ABCF->ab-angle b

Percentage Accurate: 18.9% → 51.2%

Time: 12.6s

Alternatives: 19

Speedup: 14.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.9% 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: 51.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(C + A\right) - \mathsf{hypot}\left(A - C, B\_m\right)\\ t_1 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ t_2 := \left(2 \cdot F\right) \cdot t\_1\\ t_3 := -t\_1\\ t_4 := -\sqrt{2}\\ t_5 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_6 := \frac{\sqrt{\left(2 \cdot \left(t\_5 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_5}\\ \mathbf{if}\;t\_6 \leq -\infty:\\ \;\;\;\;\sqrt{F \cdot \frac{t\_0}{\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)}} \cdot t\_4\\ \mathbf{elif}\;t\_6 \leq -1 \cdot 10^{-203}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot t\_2}}{t\_3}\\ \mathbf{elif}\;t\_6 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - 0.5 \cdot \frac{B\_m \cdot B\_m}{C}\right) + A\right) \cdot t\_2}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{B\_m}} \cdot t\_4\\ \end{array} \end{array} \]

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 (- (+ C A) (hypot (- A C) B_m)))
        (t_1 (fma (* -4.0 A) C (* B_m B_m)))
        (t_2 (* (* 2.0 F) t_1))
        (t_3 (- t_1))
        (t_4 (- (sqrt 2.0)))
        (t_5 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_6
         (/
          (sqrt
           (*
            (* 2.0 (* t_5 F))
            (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_5))))
   (if (<= t_6 (- INFINITY))
     (* (sqrt (* F (/ t_0 (fma (* C A) -4.0 (* B_m B_m))))) t_4)
     (if (<= t_6 -1e-203)
       (/ (sqrt (* t_0 t_2)) t_3)
       (if (<= t_6 INFINITY)
         (/ (sqrt (* (+ (- A (* 0.5 (/ (* B_m B_m) C))) A) t_2)) t_3)
         (* (sqrt (/ (- F) B_m)) t_4))))))

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 = (C + A) - hypot((A - C), B_m);
	double t_1 = fma((-4.0 * A), C, (B_m * B_m));
	double t_2 = (2.0 * F) * t_1;
	double t_3 = -t_1;
	double t_4 = -sqrt(2.0);
	double t_5 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_6 = sqrt(((2.0 * (t_5 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_5;
	double tmp;
	if (t_6 <= -((double) INFINITY)) {
		tmp = sqrt((F * (t_0 / fma((C * A), -4.0, (B_m * B_m))))) * t_4;
	} else if (t_6 <= -1e-203) {
		tmp = sqrt((t_0 * t_2)) / t_3;
	} else if (t_6 <= ((double) INFINITY)) {
		tmp = sqrt((((A - (0.5 * ((B_m * B_m) / C))) + A) * t_2)) / t_3;
	} else {
		tmp = sqrt((-F / B_m)) * t_4;
	}
	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(C + A) - hypot(Float64(A - C), B_m))
	t_1 = fma(Float64(-4.0 * A), C, Float64(B_m * B_m))
	t_2 = Float64(Float64(2.0 * F) * t_1)
	t_3 = Float64(-t_1)
	t_4 = Float64(-sqrt(2.0))
	t_5 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_6 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_5 * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_5))
	tmp = 0.0
	if (t_6 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(F * Float64(t_0 / fma(Float64(C * A), -4.0, Float64(B_m * B_m))))) * t_4);
	elseif (t_6 <= -1e-203)
		tmp = Float64(sqrt(Float64(t_0 * t_2)) / t_3);
	elseif (t_6 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(Float64(A - Float64(0.5 * Float64(Float64(B_m * B_m) / C))) + A) * t_2)) / t_3);
	else
		tmp = Float64(sqrt(Float64(Float64(-F) / B_m)) * t_4);
	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[(C + A), $MachinePrecision] - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * F), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = (-t$95$1)}, Block[{t$95$4 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$5 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$5 * 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$5)), $MachinePrecision]}, If[LessEqual[t$95$6, (-Infinity)], N[(N[Sqrt[N[(F * N[(t$95$0 / N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$4), $MachinePrecision], If[LessEqual[t$95$6, -1e-203], N[(N[Sqrt[N[(t$95$0 * t$95$2), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$6, Infinity], N[(N[Sqrt[N[(N[(N[(A - N[(0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + A), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[Sqrt[N[((-F) / B$95$m), $MachinePrecision]], $MachinePrecision] * t$95$4), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.1%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 63.9%

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

   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))) < -1e-203

   1. Initial program 98.5%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 98.5%

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

   if -1e-203 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 10.1%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 25.1%

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

   5. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 28.2

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

   7. Applied rewrites 28.2%

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

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

   1. Initial program 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 0.4%

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

   5. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 3.2

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

   7. Applied rewrites 3.2%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   10. Applied rewrites 4.4%

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

   11. Taylor expanded in B around inf

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

   13. Applied rewrites 18.3%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -\infty:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-203}:\\ \;\;\;\;\frac{\sqrt{\left(\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - 0.5 \cdot \frac{B \cdot B}{C}\right) + A\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 50.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * A), C, Float64(B_m * B_m))
	t_1 = Float64(Float64(2.0 * F) * t_0)
	t_2 = Float64(-t_0)
	t_3 = Float64(-sqrt(2.0))
	t_4 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_5 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_4 * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_4))
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(C + A) - hypot(Float64(A - C), B_m)) / fma(Float64(C * A), -4.0, Float64(B_m * B_m))))) * t_3);
	elseif (t_5 <= -1e-203)
		tmp = Float64(sqrt(Float64(Float64(Float64(C + A) - hypot(A, B_m)) * t_1)) / t_2);
	elseif (t_5 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(Float64(A - Float64(0.5 * Float64(Float64(B_m * B_m) / C))) + A) * t_1)) / t_2);
	else
		tmp = Float64(sqrt(Float64(Float64(-F) / B_m)) * t_3);
	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 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = (-t$95$0)}, Block[{t$95$3 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$4 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$4 * 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$4)), $MachinePrecision]}, If[LessEqual[t$95$5, (-Infinity)], N[(N[Sqrt[N[(F * N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[t$95$5, -1e-203], N[(N[Sqrt[N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(N[Sqrt[N[(N[(N[(A - N[(0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + A), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[Sqrt[N[((-F) / B$95$m), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.1%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 63.9%

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

   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))) < -1e-203

   1. Initial program 98.5%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in C around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. lower-hypot.f64 78.8

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

   7. Applied rewrites 78.8%

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

   if -1e-203 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 10.1%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 25.1%

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

   5. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 28.2

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

   7. Applied rewrites 28.2%

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

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

   1. Initial program 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 0.4%

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

   5. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 3.2

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

   7. Applied rewrites 3.2%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   10. Applied rewrites 4.4%

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

   11. Taylor expanded in B around inf

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

   13. Applied rewrites 18.3%

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

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -\infty:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-203}:\\ \;\;\;\;\frac{\sqrt{\left(\left(C + A\right) - \mathsf{hypot}\left(A, B\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - 0.5 \cdot \frac{B \cdot B}{C}\right) + A\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 48.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ t_1 := -\sqrt{2}\\ \mathbf{if}\;B\_m \leq 2.25 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - 0.5 \cdot \frac{B\_m \cdot B\_m}{C}\right) + A\right) \cdot \left(\left(2 \cdot F\right) \cdot t\_0\right)}}{-t\_0}\\ \mathbf{elif}\;B\_m \leq 5 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\_m\right)}{\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)}} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{C + A}{B\_m} - 1}{B\_m}} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 A) C (* B_m B_m))) (t_1 (- (sqrt 2.0))))
   (if (<= B_m 2.25e-29)
     (/
      (sqrt (* (+ (- A (* 0.5 (/ (* B_m B_m) C))) A) (* (* 2.0 F) t_0)))
      (- t_0))
     (if (<= B_m 5e+148)
       (*
        (sqrt
         (*
          F
          (/ (- (+ C A) (hypot (- A C) B_m)) (fma (* C A) -4.0 (* B_m B_m)))))
        t_1)
       (* (sqrt (* F (/ (- (/ (+ C A) B_m) 1.0) B_m))) t_1)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * A), C, (B_m * B_m));
	double t_1 = -sqrt(2.0);
	double tmp;
	if (B_m <= 2.25e-29) {
		tmp = sqrt((((A - (0.5 * ((B_m * B_m) / C))) + A) * ((2.0 * F) * t_0))) / -t_0;
	} else if (B_m <= 5e+148) {
		tmp = sqrt((F * (((C + A) - hypot((A - C), B_m)) / fma((C * A), -4.0, (B_m * B_m))))) * t_1;
	} else {
		tmp = sqrt((F * ((((C + A) / B_m) - 1.0) / B_m))) * t_1;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * A), C, Float64(B_m * B_m))
	t_1 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (B_m <= 2.25e-29)
		tmp = Float64(sqrt(Float64(Float64(Float64(A - Float64(0.5 * Float64(Float64(B_m * B_m) / C))) + A) * Float64(Float64(2.0 * F) * t_0))) / Float64(-t_0));
	elseif (B_m <= 5e+148)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(C + A) - hypot(Float64(A - C), B_m)) / fma(Float64(C * A), -4.0, Float64(B_m * B_m))))) * t_1);
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(C + A) / B_m) - 1.0) / B_m))) * t_1);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[B$95$m, 2.25e-29], N[(N[Sqrt[N[(N[(N[(A - N[(0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + A), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[B$95$m, 5e+148], N[(N[Sqrt[N[(F * N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[Sqrt[N[(F * N[(N[(N[(N[(C + A), $MachinePrecision] / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 2.2499999999999999e-29

   1. Initial program 20.1%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 28.3%

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

   5. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 14.2

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

   7. Applied rewrites 14.2%

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

   if 2.2499999999999999e-29 < B < 5.00000000000000024e148

   1. Initial program 38.1%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 57.3%

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

   if 5.00000000000000024e148 < B

   1. Initial program 3.3%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 3.3%

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

   5. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 2.8

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

   7. Applied rewrites 2.8%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   10. Applied rewrites 7.5%

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

   11. Taylor expanded in B around inf

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

   13. Applied rewrites 59.5%

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

4. Final simplification 26.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.25 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - 0.5 \cdot \frac{B \cdot B}{C}\right) + A\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 5 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{C + A}{B} - 1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 46.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 3.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - 0.5 \cdot \frac{B\_m \cdot B\_m}{C}\right) + A\right) \cdot \left(\left(2 \cdot F\right) \cdot t\_0\right)}}{-t\_0}\\ \mathbf{elif}\;B\_m \leq 2.6 \cdot 10^{+187}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - \mathsf{hypot}\left(A, B\_m\right)\right) \cdot F\right) \cdot 2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{C + A}{B\_m} - 1}{B\_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 A) C (* B_m B_m))))
   (if (<= B_m 3.5e-29)
     (/
      (sqrt (* (+ (- A (* 0.5 (/ (* B_m B_m) C))) A) (* (* 2.0 F) t_0)))
      (- t_0))
     (if (<= B_m 2.6e+187)
       (/ (sqrt (* (* (- A (hypot A B_m)) F) 2.0)) (- B_m))
       (* (sqrt (* F (/ (- (/ (+ C A) B_m) 1.0) B_m))) (- (sqrt 2.0)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * A), C, (B_m * B_m));
	double tmp;
	if (B_m <= 3.5e-29) {
		tmp = sqrt((((A - (0.5 * ((B_m * B_m) / C))) + A) * ((2.0 * F) * t_0))) / -t_0;
	} else if (B_m <= 2.6e+187) {
		tmp = sqrt((((A - hypot(A, B_m)) * F) * 2.0)) / -B_m;
	} else {
		tmp = sqrt((F * ((((C + A) / B_m) - 1.0) / B_m))) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * A), C, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 3.5e-29)
		tmp = Float64(sqrt(Float64(Float64(Float64(A - Float64(0.5 * Float64(Float64(B_m * B_m) / C))) + A) * Float64(Float64(2.0 * F) * t_0))) / Float64(-t_0));
	elseif (B_m <= 2.6e+187)
		tmp = Float64(sqrt(Float64(Float64(Float64(A - hypot(A, B_m)) * F) * 2.0)) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(C + A) / B_m) - 1.0) / B_m))) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 3.5e-29], N[(N[Sqrt[N[(N[(N[(A - N[(0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + A), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[B$95$m, 2.6e+187], N[(N[Sqrt[N[(N[(N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(F * N[(N[(N[(N[(C + A), $MachinePrecision] / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 3.4999999999999997e-29

   1. Initial program 20.1%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 28.3%

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

   5. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 14.2

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

   7. Applied rewrites 14.2%

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

   if 3.4999999999999997e-29 < B < 2.5999999999999999e187

   1. Initial program 32.6%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 40.1

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

   5. Applied rewrites 40.1%

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

   7. Applied rewrites 40.3%

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

   if 2.5999999999999999e187 < B

   1. Initial program 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 2.9

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

   7. Applied rewrites 2.9%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   10. Applied rewrites 4.7%

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

   11. Taylor expanded in B around inf

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

   13. Applied rewrites 62.8%

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

4. Final simplification 23.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - 0.5 \cdot \frac{B \cdot B}{C}\right) + A\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{+187}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F\right) \cdot 2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{C + A}{B} - 1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 43.6% accurate, 4.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 2.6 \cdot 10^{+18}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - 0.5 \cdot \frac{B\_m \cdot B\_m}{C}\right) + A\right) \cdot \left(\left(2 \cdot F\right) \cdot t\_0\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{C + A}{B\_m} - 1}{B\_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 A) C (* B_m B_m))))
   (if (<= B_m 2.6e+18)
     (/
      (sqrt (* (+ (- A (* 0.5 (/ (* B_m B_m) C))) A) (* (* 2.0 F) t_0)))
      (- t_0))
     (* (sqrt (* F (/ (- (/ (+ C A) B_m) 1.0) B_m))) (- (sqrt 2.0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * A), C, (B_m * B_m));
	double tmp;
	if (B_m <= 2.6e+18) {
		tmp = sqrt((((A - (0.5 * ((B_m * B_m) / C))) + A) * ((2.0 * F) * t_0))) / -t_0;
	} else {
		tmp = sqrt((F * ((((C + A) / B_m) - 1.0) / B_m))) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * A), C, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 2.6e+18)
		tmp = Float64(sqrt(Float64(Float64(Float64(A - Float64(0.5 * Float64(Float64(B_m * B_m) / C))) + A) * Float64(Float64(2.0 * F) * t_0))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(C + A) / B_m) - 1.0) / B_m))) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 2.6e+18], N[(N[Sqrt[N[(N[(N[(A - N[(0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + A), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(F * N[(N[(N[(N[(C + A), $MachinePrecision] / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 2.6e18

   1. Initial program 22.2%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 29.8%

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

   5. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 14.3

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

   7. Applied rewrites 14.3%

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

   if 2.6e18 < B

   1. Initial program 16.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 18.4%

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

   5. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 3.9

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

   7. Applied rewrites 3.9%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   10. Applied rewrites 32.6%

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

   11. Taylor expanded in B around inf

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

   13. Applied rewrites 51.7%

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

4. Final simplification 22.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.6 \cdot 10^{+18}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - 0.5 \cdot \frac{B \cdot B}{C}\right) + A\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{C + A}{B} - 1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 3.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\left(2 \cdot F\right) \cdot t\_0\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{C + A}{B\_m} - 1}{B\_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 A) C (* B_m B_m))))
   (if (<= B_m 3.2e+18)
     (/ (sqrt (* (+ A A) (* (* 2.0 F) t_0))) (- t_0))
     (* (sqrt (* F (/ (- (/ (+ C A) B_m) 1.0) B_m))) (- (sqrt 2.0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * A), C, (B_m * B_m));
	double tmp;
	if (B_m <= 3.2e+18) {
		tmp = sqrt(((A + A) * ((2.0 * F) * t_0))) / -t_0;
	} else {
		tmp = sqrt((F * ((((C + A) / B_m) - 1.0) / B_m))) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * A), C, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 3.2e+18)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(Float64(2.0 * F) * t_0))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(C + A) / B_m) - 1.0) / B_m))) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 3.2e+18], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(F * N[(N[(N[(N[(C + A), $MachinePrecision] / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 3.2e18

   1. Initial program 22.2%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 29.8%

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

   5. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. lower-neg.f64 15.1

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

   7. Applied rewrites 15.1%

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

   if 3.2e18 < B

   1. Initial program 16.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 18.4%

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

   5. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 3.9

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

   7. Applied rewrites 3.9%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   10. Applied rewrites 32.6%

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

   11. Taylor expanded in B around inf

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

   13. Applied rewrites 51.7%

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

4. Final simplification 23.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{C + A}{B} - 1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 37.0% accurate, 6.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;B\_m \leq 1.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)}\\ \mathbf{elif}\;B\_m \leq 3.4 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - B\_m\right) \cdot F\right) \cdot 2}}{-B\_m}\\ \mathbf{elif}\;B\_m \leq 2.6 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{C + A}{B\_m} - 1}{B\_m}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))))
   (if (<= B_m 1.2e-127)
     (/
      (sqrt (* (* -16.0 (* A A)) (* C F)))
      (- (fma (* -4.0 A) C (* B_m B_m))))
     (if (<= B_m 3.4e-90)
       (/ (sqrt (* (* (- A B_m) F) 2.0)) (- B_m))
       (if (<= B_m 2.6e+18)
         (* (sqrt (* F (/ -0.5 C))) t_0)
         (* (sqrt (* F (/ (- (/ (+ C A) B_m) 1.0) B_m))) t_0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0);
	double tmp;
	if (B_m <= 1.2e-127) {
		tmp = sqrt(((-16.0 * (A * A)) * (C * F))) / -fma((-4.0 * A), C, (B_m * B_m));
	} else if (B_m <= 3.4e-90) {
		tmp = sqrt((((A - B_m) * F) * 2.0)) / -B_m;
	} else if (B_m <= 2.6e+18) {
		tmp = sqrt((F * (-0.5 / C))) * t_0;
	} else {
		tmp = sqrt((F * ((((C + A) / B_m) - 1.0) / B_m))) * t_0;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (B_m <= 1.2e-127)
		tmp = Float64(sqrt(Float64(Float64(-16.0 * Float64(A * A)) * Float64(C * F))) / Float64(-fma(Float64(-4.0 * A), C, Float64(B_m * B_m))));
	elseif (B_m <= 3.4e-90)
		tmp = Float64(sqrt(Float64(Float64(Float64(A - B_m) * F) * 2.0)) / Float64(-B_m));
	elseif (B_m <= 2.6e+18)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * t_0);
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(C + A) / B_m) - 1.0) / B_m))) * t_0);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[B$95$m, 1.2e-127], N[(N[Sqrt[N[(N[(-16.0 * N[(A * A), $MachinePrecision]), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 3.4e-90], N[(N[Sqrt[N[(N[(N[(A - B$95$m), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], If[LessEqual[B$95$m, 2.6e+18], N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[N[(F * N[(N[(N[(N[(C + A), $MachinePrecision] / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 1.19999999999999991e-127

   1. Initial program 18.5%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 27.0%

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

   5. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 13.4

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

   7. Applied rewrites 13.4%

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

   8. Taylor expanded in A around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 9.7

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

   10. Applied rewrites 9.7%

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

   if 1.19999999999999991e-127 < B < 3.39999999999999994e-90

   1. Initial program 60.3%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 62.0

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

   5. Applied rewrites 62.0%

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

   7. Applied rewrites 62.6%

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

   8. Taylor expanded in A around 0

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

   10. Applied rewrites 61.9%

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

   if 3.39999999999999994e-90 < B < 2.6e18

   1. Initial program 37.2%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 41.0%

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

   5. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 9.6

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

   7. Applied rewrites 9.6%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   10. Applied rewrites 37.4%

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

   11. Taylor expanded in A around -inf

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

   13. Applied rewrites 9.4%

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

   if 2.6e18 < B

   1. Initial program 16.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 18.4%

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

   5. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 3.9

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

   7. Applied rewrites 3.9%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   10. Applied rewrites 32.6%

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

   11. Taylor expanded in B around inf

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

   13. Applied rewrites 51.7%

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

4. Final simplification 20.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 3.4 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - B\right) \cdot F\right) \cdot 2}}{-B}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{C + A}{B} - 1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 37.0% accurate, 6.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;B\_m \leq 1.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)}\\ \mathbf{elif}\;B\_m \leq 3.4 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - B\_m\right) \cdot F\right) \cdot 2}}{-B\_m}\\ \mathbf{elif}\;B\_m \leq 2.6 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(F, \frac{C + A}{B\_m}, -F\right)}{B\_m}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))))
   (if (<= B_m 1.2e-127)
     (/
      (sqrt (* (* -16.0 (* A A)) (* C F)))
      (- (fma (* -4.0 A) C (* B_m B_m))))
     (if (<= B_m 3.4e-90)
       (/ (sqrt (* (* (- A B_m) F) 2.0)) (- B_m))
       (if (<= B_m 2.6e+18)
         (* (sqrt (* F (/ -0.5 C))) t_0)
         (* (sqrt (/ (fma F (/ (+ C A) B_m) (- F)) B_m)) t_0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0);
	double tmp;
	if (B_m <= 1.2e-127) {
		tmp = sqrt(((-16.0 * (A * A)) * (C * F))) / -fma((-4.0 * A), C, (B_m * B_m));
	} else if (B_m <= 3.4e-90) {
		tmp = sqrt((((A - B_m) * F) * 2.0)) / -B_m;
	} else if (B_m <= 2.6e+18) {
		tmp = sqrt((F * (-0.5 / C))) * t_0;
	} else {
		tmp = sqrt((fma(F, ((C + A) / B_m), -F) / B_m)) * t_0;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (B_m <= 1.2e-127)
		tmp = Float64(sqrt(Float64(Float64(-16.0 * Float64(A * A)) * Float64(C * F))) / Float64(-fma(Float64(-4.0 * A), C, Float64(B_m * B_m))));
	elseif (B_m <= 3.4e-90)
		tmp = Float64(sqrt(Float64(Float64(Float64(A - B_m) * F) * 2.0)) / Float64(-B_m));
	elseif (B_m <= 2.6e+18)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * t_0);
	else
		tmp = Float64(sqrt(Float64(fma(F, Float64(Float64(C + A) / B_m), Float64(-F)) / B_m)) * t_0);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[B$95$m, 1.2e-127], N[(N[Sqrt[N[(N[(-16.0 * N[(A * A), $MachinePrecision]), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 3.4e-90], N[(N[Sqrt[N[(N[(N[(A - B$95$m), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], If[LessEqual[B$95$m, 2.6e+18], N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[N[(N[(F * N[(N[(C + A), $MachinePrecision] / B$95$m), $MachinePrecision] + (-F)), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 1.19999999999999991e-127

   1. Initial program 18.5%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 27.0%

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

   5. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 13.4

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

   7. Applied rewrites 13.4%

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

   8. Taylor expanded in A around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 9.7

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

   10. Applied rewrites 9.7%

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

   if 1.19999999999999991e-127 < B < 3.39999999999999994e-90

   1. Initial program 60.3%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 62.0

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

   5. Applied rewrites 62.0%

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

   7. Applied rewrites 62.6%

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

   8. Taylor expanded in A around 0

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

   10. Applied rewrites 61.9%

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

   if 3.39999999999999994e-90 < B < 2.6e18

   1. Initial program 37.2%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 41.0%

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

   5. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 9.6

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

   7. Applied rewrites 9.6%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   10. Applied rewrites 37.4%

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

   11. Taylor expanded in A around -inf

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

   13. Applied rewrites 9.4%

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

   if 2.6e18 < B

   1. Initial program 16.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 18.4%

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

   5. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 3.9

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

   7. Applied rewrites 3.9%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   10. Applied rewrites 32.6%

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

   11. Taylor expanded in B around inf

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

   13. Applied rewrites 51.0%

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

4. Final simplification 19.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 3.4 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - B\right) \cdot F\right) \cdot 2}}{-B}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(F, \frac{C + A}{B}, -F\right)}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 39.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;B\_m \leq 1.15 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A + A\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)}\\ \mathbf{elif}\;B\_m \leq 2.6 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{C + A}{B\_m} - 1}{B\_m}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))))
   (if (<= B_m 1.15e-26)
     (/
      (sqrt (* (* -8.0 A) (* (* C F) (+ A A))))
      (- (fma (* -4.0 A) C (* B_m B_m))))
     (if (<= B_m 2.6e+18)
       (* (sqrt (* F (/ -0.5 C))) t_0)
       (* (sqrt (* F (/ (- (/ (+ C A) B_m) 1.0) B_m))) t_0)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0);
	double tmp;
	if (B_m <= 1.15e-26) {
		tmp = sqrt(((-8.0 * A) * ((C * F) * (A + A)))) / -fma((-4.0 * A), C, (B_m * B_m));
	} else if (B_m <= 2.6e+18) {
		tmp = sqrt((F * (-0.5 / C))) * t_0;
	} else {
		tmp = sqrt((F * ((((C + A) / B_m) - 1.0) / B_m))) * t_0;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (B_m <= 1.15e-26)
		tmp = Float64(sqrt(Float64(Float64(-8.0 * A) * Float64(Float64(C * F) * Float64(A + A)))) / Float64(-fma(Float64(-4.0 * A), C, Float64(B_m * B_m))));
	elseif (B_m <= 2.6e+18)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * t_0);
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(Float64(C + A) / B_m) - 1.0) / B_m))) * t_0);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[B$95$m, 1.15e-26], N[(N[Sqrt[N[(N[(-8.0 * A), $MachinePrecision] * N[(N[(C * F), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 2.6e+18], N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[N[(F * N[(N[(N[(N[(C + A), $MachinePrecision] / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.15000000000000004e-26

   1. Initial program 20.6%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 28.7%

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

   5. Taylor expanded in C around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 11.2

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

   7. Applied rewrites 11.2%

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

   if 1.15000000000000004e-26 < B < 2.6e18

   1. Initial program 43.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 44.3%

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

   5. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 1.8

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

   7. Applied rewrites 1.8%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   10. Applied rewrites 44.6%

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

   11. Taylor expanded in A around -inf

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

   13. Applied rewrites 9.5%

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

   if 2.6e18 < B

   1. Initial program 16.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 18.4%

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

   5. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 3.9

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

   7. Applied rewrites 3.9%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   10. Applied rewrites 32.6%

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

   11. Taylor expanded in B around inf

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

   13. Applied rewrites 51.7%

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

4. Final simplification 20.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.15 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A + A\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{\frac{C + A}{B} - 1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 36.8% accurate, 7.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;B\_m \leq 1.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)}\\ \mathbf{elif}\;B\_m \leq 3.4 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - B\_m\right) \cdot F\right) \cdot 2}}{-B\_m}\\ \mathbf{elif}\;B\_m \leq 2.6 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{-1}{B\_m}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))))
   (if (<= B_m 1.2e-127)
     (/
      (sqrt (* (* -16.0 (* A A)) (* C F)))
      (- (fma (* -4.0 A) C (* B_m B_m))))
     (if (<= B_m 3.4e-90)
       (/ (sqrt (* (* (- A B_m) F) 2.0)) (- B_m))
       (if (<= B_m 2.6e+18)
         (* (sqrt (* F (/ -0.5 C))) t_0)
         (* (sqrt (* F (/ -1.0 B_m))) t_0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0);
	double tmp;
	if (B_m <= 1.2e-127) {
		tmp = sqrt(((-16.0 * (A * A)) * (C * F))) / -fma((-4.0 * A), C, (B_m * B_m));
	} else if (B_m <= 3.4e-90) {
		tmp = sqrt((((A - B_m) * F) * 2.0)) / -B_m;
	} else if (B_m <= 2.6e+18) {
		tmp = sqrt((F * (-0.5 / C))) * t_0;
	} else {
		tmp = sqrt((F * (-1.0 / B_m))) * t_0;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (B_m <= 1.2e-127)
		tmp = Float64(sqrt(Float64(Float64(-16.0 * Float64(A * A)) * Float64(C * F))) / Float64(-fma(Float64(-4.0 * A), C, Float64(B_m * B_m))));
	elseif (B_m <= 3.4e-90)
		tmp = Float64(sqrt(Float64(Float64(Float64(A - B_m) * F) * 2.0)) / Float64(-B_m));
	elseif (B_m <= 2.6e+18)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * t_0);
	else
		tmp = Float64(sqrt(Float64(F * Float64(-1.0 / B_m))) * t_0);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[B$95$m, 1.2e-127], N[(N[Sqrt[N[(N[(-16.0 * N[(A * A), $MachinePrecision]), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 3.4e-90], N[(N[Sqrt[N[(N[(N[(A - B$95$m), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], If[LessEqual[B$95$m, 2.6e+18], N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[N[(F * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 1.19999999999999991e-127

   1. Initial program 18.5%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 27.0%

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

   5. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 13.4

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

   7. Applied rewrites 13.4%

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

   8. Taylor expanded in A around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 9.7

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

   10. Applied rewrites 9.7%

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

   if 1.19999999999999991e-127 < B < 3.39999999999999994e-90

   1. Initial program 60.3%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 62.0

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

   5. Applied rewrites 62.0%

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

   7. Applied rewrites 62.6%

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

   8. Taylor expanded in A around 0

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

   10. Applied rewrites 61.9%

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

   if 3.39999999999999994e-90 < B < 2.6e18

   1. Initial program 37.2%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 41.0%

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

   5. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 9.6

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

   7. Applied rewrites 9.6%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   10. Applied rewrites 37.4%

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

   11. Taylor expanded in A around -inf

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

   13. Applied rewrites 9.4%

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

   if 2.6e18 < B

   1. Initial program 16.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 18.4%

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

   5. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 3.9

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

   7. Applied rewrites 3.9%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   10. Applied rewrites 32.6%

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

   11. Taylor expanded in B around inf

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

   13. Applied rewrites 51.3%

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

4. Final simplification 20.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 3.4 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - B\right) \cdot F\right) \cdot 2}}{-B}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{-1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 37.1% accurate, 7.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;B\_m \leq 2.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)}\\ \mathbf{elif}\;B\_m \leq 3.4 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - B\_m\right) \cdot F\right) \cdot 2}}{-B\_m}\\ \mathbf{elif}\;B\_m \leq 2.6 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{-1}{B\_m}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))))
   (if (<= B_m 2.5e-125)
     (/
      (sqrt (* -16.0 (* (* (* A A) C) F)))
      (- (fma (* -4.0 A) C (* B_m B_m))))
     (if (<= B_m 3.4e-90)
       (/ (sqrt (* (* (- A B_m) F) 2.0)) (- B_m))
       (if (<= B_m 2.6e+18)
         (* (sqrt (* F (/ -0.5 C))) t_0)
         (* (sqrt (* F (/ -1.0 B_m))) t_0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0);
	double tmp;
	if (B_m <= 2.5e-125) {
		tmp = sqrt((-16.0 * (((A * A) * C) * F))) / -fma((-4.0 * A), C, (B_m * B_m));
	} else if (B_m <= 3.4e-90) {
		tmp = sqrt((((A - B_m) * F) * 2.0)) / -B_m;
	} else if (B_m <= 2.6e+18) {
		tmp = sqrt((F * (-0.5 / C))) * t_0;
	} else {
		tmp = sqrt((F * (-1.0 / B_m))) * t_0;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (B_m <= 2.5e-125)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(Float64(Float64(A * A) * C) * F))) / Float64(-fma(Float64(-4.0 * A), C, Float64(B_m * B_m))));
	elseif (B_m <= 3.4e-90)
		tmp = Float64(sqrt(Float64(Float64(Float64(A - B_m) * F) * 2.0)) / Float64(-B_m));
	elseif (B_m <= 2.6e+18)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * t_0);
	else
		tmp = Float64(sqrt(Float64(F * Float64(-1.0 / B_m))) * t_0);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[B$95$m, 2.5e-125], N[(N[Sqrt[N[(-16.0 * N[(N[(N[(A * A), $MachinePrecision] * C), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 3.4e-90], N[(N[Sqrt[N[(N[(N[(A - B$95$m), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], If[LessEqual[B$95$m, 2.6e+18], N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[N[(F * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 2.49999999999999983e-125

   1. Initial program 18.5%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 27.0%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 8.1

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

   7. Applied rewrites 8.1%

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

   if 2.49999999999999983e-125 < B < 3.39999999999999994e-90

   1. Initial program 60.3%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 62.0

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

   5. Applied rewrites 62.0%

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

   7. Applied rewrites 62.6%

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

   8. Taylor expanded in A around 0

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

   10. Applied rewrites 61.9%

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

   if 3.39999999999999994e-90 < B < 2.6e18

   1. Initial program 37.2%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 41.0%

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

   5. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 9.6

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

   7. Applied rewrites 9.6%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   10. Applied rewrites 37.4%

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

   11. Taylor expanded in A around -inf

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

   13. Applied rewrites 9.4%

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

   if 2.6e18 < B

   1. Initial program 16.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 18.4%

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

   5. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 3.9

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

   7. Applied rewrites 3.9%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   10. Applied rewrites 32.6%

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

   11. Taylor expanded in B around inf

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

   13. Applied rewrites 51.3%

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

4. Final simplification 18.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 3.4 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - B\right) \cdot F\right) \cdot 2}}{-B}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{-1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 39.8% accurate, 9.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;B\_m \leq 2.6 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{-1}{B\_m}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))))
   (if (<= B_m 2.6e+18)
     (* (sqrt (* F (/ -0.5 C))) t_0)
     (* (sqrt (* F (/ -1.0 B_m))) t_0))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0);
	double tmp;
	if (B_m <= 2.6e+18) {
		tmp = sqrt((F * (-0.5 / C))) * t_0;
	} else {
		tmp = sqrt((F * (-1.0 / B_m))) * t_0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -sqrt(2.0d0)
    if (b_m <= 2.6d+18) then
        tmp = sqrt((f * ((-0.5d0) / c))) * t_0
    else
        tmp = sqrt((f * ((-1.0d0) / b_m))) * t_0
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = -Math.sqrt(2.0);
	double tmp;
	if (B_m <= 2.6e+18) {
		tmp = Math.sqrt((F * (-0.5 / C))) * t_0;
	} else {
		tmp = Math.sqrt((F * (-1.0 / B_m))) * t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = -math.sqrt(2.0)
	tmp = 0
	if B_m <= 2.6e+18:
		tmp = math.sqrt((F * (-0.5 / C))) * t_0
	else:
		tmp = math.sqrt((F * (-1.0 / B_m))) * t_0
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (B_m <= 2.6e+18)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * t_0);
	else
		tmp = Float64(sqrt(Float64(F * Float64(-1.0 / B_m))) * t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = -sqrt(2.0);
	tmp = 0.0;
	if (B_m <= 2.6e+18)
		tmp = sqrt((F * (-0.5 / C))) * t_0;
	else
		tmp = sqrt((F * (-1.0 / B_m))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[B$95$m, 2.6e+18], N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[N[(F * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 2.6e18

   1. Initial program 22.2%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 29.8%

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

   5. Taylor expanded in A around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 13.1

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

   7. Applied rewrites 13.1%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   10. Applied rewrites 28.2%

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

   11. Taylor expanded in A around -inf

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

   13. Applied rewrites 7.8%

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

   if 2.6e18 < B

   1. Initial program 16.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   4. Applied rewrites 18.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}} \]

   5. Taylor expanded in A around inf

      \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right)} \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \color{blue}{\left(\left(C \cdot F\right) \cdot \left(C - -1 \cdot C\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \color{blue}{\left(\left(C \cdot F\right) \cdot \left(C - -1 \cdot C\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\color{blue}{\left(C \cdot F\right)} \cdot \left(C - -1 \cdot C\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C - \color{blue}{\left(\mathsf{neg}\left(C\right)\right)}\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \color{blue}{\left(C - \left(\mathsf{neg}\left(C\right)\right)\right)}\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      9. lower-neg.f64 3.9

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C - \color{blue}{\left(-C\right)}\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

   7. Applied rewrites 3.9%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C - \left(-C\right)\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

   8. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}} \]

   10. Applied rewrites 32.6%

       \[\leadsto \color{blue}{-\sqrt{F \cdot \frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)}{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}} \cdot \sqrt{2}} \]

   11. Taylor expanded in B around inf

       \[\leadsto -\sqrt{F \cdot \frac{-1}{B}} \cdot \sqrt{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 51.3%

       \[\leadsto -\sqrt{F \cdot \frac{-1}{B}} \cdot \sqrt{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.6 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{-1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 39.8% accurate, 9.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;B\_m \leq 2.6 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{-1}{B\_m}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))))
   (if (<= B_m 2.6e+18)
     (* (sqrt (* -0.5 (/ F C))) t_0)
     (* (sqrt (* F (/ -1.0 B_m))) t_0))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0);
	double tmp;
	if (B_m <= 2.6e+18) {
		tmp = sqrt((-0.5 * (F / C))) * t_0;
	} else {
		tmp = sqrt((F * (-1.0 / B_m))) * t_0;
	}
	return tmp;
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -sqrt(2.0d0)
    if (b_m <= 2.6d+18) then
        tmp = sqrt(((-0.5d0) * (f / c))) * t_0
    else
        tmp = sqrt((f * ((-1.0d0) / b_m))) * t_0
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = -Math.sqrt(2.0);
	double tmp;
	if (B_m <= 2.6e+18) {
		tmp = Math.sqrt((-0.5 * (F / C))) * t_0;
	} else {
		tmp = Math.sqrt((F * (-1.0 / B_m))) * t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = -math.sqrt(2.0)
	tmp = 0
	if B_m <= 2.6e+18:
		tmp = math.sqrt((-0.5 * (F / C))) * t_0
	else:
		tmp = math.sqrt((F * (-1.0 / B_m))) * t_0
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (B_m <= 2.6e+18)
		tmp = Float64(sqrt(Float64(-0.5 * Float64(F / C))) * t_0);
	else
		tmp = Float64(sqrt(Float64(F * Float64(-1.0 / B_m))) * t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = -sqrt(2.0);
	tmp = 0.0;
	if (B_m <= 2.6e+18)
		tmp = sqrt((-0.5 * (F / C))) * t_0;
	else
		tmp = sqrt((F * (-1.0 / B_m))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[B$95$m, 2.6e+18], N[(N[Sqrt[N[(-0.5 * N[(F / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[N[(F * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 2.6e18

   1. Initial program 22.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   4. Applied rewrites 29.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}} \]

   5. Taylor expanded in A around inf

      \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right)} \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \color{blue}{\left(\left(C \cdot F\right) \cdot \left(C - -1 \cdot C\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \color{blue}{\left(\left(C \cdot F\right) \cdot \left(C - -1 \cdot C\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\color{blue}{\left(C \cdot F\right)} \cdot \left(C - -1 \cdot C\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C - \color{blue}{\left(\mathsf{neg}\left(C\right)\right)}\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \color{blue}{\left(C - \left(\mathsf{neg}\left(C\right)\right)\right)}\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      9. lower-neg.f64 13.1

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C - \color{blue}{\left(-C\right)}\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

   7. Applied rewrites 13.1%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C - \left(-C\right)\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

   8. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}} \]

   10. Applied rewrites 28.2%

       \[\leadsto \color{blue}{-\sqrt{F \cdot \frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)}{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}} \cdot \sqrt{2}} \]

   11. Taylor expanded in A around -inf

       \[\leadsto -\sqrt{\frac{-1}{2} \cdot \frac{F}{C}} \cdot \sqrt{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 7.8%

       \[\leadsto -\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \sqrt{2} \]

   if 2.6e18 < B

   1. Initial program 16.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   4. Applied rewrites 18.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}} \]

   5. Taylor expanded in A around inf

      \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right)} \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \color{blue}{\left(\left(C \cdot F\right) \cdot \left(C - -1 \cdot C\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \color{blue}{\left(\left(C \cdot F\right) \cdot \left(C - -1 \cdot C\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\color{blue}{\left(C \cdot F\right)} \cdot \left(C - -1 \cdot C\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C - \color{blue}{\left(\mathsf{neg}\left(C\right)\right)}\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \color{blue}{\left(C - \left(\mathsf{neg}\left(C\right)\right)\right)}\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      9. lower-neg.f64 3.9

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C - \color{blue}{\left(-C\right)}\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

   7. Applied rewrites 3.9%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C - \left(-C\right)\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

   8. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}} \]

   10. Applied rewrites 32.6%

       \[\leadsto \color{blue}{-\sqrt{F \cdot \frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)}{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}} \cdot \sqrt{2}} \]

   11. Taylor expanded in B around inf

       \[\leadsto -\sqrt{F \cdot \frac{-1}{B}} \cdot \sqrt{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 51.3%

       \[\leadsto -\sqrt{F \cdot \frac{-1}{B}} \cdot \sqrt{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.6 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{-1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 39.8% accurate, 9.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;B\_m \leq 2.6 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{B\_m}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))))
   (if (<= B_m 2.6e+18)
     (* (sqrt (* -0.5 (/ F C))) t_0)
     (* (sqrt (/ (- F) B_m)) t_0))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0);
	double tmp;
	if (B_m <= 2.6e+18) {
		tmp = sqrt((-0.5 * (F / C))) * t_0;
	} else {
		tmp = sqrt((-F / B_m)) * t_0;
	}
	return tmp;
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -sqrt(2.0d0)
    if (b_m <= 2.6d+18) then
        tmp = sqrt(((-0.5d0) * (f / c))) * t_0
    else
        tmp = sqrt((-f / b_m)) * t_0
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = -Math.sqrt(2.0);
	double tmp;
	if (B_m <= 2.6e+18) {
		tmp = Math.sqrt((-0.5 * (F / C))) * t_0;
	} else {
		tmp = Math.sqrt((-F / B_m)) * t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = -math.sqrt(2.0)
	tmp = 0
	if B_m <= 2.6e+18:
		tmp = math.sqrt((-0.5 * (F / C))) * t_0
	else:
		tmp = math.sqrt((-F / B_m)) * t_0
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (B_m <= 2.6e+18)
		tmp = Float64(sqrt(Float64(-0.5 * Float64(F / C))) * t_0);
	else
		tmp = Float64(sqrt(Float64(Float64(-F) / B_m)) * t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = -sqrt(2.0);
	tmp = 0.0;
	if (B_m <= 2.6e+18)
		tmp = sqrt((-0.5 * (F / C))) * t_0;
	else
		tmp = sqrt((-F / B_m)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[B$95$m, 2.6e+18], N[(N[Sqrt[N[(-0.5 * N[(F / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[N[((-F) / B$95$m), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 2.6e18

   1. Initial program 22.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   4. Applied rewrites 29.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}} \]

   5. Taylor expanded in A around inf

      \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right)} \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \color{blue}{\left(\left(C \cdot F\right) \cdot \left(C - -1 \cdot C\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \color{blue}{\left(\left(C \cdot F\right) \cdot \left(C - -1 \cdot C\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\color{blue}{\left(C \cdot F\right)} \cdot \left(C - -1 \cdot C\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C - \color{blue}{\left(\mathsf{neg}\left(C\right)\right)}\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \color{blue}{\left(C - \left(\mathsf{neg}\left(C\right)\right)\right)}\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      9. lower-neg.f64 13.1

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C - \color{blue}{\left(-C\right)}\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

   7. Applied rewrites 13.1%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C - \left(-C\right)\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

   8. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}} \]

   10. Applied rewrites 28.2%

       \[\leadsto \color{blue}{-\sqrt{F \cdot \frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)}{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}} \cdot \sqrt{2}} \]

   11. Taylor expanded in A around -inf

       \[\leadsto -\sqrt{\frac{-1}{2} \cdot \frac{F}{C}} \cdot \sqrt{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 7.8%

       \[\leadsto -\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \sqrt{2} \]

   if 2.6e18 < B

   1. Initial program 16.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   4. Applied rewrites 18.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}} \]

   5. Taylor expanded in A around inf

      \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right)} \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \color{blue}{\left(\left(C \cdot F\right) \cdot \left(C - -1 \cdot C\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \color{blue}{\left(\left(C \cdot F\right) \cdot \left(C - -1 \cdot C\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\color{blue}{\left(C \cdot F\right)} \cdot \left(C - -1 \cdot C\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C - \color{blue}{\left(\mathsf{neg}\left(C\right)\right)}\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \color{blue}{\left(C - \left(\mathsf{neg}\left(C\right)\right)\right)}\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      9. lower-neg.f64 3.9

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C - \color{blue}{\left(-C\right)}\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

   7. Applied rewrites 3.9%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C - \left(-C\right)\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

   8. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}} \]

   10. Applied rewrites 32.6%

       \[\leadsto \color{blue}{-\sqrt{F \cdot \frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)}{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}} \cdot \sqrt{2}} \]

   11. Taylor expanded in B around inf

       \[\leadsto -\sqrt{-1 \cdot \frac{F}{B}} \cdot \sqrt{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 51.2%

       \[\leadsto -\sqrt{\frac{-F}{B}} \cdot \sqrt{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.6 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 34.1% accurate, 10.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{\frac{-F}{B\_m}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-B\_m, F, A \cdot F\right) \cdot 2}}{-B\_m}\\ \end{array} \end{array} \]

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 -2e+45)
   (* (sqrt (/ (- F) B_m)) (- (sqrt 2.0)))
   (/ (sqrt (* (fma (- B_m) F (* A F)) 2.0)) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -2e+45) {
		tmp = sqrt((-F / B_m)) * -sqrt(2.0);
	} else {
		tmp = sqrt((fma(-B_m, F, (A * F)) * 2.0)) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -2e+45)
		tmp = Float64(sqrt(Float64(Float64(-F) / B_m)) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(sqrt(Float64(fma(Float64(-B_m), F, Float64(A * F)) * 2.0)) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -2e+45], N[(N[Sqrt[N[((-F) / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(N[((-B$95$m) * F + N[(A * F), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.9999999999999999e45

   1. Initial program 22.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   4. Applied rewrites 22.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}} \]

   5. Taylor expanded in A around inf

      \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right)} \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \color{blue}{\left(\left(C \cdot F\right) \cdot \left(C - -1 \cdot C\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \color{blue}{\left(\left(C \cdot F\right) \cdot \left(C - -1 \cdot C\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\color{blue}{\left(C \cdot F\right)} \cdot \left(C - -1 \cdot C\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C - \color{blue}{\left(\mathsf{neg}\left(C\right)\right)}\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \color{blue}{\left(C - \left(\mathsf{neg}\left(C\right)\right)\right)}\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      9. lower-neg.f64 8.9

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C - \color{blue}{\left(-C\right)}\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

   7. Applied rewrites 8.9%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C - \left(-C\right)\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

   8. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}} \]

   10. Applied rewrites 29.4%

       \[\leadsto \color{blue}{-\sqrt{F \cdot \frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)}{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}} \cdot \sqrt{2}} \]

   11. Taylor expanded in B around inf

       \[\leadsto -\sqrt{-1 \cdot \frac{F}{B}} \cdot \sqrt{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 19.0%

       \[\leadsto -\sqrt{\frac{-F}{B}} \cdot \sqrt{2} \]

   if -1.9999999999999999e45 < F

   1. Initial program 20.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      3. *-commutative N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \cdot \frac{\sqrt{2}}{B}} \]

      4. lower-*.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \cdot \frac{\sqrt{2}}{B}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \cdot \frac{\sqrt{2}}{B} \]

      6. *-commutative N/A

         \[\leadsto -\sqrt{\color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right) \cdot F}} \cdot \frac{\sqrt{2}}{B} \]

      7. lower-*.f64 N/A

         \[\leadsto -\sqrt{\color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right) \cdot F}} \cdot \frac{\sqrt{2}}{B} \]

      8. lower--.f64 N/A

         \[\leadsto -\sqrt{\color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      9. unpow2 N/A

         \[\leadsto -\sqrt{\left(A - \sqrt{\color{blue}{A \cdot A} + {B}^{2}}\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      10. unpow2 N/A

          \[\leadsto -\sqrt{\left(A - \sqrt{A \cdot A + \color{blue}{B \cdot B}}\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      11. lower-hypot.f64 N/A

          \[\leadsto -\sqrt{\left(A - \color{blue}{\mathsf{hypot}\left(A, B\right)}\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      12. lower-/.f64 N/A

          \[\leadsto -\sqrt{\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F} \cdot \color{blue}{\frac{\sqrt{2}}{B}} \]

      13. lower-sqrt.f64 19.7

          \[\leadsto -\sqrt{\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F} \cdot \frac{\color{blue}{\sqrt{2}}}{B} \]

   5. Applied rewrites 19.7%

      \[\leadsto \color{blue}{-\sqrt{\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F} \cdot \frac{\sqrt{2}}{B}} \]
   6. Step-by-step derivation

   7. Applied rewrites 19.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F\right) \cdot 2}}{-B}} \]

   8. Taylor expanded in A around 0

      \[\leadsto \frac{\sqrt{\left(-1 \cdot \left(B \cdot F\right) + A \cdot F\right) \cdot 2}}{-B} \]
   9. Step-by-step derivation

   10. Applied rewrites 16.4%

       \[\leadsto \frac{\sqrt{\mathsf{fma}\left(-B, F, A \cdot F\right) \cdot 2}}{-B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{\frac{-F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-B, F, A \cdot F\right) \cdot 2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 34.1% accurate, 10.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{\frac{-F}{B\_m}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - B\_m\right) \cdot F\right) \cdot 2}}{-B\_m}\\ \end{array} \end{array} \]

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 -2e+45)
   (* (sqrt (/ (- F) B_m)) (- (sqrt 2.0)))
   (/ (sqrt (* (* (- A B_m) F) 2.0)) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -2e+45) {
		tmp = sqrt((-F / B_m)) * -sqrt(2.0);
	} else {
		tmp = sqrt((((A - B_m) * F) * 2.0)) / -B_m;
	}
	return tmp;
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= (-2d+45)) then
        tmp = sqrt((-f / b_m)) * -sqrt(2.0d0)
    else
        tmp = sqrt((((a - b_m) * f) * 2.0d0)) / -b_m
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -2e+45) {
		tmp = Math.sqrt((-F / B_m)) * -Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt((((A - B_m) * F) * 2.0)) / -B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= -2e+45:
		tmp = math.sqrt((-F / B_m)) * -math.sqrt(2.0)
	else:
		tmp = math.sqrt((((A - B_m) * F) * 2.0)) / -B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -2e+45)
		tmp = Float64(sqrt(Float64(Float64(-F) / B_m)) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(A - B_m) * F) * 2.0)) / Float64(-B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= -2e+45)
		tmp = sqrt((-F / B_m)) * -sqrt(2.0);
	else
		tmp = sqrt((((A - B_m) * F) * 2.0)) / -B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -2e+45], N[(N[Sqrt[N[((-F) / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(A - B$95$m), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.9999999999999999e45

   1. Initial program 22.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   4. Applied rewrites 22.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}} \]

   5. Taylor expanded in A around inf

      \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right)} \cdot \left(C \cdot \left(F \cdot \left(C - -1 \cdot C\right)\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \color{blue}{\left(\left(C \cdot F\right) \cdot \left(C - -1 \cdot C\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \color{blue}{\left(\left(C \cdot F\right) \cdot \left(C - -1 \cdot C\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\color{blue}{\left(C \cdot F\right)} \cdot \left(C - -1 \cdot C\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C - \color{blue}{\left(\mathsf{neg}\left(C\right)\right)}\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \color{blue}{\left(C - \left(\mathsf{neg}\left(C\right)\right)\right)}\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

      9. lower-neg.f64 8.9

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C - \color{blue}{\left(-C\right)}\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

   7. Applied rewrites 8.9%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C - \left(-C\right)\right)\right)}}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \]

   8. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}} \]

   10. Applied rewrites 29.4%

       \[\leadsto \color{blue}{-\sqrt{F \cdot \frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)}{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}} \cdot \sqrt{2}} \]

   11. Taylor expanded in B around inf

       \[\leadsto -\sqrt{-1 \cdot \frac{F}{B}} \cdot \sqrt{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 19.0%

       \[\leadsto -\sqrt{\frac{-F}{B}} \cdot \sqrt{2} \]

   if -1.9999999999999999e45 < F

   1. Initial program 20.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      3. *-commutative N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \cdot \frac{\sqrt{2}}{B}} \]

      4. lower-*.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \cdot \frac{\sqrt{2}}{B}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -\color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \cdot \frac{\sqrt{2}}{B} \]

      6. *-commutative N/A

         \[\leadsto -\sqrt{\color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right) \cdot F}} \cdot \frac{\sqrt{2}}{B} \]

      7. lower-*.f64 N/A

         \[\leadsto -\sqrt{\color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right) \cdot F}} \cdot \frac{\sqrt{2}}{B} \]

      8. lower--.f64 N/A

         \[\leadsto -\sqrt{\color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      9. unpow2 N/A

         \[\leadsto -\sqrt{\left(A - \sqrt{\color{blue}{A \cdot A} + {B}^{2}}\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      10. unpow2 N/A

          \[\leadsto -\sqrt{\left(A - \sqrt{A \cdot A + \color{blue}{B \cdot B}}\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      11. lower-hypot.f64 N/A

          \[\leadsto -\sqrt{\left(A - \color{blue}{\mathsf{hypot}\left(A, B\right)}\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

      12. lower-/.f64 N/A

          \[\leadsto -\sqrt{\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F} \cdot \color{blue}{\frac{\sqrt{2}}{B}} \]

      13. lower-sqrt.f64 19.7

          \[\leadsto -\sqrt{\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F} \cdot \frac{\color{blue}{\sqrt{2}}}{B} \]

   5. Applied rewrites 19.7%

      \[\leadsto \color{blue}{-\sqrt{\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F} \cdot \frac{\sqrt{2}}{B}} \]
   6. Step-by-step derivation

   7. Applied rewrites 19.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F\right) \cdot 2}}{-B}} \]

   8. Taylor expanded in A around 0

      \[\leadsto \frac{\sqrt{\left(\left(A - B\right) \cdot F\right) \cdot 2}}{-B} \]
   9. Step-by-step derivation

   10. Applied rewrites 16.4%

       \[\leadsto \frac{\sqrt{\left(\left(A - B\right) \cdot F\right) \cdot 2}}{-B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{\frac{-F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - B\right) \cdot F\right) \cdot 2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 26.8% accurate, 13.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{\sqrt{\left(\left(A - B\_m\right) \cdot F\right) \cdot 2}}{-B\_m} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (/ (sqrt (* (* (- A B_m) F) 2.0)) (- B_m)))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return sqrt((((A - B_m) * F) * 2.0)) / -B_m;
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt((((a - b_m) * f) * 2.0d0)) / -b_m
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt((((A - B_m) * F) * 2.0)) / -B_m;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return math.sqrt((((A - B_m) * F) * 2.0)) / -B_m

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(sqrt(Float64(Float64(Float64(A - B_m) * F) * 2.0)) / Float64(-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((((A - B_m) * F) * 2.0)) / -B_m;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[N[(N[(N[(A - B$95$m), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]

Derivation

1. Initial program 21.0%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in C around 0

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]

   2. lower-neg.f64 N/A

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

   3. *-commutative N/A

      \[\leadsto -\color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \cdot \frac{\sqrt{2}}{B}} \]

   4. lower-*.f64 N/A

      \[\leadsto -\color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \cdot \frac{\sqrt{2}}{B}} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto -\color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \cdot \frac{\sqrt{2}}{B} \]

   6. *-commutative N/A

      \[\leadsto -\sqrt{\color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right) \cdot F}} \cdot \frac{\sqrt{2}}{B} \]

   7. lower-*.f64 N/A

      \[\leadsto -\sqrt{\color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right) \cdot F}} \cdot \frac{\sqrt{2}}{B} \]

   8. lower--.f64 N/A

      \[\leadsto -\sqrt{\color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \cdot F} \cdot \frac{\sqrt{2}}{B} \]

   9. unpow2 N/A

      \[\leadsto -\sqrt{\left(A - \sqrt{\color{blue}{A \cdot A} + {B}^{2}}\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

   10. unpow2 N/A

       \[\leadsto -\sqrt{\left(A - \sqrt{A \cdot A + \color{blue}{B \cdot B}}\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

   11. lower-hypot.f64 N/A

       \[\leadsto -\sqrt{\left(A - \color{blue}{\mathsf{hypot}\left(A, B\right)}\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

   12. lower-/.f64 N/A

       \[\leadsto -\sqrt{\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F} \cdot \color{blue}{\frac{\sqrt{2}}{B}} \]

   13. lower-sqrt.f64 15.8

       \[\leadsto -\sqrt{\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F} \cdot \frac{\color{blue}{\sqrt{2}}}{B} \]

5. Applied rewrites 15.8%

   \[\leadsto \color{blue}{-\sqrt{\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F} \cdot \frac{\sqrt{2}}{B}} \]
6. Step-by-step derivation

7. Applied rewrites 15.9%

   \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F\right) \cdot 2}}{-B}} \]

8. Taylor expanded in A around 0

   \[\leadsto \frac{\sqrt{\left(\left(A - B\right) \cdot F\right) \cdot 2}}{-B} \]
9. Step-by-step derivation

10. Applied rewrites 12.7%

    \[\leadsto \frac{\sqrt{\left(\left(A - B\right) \cdot F\right) \cdot 2}}{-B} \]
11. Add Preprocessing

Alternative 18: 26.5% accurate, 14.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{\sqrt{-2 \cdot \left(B\_m \cdot F\right)}}{-B\_m} \end{array} \]

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 (* B_m 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 * (B_m * 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) * (b_m * 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 * (B_m * 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 * (B_m * 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 Float64(sqrt(Float64(-2.0 * Float64(B_m * F))) / Float64(-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 * (B_m * 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[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]

Derivation

1. Initial program 21.0%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in C around 0

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]

   2. lower-neg.f64 N/A

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

   3. *-commutative N/A

      \[\leadsto -\color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \cdot \frac{\sqrt{2}}{B}} \]

   4. lower-*.f64 N/A

      \[\leadsto -\color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \cdot \frac{\sqrt{2}}{B}} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto -\color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \cdot \frac{\sqrt{2}}{B} \]

   6. *-commutative N/A

      \[\leadsto -\sqrt{\color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right) \cdot F}} \cdot \frac{\sqrt{2}}{B} \]

   7. lower-*.f64 N/A

      \[\leadsto -\sqrt{\color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right) \cdot F}} \cdot \frac{\sqrt{2}}{B} \]

   8. lower--.f64 N/A

      \[\leadsto -\sqrt{\color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \cdot F} \cdot \frac{\sqrt{2}}{B} \]

   9. unpow2 N/A

      \[\leadsto -\sqrt{\left(A - \sqrt{\color{blue}{A \cdot A} + {B}^{2}}\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

   10. unpow2 N/A

       \[\leadsto -\sqrt{\left(A - \sqrt{A \cdot A + \color{blue}{B \cdot B}}\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

   11. lower-hypot.f64 N/A

       \[\leadsto -\sqrt{\left(A - \color{blue}{\mathsf{hypot}\left(A, B\right)}\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]

   12. lower-/.f64 N/A

       \[\leadsto -\sqrt{\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F} \cdot \color{blue}{\frac{\sqrt{2}}{B}} \]

   13. lower-sqrt.f64 15.8

       \[\leadsto -\sqrt{\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F} \cdot \frac{\color{blue}{\sqrt{2}}}{B} \]

5. Applied rewrites 15.8%

   \[\leadsto \color{blue}{-\sqrt{\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F} \cdot \frac{\sqrt{2}}{B}} \]
6. Step-by-step derivation

7. Applied rewrites 15.9%

   \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F\right) \cdot 2}}{-B}} \]

8. Taylor expanded in A around 0

   \[\leadsto \frac{\sqrt{-2 \cdot \left(B \cdot F\right)}}{-B} \]
9. Step-by-step derivation

10. Applied rewrites 13.4%

    \[\leadsto \frac{\sqrt{-2 \cdot \left(B \cdot F\right)}}{-B} \]
11. Add Preprocessing

Alternative 19: 1.6% accurate, 18.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \sqrt{F \cdot \frac{2}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (sqrt (* F (/ 2.0 B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return sqrt((F * (2.0 / B_m)));
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt((f * (2.0d0 / b_m)))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt((F * (2.0 / B_m)));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return math.sqrt((F * (2.0 / B_m)))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return sqrt(Float64(F * Float64(2.0 / B_m)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = sqrt((F * (2.0 / B_m)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[Sqrt[N[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 21.0%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]

   2. lower-neg.f64 N/A

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)} \]

   3. *-commutative N/A

      \[\leadsto -\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]

   4. lower-*.f64 N/A

      \[\leadsto -\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]

   5. *-commutative N/A

      \[\leadsto -\color{blue}{\left(\sqrt{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

   6. unpow2 N/A

      \[\leadsto -\left(\sqrt{2} \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right) \cdot \sqrt{\frac{F}{B}} \]

   7. rem-square-sqrt N/A

      \[\leadsto -\left(\sqrt{2} \cdot \color{blue}{-1}\right) \cdot \sqrt{\frac{F}{B}} \]

   8. lower-*.f64 N/A

      \[\leadsto -\color{blue}{\left(\sqrt{2} \cdot -1\right)} \cdot \sqrt{\frac{F}{B}} \]

   9. lower-sqrt.f64 N/A

      \[\leadsto -\left(\color{blue}{\sqrt{2}} \cdot -1\right) \cdot \sqrt{\frac{F}{B}} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto -\left(\sqrt{2} \cdot -1\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

   11. lower-/.f64 2.1

       \[\leadsto -\left(\sqrt{2} \cdot -1\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 2.1%

   \[\leadsto \color{blue}{-\left(\sqrt{2} \cdot -1\right) \cdot \sqrt{\frac{F}{B}}} \]

6. Taylor expanded in B around 0

   \[\leadsto \sqrt{\frac{F}{B}} \cdot \color{blue}{\sqrt{2}} \]
7. Step-by-step derivation

8. Applied rewrites 2.1%

   \[\leadsto \sqrt{\frac{F}{B}} \cdot \color{blue}{\sqrt{2}} \]
9. Step-by-step derivation

10. Applied rewrites 2.1%

    \[\leadsto \color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]
11. Step-by-step derivation

12. Applied rewrites 2.1%

    \[\leadsto \sqrt{F \cdot \frac{2}{B}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025017 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))