + \mathsf{hypot}\left(A, B\_m\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (sqrt (* F 2.0))) (t_1 (fma (* A C) -4.0 (* B_m B_m))))
   (if (<= B_m 9.5e-60)
     (* t_0 (* -1.0 (sqrt (* (pow A -1.0) -0.5))))
     (if (<= B_m 1.62e+153)
       (*
        t_0
        (* (sqrt t_1) (/ (- (sqrt (+ (+ (hypot (- A C) B_m) A) C))) t_1)))
       (* t_0 (* -1.0 (* (pow B_m -1.0) (sqrt (+ A (hypot A B_m))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = sqrt((F * 2.0));
	double t_1 = fma((A * C), -4.0, (B_m * B_m));
	double tmp;
	if (B_m <= 9.5e-60) {
		tmp = t_0 * (-1.0 * sqrt((pow(A, -1.0) * -0.5)));
	} else if (B_m <= 1.62e+153) {
		tmp = t_0 * (sqrt(t_1) * (-sqrt(((hypot((A - C), B_m) + A) + C)) / t_1));
	} else {
		tmp = t_0 * (-1.0 * (pow(B_m, -1.0) * sqrt((A + hypot(A, B_m)))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = sqrt(Float64(F * 2.0))
	t_1 = fma(Float64(A * C), -4.0, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 9.5e-60)
		tmp = Float64(t_0 * Float64(-1.0 * sqrt(Float64((A ^ -1.0) * -0.5))));
	elseif (B_m <= 1.62e+153)
		tmp = Float64(t_0 * Float64(sqrt(t_1) * Float64(Float64(-sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C))) / t_1)));
	else
		tmp = Float64(t_0 * Float64(-1.0 * Float64((B_m ^ -1.0) * sqrt(Float64(A + hypot(A, B_m))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 9.5e-60], N[(t$95$0 * N[(-1.0 * N[Sqrt[N[(N[Power[A, -1.0], $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.62e+153], N[(t$95$0 * N[(N[Sqrt[t$95$1], $MachinePrecision] * N[((-N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(-1.0 * N[(N[Power[B$95$m, -1.0], $MachinePrecision] * N[Sqrt[N[(A + N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 9.49999999999999958e-60

   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. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

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

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 36.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 28.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-neg.f64 N/A

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

   7. Applied rewrites 28.7%

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

   8. Taylor expanded in A around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 48.0

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(-1 \cdot \sqrt{{A}^{-1} \cdot -0.5}\right) \]

   10. Applied rewrites 48.0%

       \[\leadsto \sqrt{F \cdot 2} \cdot \color{blue}{\left(-1 \cdot \sqrt{{A}^{-1} \cdot -0.5}\right)} \]

   if 9.49999999999999958e-60 < B < 1.62e153

   1. Initial program 29.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

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

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 46.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 54.9%

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

   if 1.62e153 < B

   1. Initial program 0.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

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

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 0.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 0.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-neg.f64 N/A

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

   7. Applied rewrites 0.3%

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

   8. Taylor expanded in C around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. inv-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. unpow2 N/A

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

      8. pow2 N/A

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

      9. lower-hypot.f64 77.2

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

   10. Applied rewrites 77.2%

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

Alternative 5: 57.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{F \cdot 2}\\ t_1 := \mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 9.5 \cdot 10^{-60}:\\ \;\;\;\;t\_0 \cdot \left(-1 \cdot \sqrt{{A}^{-1} \cdot -0.5}\right)\\ \mathbf{elif}\;B\_m \leq 1.26 \cdot 10^{+154}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{t\_1} \cdot \frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C}}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(-1 \cdot \frac{1}{\sqrt{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (sqrt (* F 2.0))) (t_1 (fma (* A C) -4.0 (* B_m B_m))))
   (if (<= B_m 9.5e-60)
     (* t_0 (* -1.0 (sqrt (* (pow A -1.0) -0.5))))
     (if (<= B_m 1.26e+154)
       (*
        t_0
        (* (sqrt t_1) (/ (- (sqrt (+ (+ (hypot (- A C) B_m) A) C))) t_1)))
       (* t_0 (* -1.0 (/ 1.0 (sqrt B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = sqrt((F * 2.0));
	double t_1 = fma((A * C), -4.0, (B_m * B_m));
	double tmp;
	if (B_m <= 9.5e-60) {
		tmp = t_0 * (-1.0 * sqrt((pow(A, -1.0) * -0.5)));
	} else if (B_m <= 1.26e+154) {
		tmp = t_0 * (sqrt(t_1) * (-sqrt(((hypot((A - C), B_m) + A) + C)) / t_1));
	} else {
		tmp = t_0 * (-1.0 * (1.0 / sqrt(B_m)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = sqrt(Float64(F * 2.0))
	t_1 = fma(Float64(A * C), -4.0, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 9.5e-60)
		tmp = Float64(t_0 * Float64(-1.0 * sqrt(Float64((A ^ -1.0) * -0.5))));
	elseif (B_m <= 1.26e+154)
		tmp = Float64(t_0 * Float64(sqrt(t_1) * Float64(Float64(-sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C))) / t_1)));
	else
		tmp = Float64(t_0 * Float64(-1.0 * Float64(1.0 / sqrt(B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 9.5e-60], N[(t$95$0 * N[(-1.0 * N[Sqrt[N[(N[Power[A, -1.0], $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.26e+154], N[(t$95$0 * N[(N[Sqrt[t$95$1], $MachinePrecision] * N[((-N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(-1.0 * N[(1.0 / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 9.49999999999999958e-60

   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. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

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

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 36.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 28.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-neg.f64 N/A

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

   7. Applied rewrites 28.7%

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

   8. Taylor expanded in A around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 48.0

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(-1 \cdot \sqrt{{A}^{-1} \cdot -0.5}\right) \]

   10. Applied rewrites 48.0%

       \[\leadsto \sqrt{F \cdot 2} \cdot \color{blue}{\left(-1 \cdot \sqrt{{A}^{-1} \cdot -0.5}\right)} \]

   if 9.49999999999999958e-60 < B < 1.26e154

   1. Initial program 29.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

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

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 46.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 54.9%

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

   if 1.26e154 < B

   1. Initial program 0.0%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

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

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-neg.f64 N/A

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

   7. Applied rewrites 0.0%

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

   8. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 76.4

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

   10. Applied rewrites 76.4%

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

Alternative 6: 55.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{F \cdot 2}\\ t_1 := \mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;A \leq -3.2 \cdot 10^{-35}:\\ \;\;\;\;t\_0 \cdot \left(-1 \cdot \sqrt{{A}^{-1} \cdot -0.5}\right)\\ \mathbf{elif}\;A \leq 8.2 \cdot 10^{-284}:\\ \;\;\;\;t\_0 \cdot \left(-1 \cdot \frac{1}{\sqrt{B\_m}}\right)\\ \mathbf{elif}\;A \leq 1.95 \cdot 10^{-28}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C} \cdot \frac{\sqrt{\left(t\_1 \cdot 2\right) \cdot F}}{-t\_1}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (sqrt (* F 2.0))) (t_1 (fma (* A C) -4.0 (* B_m B_m))))
   (if (<= A -3.2e-35)
     (* t_0 (* -1.0 (sqrt (* (pow A -1.0) -0.5))))
     (if (<= A 8.2e-284)
       (* t_0 (* -1.0 (/ 1.0 (sqrt B_m))))
       (if (<= A 1.95e-28)
         (*
          (sqrt (+ (+ (hypot (- A C) B_m) A) C))
          (/ (sqrt (* (* t_1 2.0) F)) (- t_1)))
         (* 0.25 (sqrt (* (/ F A) -16.0))))))))

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((F * 2.0));
	double t_1 = fma((A * C), -4.0, (B_m * B_m));
	double tmp;
	if (A <= -3.2e-35) {
		tmp = t_0 * (-1.0 * sqrt((pow(A, -1.0) * -0.5)));
	} else if (A <= 8.2e-284) {
		tmp = t_0 * (-1.0 * (1.0 / sqrt(B_m)));
	} else if (A <= 1.95e-28) {
		tmp = sqrt(((hypot((A - C), B_m) + A) + C)) * (sqrt(((t_1 * 2.0) * F)) / -t_1);
	} else {
		tmp = 0.25 * sqrt(((F / A) * -16.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = sqrt(Float64(F * 2.0))
	t_1 = fma(Float64(A * C), -4.0, Float64(B_m * B_m))
	tmp = 0.0
	if (A <= -3.2e-35)
		tmp = Float64(t_0 * Float64(-1.0 * sqrt(Float64((A ^ -1.0) * -0.5))));
	elseif (A <= 8.2e-284)
		tmp = Float64(t_0 * Float64(-1.0 * Float64(1.0 / sqrt(B_m))));
	elseif (A <= 1.95e-28)
		tmp = Float64(sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)) * Float64(sqrt(Float64(Float64(t_1 * 2.0) * F)) / Float64(-t_1)));
	else
		tmp = Float64(0.25 * sqrt(Float64(Float64(F / A) * -16.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[A, -3.2e-35], N[(t$95$0 * N[(-1.0 * N[Sqrt[N[(N[Power[A, -1.0], $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[A, 8.2e-284], N[(t$95$0 * N[(-1.0 * N[(1.0 / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[A, 1.95e-28], N[(N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(N[(t$95$1 * 2.0), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[Sqrt[N[(N[(F / A), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if A < -3.1999999999999998e-35

   1. Initial program 7.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

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

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 12.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 17.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-neg.f64 N/A

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

   7. Applied rewrites 17.0%

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

   8. Taylor expanded in A around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 53.9

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(-1 \cdot \sqrt{{A}^{-1} \cdot -0.5}\right) \]

   10. Applied rewrites 53.9%

       \[\leadsto \sqrt{F \cdot 2} \cdot \color{blue}{\left(-1 \cdot \sqrt{{A}^{-1} \cdot -0.5}\right)} \]

   if -3.1999999999999998e-35 < A < 8.19999999999999997e-284

   1. Initial program 31.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

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

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 47.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 59.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-neg.f64 N/A

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

   7. Applied rewrites 59.1%

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

   8. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 53.5

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

   10. Applied rewrites 53.5%

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

   if 8.19999999999999997e-284 < A < 1.94999999999999999e-28

   1. Initial program 34.4%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

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

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 59.3%

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

   5. Applied rewrites 59.3%

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

   if 1.94999999999999999e-28 < A

   1. Initial program 12.9%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 17.7%

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

   5. Applied rewrites 17.7%

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

   6. Taylor expanded in C around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\sqrt{\frac{F}{A}} \cdot \sqrt{-16}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\left(\sqrt{\frac{F}{A}} \cdot \sqrt{-16}\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{4} \cdot \sqrt{\frac{F}{A} \cdot -16} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \sqrt{\frac{F}{A} \cdot -16} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \sqrt{\frac{F}{A} \cdot -16} \]

      5. lower-/.f64 63.5

         \[\leadsto 0.25 \cdot \sqrt{\frac{F}{A} \cdot -16} \]

   8. Applied rewrites 63.5%

      \[\leadsto \color{blue}{0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 53.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{F \cdot 2}\\ \mathbf{if}\;B\_m \leq 5.6 \cdot 10^{+43}:\\ \;\;\;\;t\_0 \cdot \left(-1 \cdot \sqrt{{A}^{-1} \cdot -0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(-1 \cdot \frac{1}{\sqrt{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (sqrt (* F 2.0))))
   (if (<= B_m 5.6e+43)
     (* t_0 (* -1.0 (sqrt (* (pow A -1.0) -0.5))))
     (* t_0 (* -1.0 (/ 1.0 (sqrt B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = sqrt((F * 2.0));
	double tmp;
	if (B_m <= 5.6e+43) {
		tmp = t_0 * (-1.0 * sqrt((pow(A, -1.0) * -0.5)));
	} else {
		tmp = t_0 * (-1.0 * (1.0 / sqrt(B_m)));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((f * 2.0d0))
    if (b_m <= 5.6d+43) then
        tmp = t_0 * ((-1.0d0) * sqrt(((a ** (-1.0d0)) * (-0.5d0))))
    else
        tmp = t_0 * ((-1.0d0) * (1.0d0 / sqrt(b_m)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.sqrt((F * 2.0));
	double tmp;
	if (B_m <= 5.6e+43) {
		tmp = t_0 * (-1.0 * Math.sqrt((Math.pow(A, -1.0) * -0.5)));
	} else {
		tmp = t_0 * (-1.0 * (1.0 / Math.sqrt(B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = math.sqrt((F * 2.0))
	tmp = 0
	if B_m <= 5.6e+43:
		tmp = t_0 * (-1.0 * math.sqrt((math.pow(A, -1.0) * -0.5)))
	else:
		tmp = t_0 * (-1.0 * (1.0 / math.sqrt(B_m)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = sqrt(Float64(F * 2.0))
	tmp = 0.0
	if (B_m <= 5.6e+43)
		tmp = Float64(t_0 * Float64(-1.0 * sqrt(Float64((A ^ -1.0) * -0.5))));
	else
		tmp = Float64(t_0 * Float64(-1.0 * Float64(1.0 / sqrt(B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = sqrt((F * 2.0));
	tmp = 0.0;
	if (B_m <= 5.6e+43)
		tmp = t_0 * (-1.0 * sqrt(((A ^ -1.0) * -0.5)));
	else
		tmp = t_0 * (-1.0 * (1.0 / sqrt(B_m)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B$95$m, 5.6e+43], N[(t$95$0 * N[(-1.0 * N[Sqrt[N[(N[Power[A, -1.0], $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(-1.0 * N[(1.0 / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 5.60000000000000038e43

   1. Initial program 24.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

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

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 40.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 34.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-neg.f64 N/A

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

   7. Applied rewrites 34.2%

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

   8. Taylor expanded in A around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 45.8

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(-1 \cdot \sqrt{{A}^{-1} \cdot -0.5}\right) \]

   10. Applied rewrites 45.8%

       \[\leadsto \sqrt{F \cdot 2} \cdot \color{blue}{\left(-1 \cdot \sqrt{{A}^{-1} \cdot -0.5}\right)} \]

   if 5.60000000000000038e43 < B

   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. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

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

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 17.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 25.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-neg.f64 N/A

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

   7. Applied rewrites 25.5%

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

   8. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 63.9

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

   10. Applied rewrites 63.9%

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

Alternative 8: 46.8% accurate, 4.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := \sqrt{F \cdot 2}\\ \mathbf{if}\;B\_m \leq 1.8 \cdot 10^{-57}:\\ \;\;\;\;t\_1 \cdot \frac{\sqrt{t\_0} \cdot \sqrt{C + C}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(-1 \cdot \frac{1}{\sqrt{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m))) (t_1 (sqrt (* F 2.0))))
   (if (<= B_m 1.8e-57)
     (* t_1 (/ (* (sqrt t_0) (sqrt (+ C C))) (- t_0)))
     (* t_1 (* -1.0 (/ 1.0 (sqrt B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = sqrt((F * 2.0));
	double tmp;
	if (B_m <= 1.8e-57) {
		tmp = t_1 * ((sqrt(t_0) * sqrt((C + C))) / -t_0);
	} else {
		tmp = t_1 * (-1.0 * (1.0 / sqrt(B_m)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = sqrt(Float64(F * 2.0))
	tmp = 0.0
	if (B_m <= 1.8e-57)
		tmp = Float64(t_1 * Float64(Float64(sqrt(t_0) * sqrt(Float64(C + C))) / Float64(-t_0)));
	else
		tmp = Float64(t_1 * Float64(-1.0 * Float64(1.0 / sqrt(B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B$95$m, 1.8e-57], N[(t$95$1 * N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$0)), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(-1.0 * N[(1.0 / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if B < 1.8000000000000001e-57

   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. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

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

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 36.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 28.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \color{blue}{\frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}}\right) \]

      3. lift-neg.f64 N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\right) \]

      4. distribute-frac-neg N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\right)\right)}\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \color{blue}{\frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{neg}\left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)\right)}}\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{neg}\left(\color{blue}{\left(\left(A \cdot C\right) \cdot -4 + B \cdot B\right)}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{neg}\left(\left(\color{blue}{-4 \cdot \left(A \cdot C\right)} + B \cdot B\right)\right)}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\right)}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{neg}\left(\mathsf{fma}\left(-4, \color{blue}{A \cdot C}, B \cdot B\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{neg}\left(\mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{neg}\left(\mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)\right)}\right) \]

      12. lift-neg.f64 N/A

          \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\color{blue}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\right) \]

   7. Applied rewrites 28.7%

      \[\leadsto \sqrt{F \cdot 2} \cdot \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   8. Taylor expanded in A around -inf

      \[\leadsto \sqrt{F \cdot 2} \cdot \frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{\color{blue}{C} + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 37.8%

       \[\leadsto \sqrt{F \cdot 2} \cdot \frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{\color{blue}{C} + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 1.8000000000000001e-57 < B

   1. Initial program 17.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left({\color{blue}{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}^{\frac{1}{2}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot {\left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}^{\frac{1}{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}^{\frac{1}{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied rewrites 26.9%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \cdot \frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}} \cdot \frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lift-*.f64 N/A

         \[\leadsto {\color{blue}{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}^{\frac{1}{2}} \cdot \frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. unpow-prod-down N/A

         \[\leadsto \color{blue}{\left({\left(2 \cdot F\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)} \cdot \frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. associate-*l* N/A

         \[\leadsto \color{blue}{{\left(2 \cdot F\right)}^{\frac{1}{2}} \cdot \left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}} \cdot \frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\left(2 \cdot F\right)}^{\frac{1}{2}} \cdot \left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}} \cdot \frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

   5. Applied rewrites 31.7%

      \[\leadsto \color{blue}{\sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \color{blue}{\left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \color{blue}{\frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}}\right) \]

      3. lift-neg.f64 N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\right) \]

      4. distribute-frac-neg N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\right)\right)}\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \color{blue}{\frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{neg}\left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)\right)}}\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{neg}\left(\color{blue}{\left(\left(A \cdot C\right) \cdot -4 + B \cdot B\right)}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{neg}\left(\left(\color{blue}{-4 \cdot \left(A \cdot C\right)} + B \cdot B\right)\right)}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\right)}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{neg}\left(\mathsf{fma}\left(-4, \color{blue}{A \cdot C}, B \cdot B\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{neg}\left(\mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{neg}\left(\mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)\right)}\right) \]

      12. lift-neg.f64 N/A

          \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\color{blue}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\right) \]

   7. Applied rewrites 31.7%

      \[\leadsto \sqrt{F \cdot 2} \cdot \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   8. Taylor expanded in B around inf

      \[\leadsto \sqrt{F \cdot 2} \cdot \color{blue}{\left(-1 \cdot \sqrt{\frac{1}{B}}\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(-1 \cdot \color{blue}{\sqrt{\frac{1}{B}}}\right) \]

      2. sqrt-div N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(-1 \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{B}}}\right) \]

      3. metadata-eval N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(-1 \cdot \frac{1}{\sqrt{\color{blue}{B}}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(-1 \cdot \frac{1}{\color{blue}{\sqrt{B}}}\right) \]

      5. lower-sqrt.f64 53.0

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(-1 \cdot \frac{1}{\sqrt{B}}\right) \]

   10. Applied rewrites 53.0%

       \[\leadsto \sqrt{F \cdot 2} \cdot \color{blue}{\left(-1 \cdot \frac{1}{\sqrt{B}}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 45.8% accurate, 9.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 5.5 \cdot 10^{-61}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot 2} \cdot \left(-1 \cdot \frac{1}{\sqrt{B\_m}}\right)\\ \end{array} \end{array} \]

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 5.5e-61)
   (* 0.25 (sqrt (* (/ F A) -16.0)))
   (* (sqrt (* F 2.0)) (* -1.0 (/ 1.0 (sqrt B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 5.5e-61) {
		tmp = 0.25 * sqrt(((F / A) * -16.0));
	} else {
		tmp = sqrt((F * 2.0)) * (-1.0 * (1.0 / sqrt(B_m)));
	}
	return tmp;
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 5.5d-61) then
        tmp = 0.25d0 * sqrt(((f / a) * (-16.0d0)))
    else
        tmp = sqrt((f * 2.0d0)) * ((-1.0d0) * (1.0d0 / sqrt(b_m)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 5.5e-61) {
		tmp = 0.25 * Math.sqrt(((F / A) * -16.0));
	} else {
		tmp = Math.sqrt((F * 2.0)) * (-1.0 * (1.0 / Math.sqrt(B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 5.5e-61:
		tmp = 0.25 * math.sqrt(((F / A) * -16.0))
	else:
		tmp = math.sqrt((F * 2.0)) * (-1.0 * (1.0 / math.sqrt(B_m)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 5.5e-61)
		tmp = Float64(0.25 * sqrt(Float64(Float64(F / A) * -16.0)));
	else
		tmp = Float64(sqrt(Float64(F * 2.0)) * Float64(-1.0 * Float64(1.0 / sqrt(B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 5.5e-61)
		tmp = 0.25 * sqrt(((F / A) * -16.0));
	else
		tmp = sqrt((F * 2.0)) * (-1.0 * (1.0 / sqrt(B_m)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 5.5e-61], N[(0.25 * N[Sqrt[N[(N[(F / A), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] * N[(-1.0 * N[(1.0 / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 5.4999999999999997e-61

   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. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{-\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. pow1/2 N/A

         \[\leadsto \frac{-\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}^{\frac{1}{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{-\color{blue}{e^{\log \left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right) \cdot \frac{1}{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-exp.f64 N/A

         \[\leadsto \frac{-\color{blue}{e^{\log \left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right) \cdot \frac{1}{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-e^{\color{blue}{\log \left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right) \cdot \frac{1}{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied rewrites 29.6%

      \[\leadsto \frac{-\color{blue}{e^{\log \left(\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)\right) \cdot 0.5}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 25.7%

      \[\leadsto \frac{-e^{\log \color{blue}{\left(\mathsf{fma}\left(\mathsf{hypot}\left(A - C, B\right), \left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot 2\right) \cdot F, \left(\left(A + C\right) \cdot 2\right) \cdot \left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot F\right)\right)\right)} \cdot 0.5}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in C around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\sqrt{\frac{F}{A}} \cdot \sqrt{-16}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\left(\sqrt{\frac{F}{A}} \cdot \sqrt{-16}\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{4} \cdot \sqrt{\frac{F}{A} \cdot -16} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \sqrt{\frac{F}{A} \cdot -16} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \sqrt{\frac{F}{A} \cdot -16} \]

      5. lower-/.f64 35.4

         \[\leadsto 0.25 \cdot \sqrt{\frac{F}{A} \cdot -16} \]

   8. Applied rewrites 35.4%

      \[\leadsto \color{blue}{0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}} \]

   if 5.4999999999999997e-61 < B

   1. Initial program 17.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left({\color{blue}{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}^{\frac{1}{2}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot {\left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}^{\frac{1}{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}^{\frac{1}{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied rewrites 26.9%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \cdot \frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}} \cdot \frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lift-*.f64 N/A

         \[\leadsto {\color{blue}{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}^{\frac{1}{2}} \cdot \frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. unpow-prod-down N/A

         \[\leadsto \color{blue}{\left({\left(2 \cdot F\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)} \cdot \frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. associate-*l* N/A

         \[\leadsto \color{blue}{{\left(2 \cdot F\right)}^{\frac{1}{2}} \cdot \left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}} \cdot \frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\left(2 \cdot F\right)}^{\frac{1}{2}} \cdot \left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}} \cdot \frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

   5. Applied rewrites 31.7%

      \[\leadsto \color{blue}{\sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \color{blue}{\left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \color{blue}{\frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}}\right) \]

      3. lift-neg.f64 N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\right) \]

      4. distribute-frac-neg N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\right)\right)}\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \color{blue}{\frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{neg}\left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)\right)}}\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{neg}\left(\color{blue}{\left(\left(A \cdot C\right) \cdot -4 + B \cdot B\right)}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{neg}\left(\left(\color{blue}{-4 \cdot \left(A \cdot C\right)} + B \cdot B\right)\right)}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\right)}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{neg}\left(\mathsf{fma}\left(-4, \color{blue}{A \cdot C}, B \cdot B\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{neg}\left(\mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{neg}\left(\mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)\right)}\right) \]

      12. lift-neg.f64 N/A

          \[\leadsto \sqrt{F \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\color{blue}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\right) \]

   7. Applied rewrites 31.7%

      \[\leadsto \sqrt{F \cdot 2} \cdot \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   8. Taylor expanded in B around inf

      \[\leadsto \sqrt{F \cdot 2} \cdot \color{blue}{\left(-1 \cdot \sqrt{\frac{1}{B}}\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(-1 \cdot \color{blue}{\sqrt{\frac{1}{B}}}\right) \]

      2. sqrt-div N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(-1 \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{B}}}\right) \]

      3. metadata-eval N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(-1 \cdot \frac{1}{\sqrt{\color{blue}{B}}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(-1 \cdot \frac{1}{\color{blue}{\sqrt{B}}}\right) \]

      5. lower-sqrt.f64 52.7

         \[\leadsto \sqrt{F \cdot 2} \cdot \left(-1 \cdot \frac{1}{\sqrt{B}}\right) \]

   10. Applied rewrites 52.7%

       \[\leadsto \sqrt{F \cdot 2} \cdot \color{blue}{\left(-1 \cdot \frac{1}{\sqrt{B}}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 38.1% accurate, 12.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 4.4 \cdot 10^{-60}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \sqrt{\frac{F}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 4.4e-60)
   (* 0.25 (sqrt (* (/ F A) -16.0)))
   (* -1.0 (sqrt (* (/ F B_m) 2.0)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 4.4e-60) {
		tmp = 0.25 * sqrt(((F / A) * -16.0));
	} else {
		tmp = -1.0 * sqrt(((F / B_m) * 2.0));
	}
	return tmp;
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 4.4d-60) then
        tmp = 0.25d0 * sqrt(((f / a) * (-16.0d0)))
    else
        tmp = (-1.0d0) * sqrt(((f / b_m) * 2.0d0))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 4.4e-60) {
		tmp = 0.25 * Math.sqrt(((F / A) * -16.0));
	} else {
		tmp = -1.0 * Math.sqrt(((F / B_m) * 2.0));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 4.4e-60:
		tmp = 0.25 * math.sqrt(((F / A) * -16.0))
	else:
		tmp = -1.0 * math.sqrt(((F / B_m) * 2.0))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 4.4e-60)
		tmp = Float64(0.25 * sqrt(Float64(Float64(F / A) * -16.0)));
	else
		tmp = Float64(-1.0 * sqrt(Float64(Float64(F / B_m) * 2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 4.4e-60)
		tmp = 0.25 * sqrt(((F / A) * -16.0));
	else
		tmp = -1.0 * sqrt(((F / B_m) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 4.4e-60], N[(0.25 * N[Sqrt[N[(N[(F / A), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 4.3999999999999998e-60

   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. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{-\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. pow1/2 N/A

         \[\leadsto \frac{-\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}^{\frac{1}{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{-\color{blue}{e^{\log \left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right) \cdot \frac{1}{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-exp.f64 N/A

         \[\leadsto \frac{-\color{blue}{e^{\log \left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right) \cdot \frac{1}{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-e^{\color{blue}{\log \left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right) \cdot \frac{1}{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied rewrites 29.6%

      \[\leadsto \frac{-\color{blue}{e^{\log \left(\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)\right) \cdot 0.5}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 25.7%

      \[\leadsto \frac{-e^{\log \color{blue}{\left(\mathsf{fma}\left(\mathsf{hypot}\left(A - C, B\right), \left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot 2\right) \cdot F, \left(\left(A + C\right) \cdot 2\right) \cdot \left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot F\right)\right)\right)} \cdot 0.5}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in C around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\sqrt{\frac{F}{A}} \cdot \sqrt{-16}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\left(\sqrt{\frac{F}{A}} \cdot \sqrt{-16}\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{4} \cdot \sqrt{\frac{F}{A} \cdot -16} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \sqrt{\frac{F}{A} \cdot -16} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \sqrt{\frac{F}{A} \cdot -16} \]

      5. lower-/.f64 35.5

         \[\leadsto 0.25 \cdot \sqrt{\frac{F}{A} \cdot -16} \]

   8. Applied rewrites 35.5%

      \[\leadsto \color{blue}{0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}} \]

   if 4.3999999999999998e-60 < B

   1. Initial program 17.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{-\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. pow1/2 N/A

         \[\leadsto \frac{-\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}^{\frac{1}{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{-\color{blue}{e^{\log \left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right) \cdot \frac{1}{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-exp.f64 N/A

         \[\leadsto \frac{-\color{blue}{e^{\log \left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right) \cdot \frac{1}{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-e^{\color{blue}{\log \left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right) \cdot \frac{1}{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied rewrites 20.0%

      \[\leadsto \frac{-\color{blue}{e^{\log \left(\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)\right) \cdot 0.5}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 18.8%

      \[\leadsto \frac{-e^{\log \color{blue}{\left(\mathsf{fma}\left(\mathsf{hypot}\left(A - C, B\right), \left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot 2\right) \cdot F, \left(\left(A + C\right) \cdot 2\right) \cdot \left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot F\right)\right)\right)} \cdot 0.5}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F}{B} \cdot 2} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F}{B} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F}{B} \cdot 2} \]

      5. lower-/.f64 39.8

         \[\leadsto -1 \cdot \sqrt{\frac{F}{B} \cdot 2} \]

   8. Applied rewrites 39.8%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{B} \cdot 2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 26.9% accurate, 15.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -1 \cdot \sqrt{\frac{F}{B\_m} \cdot 2} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (* -1.0 (sqrt (* (/ F B_m) 2.0))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -1.0 * sqrt(((F / B_m) * 2.0));
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (-1.0d0) * sqrt(((f / b_m) * 2.0d0))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -1.0 * Math.sqrt(((F / B_m) * 2.0));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -1.0 * math.sqrt(((F / B_m) * 2.0))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-1.0 * sqrt(Float64(Float64(F / B_m) * 2.0)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -1.0 * sqrt(((F / B_m) * 2.0));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(-1.0 * N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.6%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{-\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. pow1/2 N/A

      \[\leadsto \frac{-\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}^{\frac{1}{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. pow-to-exp N/A

      \[\leadsto \frac{-\color{blue}{e^{\log \left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right) \cdot \frac{1}{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. lower-exp.f64 N/A

      \[\leadsto \frac{-\color{blue}{e^{\log \left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right) \cdot \frac{1}{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{-e^{\color{blue}{\log \left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right) \cdot \frac{1}{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

3. Applied rewrites 23.9%

   \[\leadsto \frac{-\color{blue}{e^{\log \left(\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)\right) \cdot 0.5}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

5. Applied rewrites 21.6%

   \[\leadsto \frac{-e^{\log \color{blue}{\left(\mathsf{fma}\left(\mathsf{hypot}\left(A - C, B\right), \left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot 2\right) \cdot F, \left(\left(A + C\right) \cdot 2\right) \cdot \left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot F\right)\right)\right)} \cdot 0.5}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

6. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   2. sqrt-unprod N/A

      \[\leadsto -1 \cdot \sqrt{\frac{F}{B} \cdot 2} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto -1 \cdot \sqrt{\frac{F}{B} \cdot 2} \]

   4. lower-*.f64 N/A

      \[\leadsto -1 \cdot \sqrt{\frac{F}{B} \cdot 2} \]

   5. lower-/.f64 26.9

      \[\leadsto -1 \cdot \sqrt{\frac{F}{B} \cdot 2} \]

8. Applied rewrites 26.9%

   \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{B} \cdot 2}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025106 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))