ABCF->ab-angle a

Percentage Accurate: 19.0% → 63.1%

Time: 11.4s

Alternatives: 13

Speedup: 20.5×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.0% 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: 63.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := B\_m \cdot B\_m - t\_0\\ t_2 := t\_1 \cdot F\\ t_3 := {B\_m}^{2} - t\_0\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(t\_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_3}\\ \mathbf{if}\;t\_4 \leq -1 \cdot 10^{-197}:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot \left(\sqrt{t\_1} \cdot \left(-\sqrt{F}\right)\right)\right) \cdot \sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\_m\right)}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot t\_2\right) \cdot \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, 2 \cdot C\right)}}{\left(-B\_m\right) \cdot B\_m + t\_0}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot \sqrt{t\_2}\right) \cdot \left(-\sqrt{C \cdot 2}\right)}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right)\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C))
        (t_1 (- (* B_m B_m) t_0))
        (t_2 (* t_1 F))
        (t_3 (- (pow B_m 2.0) t_0))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* t_3 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_3))))
   (if (<= t_4 -1e-197)
     (/
      (*
       (* (sqrt 2.0) (* (sqrt t_1) (- (sqrt F))))
       (sqrt (+ (+ A C) (hypot (- A C) B_m))))
      t_3)
     (if (<= t_4 0.0)
       (/
        (sqrt (* (* 2.0 t_2) (fma -0.5 (/ (* B_m B_m) A) (* 2.0 C))))
        (+ (* (- B_m) B_m) t_0))
       (if (<= t_4 INFINITY)
         (/ (* (* (sqrt 2.0) (sqrt t_2)) (- (sqrt (* C 2.0)))) t_3)
         (*
          (/ (sqrt 2.0) (- B_m))
          (* (sqrt F) (sqrt (+ 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 = (4.0 * A) * C;
	double t_1 = (B_m * B_m) - t_0;
	double t_2 = t_1 * F;
	double t_3 = pow(B_m, 2.0) - t_0;
	double t_4 = sqrt(((2.0 * (t_3 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_3;
	double tmp;
	if (t_4 <= -1e-197) {
		tmp = ((sqrt(2.0) * (sqrt(t_1) * -sqrt(F))) * sqrt(((A + C) + hypot((A - C), B_m)))) / t_3;
	} else if (t_4 <= 0.0) {
		tmp = sqrt(((2.0 * t_2) * fma(-0.5, ((B_m * B_m) / A), (2.0 * C)))) / ((-B_m * B_m) + t_0);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = ((sqrt(2.0) * sqrt(t_2)) * -sqrt((C * 2.0))) / t_3;
	} else {
		tmp = (sqrt(2.0) / -B_m) * (sqrt(F) * sqrt((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 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(Float64(B_m * B_m) - t_0)
	t_2 = Float64(t_1 * F)
	t_3 = Float64((B_m ^ 2.0) - t_0)
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_3 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_3))
	tmp = 0.0
	if (t_4 <= -1e-197)
		tmp = Float64(Float64(Float64(sqrt(2.0) * Float64(sqrt(t_1) * Float64(-sqrt(F)))) * sqrt(Float64(Float64(A + C) + hypot(Float64(A - C), B_m)))) / t_3);
	elseif (t_4 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(2.0 * t_2) * fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(2.0 * C)))) / Float64(Float64(Float64(-B_m) * B_m) + t_0));
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(Float64(sqrt(2.0) * sqrt(t_2)) * Float64(-sqrt(Float64(C * 2.0)))) / t_3);
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * Float64(sqrt(F) * sqrt(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[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * F), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$3)), $MachinePrecision]}, If[LessEqual[t$95$4, -1e-197], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[t$95$1], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(N[Sqrt[N[(N[(2.0 * t$95$2), $MachinePrecision] * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[((-B$95$m) * B$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $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))) < -9.9999999999999999e-198

   1. Initial program 35.7%

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

   4. Applied rewrites 59.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 59.6

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

   6. Applied rewrites 59.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sqrt-prod N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower--.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-sqrt.f64 73.7

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

   8. Applied rewrites 73.7%

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

   if -9.9999999999999999e-198 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.4%

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

   4. Applied rewrites 7.8%

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

   5. Taylor expanded in B around inf

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

   7. Applied rewrites 3.2%

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

   9. Applied rewrites 4.0%

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

   10. Taylor expanded in A around -inf

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

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. lower-*.f64 45.7

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

   12. Applied rewrites 45.7%

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

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

   1. Initial program 46.5%

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

   4. Applied rewrites 94.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 94.3

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

   6. Applied rewrites 94.3%

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

   7. Taylor expanded in A around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow2 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. sqrt-pow2 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 42.7

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

   9. Applied rewrites 42.7%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 0.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 0.0

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in A around 0

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

      1. sqrt-prod N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

   9. Applied rewrites 14.7%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-+.f64 N/A

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

       4. lift-hypot.f64 N/A

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

       5. sqrt-prod N/A

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

       6. pow2 N/A

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

       7. pow2 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-sqrt.f64 N/A

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

       10. pow2 N/A

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

       11. pow2 N/A

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

       12. lower-sqrt.f64 N/A

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

       13. lift-hypot.f64 N/A

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

       14. lift-+.f64 26.7

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

   11. Applied rewrites 26.7%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-197}:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot \left(\sqrt{B \cdot B - \left(4 \cdot A\right) \cdot C} \cdot \left(-\sqrt{F}\right)\right)\right) \cdot \sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 0:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, 2 \cdot C\right)}}{\left(-B\right) \cdot B + \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot \sqrt{\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F}\right) \cdot \left(-\sqrt{C \cdot 2}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 56.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := 2 \cdot \left(\left(B\_m \cdot B\_m - t\_0\right) \cdot F\right)\\ \mathbf{if}\;B\_m \leq 3 \cdot 10^{-139}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(2 \cdot C\right)}}{\left(-B\_m\right) \cdot B\_m + t\_0}\\ \mathbf{elif}\;B\_m \leq 1.3 \cdot 10^{+54}:\\ \;\;\;\;\frac{\sqrt{t\_1} \cdot \left(-\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, 2 \cdot C\right)}\right)}{{B\_m}^{2} - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right)\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C)) (t_1 (* 2.0 (* (- (* B_m B_m) t_0) F))))
   (if (<= B_m 3e-139)
     (/ (sqrt (* t_1 (* 2.0 C))) (+ (* (- B_m) B_m) t_0))
     (if (<= B_m 1.3e+54)
       (/
        (* (sqrt t_1) (- (sqrt (fma -0.5 (/ (* B_m B_m) A) (* 2.0 C)))))
        (- (pow B_m 2.0) t_0))
       (* (/ (sqrt 2.0) (- B_m)) (* (sqrt F) (sqrt (+ 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 = (4.0 * A) * C;
	double t_1 = 2.0 * (((B_m * B_m) - t_0) * F);
	double tmp;
	if (B_m <= 3e-139) {
		tmp = sqrt((t_1 * (2.0 * C))) / ((-B_m * B_m) + t_0);
	} else if (B_m <= 1.3e+54) {
		tmp = (sqrt(t_1) * -sqrt(fma(-0.5, ((B_m * B_m) / A), (2.0 * C)))) / (pow(B_m, 2.0) - t_0);
	} else {
		tmp = (sqrt(2.0) / -B_m) * (sqrt(F) * sqrt((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 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(2.0 * Float64(Float64(Float64(B_m * B_m) - t_0) * F))
	tmp = 0.0
	if (B_m <= 3e-139)
		tmp = Float64(sqrt(Float64(t_1 * Float64(2.0 * C))) / Float64(Float64(Float64(-B_m) * B_m) + t_0));
	elseif (B_m <= 1.3e+54)
		tmp = Float64(Float64(sqrt(t_1) * Float64(-sqrt(fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(2.0 * C))))) / Float64((B_m ^ 2.0) - t_0));
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * Float64(sqrt(F) * sqrt(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[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 3e-139], N[(N[Sqrt[N[(t$95$1 * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[((-B$95$m) * B$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.3e+54], N[(N[(N[Sqrt[t$95$1], $MachinePrecision] * (-N[Sqrt[N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 2.9999999999999999e-139

   1. Initial program 17.4%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 15.9

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

   5. Applied rewrites 15.9%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 15.9

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 15.9

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

   7. Applied rewrites 15.9%

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

   if 2.9999999999999999e-139 < B < 1.30000000000000003e54

   1. Initial program 25.5%

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

   4. Applied rewrites 40.0%

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

   5. Taylor expanded in A around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 23.5

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

   7. Applied rewrites 23.5%

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

   if 1.30000000000000003e54 < B

   1. Initial program 8.4%

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

   4. Applied rewrites 17.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 17.7

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

   6. Applied rewrites 17.7%

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

   7. Taylor expanded in A around 0

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

      1. sqrt-prod N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

   9. Applied rewrites 43.3%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-+.f64 N/A

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

       4. lift-hypot.f64 N/A

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

       5. sqrt-prod N/A

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

       6. pow2 N/A

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

       7. pow2 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-sqrt.f64 N/A

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

       10. pow2 N/A

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

       11. pow2 N/A

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

       12. lower-sqrt.f64 N/A

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

       13. lift-hypot.f64 N/A

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

       14. lift-+.f64 73.0

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

   11. Applied rewrites 73.0%

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

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3 \cdot 10^{-139}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(-B\right) \cdot B + \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{+54}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, 2 \cdot C\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \left(-B\_m\right) \cdot B\_m + t\_0\\ t_2 := 2 \cdot \left(\left(B\_m \cdot B\_m - t\_0\right) \cdot F\right)\\ \mathbf{if}\;B\_m \leq 3.8 \cdot 10^{-139}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(2 \cdot C\right)}}{t\_1}\\ \mathbf{elif}\;B\_m \leq 1.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sqrt{t\_2} \cdot \left(-\sqrt{2 \cdot C}\right)}{{B\_m}^{2} - t\_0}\\ \mathbf{elif}\;B\_m \leq 2.1 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, 2 \cdot C\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right)\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C))
        (t_1 (+ (* (- B_m) B_m) t_0))
        (t_2 (* 2.0 (* (- (* B_m B_m) t_0) F))))
   (if (<= B_m 3.8e-139)
     (/ (sqrt (* t_2 (* 2.0 C))) t_1)
     (if (<= B_m 1.5e-11)
       (/ (* (sqrt t_2) (- (sqrt (* 2.0 C)))) (- (pow B_m 2.0) t_0))
       (if (<= B_m 2.1e+51)
         (/ (sqrt (* t_2 (fma -0.5 (/ (* B_m B_m) A) (* 2.0 C)))) t_1)
         (*
          (/ (sqrt 2.0) (- B_m))
          (* (sqrt F) (sqrt (+ 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 = (4.0 * A) * C;
	double t_1 = (-B_m * B_m) + t_0;
	double t_2 = 2.0 * (((B_m * B_m) - t_0) * F);
	double tmp;
	if (B_m <= 3.8e-139) {
		tmp = sqrt((t_2 * (2.0 * C))) / t_1;
	} else if (B_m <= 1.5e-11) {
		tmp = (sqrt(t_2) * -sqrt((2.0 * C))) / (pow(B_m, 2.0) - t_0);
	} else if (B_m <= 2.1e+51) {
		tmp = sqrt((t_2 * fma(-0.5, ((B_m * B_m) / A), (2.0 * C)))) / t_1;
	} else {
		tmp = (sqrt(2.0) / -B_m) * (sqrt(F) * sqrt((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 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(Float64(Float64(-B_m) * B_m) + t_0)
	t_2 = Float64(2.0 * Float64(Float64(Float64(B_m * B_m) - t_0) * F))
	tmp = 0.0
	if (B_m <= 3.8e-139)
		tmp = Float64(sqrt(Float64(t_2 * Float64(2.0 * C))) / t_1);
	elseif (B_m <= 1.5e-11)
		tmp = Float64(Float64(sqrt(t_2) * Float64(-sqrt(Float64(2.0 * C)))) / Float64((B_m ^ 2.0) - t_0));
	elseif (B_m <= 2.1e+51)
		tmp = Float64(sqrt(Float64(t_2 * fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(2.0 * C)))) / t_1);
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * Float64(sqrt(F) * sqrt(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[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[((-B$95$m) * B$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 3.8e-139], N[(N[Sqrt[N[(t$95$2 * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[B$95$m, 1.5e-11], N[(N[(N[Sqrt[t$95$2], $MachinePrecision] * (-N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 2.1e+51], N[(N[Sqrt[N[(t$95$2 * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 3.80000000000000008e-139

   1. Initial program 17.4%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 15.9

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

   5. Applied rewrites 15.9%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 15.9

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 15.9

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

   7. Applied rewrites 15.9%

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

   if 3.80000000000000008e-139 < B < 1.5e-11

   1. Initial program 27.5%

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

   4. Applied rewrites 35.1%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 20.5

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

   7. Applied rewrites 20.5%

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

   if 1.5e-11 < B < 2.1000000000000001e51

   1. Initial program 17.7%

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

   4. Applied rewrites 40.2%

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

   5. Taylor expanded in B around inf

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

   7. Applied rewrites 20.6%

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

   9. Applied rewrites 20.7%

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

   10. Taylor expanded in A around -inf

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

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. lower-*.f64 39.0

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

   12. Applied rewrites 39.0%

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

   if 2.1000000000000001e51 < B

   1. Initial program 10.0%

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

   4. Applied rewrites 20.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 20.7

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

   6. Applied rewrites 20.7%

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

   7. Taylor expanded in A around 0

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

      1. sqrt-prod N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

   9. Applied rewrites 43.5%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-+.f64 N/A

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

       4. lift-hypot.f64 N/A

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

       5. sqrt-prod N/A

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

       6. pow2 N/A

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

       7. pow2 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-sqrt.f64 N/A

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

       10. pow2 N/A

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

       11. pow2 N/A

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

       12. lower-sqrt.f64 N/A

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

       13. lift-hypot.f64 N/A

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

       14. lift-+.f64 72.2

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

   11. Applied rewrites 72.2%

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

4. Final simplification 29.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.8 \cdot 10^{-139}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(-B\right) \cdot B + \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq 1.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq 2.1 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, 2 \cdot C\right)}}{\left(-B\right) \cdot B + \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \left(-B\_m\right) \cdot B\_m + t\_0\\ t_2 := 2 \cdot \left(\left(B\_m \cdot B\_m - t\_0\right) \cdot F\right)\\ \mathbf{if}\;B\_m \leq 5.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(2 \cdot C\right)}}{t\_1}\\ \mathbf{elif}\;B\_m \leq 290000000:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)}{B\_m \cdot B\_m - 4 \cdot \left(A \cdot C\right)} \cdot 2}\\ \mathbf{elif}\;B\_m \leq 2.1 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, 2 \cdot C\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right)\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C))
        (t_1 (+ (* (- B_m) B_m) t_0))
        (t_2 (* 2.0 (* (- (* B_m B_m) t_0) F))))
   (if (<= B_m 5.8e-33)
     (/ (sqrt (* t_2 (* 2.0 C))) t_1)
     (if (<= B_m 290000000.0)
       (-
        (sqrt
         (*
          (/
           (* F (+ A (+ C (hypot B_m (- A C)))))
           (- (* B_m B_m) (* 4.0 (* A C))))
          2.0)))
       (if (<= B_m 2.1e+51)
         (/ (sqrt (* t_2 (fma -0.5 (/ (* B_m B_m) A) (* 2.0 C)))) t_1)
         (*
          (/ (sqrt 2.0) (- B_m))
          (* (sqrt F) (sqrt (+ 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 = (4.0 * A) * C;
	double t_1 = (-B_m * B_m) + t_0;
	double t_2 = 2.0 * (((B_m * B_m) - t_0) * F);
	double tmp;
	if (B_m <= 5.8e-33) {
		tmp = sqrt((t_2 * (2.0 * C))) / t_1;
	} else if (B_m <= 290000000.0) {
		tmp = -sqrt((((F * (A + (C + hypot(B_m, (A - C))))) / ((B_m * B_m) - (4.0 * (A * C)))) * 2.0));
	} else if (B_m <= 2.1e+51) {
		tmp = sqrt((t_2 * fma(-0.5, ((B_m * B_m) / A), (2.0 * C)))) / t_1;
	} else {
		tmp = (sqrt(2.0) / -B_m) * (sqrt(F) * sqrt((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 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(Float64(Float64(-B_m) * B_m) + t_0)
	t_2 = Float64(2.0 * Float64(Float64(Float64(B_m * B_m) - t_0) * F))
	tmp = 0.0
	if (B_m <= 5.8e-33)
		tmp = Float64(sqrt(Float64(t_2 * Float64(2.0 * C))) / t_1);
	elseif (B_m <= 290000000.0)
		tmp = Float64(-sqrt(Float64(Float64(Float64(F * Float64(A + Float64(C + hypot(B_m, Float64(A - C))))) / Float64(Float64(B_m * B_m) - Float64(4.0 * Float64(A * C)))) * 2.0)));
	elseif (B_m <= 2.1e+51)
		tmp = Float64(sqrt(Float64(t_2 * fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(2.0 * C)))) / t_1);
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * Float64(sqrt(F) * sqrt(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[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[((-B$95$m) * B$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 5.8e-33], N[(N[Sqrt[N[(t$95$2 * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[B$95$m, 290000000.0], (-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[(N[(B$95$m * B$95$m), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), If[LessEqual[B$95$m, 2.1e+51], N[(N[Sqrt[N[(t$95$2 * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 5.80000000000000005e-33

   1. Initial program 18.3%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 17.4

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

   5. Applied rewrites 17.4%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 17.4

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 17.4

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

   7. Applied rewrites 17.4%

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

   if 5.80000000000000005e-33 < B < 2.9e8

   1. Initial program 49.0%

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

   3. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 64.5%

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

   if 2.9e8 < B < 2.1000000000000001e51

   1. Initial program 11.6%

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

   4. Applied rewrites 38.2%

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

   5. Taylor expanded in B around inf

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

   7. Applied rewrites 14.5%

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

   9. Applied rewrites 14.6%

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

   10. Taylor expanded in A around -inf

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

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. lower-*.f64 37.0

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

   12. Applied rewrites 37.0%

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

   if 2.1000000000000001e51 < B

   1. Initial program 10.0%

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

   4. Applied rewrites 20.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 20.7

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

   6. Applied rewrites 20.7%

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

   7. Taylor expanded in A around 0

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

      1. sqrt-prod N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

   9. Applied rewrites 43.5%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-+.f64 N/A

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

       4. lift-hypot.f64 N/A

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

       5. sqrt-prod N/A

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

       6. pow2 N/A

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

       7. pow2 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-sqrt.f64 N/A

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

       10. pow2 N/A

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

       11. pow2 N/A

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

       12. lower-sqrt.f64 N/A

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

       13. lift-hypot.f64 N/A

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

       14. lift-+.f64 72.2

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

   11. Applied rewrites 72.2%

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

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(-B\right) \cdot B + \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq 290000000:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}\\ \mathbf{elif}\;B \leq 2.1 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, 2 \cdot C\right)}}{\left(-B\right) \cdot B + \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 2.1 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B\_m \cdot B\_m - t\_0\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, 2 \cdot C\right)}}{\left(-B\_m\right) \cdot B\_m + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right)\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C)))
   (if (<= B_m 2.1e+51)
     (/
      (sqrt
       (*
        (* 2.0 (* (- (* B_m B_m) t_0) F))
        (fma -0.5 (/ (* B_m B_m) A) (* 2.0 C))))
      (+ (* (- B_m) B_m) t_0))
     (* (/ (sqrt 2.0) (- B_m)) (* (sqrt F) (sqrt (+ 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 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 2.1e+51) {
		tmp = sqrt(((2.0 * (((B_m * B_m) - t_0) * F)) * fma(-0.5, ((B_m * B_m) / A), (2.0 * C)))) / ((-B_m * B_m) + t_0);
	} else {
		tmp = (sqrt(2.0) / -B_m) * (sqrt(F) * sqrt((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 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if (B_m <= 2.1e+51)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64(Float64(B_m * B_m) - t_0) * F)) * fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(2.0 * C)))) / Float64(Float64(Float64(-B_m) * B_m) + t_0));
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * Float64(sqrt(F) * sqrt(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[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[B$95$m, 2.1e+51], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[((-B$95$m) * B$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 2.1000000000000001e51

   1. Initial program 18.7%

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

   4. Applied rewrites 32.1%

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

   5. Taylor expanded in B around inf

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

   7. Applied rewrites 3.3%

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

   9. Applied rewrites 3.8%

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

   10. Taylor expanded in A around -inf

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

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. lower-*.f64 18.8

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

   12. Applied rewrites 18.8%

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

   if 2.1000000000000001e51 < B

   1. Initial program 10.0%

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

   4. Applied rewrites 20.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 20.7

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

   6. Applied rewrites 20.7%

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

   7. Taylor expanded in A around 0

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

      1. sqrt-prod N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

   9. Applied rewrites 43.5%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-+.f64 N/A

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

       4. lift-hypot.f64 N/A

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

       5. sqrt-prod N/A

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

       6. pow2 N/A

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

       7. pow2 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-sqrt.f64 N/A

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

       10. pow2 N/A

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

       11. pow2 N/A

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

       12. lower-sqrt.f64 N/A

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

       13. lift-hypot.f64 N/A

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

       14. lift-+.f64 72.2

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

   11. Applied rewrites 72.2%

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

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.1 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, 2 \cdot C\right)}}{\left(-B\right) \cdot B + \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 2.15 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B\_m \cdot B\_m - t\_0\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, 2 \cdot C\right)}}{\left(-B\_m\right) \cdot B\_m + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \mathsf{fma}\left(\sqrt{B\_m}, \sqrt{F}, 0.5 \cdot \left(\sqrt{\frac{F}{B\_m}} \cdot C\right)\right)\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C)))
   (if (<= B_m 2.15e+51)
     (/
      (sqrt
       (*
        (* 2.0 (* (- (* B_m B_m) t_0) F))
        (fma -0.5 (/ (* B_m B_m) A) (* 2.0 C))))
      (+ (* (- B_m) B_m) t_0))
     (*
      (/ (sqrt 2.0) (- B_m))
      (fma (sqrt B_m) (sqrt F) (* 0.5 (* (sqrt (/ F 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 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 2.15e+51) {
		tmp = sqrt(((2.0 * (((B_m * B_m) - t_0) * F)) * fma(-0.5, ((B_m * B_m) / A), (2.0 * C)))) / ((-B_m * B_m) + t_0);
	} else {
		tmp = (sqrt(2.0) / -B_m) * fma(sqrt(B_m), sqrt(F), (0.5 * (sqrt((F / 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 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if (B_m <= 2.15e+51)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64(Float64(B_m * B_m) - t_0) * F)) * fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(2.0 * C)))) / Float64(Float64(Float64(-B_m) * B_m) + t_0));
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * fma(sqrt(B_m), sqrt(F), Float64(0.5 * Float64(sqrt(Float64(F / 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[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[B$95$m, 2.15e+51], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[((-B$95$m) * B$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[(N[Sqrt[B$95$m], $MachinePrecision] * N[Sqrt[F], $MachinePrecision] + N[(0.5 * N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 2.1499999999999999e51

   1. Initial program 18.7%

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

   4. Applied rewrites 32.1%

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

   5. Taylor expanded in B around inf

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

   7. Applied rewrites 3.3%

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

   9. Applied rewrites 3.8%

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

   10. Taylor expanded in A around -inf

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

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. lower-*.f64 18.8

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

   12. Applied rewrites 18.8%

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

   if 2.1499999999999999e51 < B

   1. Initial program 10.0%

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

   4. Applied rewrites 20.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 20.7

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

   6. Applied rewrites 20.7%

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

   7. Taylor expanded in A around 0

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

      1. sqrt-prod N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

   9. Applied rewrites 43.5%

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

   10. Taylor expanded in C around 0

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

       1. sqrt-prod N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lower-/.f64 64.1

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

   12. Applied rewrites 64.1%

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

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.15 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, 2 \cdot C\right)}}{\left(-B\right) \cdot B + \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \mathsf{fma}\left(\sqrt{B}, \sqrt{F}, 0.5 \cdot \left(\sqrt{\frac{F}{B}} \cdot C\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.1% accurate, 5.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 2.6 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B\_m \cdot B\_m - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(-B\_m\right) \cdot B\_m + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \mathsf{fma}\left(\sqrt{B\_m}, \sqrt{F}, 0.5 \cdot \left(\sqrt{\frac{F}{B\_m}} \cdot C\right)\right)\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C)))
   (if (<= B_m 2.6e+51)
     (/
      (sqrt (* (* 2.0 (* (- (* B_m B_m) t_0) F)) (* 2.0 C)))
      (+ (* (- B_m) B_m) t_0))
     (*
      (/ (sqrt 2.0) (- B_m))
      (fma (sqrt B_m) (sqrt F) (* 0.5 (* (sqrt (/ F 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 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 2.6e+51) {
		tmp = sqrt(((2.0 * (((B_m * B_m) - t_0) * F)) * (2.0 * C))) / ((-B_m * B_m) + t_0);
	} else {
		tmp = (sqrt(2.0) / -B_m) * fma(sqrt(B_m), sqrt(F), (0.5 * (sqrt((F / 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 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if (B_m <= 2.6e+51)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64(Float64(B_m * B_m) - t_0) * F)) * Float64(2.0 * C))) / Float64(Float64(Float64(-B_m) * B_m) + t_0));
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * fma(sqrt(B_m), sqrt(F), Float64(0.5 * Float64(sqrt(Float64(F / 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[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[B$95$m, 2.6e+51], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[((-B$95$m) * B$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[(N[Sqrt[B$95$m], $MachinePrecision] * N[Sqrt[F], $MachinePrecision] + N[(0.5 * N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 2.6000000000000001e51

   1. Initial program 18.7%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 17.2

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

   5. Applied rewrites 17.2%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 17.2

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 17.2

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

   7. Applied rewrites 17.2%

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

   if 2.6000000000000001e51 < B

   1. Initial program 10.0%

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

   4. Applied rewrites 20.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 20.7

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

   6. Applied rewrites 20.7%

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

   7. Taylor expanded in A around 0

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

      1. sqrt-prod N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

   9. Applied rewrites 43.5%

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

   10. Taylor expanded in C around 0

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

       1. sqrt-prod N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lower-/.f64 64.1

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

   12. Applied rewrites 64.1%

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

4. Final simplification 27.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.6 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(-B\right) \cdot B + \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \mathsf{fma}\left(\sqrt{B}, \sqrt{F}, 0.5 \cdot \left(\sqrt{\frac{F}{B}} \cdot C\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 2.8 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B\_m \cdot B\_m - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(-B\_m\right) \cdot B\_m + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{F \cdot \left(C + B\_m\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)))
   (if (<= B_m 2.8e+51)
     (/
      (sqrt (* (* 2.0 (* (- (* B_m B_m) t_0) F)) (* 2.0 C)))
      (+ (* (- B_m) B_m) t_0))
     (* (/ (sqrt 2.0) (- B_m)) (sqrt (* F (+ C B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 2.8e+51) {
		tmp = sqrt(((2.0 * (((B_m * B_m) - t_0) * F)) * (2.0 * C))) / ((-B_m * B_m) + t_0);
	} else {
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (C + B_m)));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (4.0d0 * a) * c
    if (b_m <= 2.8d+51) then
        tmp = sqrt(((2.0d0 * (((b_m * b_m) - t_0) * f)) * (2.0d0 * c))) / ((-b_m * b_m) + t_0)
    else
        tmp = (sqrt(2.0d0) / -b_m) * sqrt((f * (c + b_m)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 2.8e+51) {
		tmp = Math.sqrt(((2.0 * (((B_m * B_m) - t_0) * F)) * (2.0 * C))) / ((-B_m * B_m) + t_0);
	} else {
		tmp = (Math.sqrt(2.0) / -B_m) * Math.sqrt((F * (C + B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = (4.0 * A) * C
	tmp = 0
	if B_m <= 2.8e+51:
		tmp = math.sqrt(((2.0 * (((B_m * B_m) - t_0) * F)) * (2.0 * C))) / ((-B_m * B_m) + t_0)
	else:
		tmp = (math.sqrt(2.0) / -B_m) * math.sqrt((F * (C + B_m)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if (B_m <= 2.8e+51)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64(Float64(B_m * B_m) - t_0) * F)) * Float64(2.0 * C))) / Float64(Float64(Float64(-B_m) * B_m) + t_0));
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(F * Float64(C + B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	tmp = 0.0;
	if (B_m <= 2.8e+51)
		tmp = sqrt(((2.0 * (((B_m * B_m) - t_0) * F)) * (2.0 * C))) / ((-B_m * B_m) + t_0);
	else
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (C + B_m)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[B$95$m, 2.8e+51], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[((-B$95$m) * B$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(F * N[(C + B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 2.80000000000000005e51

   1. Initial program 18.7%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 17.2

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

   5. Applied rewrites 17.2%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 17.2

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 17.2

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

   7. Applied rewrites 17.2%

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

   if 2.80000000000000005e51 < B

   1. Initial program 10.0%

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

   4. Applied rewrites 20.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 20.7

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

   6. Applied rewrites 20.7%

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

   7. Taylor expanded in A around 0

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

      1. sqrt-prod N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

   9. Applied rewrites 43.5%

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

   10. Taylor expanded in B around inf

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

   12. Applied rewrites 40.3%

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

4. Final simplification 22.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.8 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(-B\right) \cdot B + \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(C + B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.1% accurate, 8.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 2.4 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-16}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{F \cdot \left(C + B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 2.4e-290)
   (sqrt (/ (- F) A))
   (if (<= F 2.8e-16)
     (* (/ (sqrt 2.0) (- B_m)) (sqrt (* F (+ C B_m))))
     (- (sqrt (* (/ F B_m) 2.0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 2.4e-290) {
		tmp = sqrt((-F / A));
	} else if (F <= 2.8e-16) {
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (C + B_m)));
	} else {
		tmp = -sqrt(((F / B_m) * 2.0));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= 2.4d-290) then
        tmp = sqrt((-f / a))
    else if (f <= 2.8d-16) then
        tmp = (sqrt(2.0d0) / -b_m) * sqrt((f * (c + b_m)))
    else
        tmp = -sqrt(((f / b_m) * 2.0d0))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 2.4e-290) {
		tmp = Math.sqrt((-F / A));
	} else if (F <= 2.8e-16) {
		tmp = (Math.sqrt(2.0) / -B_m) * Math.sqrt((F * (C + B_m)));
	} else {
		tmp = -Math.sqrt(((F / B_m) * 2.0));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= 2.4e-290:
		tmp = math.sqrt((-F / A))
	elif F <= 2.8e-16:
		tmp = (math.sqrt(2.0) / -B_m) * math.sqrt((F * (C + B_m)))
	else:
		tmp = -math.sqrt(((F / B_m) * 2.0))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 2.4e-290)
		tmp = sqrt(Float64(Float64(-F) / A));
	elseif (F <= 2.8e-16)
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(F * Float64(C + B_m))));
	else
		tmp = Float64(-sqrt(Float64(Float64(F / B_m) * 2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 2.4e-290)
		tmp = sqrt((-F / A));
	elseif (F <= 2.8e-16)
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (C + B_m)));
	else
		tmp = -sqrt(((F / B_m) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 2.4e-290], N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision], If[LessEqual[F, 2.8e-16], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(F * N[(C + B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if F < 2.4000000000000001e-290

   1. Initial program 25.4%

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

   3. Taylor expanded in F around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 41.0%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 26.9

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

   8. Applied rewrites 26.9%

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

   if 2.4000000000000001e-290 < F < 2.8000000000000001e-16

   1. Initial program 22.6%

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

   4. Applied rewrites 36.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 36.8

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

   6. Applied rewrites 36.8%

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

   7. Taylor expanded in A around 0

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

      1. sqrt-prod N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

   9. Applied rewrites 23.2%

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

   10. Taylor expanded in B around inf

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

   12. Applied rewrites 19.5%

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

   if 2.8000000000000001e-16 < F

   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. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. 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 17.1

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

   5. Applied rewrites 17.1%

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

4. Final simplification 19.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 2.4 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-16}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(C + B\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.8% accurate, 8.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 2.4 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;F \leq 6.4 \cdot 10^{-30}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{F \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 2.4e-290)
   (sqrt (/ (- F) A))
   (if (<= F 6.4e-30)
     (* (/ (sqrt 2.0) (- B_m)) (sqrt (* F B_m)))
     (- (sqrt (* (/ F B_m) 2.0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 2.4e-290) {
		tmp = sqrt((-F / A));
	} else if (F <= 6.4e-30) {
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * B_m));
	} else {
		tmp = -sqrt(((F / B_m) * 2.0));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= 2.4d-290) then
        tmp = sqrt((-f / a))
    else if (f <= 6.4d-30) then
        tmp = (sqrt(2.0d0) / -b_m) * sqrt((f * b_m))
    else
        tmp = -sqrt(((f / b_m) * 2.0d0))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 2.4e-290) {
		tmp = Math.sqrt((-F / A));
	} else if (F <= 6.4e-30) {
		tmp = (Math.sqrt(2.0) / -B_m) * Math.sqrt((F * B_m));
	} else {
		tmp = -Math.sqrt(((F / B_m) * 2.0));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= 2.4e-290:
		tmp = math.sqrt((-F / A))
	elif F <= 6.4e-30:
		tmp = (math.sqrt(2.0) / -B_m) * math.sqrt((F * B_m))
	else:
		tmp = -math.sqrt(((F / B_m) * 2.0))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 2.4e-290)
		tmp = sqrt(Float64(Float64(-F) / A));
	elseif (F <= 6.4e-30)
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(F * B_m)));
	else
		tmp = Float64(-sqrt(Float64(Float64(F / B_m) * 2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 2.4e-290)
		tmp = sqrt((-F / A));
	elseif (F <= 6.4e-30)
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * B_m));
	else
		tmp = -sqrt(((F / B_m) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 2.4e-290], N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision], If[LessEqual[F, 6.4e-30], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(F * B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if F < 2.4000000000000001e-290

   1. Initial program 25.4%

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

   3. Taylor expanded in F around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 41.0%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 26.9

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

   8. Applied rewrites 26.9%

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

   if 2.4000000000000001e-290 < F < 6.400000000000001e-30

   1. Initial program 21.5%

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

   4. Applied rewrites 36.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 36.3

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

   6. Applied rewrites 36.3%

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

   7. Taylor expanded in A around 0

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

      1. sqrt-prod N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

   9. Applied rewrites 23.0%

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

   10. Taylor expanded in B around inf

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

   12. Applied rewrites 19.8%

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

   if 6.400000000000001e-30 < F

   1. Initial program 10.8%

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

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. 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 17.4

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

   5. Applied rewrites 17.4%

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

4. Final simplification 19.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 2.4 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;F \leq 6.4 \cdot 10^{-30}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot B}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 39.0% accurate, 14.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 7 \cdot 10^{-33}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 7e-33) (sqrt (/ (- F) A)) (- (sqrt (* (/ F B_m) 2.0)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 7e-33) {
		tmp = sqrt((-F / A));
	} else {
		tmp = -sqrt(((F / B_m) * 2.0));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 7d-33) then
        tmp = sqrt((-f / a))
    else
        tmp = -sqrt(((f / b_m) * 2.0d0))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 7e-33) {
		tmp = Math.sqrt((-F / A));
	} else {
		tmp = -Math.sqrt(((F / B_m) * 2.0));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 7e-33:
		tmp = math.sqrt((-F / A))
	else:
		tmp = -math.sqrt(((F / B_m) * 2.0))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 7e-33)
		tmp = sqrt(Float64(Float64(-F) / A));
	else
		tmp = Float64(-sqrt(Float64(Float64(F / B_m) * 2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 7e-33)
		tmp = sqrt((-F / A));
	else
		tmp = -sqrt(((F / B_m) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 7e-33], N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision], (-N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < 6.9999999999999997e-33

   1. Initial program 18.3%

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

   3. Taylor expanded in F around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 9.5%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 14.5

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

   8. Applied rewrites 14.5%

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

   if 6.9999999999999997e-33 < B

   1. Initial program 13.1%

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

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. 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 40.3

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

   5. Applied rewrites 40.3%

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

4. Final simplification 21.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7 \cdot 10^{-33}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 1.6% accurate, 18.2× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (sqrt (* -2.0 (/ F B_m))))

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.9%

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

3. Taylor expanded in F around -inf

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. sqrt-unprod N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-*.f64 N/A

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

5. Applied rewrites 8.0%

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

6. Taylor expanded in B around -inf

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

   1. lower-*.f64 N/A

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

   2. lift-/.f64 2.0

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

8. Applied rewrites 2.0%

   \[\leadsto \sqrt{-2 \cdot \frac{F}{B}} \]
9. Add Preprocessing

Alternative 13: 20.3% accurate, 20.5× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (sqrt (/ (- F) A)))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return sqrt((-F / A));
}

Fortran

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

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

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt((-f / a))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt((-F / A));
}

Python

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

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return sqrt(Float64(Float64(-F) / A))
end

MATLAB

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

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 16.9%

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

3. Taylor expanded in F around -inf

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. sqrt-unprod N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-*.f64 N/A

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

5. Applied rewrites 8.0%

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

6. Taylor expanded in A around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 11.4

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

8. Applied rewrites 11.4%

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

9. Final simplification 11.4%

   \[\leadsto \sqrt{\frac{-F}{A}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025051 
(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))))