ABCF->ab-angle a

Percentage Accurate: 18.2% → 54.5%

Time: 9.0s

Alternatives: 8

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: 18.2% 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: 54.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_2 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{-\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_2}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{-207}:\\ \;\;\;\;\sqrt{F \cdot 2} \cdot \left(\sqrt{t\_0} \cdot \frac{-\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C}}{t\_0}\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot F} \cdot \left(-\sqrt{\mathsf{fma}\left(-1, \frac{B\_m \cdot B\_m}{A}, 4 \cdot C\right)}\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)}\right)\\ \end{array} \end{array} \]

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 (fma (* A C) -4.0 (* B_m B_m)))
        (t_1 (fma -4.0 (* C A) (* B_m B_m)))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_2 F))
             (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0)))))))
          t_2)))
   (if (<= t_3 -1e-207)
     (*
      (sqrt (* F 2.0))
      (* (sqrt t_0) (/ (- (sqrt (+ (+ (hypot (- A C) B_m) A) C))) t_0)))
     (if (<= t_3 INFINITY)
       (/
        (* (sqrt (* t_1 F)) (- (sqrt (fma -1.0 (/ (* B_m B_m) A) (* 4.0 C)))))
        t_1)
       (* -1.0 (* (/ (sqrt 2.0) B_m) (sqrt (* F (+ C (hypot B_m C))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((A * C), -4.0, (B_m * B_m));
	double t_1 = fma(-4.0, (C * A), (B_m * B_m));
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_3 = -sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_2;
	double tmp;
	if (t_3 <= -1e-207) {
		tmp = sqrt((F * 2.0)) * (sqrt(t_0) * (-sqrt(((hypot((A - C), B_m) + A) + C)) / t_0));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt((t_1 * F)) * -sqrt(fma(-1.0, ((B_m * B_m) / A), (4.0 * C)))) / t_1;
	} else {
		tmp = -1.0 * ((sqrt(2.0) / B_m) * sqrt((F * (C + hypot(B_m, C)))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(A * C), -4.0, Float64(B_m * B_m))
	t_1 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))))))) / t_2)
	tmp = 0.0
	if (t_3 <= -1e-207)
		tmp = Float64(sqrt(Float64(F * 2.0)) * Float64(sqrt(t_0) * Float64(Float64(-sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C))) / t_0)));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(t_1 * F)) * Float64(-sqrt(fma(-1.0, Float64(Float64(B_m * B_m) / A), Float64(4.0 * C))))) / t_1);
	else
		tmp = Float64(-1.0 * Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(F * Float64(C + hypot(B_m, C))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, -1e-207], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[t$95$0], $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$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(t$95$1 * F), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(-1.0 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$1), $MachinePrecision], N[(-1.0 * N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.99999999999999925e-208

   1. Initial program 41.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 65.4%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 80.9%

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

   if -9.99999999999999925e-208 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 18.0%

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

   3. Applied rewrites 29.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-unprod N/A

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

   5. Applied rewrites 37.0%

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

   7. Applied rewrites 37.1%

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

   8. Taylor expanded in A around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 56.7

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

   10. Applied rewrites 56.7%

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

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

   1. Initial program 0.0%

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

   3. Applied rewrites 1.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-unprod N/A

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

   5. Applied rewrites 0.1%

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

   7. Applied rewrites 0.1%

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

   8. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. pow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 32.9

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

   10. Applied rewrites 32.9%

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

Alternative 2: 50.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := \sqrt{t\_0 \cdot F}\\ t_2 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{-\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_2}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{t\_1 \cdot \left(-\sqrt{4 \cdot C}\right)}{t\_0}\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-207}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right) + A\right) \cdot \left(\left(F + F\right) \cdot t\_0\right)}}{t\_0}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{t\_1 \cdot \left(-\sqrt{\mathsf{fma}\left(-1, \frac{B\_m \cdot B\_m}{A}, 4 \cdot C\right)}\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)}\right)\\ \end{array} \end{array} \]

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 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (sqrt (* t_0 F)))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_2 F))
             (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0)))))))
          t_2)))
   (if (<= t_3 (- INFINITY))
     (/ (* t_1 (- (sqrt (* 4.0 C)))) t_0)
     (if (<= t_3 -1e-207)
       (/ (- (sqrt (* (+ (+ C (hypot B_m (- A C))) A) (* (+ F F) t_0)))) t_0)
       (if (<= t_3 INFINITY)
         (/ (* t_1 (- (sqrt (fma -1.0 (/ (* B_m B_m) A) (* 4.0 C))))) t_0)
         (* -1.0 (* (/ (sqrt 2.0) B_m) (sqrt (* F (+ C (hypot B_m C)))))))))))

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 * F));
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_3 = -sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_2;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (t_1 * -sqrt((4.0 * C))) / t_0;
	} else if (t_3 <= -1e-207) {
		tmp = -sqrt((((C + hypot(B_m, (A - C))) + A) * ((F + F) * t_0))) / t_0;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (t_1 * -sqrt(fma(-1.0, ((B_m * B_m) / A), (4.0 * C)))) / t_0;
	} else {
		tmp = -1.0 * ((sqrt(2.0) / B_m) * sqrt((F * (C + hypot(B_m, C)))));
	}
	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(t_0 * F))
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))))))) / t_2)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(t_1 * Float64(-sqrt(Float64(4.0 * C)))) / t_0);
	elseif (t_3 <= -1e-207)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(C + hypot(B_m, Float64(A - C))) + A) * Float64(Float64(F + F) * t_0)))) / t_0);
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(t_1 * Float64(-sqrt(fma(-1.0, Float64(Float64(B_m * B_m) / A), Float64(4.0 * C))))) / t_0);
	else
		tmp = Float64(-1.0 * Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(F * Float64(C + hypot(B_m, C))))));
	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[(t$95$0 * F), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(t$95$1 * (-N[Sqrt[N[(4.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[t$95$3, -1e-207], N[((-N[Sqrt[N[(N[(N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + A), $MachinePrecision] * N[(N[(F + F), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(t$95$1 * (-N[Sqrt[N[(-1.0 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(-1.0 * N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.2%

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

   3. Applied rewrites 18.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-unprod N/A

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

   5. Applied rewrites 42.7%

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

   7. Applied rewrites 42.7%

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

   8. Taylor expanded in A around -inf

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

      1. lower-*.f64 49.4

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

   10. Applied rewrites 49.4%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.99999999999999925e-208

   1. Initial program 97.6%

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

   3. Applied rewrites 97.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 97.6

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

   5. Applied rewrites 97.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 97.6

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

      7. lift-hypot.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-hypot.f64 97.6

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

   7. Applied rewrites 97.6%

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

   if -9.99999999999999925e-208 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 18.0%

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

   3. Applied rewrites 29.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-unprod N/A

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

   5. Applied rewrites 37.0%

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

   7. Applied rewrites 37.1%

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

   8. Taylor expanded in A around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 56.7

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

   10. Applied rewrites 56.7%

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

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

   1. Initial program 0.0%

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

   3. Applied rewrites 1.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-unprod N/A

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

   5. Applied rewrites 0.1%

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

   7. Applied rewrites 0.1%

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

   8. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. pow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 32.9

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

   10. Applied rewrites 32.9%

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

Alternative 3: 49.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{-\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_1}\\ t_3 := \sqrt{t\_0 \cdot F}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+158}:\\ \;\;\;\;\frac{t\_3 \cdot \left(-\sqrt{4 \cdot C}\right)}{t\_0}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-207}:\\ \;\;\;\;-1 \cdot \sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)} \cdot 2}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{t\_3 \cdot \left(-\sqrt{\mathsf{fma}\left(-1, \frac{B\_m \cdot B\_m}{A}, 4 \cdot C\right)}\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)}\right)\\ \end{array} \end{array} \]

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 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_2
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_1 F))
             (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0)))))))
          t_1))
        (t_3 (sqrt (* t_0 F))))
   (if (<= t_2 -1e+158)
     (/ (* t_3 (- (sqrt (* 4.0 C)))) t_0)
     (if (<= t_2 -1e-207)
       (*
        -1.0
        (sqrt
         (*
          (/
           (* F (+ A (+ C (hypot B_m (- A C)))))
           (fma -4.0 (* A C) (* B_m B_m)))
          2.0)))
       (if (<= t_2 INFINITY)
         (/ (* t_3 (- (sqrt (fma -1.0 (/ (* B_m B_m) A) (* 4.0 C))))) t_0)
         (* -1.0 (* (/ (sqrt 2.0) B_m) (sqrt (* F (+ C (hypot B_m C)))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_2 = -sqrt(((2.0 * (t_1 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_1;
	double t_3 = sqrt((t_0 * F));
	double tmp;
	if (t_2 <= -1e+158) {
		tmp = (t_3 * -sqrt((4.0 * C))) / t_0;
	} else if (t_2 <= -1e-207) {
		tmp = -1.0 * sqrt((((F * (A + (C + hypot(B_m, (A - C))))) / fma(-4.0, (A * C), (B_m * B_m))) * 2.0));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (t_3 * -sqrt(fma(-1.0, ((B_m * B_m) / A), (4.0 * C)))) / t_0;
	} else {
		tmp = -1.0 * ((sqrt(2.0) / B_m) * sqrt((F * (C + hypot(B_m, C)))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_2 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))))))) / t_1)
	t_3 = sqrt(Float64(t_0 * F))
	tmp = 0.0
	if (t_2 <= -1e+158)
		tmp = Float64(Float64(t_3 * Float64(-sqrt(Float64(4.0 * C)))) / t_0);
	elseif (t_2 <= -1e-207)
		tmp = Float64(-1.0 * sqrt(Float64(Float64(Float64(F * Float64(A + Float64(C + hypot(B_m, Float64(A - C))))) / fma(-4.0, Float64(A * C), Float64(B_m * B_m))) * 2.0)));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(t_3 * Float64(-sqrt(fma(-1.0, Float64(Float64(B_m * B_m) / A), Float64(4.0 * C))))) / t_0);
	else
		tmp = Float64(-1.0 * Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(F * Float64(C + hypot(B_m, C))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$0 * F), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, -1e+158], N[(N[(t$95$3 * (-N[Sqrt[N[(4.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[t$95$2, -1e-207], N[(-1.0 * N[Sqrt[N[(N[(N[(F * N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(t$95$3 * (-N[Sqrt[N[(-1.0 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(-1.0 * N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 9.4%

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

   3. Applied rewrites 23.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-unprod N/A

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

   5. Applied rewrites 46.3%

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

   7. Applied rewrites 46.4%

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

   8. Taylor expanded in A around -inf

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

      1. lower-*.f64 51.5

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

   10. Applied rewrites 51.5%

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

   if -9.99999999999999953e157 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.99999999999999925e-208

   1. Initial program 97.5%

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

   3. Applied rewrites 97.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-unprod N/A

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

   5. Applied rewrites 97.7%

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

   7. Applied rewrites 97.9%

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

   8. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

   10. Applied rewrites 88.9%

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

   if -9.99999999999999925e-208 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 18.0%

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

   3. Applied rewrites 29.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-unprod N/A

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

   5. Applied rewrites 37.0%

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

   7. Applied rewrites 37.1%

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

   8. Taylor expanded in A around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 56.7

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

   10. Applied rewrites 56.7%

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

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

   1. Initial program 0.0%

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

   3. Applied rewrites 1.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-unprod N/A

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

   5. Applied rewrites 0.1%

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

   7. Applied rewrites 0.1%

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

   8. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. pow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 32.9

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

   10. Applied rewrites 32.9%

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

Alternative 4: 47.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := -1 \cdot \left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)}\right)\\ t_2 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{-\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_2}\\ t_4 := \sqrt{t\_0 \cdot F}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+176}:\\ \;\;\;\;\frac{t\_4 \cdot \left(-\sqrt{4 \cdot C}\right)}{t\_0}\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-207}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{t\_4 \cdot \left(-\sqrt{\mathsf{fma}\left(-1, \frac{B\_m \cdot B\_m}{A}, 4 \cdot C\right)}\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* -1.0 (* (/ (sqrt 2.0) B_m) (sqrt (* F (+ C (hypot B_m C)))))))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_2 F))
             (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0)))))))
          t_2))
        (t_4 (sqrt (* t_0 F))))
   (if (<= t_3 -2e+176)
     (/ (* t_4 (- (sqrt (* 4.0 C)))) t_0)
     (if (<= t_3 -1e-207)
       t_1
       (if (<= t_3 INFINITY)
         (/ (* t_4 (- (sqrt (fma -1.0 (/ (* B_m B_m) A) (* 4.0 C))))) t_0)
         t_1)))))

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 = -1.0 * ((sqrt(2.0) / B_m) * sqrt((F * (C + hypot(B_m, C)))));
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_3 = -sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_2;
	double t_4 = sqrt((t_0 * F));
	double tmp;
	if (t_3 <= -2e+176) {
		tmp = (t_4 * -sqrt((4.0 * C))) / t_0;
	} else if (t_3 <= -1e-207) {
		tmp = t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (t_4 * -sqrt(fma(-1.0, ((B_m * B_m) / A), (4.0 * C)))) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64(-1.0 * Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(F * Float64(C + hypot(B_m, C))))))
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))))))) / t_2)
	t_4 = sqrt(Float64(t_0 * F))
	tmp = 0.0
	if (t_3 <= -2e+176)
		tmp = Float64(Float64(t_4 * Float64(-sqrt(Float64(4.0 * C)))) / t_0);
	elseif (t_3 <= -1e-207)
		tmp = t_1;
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(t_4 * Float64(-sqrt(fma(-1.0, Float64(Float64(B_m * B_m) / A), Float64(4.0 * C))))) / t_0);
	else
		tmp = t_1;
	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[(-1.0 * N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t$95$0 * F), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, -2e+176], N[(N[(t$95$4 * (-N[Sqrt[N[(4.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[t$95$3, -1e-207], t$95$1, If[LessEqual[t$95$3, Infinity], N[(N[(t$95$4 * (-N[Sqrt[N[(-1.0 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -2e176

   1. Initial program 8.1%

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

   3. Applied rewrites 22.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-unprod N/A

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

   5. Applied rewrites 45.5%

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

   7. Applied rewrites 45.6%

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

   8. Taylor expanded in A around -inf

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

      1. lower-*.f64 51.2

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

   10. Applied rewrites 51.2%

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

   if -2e176 < (/.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))) < -9.99999999999999925e-208 or +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 22.1%

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

   3. Applied rewrites 23.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-unprod N/A

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

   5. Applied rewrites 22.2%

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

   7. Applied rewrites 22.3%

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

   8. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. pow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 42.4

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

   10. Applied rewrites 42.4%

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

   if -9.99999999999999925e-208 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 18.0%

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

   3. Applied rewrites 29.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-unprod N/A

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

   5. Applied rewrites 37.0%

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

   7. Applied rewrites 37.1%

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

   8. Taylor expanded in A around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 56.7

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

   10. Applied rewrites 56.7%

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

Alternative 5: 46.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{-\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_1}\\ t_3 := \sqrt{t\_0 \cdot F}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+64}:\\ \;\;\;\;\frac{t\_3 \cdot \left(-\sqrt{4 \cdot C}\right)}{t\_0}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-207}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(A, B\_m\right)\right)}\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{t\_3 \cdot \left(-\sqrt{\mathsf{fma}\left(-1, \frac{B\_m \cdot B\_m}{A}, 4 \cdot C\right)}\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \sqrt{\frac{F}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_2
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_1 F))
             (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0)))))))
          t_1))
        (t_3 (sqrt (* t_0 F))))
   (if (<= t_2 -5e+64)
     (/ (* t_3 (- (sqrt (* 4.0 C)))) t_0)
     (if (<= t_2 -1e-207)
       (* -1.0 (* (/ (sqrt 2.0) B_m) (sqrt (* F (+ A (hypot A B_m))))))
       (if (<= t_2 INFINITY)
         (/ (* t_3 (- (sqrt (fma -1.0 (/ (* B_m B_m) A) (* 4.0 C))))) t_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 t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_2 = -sqrt(((2.0 * (t_1 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_1;
	double t_3 = sqrt((t_0 * F));
	double tmp;
	if (t_2 <= -5e+64) {
		tmp = (t_3 * -sqrt((4.0 * C))) / t_0;
	} else if (t_2 <= -1e-207) {
		tmp = -1.0 * ((sqrt(2.0) / B_m) * sqrt((F * (A + hypot(A, B_m)))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (t_3 * -sqrt(fma(-1.0, ((B_m * B_m) / A), (4.0 * C)))) / t_0;
	} else {
		tmp = -1.0 * 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)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_2 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))))))) / t_1)
	t_3 = sqrt(Float64(t_0 * F))
	tmp = 0.0
	if (t_2 <= -5e+64)
		tmp = Float64(Float64(t_3 * Float64(-sqrt(Float64(4.0 * C)))) / t_0);
	elseif (t_2 <= -1e-207)
		tmp = Float64(-1.0 * Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(F * Float64(A + hypot(A, B_m))))));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(t_3 * Float64(-sqrt(fma(-1.0, Float64(Float64(B_m * B_m) / A), Float64(4.0 * C))))) / t_0);
	else
		tmp = Float64(-1.0 * sqrt(Float64(Float64(F / B_m) * 2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$0 * F), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, -5e+64], N[(N[(t$95$3 * (-N[Sqrt[N[(4.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[t$95$2, -1e-207], N[(-1.0 * N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(A + N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(t$95$3 * (-N[Sqrt[N[(-1.0 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(-1.0 * N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5e64

   1. Initial program 19.2%

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

   3. Applied rewrites 32.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-unprod N/A

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

   5. Applied rewrites 52.1%

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

   7. Applied rewrites 52.1%

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

   8. Taylor expanded in A around -inf

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

      1. lower-*.f64 51.8

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

   10. Applied rewrites 51.8%

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

   if -5e64 < (/.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))) < -9.99999999999999925e-208

   1. Initial program 97.5%

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

   3. Applied rewrites 97.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-unprod N/A

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

   5. Applied rewrites 97.6%

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

   7. Applied rewrites 97.9%

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

   8. Taylor expanded in C around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. pow2 N/A

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

      10. lower-hypot.f64 66.8

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

   10. Applied rewrites 66.8%

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

   if -9.99999999999999925e-208 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 18.0%

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

   3. Applied rewrites 29.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-unprod N/A

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

   5. Applied rewrites 37.0%

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

   7. Applied rewrites 37.1%

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

   8. Taylor expanded in A around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 56.7

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

   10. Applied rewrites 56.7%

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

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

   1. Initial program 0.0%

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

   3. Applied rewrites 1.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-unprod N/A

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

   5. Applied rewrites 0.1%

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

   7. Applied rewrites 0.1%

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

   8. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. 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 33.2

         \[\leadsto -1 \cdot \sqrt{\frac{F}{B} \cdot 2} \]

   10. Applied rewrites 33.2%

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

Alternative 6: 46.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{-\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_1}\\ t_3 := \sqrt{t\_0 \cdot F}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+64}:\\ \;\;\;\;\frac{t\_3 \cdot \left(-\sqrt{4 \cdot C}\right)}{t\_0}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-207}:\\ \;\;\;\;\frac{t\_3 \cdot \left(-\sqrt{B\_m \cdot 2}\right)}{t\_0}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{t\_3 \cdot \left(-\sqrt{\mathsf{fma}\left(-1, \frac{B\_m \cdot B\_m}{A}, 4 \cdot C\right)}\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \sqrt{\frac{F}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_2
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_1 F))
             (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0)))))))
          t_1))
        (t_3 (sqrt (* t_0 F))))
   (if (<= t_2 -5e+64)
     (/ (* t_3 (- (sqrt (* 4.0 C)))) t_0)
     (if (<= t_2 -1e-207)
       (/ (* t_3 (- (sqrt (* B_m 2.0)))) t_0)
       (if (<= t_2 INFINITY)
         (/ (* t_3 (- (sqrt (fma -1.0 (/ (* B_m B_m) A) (* 4.0 C))))) t_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 t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_2 = -sqrt(((2.0 * (t_1 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_1;
	double t_3 = sqrt((t_0 * F));
	double tmp;
	if (t_2 <= -5e+64) {
		tmp = (t_3 * -sqrt((4.0 * C))) / t_0;
	} else if (t_2 <= -1e-207) {
		tmp = (t_3 * -sqrt((B_m * 2.0))) / t_0;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (t_3 * -sqrt(fma(-1.0, ((B_m * B_m) / A), (4.0 * C)))) / t_0;
	} else {
		tmp = -1.0 * 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)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_2 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))))))) / t_1)
	t_3 = sqrt(Float64(t_0 * F))
	tmp = 0.0
	if (t_2 <= -5e+64)
		tmp = Float64(Float64(t_3 * Float64(-sqrt(Float64(4.0 * C)))) / t_0);
	elseif (t_2 <= -1e-207)
		tmp = Float64(Float64(t_3 * Float64(-sqrt(Float64(B_m * 2.0)))) / t_0);
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(t_3 * Float64(-sqrt(fma(-1.0, Float64(Float64(B_m * B_m) / A), Float64(4.0 * C))))) / t_0);
	else
		tmp = Float64(-1.0 * sqrt(Float64(Float64(F / B_m) * 2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$0 * F), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, -5e+64], N[(N[(t$95$3 * (-N[Sqrt[N[(4.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[t$95$2, -1e-207], N[(N[(t$95$3 * (-N[Sqrt[N[(B$95$m * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(t$95$3 * (-N[Sqrt[N[(-1.0 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(-1.0 * N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5e64

   1. Initial program 19.2%

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

   3. Applied rewrites 32.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-unprod N/A

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

   5. Applied rewrites 52.1%

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

   7. Applied rewrites 52.1%

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

   8. Taylor expanded in A around -inf

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

      1. lower-*.f64 51.8

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

   10. Applied rewrites 51.8%

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

   if -5e64 < (/.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))) < -9.99999999999999925e-208

   1. Initial program 97.5%

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

   3. Applied rewrites 97.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-unprod N/A

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

   5. Applied rewrites 97.6%

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

   7. Applied rewrites 97.9%

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

   8. Taylor expanded in B around inf

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

   10. Applied rewrites 65.6%

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

   if -9.99999999999999925e-208 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 18.0%

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

   3. Applied rewrites 29.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-unprod N/A

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

   5. Applied rewrites 37.0%

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

   7. Applied rewrites 37.1%

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

   8. Taylor expanded in A around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 56.7

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

   10. Applied rewrites 56.7%

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

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

   1. Initial program 0.0%

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

   3. Applied rewrites 1.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-unprod N/A

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

   5. Applied rewrites 0.1%

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

   7. Applied rewrites 0.1%

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

   8. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. 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 33.2

         \[\leadsto -1 \cdot \sqrt{\frac{F}{B} \cdot 2} \]

   10. Applied rewrites 33.2%

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

Alternative 7: 43.8% accurate, 5.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 1.65 \cdot 10^{-13}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot F} \cdot \left(-\sqrt{4 \cdot C}\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \sqrt{\frac{F}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m))))
   (if (<= B_m 1.65e-13)
     (/ (* (sqrt (* t_0 F)) (- (sqrt (* 4.0 C)))) t_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 t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double tmp;
	if (B_m <= 1.65e-13) {
		tmp = (sqrt((t_0 * F)) * -sqrt((4.0 * C))) / t_0;
	} else {
		tmp = -1.0 * 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)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 1.65e-13)
		tmp = Float64(Float64(sqrt(Float64(t_0 * F)) * Float64(-sqrt(Float64(4.0 * C)))) / t_0);
	else
		tmp = Float64(-1.0 * sqrt(Float64(Float64(F / B_m) * 2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.65e-13], N[(N[(N[Sqrt[N[(t$95$0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(4.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $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 < 1.65e-13

   1. Initial program 21.6%

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

   3. Applied rewrites 31.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-unprod N/A

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

   5. Applied rewrites 37.6%

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

   7. Applied rewrites 37.6%

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

   8. Taylor expanded in A around -inf

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

      1. lower-*.f64 43.2

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

   10. Applied rewrites 43.2%

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

   if 1.65e-13 < B

   1. Initial program 15.3%

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

   3. Applied rewrites 18.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-unprod N/A

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

   5. Applied rewrites 24.2%

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

   7. Applied rewrites 24.3%

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

   8. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. 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 44.4

         \[\leadsto -1 \cdot \sqrt{\frac{F}{B} \cdot 2} \]

   10. Applied rewrites 44.4%

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

Alternative 8: 28.1% 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 18.2%

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

3. Applied rewrites 24.5%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. sqrt-unprod N/A

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

5. Applied rewrites 30.4%

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

7. Applied rewrites 30.5%

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

8. Taylor expanded in B around inf

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

   1. lower-*.f64 N/A

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

   2. 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 28.1

      \[\leadsto -1 \cdot \sqrt{\frac{F}{B} \cdot 2} \]

10. Applied rewrites 28.1%

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

Reproduce

herbie shell --seed 2025107 
(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))))