ABCF->ab-angle a

Percentage Accurate: 18.8% → 65.2%

Time: 13.3s

Alternatives: 29

Speedup: 16.9×

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

real(8) function code(a, b, c, f)
    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: 18.8% 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

real(8) function code(a, b, c, f)
    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: 65.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(A \cdot 4\right)\\ t_1 := t\_0 - {B\_m}^{2}\\ t_2 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_3 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_0\right)\right) \cdot 2\right)}}{t\_1}\\ t_4 := \mathsf{hypot}\left(A - C, B\_m\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{-\sqrt{C \cdot 2}}{{\left({\left(\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\right)}^{-0.5} \cdot \sqrt{F \cdot 2}\right)}^{-1}}\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\sqrt{C - \frac{{t\_4}^{2} - A \cdot A}{A - t\_4}} \cdot \frac{\sqrt{t\_2 \cdot \left(F \cdot 2\right)}}{-t\_2}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{\left(t\_2 \cdot 2\right) \cdot \left(\left(\frac{B\_m \cdot B\_m}{A} \cdot -0.5 + C\right) + C\right)}}{t\_1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot F} \cdot \sqrt{\left(\left(t\_4 + A\right) + C\right) \cdot 2}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{B\_m} \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot 2}\right) \cdot \sqrt{F}\\ \end{array} \end{array} \]

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 4.0)))
        (t_1 (- t_0 (pow B_m 2.0)))
        (t_2 (fma -4.0 (* C A) (* B_m B_m)))
        (t_3
         (/
          (sqrt
           (*
            (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))
            (* (* F (- (pow B_m 2.0) t_0)) 2.0)))
          t_1))
        (t_4 (hypot (- A C) B_m)))
   (if (<= t_3 (- INFINITY))
     (/
      (- (sqrt (* C 2.0)))
      (pow
       (* (pow (fma (* C A) -4.0 (* B_m B_m)) -0.5) (sqrt (* F 2.0)))
       -1.0))
     (if (<= t_3 -2e-227)
       (*
        (sqrt (- C (/ (- (pow t_4 2.0) (* A A)) (- A t_4))))
        (/ (sqrt (* t_2 (* F 2.0))) (- t_2)))
       (if (<= t_3 0.0)
         (/
          (*
           (sqrt F)
           (sqrt (* (* t_2 2.0) (+ (+ (* (/ (* B_m B_m) A) -0.5) C) C))))
          t_1)
         (if (<= t_3 INFINITY)
           (/ (* (sqrt (* t_2 F)) (sqrt (* (+ (+ t_4 A) C) 2.0))) t_1)
           (*
            (* (/ -1.0 B_m) (sqrt (* (+ (hypot C B_m) C) 2.0)))
            (sqrt F))))))))

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 * 4.0);
	double t_1 = t_0 - pow(B_m, 2.0);
	double t_2 = fma(-4.0, (C * A), (B_m * B_m));
	double t_3 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * ((F * (pow(B_m, 2.0) - t_0)) * 2.0))) / t_1;
	double t_4 = hypot((A - C), B_m);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = -sqrt((C * 2.0)) / pow((pow(fma((C * A), -4.0, (B_m * B_m)), -0.5) * sqrt((F * 2.0))), -1.0);
	} else if (t_3 <= -2e-227) {
		tmp = sqrt((C - ((pow(t_4, 2.0) - (A * A)) / (A - t_4)))) * (sqrt((t_2 * (F * 2.0))) / -t_2);
	} else if (t_3 <= 0.0) {
		tmp = (sqrt(F) * sqrt(((t_2 * 2.0) * (((((B_m * B_m) / A) * -0.5) + C) + C)))) / t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt((t_2 * F)) * sqrt((((t_4 + A) + C) * 2.0))) / t_1;
	} else {
		tmp = ((-1.0 / B_m) * sqrt(((hypot(C, B_m) + C) * 2.0))) * sqrt(F);
	}
	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(C * Float64(A * 4.0))
	t_1 = Float64(t_0 - (B_m ^ 2.0))
	t_2 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_3 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_0)) * 2.0))) / t_1)
	t_4 = hypot(Float64(A - C), B_m)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(-sqrt(Float64(C * 2.0))) / (Float64((fma(Float64(C * A), -4.0, Float64(B_m * B_m)) ^ -0.5) * sqrt(Float64(F * 2.0))) ^ -1.0));
	elseif (t_3 <= -2e-227)
		tmp = Float64(sqrt(Float64(C - Float64(Float64((t_4 ^ 2.0) - Float64(A * A)) / Float64(A - t_4)))) * Float64(sqrt(Float64(t_2 * Float64(F * 2.0))) / Float64(-t_2)));
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(Float64(t_2 * 2.0) * Float64(Float64(Float64(Float64(Float64(B_m * B_m) / A) * -0.5) + C) + C)))) / t_1);
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(t_2 * F)) * sqrt(Float64(Float64(Float64(t_4 + A) + C) * 2.0))) / t_1);
	else
		tmp = Float64(Float64(Float64(-1.0 / B_m) * sqrt(Float64(Float64(hypot(C, B_m) + C) * 2.0))) * sqrt(F));
	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[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[((-N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision]) / N[Power[N[(N[Power[N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -2e-227], N[(N[Sqrt[N[(C - N[(N[(N[Power[t$95$4, 2.0], $MachinePrecision] - N[(A * A), $MachinePrecision]), $MachinePrecision] / N[(A - t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(t$95$2 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$2)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(N[(t$95$2 * 2.0), $MachinePrecision] * N[(N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(t$95$2 * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(N[(t$95$4 + A), $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[(-1.0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.0%

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

   4. Applied rewrites 33.8%

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

   5. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 22.4

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

   7. Applied rewrites 22.4%

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

   9. Applied rewrites 32.5%

      \[\leadsto \color{blue}{\frac{-\sqrt{C \cdot 2}}{{\left(\sqrt{F \cdot 2} \cdot {\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right)}^{-0.5}\right)}^{-1}}} \]

   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))) < -1.99999999999999989e-227

   1. Initial program 97.2%

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

   4. Applied rewrites 97.2%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 90.3

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

   6. Applied rewrites 90.3%

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

   if -1.99999999999999989e-227 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. sqrt-prod N/A

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

      9. pow1/2 N/A

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

   4. Applied rewrites 16.4%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 29.3

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

   7. Applied rewrites 29.3%

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

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

   1. Initial program 34.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 86.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 16.9

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

   5. Applied rewrites 16.9%

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

   7. Applied rewrites 24.8%

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

   9. Applied rewrites 24.9%

      \[\leadsto \left(\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2} \cdot \frac{-1}{B}\right) \cdot \sqrt{\color{blue}{F}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{-\sqrt{C \cdot 2}}{{\left({\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right)}^{-0.5} \cdot \sqrt{F \cdot 2}\right)}^{-1}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\sqrt{C - \frac{{\left(\mathsf{hypot}\left(A - C, B\right)\right)}^{2} - A \cdot A}{A - \mathsf{hypot}\left(A - C, B\right)}} \cdot \frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2\right) \cdot \left(\left(\frac{B \cdot B}{A} \cdot -0.5 + C\right) + C\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F} \cdot \sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot 2}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{B} \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}\right) \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := \left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C\\ t_2 := C \cdot \left(A \cdot 4\right)\\ t_3 := t\_2 - {B\_m}^{2}\\ t_4 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_2\right)\right) \cdot 2\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+202}:\\ \;\;\;\;\frac{-\sqrt{C \cdot 2}}{{\left({\left(\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\right)}^{-0.5} \cdot \sqrt{F \cdot 2}\right)}^{-1}}\\ \mathbf{elif}\;t\_4 \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{-1}{t\_0} \cdot \sqrt{t\_1 \cdot \left(t\_0 \cdot \left(F \cdot 2\right)\right)}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{\left(t\_0 \cdot 2\right) \cdot \left(\left(\frac{B\_m \cdot B\_m}{A} \cdot -0.5 + C\right) + C\right)}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot F} \cdot \sqrt{t\_1 \cdot 2}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{B\_m} \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot 2}\right) \cdot \sqrt{F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (+ (+ (hypot (- A C) B_m) A) C))
        (t_2 (* C (* A 4.0)))
        (t_3 (- t_2 (pow B_m 2.0)))
        (t_4
         (/
          (sqrt
           (*
            (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))
            (* (* F (- (pow B_m 2.0) t_2)) 2.0)))
          t_3)))
   (if (<= t_4 -5e+202)
     (/
      (- (sqrt (* C 2.0)))
      (pow
       (* (pow (fma (* C A) -4.0 (* B_m B_m)) -0.5) (sqrt (* F 2.0)))
       -1.0))
     (if (<= t_4 -2e-227)
       (* (/ -1.0 t_0) (sqrt (* t_1 (* t_0 (* F 2.0)))))
       (if (<= t_4 0.0)
         (/
          (*
           (sqrt F)
           (sqrt (* (* t_0 2.0) (+ (+ (* (/ (* B_m B_m) A) -0.5) C) C))))
          t_3)
         (if (<= t_4 INFINITY)
           (/ (* (sqrt (* t_0 F)) (sqrt (* t_1 2.0))) t_3)
           (*
            (* (/ -1.0 B_m) (sqrt (* (+ (hypot C B_m) C) 2.0)))
            (sqrt F))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = (hypot((A - C), B_m) + A) + C;
	double t_2 = C * (A * 4.0);
	double t_3 = t_2 - pow(B_m, 2.0);
	double t_4 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * ((F * (pow(B_m, 2.0) - t_2)) * 2.0))) / t_3;
	double tmp;
	if (t_4 <= -5e+202) {
		tmp = -sqrt((C * 2.0)) / pow((pow(fma((C * A), -4.0, (B_m * B_m)), -0.5) * sqrt((F * 2.0))), -1.0);
	} else if (t_4 <= -2e-227) {
		tmp = (-1.0 / t_0) * sqrt((t_1 * (t_0 * (F * 2.0))));
	} else if (t_4 <= 0.0) {
		tmp = (sqrt(F) * sqrt(((t_0 * 2.0) * (((((B_m * B_m) / A) * -0.5) + C) + C)))) / t_3;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt((t_0 * F)) * sqrt((t_1 * 2.0))) / t_3;
	} else {
		tmp = ((-1.0 / B_m) * sqrt(((hypot(C, B_m) + C) * 2.0))) * sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)
	t_2 = Float64(C * Float64(A * 4.0))
	t_3 = Float64(t_2 - (B_m ^ 2.0))
	t_4 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_2)) * 2.0))) / t_3)
	tmp = 0.0
	if (t_4 <= -5e+202)
		tmp = Float64(Float64(-sqrt(Float64(C * 2.0))) / (Float64((fma(Float64(C * A), -4.0, Float64(B_m * B_m)) ^ -0.5) * sqrt(Float64(F * 2.0))) ^ -1.0));
	elseif (t_4 <= -2e-227)
		tmp = Float64(Float64(-1.0 / t_0) * sqrt(Float64(t_1 * Float64(t_0 * Float64(F * 2.0)))));
	elseif (t_4 <= 0.0)
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(Float64(t_0 * 2.0) * Float64(Float64(Float64(Float64(Float64(B_m * B_m) / A) * -0.5) + C) + C)))) / t_3);
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(t_0 * F)) * sqrt(Float64(t_1 * 2.0))) / t_3);
	else
		tmp = Float64(Float64(Float64(-1.0 / B_m) * sqrt(Float64(Float64(hypot(C, B_m) + C) * 2.0))) * sqrt(F));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]}, Block[{t$95$2 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, -5e+202], N[((-N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision]) / N[Power[N[(N[Power[N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -2e-227], N[(N[(-1.0 / t$95$0), $MachinePrecision] * N[Sqrt[N[(t$95$1 * N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(N[(t$95$0 * 2.0), $MachinePrecision] * N[(N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Sqrt[N[(t$95$0 * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$1 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(N[(-1.0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.9999999999999999e202

   1. Initial program 4.7%

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

   4. Applied rewrites 34.9%

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

   5. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 22.1

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

   7. Applied rewrites 22.1%

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

   9. Applied rewrites 32.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{C \cdot 2}}{{\left(\sqrt{F \cdot 2} \cdot {\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right)}^{-0.5}\right)}^{-1}}} \]

   if -4.9999999999999999e202 < (/.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))) < -1.99999999999999989e-227

   1. Initial program 97.1%

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

   4. Applied rewrites 97.3%

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

   if -1.99999999999999989e-227 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. sqrt-prod N/A

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

      9. pow1/2 N/A

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

   4. Applied rewrites 16.4%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 29.3

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

   7. Applied rewrites 29.3%

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

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

   1. Initial program 34.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 86.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 16.9

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

   5. Applied rewrites 16.9%

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

   7. Applied rewrites 24.8%

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

   9. Applied rewrites 24.9%

      \[\leadsto \left(\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2} \cdot \frac{-1}{B}\right) \cdot \sqrt{\color{blue}{F}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -5 \cdot 10^{+202}:\\ \;\;\;\;\frac{-\sqrt{C \cdot 2}}{{\left({\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right)}^{-0.5} \cdot \sqrt{F \cdot 2}\right)}^{-1}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2\right) \cdot \left(\left(\frac{B \cdot B}{A} \cdot -0.5 + C\right) + C\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F} \cdot \sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot 2}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{B} \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}\right) \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := C \cdot \left(A \cdot 4\right)\\ t_2 := t\_1 - {B\_m}^{2}\\ t_3 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_1\right)\right) \cdot 2\right)}}{t\_2}\\ t_4 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_5 := \left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+202}:\\ \;\;\;\;\frac{-\sqrt{C \cdot 2}}{{\left({t\_4}^{-0.5} \cdot \sqrt{F \cdot 2}\right)}^{-1}}\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{-1}{t\_0} \cdot \sqrt{t\_5 \cdot \left(t\_0 \cdot \left(F \cdot 2\right)\right)}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{\left(t\_0 \cdot 2\right) \cdot \left(\left(\frac{B\_m \cdot B\_m}{A} \cdot -0.5 + C\right) + C\right)}}{t\_2}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{t\_5} \cdot \frac{\left(-\sqrt{2}\right) \cdot \sqrt{t\_4 \cdot F}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{B\_m} \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot 2}\right) \cdot \sqrt{F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (* C (* A 4.0)))
        (t_2 (- t_1 (pow B_m 2.0)))
        (t_3
         (/
          (sqrt
           (*
            (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))
            (* (* F (- (pow B_m 2.0) t_1)) 2.0)))
          t_2))
        (t_4 (fma (* C A) -4.0 (* B_m B_m)))
        (t_5 (+ (+ (hypot (- A C) B_m) A) C)))
   (if (<= t_3 -5e+202)
     (/ (- (sqrt (* C 2.0))) (pow (* (pow t_4 -0.5) (sqrt (* F 2.0))) -1.0))
     (if (<= t_3 -2e-227)
       (* (/ -1.0 t_0) (sqrt (* t_5 (* t_0 (* F 2.0)))))
       (if (<= t_3 0.0)
         (/
          (*
           (sqrt F)
           (sqrt (* (* t_0 2.0) (+ (+ (* (/ (* B_m B_m) A) -0.5) C) C))))
          t_2)
         (if (<= t_3 INFINITY)
           (* (sqrt t_5) (/ (* (- (sqrt 2.0)) (sqrt (* t_4 F))) t_0))
           (*
            (* (/ -1.0 B_m) (sqrt (* (+ (hypot C B_m) C) 2.0)))
            (sqrt F))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = C * (A * 4.0);
	double t_2 = t_1 - pow(B_m, 2.0);
	double t_3 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * ((F * (pow(B_m, 2.0) - t_1)) * 2.0))) / t_2;
	double t_4 = fma((C * A), -4.0, (B_m * B_m));
	double t_5 = (hypot((A - C), B_m) + A) + C;
	double tmp;
	if (t_3 <= -5e+202) {
		tmp = -sqrt((C * 2.0)) / pow((pow(t_4, -0.5) * sqrt((F * 2.0))), -1.0);
	} else if (t_3 <= -2e-227) {
		tmp = (-1.0 / t_0) * sqrt((t_5 * (t_0 * (F * 2.0))));
	} else if (t_3 <= 0.0) {
		tmp = (sqrt(F) * sqrt(((t_0 * 2.0) * (((((B_m * B_m) / A) * -0.5) + C) + C)))) / t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(t_5) * ((-sqrt(2.0) * sqrt((t_4 * F))) / t_0);
	} else {
		tmp = ((-1.0 / B_m) * sqrt(((hypot(C, B_m) + C) * 2.0))) * sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64(C * Float64(A * 4.0))
	t_2 = Float64(t_1 - (B_m ^ 2.0))
	t_3 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_1)) * 2.0))) / t_2)
	t_4 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_5 = Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)
	tmp = 0.0
	if (t_3 <= -5e+202)
		tmp = Float64(Float64(-sqrt(Float64(C * 2.0))) / (Float64((t_4 ^ -0.5) * sqrt(Float64(F * 2.0))) ^ -1.0));
	elseif (t_3 <= -2e-227)
		tmp = Float64(Float64(-1.0 / t_0) * sqrt(Float64(t_5 * Float64(t_0 * Float64(F * 2.0)))));
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(Float64(t_0 * 2.0) * Float64(Float64(Float64(Float64(Float64(B_m * B_m) / A) * -0.5) + C) + C)))) / t_2);
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(t_5) * Float64(Float64(Float64(-sqrt(2.0)) * sqrt(Float64(t_4 * F))) / t_0));
	else
		tmp = Float64(Float64(Float64(-1.0 / B_m) * sqrt(Float64(Float64(hypot(C, B_m) + C) * 2.0))) * sqrt(F));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+202], N[((-N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision]) / N[Power[N[(N[Power[t$95$4, -0.5], $MachinePrecision] * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -2e-227], N[(N[(-1.0 / t$95$0), $MachinePrecision] * N[Sqrt[N[(t$95$5 * N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(N[(t$95$0 * 2.0), $MachinePrecision] * N[(N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[t$95$5], $MachinePrecision] * N[(N[((-N[Sqrt[2.0], $MachinePrecision]) * N[Sqrt[N[(t$95$4 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.9999999999999999e202

   1. Initial program 4.7%

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

   4. Applied rewrites 34.9%

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

   5. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 22.1

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

   7. Applied rewrites 22.1%

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

   9. Applied rewrites 32.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{C \cdot 2}}{{\left(\sqrt{F \cdot 2} \cdot {\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right)}^{-0.5}\right)}^{-1}}} \]

   if -4.9999999999999999e202 < (/.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))) < -1.99999999999999989e-227

   1. Initial program 97.1%

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

   4. Applied rewrites 97.3%

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

   if -1.99999999999999989e-227 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. sqrt-prod N/A

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

      9. pow1/2 N/A

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

   4. Applied rewrites 16.4%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 29.3

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

   7. Applied rewrites 29.3%

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

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

   1. Initial program 34.1%

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

   4. Applied rewrites 86.2%

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

   6. Applied rewrites 86.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 16.9

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

   5. Applied rewrites 16.9%

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

   7. Applied rewrites 24.8%

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

   9. Applied rewrites 24.9%

      \[\leadsto \left(\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2} \cdot \frac{-1}{B}\right) \cdot \sqrt{\color{blue}{F}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -5 \cdot 10^{+202}:\\ \;\;\;\;\frac{-\sqrt{C \cdot 2}}{{\left({\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right)}^{-0.5} \cdot \sqrt{F \cdot 2}\right)}^{-1}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2\right) \cdot \left(\left(\frac{B \cdot B}{A} \cdot -0.5 + C\right) + C\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C} \cdot \frac{\left(-\sqrt{2}\right) \cdot \sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{B} \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}\right) \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C\\ t_1 := C \cdot \left(A \cdot 4\right)\\ t_2 := t\_1 - {B\_m}^{2}\\ t_3 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_1\right)\right) \cdot 2\right)}}{t\_2}\\ t_4 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_5 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+202}:\\ \;\;\;\;\left({t\_4}^{-0.5} \cdot \sqrt{F \cdot 2}\right) \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{-1}{t\_5} \cdot \sqrt{t\_0 \cdot \left(t\_5 \cdot \left(F \cdot 2\right)\right)}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{\left(t\_5 \cdot 2\right) \cdot \left(\left(\frac{B\_m \cdot B\_m}{A} \cdot -0.5 + C\right) + C\right)}}{t\_2}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{t\_0} \cdot \frac{\left(-\sqrt{2}\right) \cdot \sqrt{t\_4 \cdot F}}{t\_5}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{B\_m} \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot 2}\right) \cdot \sqrt{F}\\ \end{array} \end{array} \]

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 (+ (+ (hypot (- A C) B_m) A) C))
        (t_1 (* C (* A 4.0)))
        (t_2 (- t_1 (pow B_m 2.0)))
        (t_3
         (/
          (sqrt
           (*
            (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))
            (* (* F (- (pow B_m 2.0) t_1)) 2.0)))
          t_2))
        (t_4 (fma (* C A) -4.0 (* B_m B_m)))
        (t_5 (fma -4.0 (* C A) (* B_m B_m))))
   (if (<= t_3 -5e+202)
     (* (* (pow t_4 -0.5) (sqrt (* F 2.0))) (- (sqrt (* C 2.0))))
     (if (<= t_3 -2e-227)
       (* (/ -1.0 t_5) (sqrt (* t_0 (* t_5 (* F 2.0)))))
       (if (<= t_3 0.0)
         (/
          (*
           (sqrt F)
           (sqrt (* (* t_5 2.0) (+ (+ (* (/ (* B_m B_m) A) -0.5) C) C))))
          t_2)
         (if (<= t_3 INFINITY)
           (* (sqrt t_0) (/ (* (- (sqrt 2.0)) (sqrt (* t_4 F))) t_5))
           (*
            (* (/ -1.0 B_m) (sqrt (* (+ (hypot C B_m) C) 2.0)))
            (sqrt F))))))))

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 = (hypot((A - C), B_m) + A) + C;
	double t_1 = C * (A * 4.0);
	double t_2 = t_1 - pow(B_m, 2.0);
	double t_3 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * ((F * (pow(B_m, 2.0) - t_1)) * 2.0))) / t_2;
	double t_4 = fma((C * A), -4.0, (B_m * B_m));
	double t_5 = fma(-4.0, (C * A), (B_m * B_m));
	double tmp;
	if (t_3 <= -5e+202) {
		tmp = (pow(t_4, -0.5) * sqrt((F * 2.0))) * -sqrt((C * 2.0));
	} else if (t_3 <= -2e-227) {
		tmp = (-1.0 / t_5) * sqrt((t_0 * (t_5 * (F * 2.0))));
	} else if (t_3 <= 0.0) {
		tmp = (sqrt(F) * sqrt(((t_5 * 2.0) * (((((B_m * B_m) / A) * -0.5) + C) + C)))) / t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(t_0) * ((-sqrt(2.0) * sqrt((t_4 * F))) / t_5);
	} else {
		tmp = ((-1.0 / B_m) * sqrt(((hypot(C, B_m) + C) * 2.0))) * sqrt(F);
	}
	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(hypot(Float64(A - C), B_m) + A) + C)
	t_1 = Float64(C * Float64(A * 4.0))
	t_2 = Float64(t_1 - (B_m ^ 2.0))
	t_3 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_1)) * 2.0))) / t_2)
	t_4 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_5 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	tmp = 0.0
	if (t_3 <= -5e+202)
		tmp = Float64(Float64((t_4 ^ -0.5) * sqrt(Float64(F * 2.0))) * Float64(-sqrt(Float64(C * 2.0))));
	elseif (t_3 <= -2e-227)
		tmp = Float64(Float64(-1.0 / t_5) * sqrt(Float64(t_0 * Float64(t_5 * Float64(F * 2.0)))));
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(Float64(t_5 * 2.0) * Float64(Float64(Float64(Float64(Float64(B_m * B_m) / A) * -0.5) + C) + C)))) / t_2);
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(t_0) * Float64(Float64(Float64(-sqrt(2.0)) * sqrt(Float64(t_4 * F))) / t_5));
	else
		tmp = Float64(Float64(Float64(-1.0 / B_m) * sqrt(Float64(Float64(hypot(C, B_m) + C) * 2.0))) * sqrt(F));
	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[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]}, Block[{t$95$1 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+202], N[(N[(N[Power[t$95$4, -0.5], $MachinePrecision] * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$3, -2e-227], N[(N[(-1.0 / t$95$5), $MachinePrecision] * N[Sqrt[N[(t$95$0 * N[(t$95$5 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(N[(t$95$5 * 2.0), $MachinePrecision] * N[(N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[t$95$0], $MachinePrecision] * N[(N[((-N[Sqrt[2.0], $MachinePrecision]) * N[Sqrt[N[(t$95$4 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.9999999999999999e202

   1. Initial program 4.7%

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

   4. Applied rewrites 34.9%

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

   5. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 22.1

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

   7. Applied rewrites 22.1%

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

   9. Applied rewrites 32.0%

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

   if -4.9999999999999999e202 < (/.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))) < -1.99999999999999989e-227

   1. Initial program 97.1%

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

   4. Applied rewrites 97.3%

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

   if -1.99999999999999989e-227 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. sqrt-prod N/A

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

      9. pow1/2 N/A

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

   4. Applied rewrites 16.4%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 29.3

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

   7. Applied rewrites 29.3%

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

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

   1. Initial program 34.1%

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

   4. Applied rewrites 86.2%

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

   6. Applied rewrites 86.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 16.9

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

   5. Applied rewrites 16.9%

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

   7. Applied rewrites 24.8%

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

   9. Applied rewrites 24.9%

      \[\leadsto \left(\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2} \cdot \frac{-1}{B}\right) \cdot \sqrt{\color{blue}{F}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -5 \cdot 10^{+202}:\\ \;\;\;\;\left({\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right)}^{-0.5} \cdot \sqrt{F \cdot 2}\right) \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2\right) \cdot \left(\left(\frac{B \cdot B}{A} \cdot -0.5 + C\right) + C\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C} \cdot \frac{\left(-\sqrt{2}\right) \cdot \sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{B} \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}\right) \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := -\sqrt{C \cdot 2}\\ t_2 := C \cdot \left(A \cdot 4\right)\\ t_3 := t\_2 - {B\_m}^{2}\\ t_4 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_2\right)\right) \cdot 2\right)}}{t\_3}\\ t_5 := t\_0 \cdot \left(F \cdot 2\right)\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+202}:\\ \;\;\;\;\left({\left(\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\right)}^{-0.5} \cdot \sqrt{F \cdot 2}\right) \cdot t\_1\\ \mathbf{elif}\;t\_4 \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{-1}{t\_0} \cdot \sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C\right) \cdot t\_5}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{\left(t\_0 \cdot 2\right) \cdot \left(\left(\frac{B\_m \cdot B\_m}{A} \cdot -0.5 + C\right) + C\right)}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_5}}{t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{B\_m} \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot 2}\right) \cdot \sqrt{F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (- (sqrt (* C 2.0))))
        (t_2 (* C (* A 4.0)))
        (t_3 (- t_2 (pow B_m 2.0)))
        (t_4
         (/
          (sqrt
           (*
            (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))
            (* (* F (- (pow B_m 2.0) t_2)) 2.0)))
          t_3))
        (t_5 (* t_0 (* F 2.0))))
   (if (<= t_4 -5e+202)
     (* (* (pow (fma (* C A) -4.0 (* B_m B_m)) -0.5) (sqrt (* F 2.0))) t_1)
     (if (<= t_4 -2e-227)
       (* (/ -1.0 t_0) (sqrt (* (+ (+ (hypot (- A C) B_m) A) C) t_5)))
       (if (<= t_4 0.0)
         (/
          (*
           (sqrt F)
           (sqrt (* (* t_0 2.0) (+ (+ (* (/ (* B_m B_m) A) -0.5) C) C))))
          t_3)
         (if (<= t_4 INFINITY)
           (* (/ (sqrt t_5) t_0) t_1)
           (*
            (* (/ -1.0 B_m) (sqrt (* (+ (hypot C B_m) C) 2.0)))
            (sqrt F))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = -sqrt((C * 2.0));
	double t_2 = C * (A * 4.0);
	double t_3 = t_2 - pow(B_m, 2.0);
	double t_4 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * ((F * (pow(B_m, 2.0) - t_2)) * 2.0))) / t_3;
	double t_5 = t_0 * (F * 2.0);
	double tmp;
	if (t_4 <= -5e+202) {
		tmp = (pow(fma((C * A), -4.0, (B_m * B_m)), -0.5) * sqrt((F * 2.0))) * t_1;
	} else if (t_4 <= -2e-227) {
		tmp = (-1.0 / t_0) * sqrt((((hypot((A - C), B_m) + A) + C) * t_5));
	} else if (t_4 <= 0.0) {
		tmp = (sqrt(F) * sqrt(((t_0 * 2.0) * (((((B_m * B_m) / A) * -0.5) + C) + C)))) / t_3;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt(t_5) / t_0) * t_1;
	} else {
		tmp = ((-1.0 / B_m) * sqrt(((hypot(C, B_m) + C) * 2.0))) * sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64(-sqrt(Float64(C * 2.0)))
	t_2 = Float64(C * Float64(A * 4.0))
	t_3 = Float64(t_2 - (B_m ^ 2.0))
	t_4 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_2)) * 2.0))) / t_3)
	t_5 = Float64(t_0 * Float64(F * 2.0))
	tmp = 0.0
	if (t_4 <= -5e+202)
		tmp = Float64(Float64((fma(Float64(C * A), -4.0, Float64(B_m * B_m)) ^ -0.5) * sqrt(Float64(F * 2.0))) * t_1);
	elseif (t_4 <= -2e-227)
		tmp = Float64(Float64(-1.0 / t_0) * sqrt(Float64(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C) * t_5)));
	elseif (t_4 <= 0.0)
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(Float64(t_0 * 2.0) * Float64(Float64(Float64(Float64(Float64(B_m * B_m) / A) * -0.5) + C) + C)))) / t_3);
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(t_5) / t_0) * t_1);
	else
		tmp = Float64(Float64(Float64(-1.0 / B_m) * sqrt(Float64(Float64(hypot(C, B_m) + C) * 2.0))) * sqrt(F));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision])}, Block[{t$95$2 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -5e+202], N[(N[(N[Power[N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$4, -2e-227], N[(N[(-1.0 / t$95$0), $MachinePrecision] * N[Sqrt[N[(N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision] * t$95$5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(N[(t$95$0 * 2.0), $MachinePrecision] * N[(N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Sqrt[t$95$5], $MachinePrecision] / t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(-1.0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.9999999999999999e202

   1. Initial program 4.7%

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

   4. Applied rewrites 34.9%

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

   5. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 22.1

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

   7. Applied rewrites 22.1%

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

   9. Applied rewrites 32.0%

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

   if -4.9999999999999999e202 < (/.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))) < -1.99999999999999989e-227

   1. Initial program 97.1%

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

   4. Applied rewrites 97.3%

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

   if -1.99999999999999989e-227 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. sqrt-prod N/A

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

      9. pow1/2 N/A

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

   4. Applied rewrites 16.4%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 29.3

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

   7. Applied rewrites 29.3%

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

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

   1. Initial program 34.1%

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

   4. Applied rewrites 86.2%

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

   5. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 39.7

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

   7. Applied rewrites 39.7%

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

   9. Applied rewrites 39.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 16.9

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

   5. Applied rewrites 16.9%

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

   7. Applied rewrites 24.8%

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

   9. Applied rewrites 24.9%

      \[\leadsto \left(\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2} \cdot \frac{-1}{B}\right) \cdot \sqrt{\color{blue}{F}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -5 \cdot 10^{+202}:\\ \;\;\;\;\left({\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right)}^{-0.5} \cdot \sqrt{F \cdot 2}\right) \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2\right) \cdot \left(\left(\frac{B \cdot B}{A} \cdot -0.5 + C\right) + C\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{B} \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}\right) \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := -\sqrt{C \cdot 2}\\ t_2 := C \cdot \left(A \cdot 4\right)\\ t_3 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_2\right)\right) \cdot 2\right)}}{t\_2 - {B\_m}^{2}}\\ t_4 := t\_0 \cdot \left(F \cdot 2\right)\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+202}:\\ \;\;\;\;\left({\left(\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\right)}^{-0.5} \cdot \sqrt{F \cdot 2}\right) \cdot t\_1\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{-1}{t\_0} \cdot \sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C\right) \cdot t\_4}\\ \mathbf{elif}\;t\_3 \leq 5:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot F\right) \cdot \left(\left(\frac{B\_m \cdot B\_m}{A} \cdot -0.5 + C\right) + C\right)}}{-t\_0} \cdot \sqrt{2}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_4}}{t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{B\_m} \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot 2}\right) \cdot \sqrt{F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (- (sqrt (* C 2.0))))
        (t_2 (* C (* A 4.0)))
        (t_3
         (/
          (sqrt
           (*
            (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))
            (* (* F (- (pow B_m 2.0) t_2)) 2.0)))
          (- t_2 (pow B_m 2.0))))
        (t_4 (* t_0 (* F 2.0))))
   (if (<= t_3 -5e+202)
     (* (* (pow (fma (* C A) -4.0 (* B_m B_m)) -0.5) (sqrt (* F 2.0))) t_1)
     (if (<= t_3 -2e-227)
       (* (/ -1.0 t_0) (sqrt (* (+ (+ (hypot (- A C) B_m) A) C) t_4)))
       (if (<= t_3 5.0)
         (*
          (/
           (sqrt (* (* t_0 F) (+ (+ (* (/ (* B_m B_m) A) -0.5) C) C)))
           (- t_0))
          (sqrt 2.0))
         (if (<= t_3 INFINITY)
           (* (/ (sqrt t_4) t_0) t_1)
           (*
            (* (/ -1.0 B_m) (sqrt (* (+ (hypot C B_m) C) 2.0)))
            (sqrt F))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = -sqrt((C * 2.0));
	double t_2 = C * (A * 4.0);
	double t_3 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * ((F * (pow(B_m, 2.0) - t_2)) * 2.0))) / (t_2 - pow(B_m, 2.0));
	double t_4 = t_0 * (F * 2.0);
	double tmp;
	if (t_3 <= -5e+202) {
		tmp = (pow(fma((C * A), -4.0, (B_m * B_m)), -0.5) * sqrt((F * 2.0))) * t_1;
	} else if (t_3 <= -2e-227) {
		tmp = (-1.0 / t_0) * sqrt((((hypot((A - C), B_m) + A) + C) * t_4));
	} else if (t_3 <= 5.0) {
		tmp = (sqrt(((t_0 * F) * (((((B_m * B_m) / A) * -0.5) + C) + C))) / -t_0) * sqrt(2.0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt(t_4) / t_0) * t_1;
	} else {
		tmp = ((-1.0 / B_m) * sqrt(((hypot(C, B_m) + C) * 2.0))) * sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64(-sqrt(Float64(C * 2.0)))
	t_2 = Float64(C * Float64(A * 4.0))
	t_3 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_2)) * 2.0))) / Float64(t_2 - (B_m ^ 2.0)))
	t_4 = Float64(t_0 * Float64(F * 2.0))
	tmp = 0.0
	if (t_3 <= -5e+202)
		tmp = Float64(Float64((fma(Float64(C * A), -4.0, Float64(B_m * B_m)) ^ -0.5) * sqrt(Float64(F * 2.0))) * t_1);
	elseif (t_3 <= -2e-227)
		tmp = Float64(Float64(-1.0 / t_0) * sqrt(Float64(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C) * t_4)));
	elseif (t_3 <= 5.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(t_0 * F) * Float64(Float64(Float64(Float64(Float64(B_m * B_m) / A) * -0.5) + C) + C))) / Float64(-t_0)) * sqrt(2.0));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(t_4) / t_0) * t_1);
	else
		tmp = Float64(Float64(Float64(-1.0 / B_m) * sqrt(Float64(Float64(hypot(C, B_m) + C) * 2.0))) * sqrt(F));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision])}, Block[{t$95$2 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+202], N[(N[(N[Power[N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$3, -2e-227], N[(N[(-1.0 / t$95$0), $MachinePrecision] * N[Sqrt[N[(N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision] * t$95$4), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5.0], N[(N[(N[Sqrt[N[(N[(t$95$0 * F), $MachinePrecision] * N[(N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[t$95$4], $MachinePrecision] / t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(-1.0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.9999999999999999e202

   1. Initial program 4.7%

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

   4. Applied rewrites 34.9%

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

   5. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 22.1

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

   7. Applied rewrites 22.1%

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

   9. Applied rewrites 32.0%

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

   if -4.9999999999999999e202 < (/.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))) < -1.99999999999999989e-227

   1. Initial program 97.1%

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

   4. Applied rewrites 97.3%

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

   if -1.99999999999999989e-227 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 5

   1. Initial program 7.6%

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

   4. Applied rewrites 11.9%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 29.5

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(C + -0.5 \cdot \frac{\color{blue}{B \cdot B}}{A}\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 29.5%

      \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\color{blue}{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)} + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 5 < (/.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 27.5%

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 42.5

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

   7. Applied rewrites 42.5%

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

   9. Applied rewrites 42.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 16.9

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

   5. Applied rewrites 16.9%

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

   7. Applied rewrites 24.8%

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

   9. Applied rewrites 24.9%

      \[\leadsto \left(\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2} \cdot \frac{-1}{B}\right) \cdot \sqrt{\color{blue}{F}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -5 \cdot 10^{+202}:\\ \;\;\;\;\left({\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right)}^{-0.5} \cdot \sqrt{F \cdot 2}\right) \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 5:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F\right) \cdot \left(\left(\frac{B \cdot B}{A} \cdot -0.5 + C\right) + C\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{B} \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}\right) \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := t\_0 \cdot \left(F \cdot 2\right)\\ t_2 := -t\_0\\ t_3 := C \cdot \left(A \cdot 4\right)\\ t_4 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_3\right)\right) \cdot 2\right)}}{t\_3 - {B\_m}^{2}}\\ t_5 := -\sqrt{C \cdot 2}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\left({\left(\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\right)}^{-0.5} \cdot \sqrt{F \cdot 2}\right) \cdot t\_5\\ \mathbf{elif}\;t\_4 \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C\right) \cdot t\_1}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq 5:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot F\right) \cdot \left(\left(\frac{B\_m \cdot B\_m}{A} \cdot -0.5 + C\right) + C\right)}}{t\_2} \cdot \sqrt{2}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_1}}{t\_0} \cdot t\_5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{B\_m} \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot 2}\right) \cdot \sqrt{F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (* t_0 (* F 2.0)))
        (t_2 (- t_0))
        (t_3 (* C (* A 4.0)))
        (t_4
         (/
          (sqrt
           (*
            (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))
            (* (* F (- (pow B_m 2.0) t_3)) 2.0)))
          (- t_3 (pow B_m 2.0))))
        (t_5 (- (sqrt (* C 2.0)))))
   (if (<= t_4 (- INFINITY))
     (* (* (pow (fma (* C A) -4.0 (* B_m B_m)) -0.5) (sqrt (* F 2.0))) t_5)
     (if (<= t_4 -2e-227)
       (/ (sqrt (* (+ (+ (hypot (- A C) B_m) A) C) t_1)) t_2)
       (if (<= t_4 5.0)
         (*
          (/ (sqrt (* (* t_0 F) (+ (+ (* (/ (* B_m B_m) A) -0.5) C) C))) t_2)
          (sqrt 2.0))
         (if (<= t_4 INFINITY)
           (* (/ (sqrt t_1) t_0) t_5)
           (*
            (* (/ -1.0 B_m) (sqrt (* (+ (hypot C B_m) C) 2.0)))
            (sqrt F))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = t_0 * (F * 2.0);
	double t_2 = -t_0;
	double t_3 = C * (A * 4.0);
	double t_4 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * ((F * (pow(B_m, 2.0) - t_3)) * 2.0))) / (t_3 - pow(B_m, 2.0));
	double t_5 = -sqrt((C * 2.0));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = (pow(fma((C * A), -4.0, (B_m * B_m)), -0.5) * sqrt((F * 2.0))) * t_5;
	} else if (t_4 <= -2e-227) {
		tmp = sqrt((((hypot((A - C), B_m) + A) + C) * t_1)) / t_2;
	} else if (t_4 <= 5.0) {
		tmp = (sqrt(((t_0 * F) * (((((B_m * B_m) / A) * -0.5) + C) + C))) / t_2) * sqrt(2.0);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt(t_1) / t_0) * t_5;
	} else {
		tmp = ((-1.0 / B_m) * sqrt(((hypot(C, B_m) + C) * 2.0))) * sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64(t_0 * Float64(F * 2.0))
	t_2 = Float64(-t_0)
	t_3 = Float64(C * Float64(A * 4.0))
	t_4 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_3)) * 2.0))) / Float64(t_3 - (B_m ^ 2.0)))
	t_5 = Float64(-sqrt(Float64(C * 2.0)))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64((fma(Float64(C * A), -4.0, Float64(B_m * B_m)) ^ -0.5) * sqrt(Float64(F * 2.0))) * t_5);
	elseif (t_4 <= -2e-227)
		tmp = Float64(sqrt(Float64(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C) * t_1)) / t_2);
	elseif (t_4 <= 5.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(t_0 * F) * Float64(Float64(Float64(Float64(Float64(B_m * B_m) / A) * -0.5) + C) + C))) / t_2) * sqrt(2.0));
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(t_1) / t_0) * t_5);
	else
		tmp = Float64(Float64(Float64(-1.0 / B_m) * sqrt(Float64(Float64(hypot(C, B_m) + C) * 2.0))) * sqrt(F));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-t$95$0)}, Block[{t$95$3 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$3 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = (-N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision])}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[Power[N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision], If[LessEqual[t$95$4, -2e-227], N[(N[Sqrt[N[(N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, 5.0], N[(N[(N[Sqrt[N[(N[(t$95$0 * F), $MachinePrecision] * N[(N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Sqrt[t$95$1], $MachinePrecision] / t$95$0), $MachinePrecision] * t$95$5), $MachinePrecision], N[(N[(N[(-1.0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.0%

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

   4. Applied rewrites 33.8%

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

   5. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 22.4

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

   7. Applied rewrites 22.4%

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

   9. Applied rewrites 32.5%

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

   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))) < -1.99999999999999989e-227

   1. Initial program 97.2%

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

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      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 97.2%

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

   if -1.99999999999999989e-227 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 5

   1. Initial program 7.6%

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

   4. Applied rewrites 11.9%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 29.5

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(C + -0.5 \cdot \frac{\color{blue}{B \cdot B}}{A}\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 29.5%

      \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\color{blue}{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)} + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 5 < (/.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 27.5%

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 42.5

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

   7. Applied rewrites 42.5%

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

   9. Applied rewrites 42.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 16.9

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

   5. Applied rewrites 16.9%

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

   7. Applied rewrites 24.8%

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

   9. Applied rewrites 24.9%

      \[\leadsto \left(\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2} \cdot \frac{-1}{B}\right) \cdot \sqrt{\color{blue}{F}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\left({\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right)}^{-0.5} \cdot \sqrt{F \cdot 2}\right) \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 5:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F\right) \cdot \left(\left(\frac{B \cdot B}{A} \cdot -0.5 + C\right) + C\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{B} \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}\right) \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := C \cdot \left(A \cdot 4\right)\\ t_2 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_1\right)\right) \cdot 2\right)}}{t\_1 - {B\_m}^{2}}\\ t_3 := -\sqrt{C \cdot 2}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left({\left(\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\right)}^{-0.5} \cdot \sqrt{F \cdot 2}\right) \cdot t\_3\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{F}\right) \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C}\right)\\ \mathbf{elif}\;t\_2 \leq 5:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot F\right) \cdot \left(\left(\frac{B\_m \cdot B\_m}{A} \cdot -0.5 + C\right) + C\right)}}{-t\_0} \cdot \sqrt{2}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(F \cdot 2\right)}}{t\_0} \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{B\_m} \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot 2}\right) \cdot \sqrt{F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (* C (* A 4.0)))
        (t_2
         (/
          (sqrt
           (*
            (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))
            (* (* F (- (pow B_m 2.0) t_1)) 2.0)))
          (- t_1 (pow B_m 2.0))))
        (t_3 (- (sqrt (* C 2.0)))))
   (if (<= t_2 (- INFINITY))
     (* (* (pow (fma (* C A) -4.0 (* B_m B_m)) -0.5) (sqrt (* F 2.0))) t_3)
     (if (<= t_2 -2e-227)
       (*
        (* (/ (sqrt 2.0) B_m) (sqrt F))
        (- (sqrt (+ (+ (hypot (- A C) B_m) A) C))))
       (if (<= t_2 5.0)
         (*
          (/
           (sqrt (* (* t_0 F) (+ (+ (* (/ (* B_m B_m) A) -0.5) C) C)))
           (- t_0))
          (sqrt 2.0))
         (if (<= t_2 INFINITY)
           (* (/ (sqrt (* t_0 (* F 2.0))) t_0) t_3)
           (*
            (* (/ -1.0 B_m) (sqrt (* (+ (hypot C B_m) C) 2.0)))
            (sqrt F))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = C * (A * 4.0);
	double t_2 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * ((F * (pow(B_m, 2.0) - t_1)) * 2.0))) / (t_1 - pow(B_m, 2.0));
	double t_3 = -sqrt((C * 2.0));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (pow(fma((C * A), -4.0, (B_m * B_m)), -0.5) * sqrt((F * 2.0))) * t_3;
	} else if (t_2 <= -2e-227) {
		tmp = ((sqrt(2.0) / B_m) * sqrt(F)) * -sqrt(((hypot((A - C), B_m) + A) + C));
	} else if (t_2 <= 5.0) {
		tmp = (sqrt(((t_0 * F) * (((((B_m * B_m) / A) * -0.5) + C) + C))) / -t_0) * sqrt(2.0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (sqrt((t_0 * (F * 2.0))) / t_0) * t_3;
	} else {
		tmp = ((-1.0 / B_m) * sqrt(((hypot(C, B_m) + C) * 2.0))) * sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64(C * Float64(A * 4.0))
	t_2 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_1)) * 2.0))) / Float64(t_1 - (B_m ^ 2.0)))
	t_3 = Float64(-sqrt(Float64(C * 2.0)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64((fma(Float64(C * A), -4.0, Float64(B_m * B_m)) ^ -0.5) * sqrt(Float64(F * 2.0))) * t_3);
	elseif (t_2 <= -2e-227)
		tmp = Float64(Float64(Float64(sqrt(2.0) / B_m) * sqrt(F)) * Float64(-sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C))));
	elseif (t_2 <= 5.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(t_0 * F) * Float64(Float64(Float64(Float64(Float64(B_m * B_m) / A) * -0.5) + C) + C))) / Float64(-t_0)) * sqrt(2.0));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(t_0 * Float64(F * 2.0))) / t_0) * t_3);
	else
		tmp = Float64(Float64(Float64(-1.0 / B_m) * sqrt(Float64(Float64(hypot(C, B_m) + C) * 2.0))) * sqrt(F));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = (-N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision])}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[Power[N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[t$95$2, -2e-227], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$2, 5.0], N[(N[(N[Sqrt[N[(N[(t$95$0 * F), $MachinePrecision] * N[(N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[Sqrt[N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] * t$95$3), $MachinePrecision], N[(N[(N[(-1.0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.0%

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

   4. Applied rewrites 33.8%

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

   5. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 22.4

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

   7. Applied rewrites 22.4%

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

   9. Applied rewrites 32.5%

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

   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))) < -1.99999999999999989e-227

   1. Initial program 97.2%

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

   4. Applied rewrites 97.2%

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

   5. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 38.9

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

   7. Applied rewrites 38.9%

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

   if -1.99999999999999989e-227 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 5

   1. Initial program 7.6%

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

   4. Applied rewrites 11.9%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 29.5

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(C + -0.5 \cdot \frac{\color{blue}{B \cdot B}}{A}\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 29.5%

      \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\color{blue}{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)} + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 5 < (/.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 27.5%

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 42.5

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

   7. Applied rewrites 42.5%

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

   9. Applied rewrites 42.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 16.9

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

   5. Applied rewrites 16.9%

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

   7. Applied rewrites 24.8%

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

   9. Applied rewrites 24.9%

      \[\leadsto \left(\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2} \cdot \frac{-1}{B}\right) \cdot \sqrt{\color{blue}{F}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\left({\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right)}^{-0.5} \cdot \sqrt{F \cdot 2}\right) \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F}\right) \cdot \left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 5:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F\right) \cdot \left(\left(\frac{B \cdot B}{A} \cdot -0.5 + C\right) + C\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{B} \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}\right) \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := -\sqrt{C \cdot 2}\\ t_2 := C \cdot \left(A \cdot 4\right)\\ t_3 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_2\right)\right) \cdot 2\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+202}:\\ \;\;\;\;\left({\left(\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\right)}^{-0.5} \cdot \sqrt{F \cdot 2}\right) \cdot t\_1\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{\frac{\left(\left(\mathsf{hypot}\left(A - C, B\_m\right) + C\right) + A\right) \cdot F}{t\_0}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_3 \leq 5:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot F\right) \cdot \left(\left(\frac{B\_m \cdot B\_m}{A} \cdot -0.5 + C\right) + C\right)}}{-t\_0} \cdot \sqrt{2}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(F \cdot 2\right)}}{t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{B\_m} \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot 2}\right) \cdot \sqrt{F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (- (sqrt (* C 2.0))))
        (t_2 (* C (* A 4.0)))
        (t_3
         (/
          (sqrt
           (*
            (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))
            (* (* F (- (pow B_m 2.0) t_2)) 2.0)))
          (- t_2 (pow B_m 2.0)))))
   (if (<= t_3 -5e+202)
     (* (* (pow (fma (* C A) -4.0 (* B_m B_m)) -0.5) (sqrt (* F 2.0))) t_1)
     (if (<= t_3 -2e-153)
       (* (sqrt (/ (* (+ (+ (hypot (- A C) B_m) C) A) F) t_0)) (- (sqrt 2.0)))
       (if (<= t_3 5.0)
         (*
          (/
           (sqrt (* (* t_0 F) (+ (+ (* (/ (* B_m B_m) A) -0.5) C) C)))
           (- t_0))
          (sqrt 2.0))
         (if (<= t_3 INFINITY)
           (* (/ (sqrt (* t_0 (* F 2.0))) t_0) t_1)
           (*
            (* (/ -1.0 B_m) (sqrt (* (+ (hypot C B_m) C) 2.0)))
            (sqrt F))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = -sqrt((C * 2.0));
	double t_2 = C * (A * 4.0);
	double t_3 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * ((F * (pow(B_m, 2.0) - t_2)) * 2.0))) / (t_2 - pow(B_m, 2.0));
	double tmp;
	if (t_3 <= -5e+202) {
		tmp = (pow(fma((C * A), -4.0, (B_m * B_m)), -0.5) * sqrt((F * 2.0))) * t_1;
	} else if (t_3 <= -2e-153) {
		tmp = sqrt(((((hypot((A - C), B_m) + C) + A) * F) / t_0)) * -sqrt(2.0);
	} else if (t_3 <= 5.0) {
		tmp = (sqrt(((t_0 * F) * (((((B_m * B_m) / A) * -0.5) + C) + C))) / -t_0) * sqrt(2.0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt((t_0 * (F * 2.0))) / t_0) * t_1;
	} else {
		tmp = ((-1.0 / B_m) * sqrt(((hypot(C, B_m) + C) * 2.0))) * sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64(-sqrt(Float64(C * 2.0)))
	t_2 = Float64(C * Float64(A * 4.0))
	t_3 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_2)) * 2.0))) / Float64(t_2 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_3 <= -5e+202)
		tmp = Float64(Float64((fma(Float64(C * A), -4.0, Float64(B_m * B_m)) ^ -0.5) * sqrt(Float64(F * 2.0))) * t_1);
	elseif (t_3 <= -2e-153)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(hypot(Float64(A - C), B_m) + C) + A) * F) / t_0)) * Float64(-sqrt(2.0)));
	elseif (t_3 <= 5.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(t_0 * F) * Float64(Float64(Float64(Float64(Float64(B_m * B_m) / A) * -0.5) + C) + C))) / Float64(-t_0)) * sqrt(2.0));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(t_0 * Float64(F * 2.0))) / t_0) * t_1);
	else
		tmp = Float64(Float64(Float64(-1.0 / B_m) * sqrt(Float64(Float64(hypot(C, B_m) + C) * 2.0))) * sqrt(F));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision])}, Block[{t$95$2 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+202], N[(N[(N[Power[N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$3, -2e-153], N[(N[Sqrt[N[(N[(N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] + A), $MachinePrecision] * F), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$3, 5.0], N[(N[(N[Sqrt[N[(N[(t$95$0 * F), $MachinePrecision] * N[(N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(-1.0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.9999999999999999e202

   1. Initial program 4.7%

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

   4. Applied rewrites 34.9%

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

   5. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 22.1

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

   7. Applied rewrites 22.1%

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

   9. Applied rewrites 32.0%

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

   if -4.9999999999999999e202 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -2.00000000000000008e-153

   1. Initial program 97.0%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 97.0%

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

   if -2.00000000000000008e-153 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 5

   1. Initial program 11.3%

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

   4. Applied rewrites 15.4%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 28.5

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(C + -0.5 \cdot \frac{\color{blue}{B \cdot B}}{A}\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 28.5%

      \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\color{blue}{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)} + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 5 < (/.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 27.5%

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 42.5

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

   7. Applied rewrites 42.5%

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

   9. Applied rewrites 42.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 16.9

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

   5. Applied rewrites 16.9%

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

   7. Applied rewrites 24.8%

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

   9. Applied rewrites 24.9%

      \[\leadsto \left(\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2} \cdot \frac{-1}{B}\right) \cdot \sqrt{\color{blue}{F}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -5 \cdot 10^{+202}:\\ \;\;\;\;\left({\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right)}^{-0.5} \cdot \sqrt{F \cdot 2}\right) \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{\frac{\left(\left(\mathsf{hypot}\left(A - C, B\right) + C\right) + A\right) \cdot F}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 5:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F\right) \cdot \left(\left(\frac{B \cdot B}{A} \cdot -0.5 + C\right) + C\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{B} \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}\right) \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(C, B\_m\right) + C\\ t_1 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_2 := C \cdot \left(A \cdot 4\right)\\ t_3 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_2\right)\right) \cdot 2\right)}}{t\_2 - {B\_m}^{2}}\\ t_4 := -\sqrt{C \cdot 2}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\left({\left(\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\right)}^{-0.5} \cdot \sqrt{F \cdot 2}\right) \cdot t\_4\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot F\right) \cdot 2}}{-B\_m}\\ \mathbf{elif}\;t\_3 \leq 5:\\ \;\;\;\;\frac{\sqrt{\left(t\_1 \cdot F\right) \cdot \left(\left(\frac{B\_m \cdot B\_m}{A} \cdot -0.5 + C\right) + C\right)}}{-t\_1} \cdot \sqrt{2}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(F \cdot 2\right)}}{t\_1} \cdot t\_4\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{B\_m} \cdot \sqrt{t\_0 \cdot 2}\right) \cdot \sqrt{F}\\ \end{array} \end{array} \]

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 (+ (hypot C B_m) C))
        (t_1 (fma -4.0 (* C A) (* B_m B_m)))
        (t_2 (* C (* A 4.0)))
        (t_3
         (/
          (sqrt
           (*
            (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))
            (* (* F (- (pow B_m 2.0) t_2)) 2.0)))
          (- t_2 (pow B_m 2.0))))
        (t_4 (- (sqrt (* C 2.0)))))
   (if (<= t_3 (- INFINITY))
     (* (* (pow (fma (* C A) -4.0 (* B_m B_m)) -0.5) (sqrt (* F 2.0))) t_4)
     (if (<= t_3 -2e-227)
       (/ (sqrt (* (* t_0 F) 2.0)) (- B_m))
       (if (<= t_3 5.0)
         (*
          (/
           (sqrt (* (* t_1 F) (+ (+ (* (/ (* B_m B_m) A) -0.5) C) C)))
           (- t_1))
          (sqrt 2.0))
         (if (<= t_3 INFINITY)
           (* (/ (sqrt (* t_1 (* F 2.0))) t_1) t_4)
           (* (* (/ -1.0 B_m) (sqrt (* t_0 2.0))) (sqrt F))))))))

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 = hypot(C, B_m) + C;
	double t_1 = fma(-4.0, (C * A), (B_m * B_m));
	double t_2 = C * (A * 4.0);
	double t_3 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * ((F * (pow(B_m, 2.0) - t_2)) * 2.0))) / (t_2 - pow(B_m, 2.0));
	double t_4 = -sqrt((C * 2.0));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (pow(fma((C * A), -4.0, (B_m * B_m)), -0.5) * sqrt((F * 2.0))) * t_4;
	} else if (t_3 <= -2e-227) {
		tmp = sqrt(((t_0 * F) * 2.0)) / -B_m;
	} else if (t_3 <= 5.0) {
		tmp = (sqrt(((t_1 * F) * (((((B_m * B_m) / A) * -0.5) + C) + C))) / -t_1) * sqrt(2.0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt((t_1 * (F * 2.0))) / t_1) * t_4;
	} else {
		tmp = ((-1.0 / B_m) * sqrt((t_0 * 2.0))) * sqrt(F);
	}
	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(hypot(C, B_m) + C)
	t_1 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_2 = Float64(C * Float64(A * 4.0))
	t_3 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_2)) * 2.0))) / Float64(t_2 - (B_m ^ 2.0)))
	t_4 = Float64(-sqrt(Float64(C * 2.0)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64((fma(Float64(C * A), -4.0, Float64(B_m * B_m)) ^ -0.5) * sqrt(Float64(F * 2.0))) * t_4);
	elseif (t_3 <= -2e-227)
		tmp = Float64(sqrt(Float64(Float64(t_0 * F) * 2.0)) / Float64(-B_m));
	elseif (t_3 <= 5.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(t_1 * F) * Float64(Float64(Float64(Float64(Float64(B_m * B_m) / A) * -0.5) + C) + C))) / Float64(-t_1)) * sqrt(2.0));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(t_1 * Float64(F * 2.0))) / t_1) * t_4);
	else
		tmp = Float64(Float64(Float64(-1.0 / B_m) * sqrt(Float64(t_0 * 2.0))) * sqrt(F));
	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[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = (-N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision])}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[Power[N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision], If[LessEqual[t$95$3, -2e-227], N[(N[Sqrt[N[(N[(t$95$0 * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], If[LessEqual[t$95$3, 5.0], N[(N[(N[Sqrt[N[(N[(t$95$1 * F), $MachinePrecision] * N[(N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(t$95$1 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision] * t$95$4), $MachinePrecision], N[(N[(N[(-1.0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.0%

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

   4. Applied rewrites 33.8%

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

   5. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 22.4

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

   7. Applied rewrites 22.4%

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

   9. Applied rewrites 32.5%

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

   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))) < -1.99999999999999989e-227

   1. Initial program 97.2%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 36.1

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

   5. Applied rewrites 36.1%

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

   7. Applied rewrites 36.2%

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

   if -1.99999999999999989e-227 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 5

   1. Initial program 7.6%

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

   4. Applied rewrites 11.9%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 29.5

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(C + -0.5 \cdot \frac{\color{blue}{B \cdot B}}{A}\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 29.5%

      \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\color{blue}{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)} + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 5 < (/.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 27.5%

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 42.5

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

   7. Applied rewrites 42.5%

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

   9. Applied rewrites 42.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 16.9

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

   5. Applied rewrites 16.9%

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

   7. Applied rewrites 24.8%

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

   9. Applied rewrites 24.9%

      \[\leadsto \left(\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2} \cdot \frac{-1}{B}\right) \cdot \sqrt{\color{blue}{F}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\left({\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right)}^{-0.5} \cdot \sqrt{F \cdot 2}\right) \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F\right) \cdot 2}}{-B}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 5:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F\right) \cdot \left(\left(\frac{B \cdot B}{A} \cdot -0.5 + C\right) + C\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{B} \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}\right) \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(C, B\_m\right) + C\\ t_1 := C \cdot \left(A \cdot 4\right)\\ t_2 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_1\right)\right) \cdot 2\right)}}{t\_1 - {B\_m}^{2}}\\ t_3 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right) \cdot 2} \cdot \sqrt{F}}{t\_3} \cdot \left(\sqrt{C} \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot F\right) \cdot 2}}{-B\_m}\\ \mathbf{elif}\;t\_2 \leq 5:\\ \;\;\;\;\frac{\sqrt{\left(t\_3 \cdot F\right) \cdot \left(\left(\frac{B\_m \cdot B\_m}{A} \cdot -0.5 + C\right) + C\right)}}{-t\_3} \cdot \sqrt{2}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_3 \cdot \left(F \cdot 2\right)}}{t\_3} \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{B\_m} \cdot \sqrt{t\_0 \cdot 2}\right) \cdot \sqrt{F}\\ \end{array} \end{array} \]

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 (+ (hypot C B_m) C))
        (t_1 (* C (* A 4.0)))
        (t_2
         (/
          (sqrt
           (*
            (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))
            (* (* F (- (pow B_m 2.0) t_1)) 2.0)))
          (- t_1 (pow B_m 2.0))))
        (t_3 (fma -4.0 (* C A) (* B_m B_m))))
   (if (<= t_2 (- INFINITY))
     (*
      (/ (* (sqrt (* (fma (* C A) -4.0 (* B_m B_m)) 2.0)) (sqrt F)) t_3)
      (* (sqrt C) (- (sqrt 2.0))))
     (if (<= t_2 -2e-227)
       (/ (sqrt (* (* t_0 F) 2.0)) (- B_m))
       (if (<= t_2 5.0)
         (*
          (/
           (sqrt (* (* t_3 F) (+ (+ (* (/ (* B_m B_m) A) -0.5) C) C)))
           (- t_3))
          (sqrt 2.0))
         (if (<= t_2 INFINITY)
           (* (/ (sqrt (* t_3 (* F 2.0))) t_3) (- (sqrt (* C 2.0))))
           (* (* (/ -1.0 B_m) (sqrt (* t_0 2.0))) (sqrt F))))))))

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 = hypot(C, B_m) + C;
	double t_1 = C * (A * 4.0);
	double t_2 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * ((F * (pow(B_m, 2.0) - t_1)) * 2.0))) / (t_1 - pow(B_m, 2.0));
	double t_3 = fma(-4.0, (C * A), (B_m * B_m));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = ((sqrt((fma((C * A), -4.0, (B_m * B_m)) * 2.0)) * sqrt(F)) / t_3) * (sqrt(C) * -sqrt(2.0));
	} else if (t_2 <= -2e-227) {
		tmp = sqrt(((t_0 * F) * 2.0)) / -B_m;
	} else if (t_2 <= 5.0) {
		tmp = (sqrt(((t_3 * F) * (((((B_m * B_m) / A) * -0.5) + C) + C))) / -t_3) * sqrt(2.0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (sqrt((t_3 * (F * 2.0))) / t_3) * -sqrt((C * 2.0));
	} else {
		tmp = ((-1.0 / B_m) * sqrt((t_0 * 2.0))) * sqrt(F);
	}
	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(hypot(C, B_m) + C)
	t_1 = Float64(C * Float64(A * 4.0))
	t_2 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_1)) * 2.0))) / Float64(t_1 - (B_m ^ 2.0)))
	t_3 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(sqrt(Float64(fma(Float64(C * A), -4.0, Float64(B_m * B_m)) * 2.0)) * sqrt(F)) / t_3) * Float64(sqrt(C) * Float64(-sqrt(2.0))));
	elseif (t_2 <= -2e-227)
		tmp = Float64(sqrt(Float64(Float64(t_0 * F) * 2.0)) / Float64(-B_m));
	elseif (t_2 <= 5.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(t_3 * F) * Float64(Float64(Float64(Float64(Float64(B_m * B_m) / A) * -0.5) + C) + C))) / Float64(-t_3)) * sqrt(2.0));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(t_3 * Float64(F * 2.0))) / t_3) * Float64(-sqrt(Float64(C * 2.0))));
	else
		tmp = Float64(Float64(Float64(-1.0 / B_m) * sqrt(Float64(t_0 * 2.0))) * sqrt(F));
	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[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]}, Block[{t$95$1 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(N[Sqrt[N[(N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] * N[(N[Sqrt[C], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e-227], N[(N[Sqrt[N[(N[(t$95$0 * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], If[LessEqual[t$95$2, 5.0], N[(N[(N[Sqrt[N[(N[(t$95$3 * F), $MachinePrecision] * N[(N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$3)), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[Sqrt[N[(t$95$3 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision] * (-N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(N[(-1.0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.0%

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

   4. Applied rewrites 33.8%

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

   5. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 22.4

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

   7. Applied rewrites 22.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. sqrt-prod N/A

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

      6. lift-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. sqrt-prod N/A

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

      17. associate-*r* N/A

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

   9. Applied rewrites 32.4%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.99999999999999989e-227

   1. Initial program 97.2%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 36.1

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

   5. Applied rewrites 36.1%

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

   7. Applied rewrites 36.2%

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

   if -1.99999999999999989e-227 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 5

   1. Initial program 7.6%

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

   4. Applied rewrites 11.9%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 29.5

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(C + -0.5 \cdot \frac{\color{blue}{B \cdot B}}{A}\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 29.5%

      \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\color{blue}{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)} + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 5 < (/.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 27.5%

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 42.5

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

   7. Applied rewrites 42.5%

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

   9. Applied rewrites 42.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 16.9

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

   5. Applied rewrites 16.9%

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

   7. Applied rewrites 24.8%

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

   9. Applied rewrites 24.9%

      \[\leadsto \left(\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2} \cdot \frac{-1}{B}\right) \cdot \sqrt{\color{blue}{F}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2} \cdot \sqrt{F}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{C} \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F\right) \cdot 2}}{-B}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 5:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F\right) \cdot \left(\left(\frac{B \cdot B}{A} \cdot -0.5 + C\right) + C\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{B} \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}\right) \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(C, B\_m\right) + C\\ t_1 := C \cdot \left(A \cdot 4\right)\\ t_2 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_1\right)\right) \cdot 2\right)}}{t\_1 - {B\_m}^{2}}\\ t_3 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right) \cdot 2} \cdot \sqrt{F}}{t\_3} \cdot \left(\sqrt{C} \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot F\right) \cdot 2}}{-B\_m}\\ \mathbf{elif}\;t\_2 \leq 5:\\ \;\;\;\;\frac{\sqrt{\left(t\_3 \cdot F\right) \cdot \left(\left(\frac{B\_m \cdot B\_m}{A} \cdot -0.5 + C\right) + C\right)}}{-t\_3} \cdot \sqrt{2}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_3 \cdot \left(F \cdot 2\right)}}{t\_3} \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot 2}}{-B\_m} \cdot \sqrt{F}\\ \end{array} \end{array} \]

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 (+ (hypot C B_m) C))
        (t_1 (* C (* A 4.0)))
        (t_2
         (/
          (sqrt
           (*
            (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))
            (* (* F (- (pow B_m 2.0) t_1)) 2.0)))
          (- t_1 (pow B_m 2.0))))
        (t_3 (fma -4.0 (* C A) (* B_m B_m))))
   (if (<= t_2 (- INFINITY))
     (*
      (/ (* (sqrt (* (fma (* C A) -4.0 (* B_m B_m)) 2.0)) (sqrt F)) t_3)
      (* (sqrt C) (- (sqrt 2.0))))
     (if (<= t_2 -2e-227)
       (/ (sqrt (* (* t_0 F) 2.0)) (- B_m))
       (if (<= t_2 5.0)
         (*
          (/
           (sqrt (* (* t_3 F) (+ (+ (* (/ (* B_m B_m) A) -0.5) C) C)))
           (- t_3))
          (sqrt 2.0))
         (if (<= t_2 INFINITY)
           (* (/ (sqrt (* t_3 (* F 2.0))) t_3) (- (sqrt (* C 2.0))))
           (* (/ (sqrt (* t_0 2.0)) (- B_m)) (sqrt F))))))))

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 = hypot(C, B_m) + C;
	double t_1 = C * (A * 4.0);
	double t_2 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * ((F * (pow(B_m, 2.0) - t_1)) * 2.0))) / (t_1 - pow(B_m, 2.0));
	double t_3 = fma(-4.0, (C * A), (B_m * B_m));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = ((sqrt((fma((C * A), -4.0, (B_m * B_m)) * 2.0)) * sqrt(F)) / t_3) * (sqrt(C) * -sqrt(2.0));
	} else if (t_2 <= -2e-227) {
		tmp = sqrt(((t_0 * F) * 2.0)) / -B_m;
	} else if (t_2 <= 5.0) {
		tmp = (sqrt(((t_3 * F) * (((((B_m * B_m) / A) * -0.5) + C) + C))) / -t_3) * sqrt(2.0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (sqrt((t_3 * (F * 2.0))) / t_3) * -sqrt((C * 2.0));
	} else {
		tmp = (sqrt((t_0 * 2.0)) / -B_m) * sqrt(F);
	}
	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(hypot(C, B_m) + C)
	t_1 = Float64(C * Float64(A * 4.0))
	t_2 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_1)) * 2.0))) / Float64(t_1 - (B_m ^ 2.0)))
	t_3 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(sqrt(Float64(fma(Float64(C * A), -4.0, Float64(B_m * B_m)) * 2.0)) * sqrt(F)) / t_3) * Float64(sqrt(C) * Float64(-sqrt(2.0))));
	elseif (t_2 <= -2e-227)
		tmp = Float64(sqrt(Float64(Float64(t_0 * F) * 2.0)) / Float64(-B_m));
	elseif (t_2 <= 5.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(t_3 * F) * Float64(Float64(Float64(Float64(Float64(B_m * B_m) / A) * -0.5) + C) + C))) / Float64(-t_3)) * sqrt(2.0));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(t_3 * Float64(F * 2.0))) / t_3) * Float64(-sqrt(Float64(C * 2.0))));
	else
		tmp = Float64(Float64(sqrt(Float64(t_0 * 2.0)) / Float64(-B_m)) * sqrt(F));
	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[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]}, Block[{t$95$1 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(N[Sqrt[N[(N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] * N[(N[Sqrt[C], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e-227], N[(N[Sqrt[N[(N[(t$95$0 * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], If[LessEqual[t$95$2, 5.0], N[(N[(N[Sqrt[N[(N[(t$95$3 * F), $MachinePrecision] * N[(N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$3)), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[Sqrt[N[(t$95$3 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision] * (-N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.0%

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

   4. Applied rewrites 33.8%

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

   5. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 22.4

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

   7. Applied rewrites 22.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. sqrt-prod N/A

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

      6. lift-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. sqrt-prod N/A

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

      17. associate-*r* N/A

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

   9. Applied rewrites 32.4%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.99999999999999989e-227

   1. Initial program 97.2%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 36.1

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

   5. Applied rewrites 36.1%

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

   7. Applied rewrites 36.2%

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

   if -1.99999999999999989e-227 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 5

   1. Initial program 7.6%

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

   4. Applied rewrites 11.9%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 29.5

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(C + -0.5 \cdot \frac{\color{blue}{B \cdot B}}{A}\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 29.5%

      \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\color{blue}{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)} + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 5 < (/.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 27.5%

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 42.5

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

   7. Applied rewrites 42.5%

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

   9. Applied rewrites 42.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 16.9

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

   5. Applied rewrites 16.9%

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

   7. Applied rewrites 24.8%

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

   9. Applied rewrites 24.9%

      \[\leadsto \frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \color{blue}{\sqrt{F}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2} \cdot \sqrt{F}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{C} \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F\right) \cdot 2}}{-B}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 5:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F\right) \cdot \left(\left(\frac{B \cdot B}{A} \cdot -0.5 + C\right) + C\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(A \cdot 4\right)\\ t_1 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_0\right)\right) \cdot 2\right)}}{t\_0 - {B\_m}^{2}}\\ t_2 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right) \cdot 2} \cdot \sqrt{F}}{t\_2} \cdot \left(\sqrt{C} \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot F\right) \cdot 2}}{-B\_m}\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;\frac{\sqrt{\left(t\_2 \cdot F\right) \cdot \left(\left(\frac{B\_m \cdot B\_m}{A} \cdot -0.5 + C\right) + C\right)}}{-t\_2} \cdot \sqrt{2}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(F \cdot 2\right)}}{t\_2} \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-{B\_m}^{-0.5}\right) \cdot \sqrt{F \cdot 2}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* C (* A 4.0)))
        (t_1
         (/
          (sqrt
           (*
            (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))
            (* (* F (- (pow B_m 2.0) t_0)) 2.0)))
          (- t_0 (pow B_m 2.0))))
        (t_2 (fma -4.0 (* C A) (* B_m B_m))))
   (if (<= t_1 (- INFINITY))
     (*
      (/ (* (sqrt (* (fma (* C A) -4.0 (* B_m B_m)) 2.0)) (sqrt F)) t_2)
      (* (sqrt C) (- (sqrt 2.0))))
     (if (<= t_1 -2e-227)
       (/ (sqrt (* (* (+ (hypot C B_m) C) F) 2.0)) (- B_m))
       (if (<= t_1 5.0)
         (*
          (/
           (sqrt (* (* t_2 F) (+ (+ (* (/ (* B_m B_m) A) -0.5) C) C)))
           (- t_2))
          (sqrt 2.0))
         (if (<= t_1 INFINITY)
           (* (/ (sqrt (* t_2 (* F 2.0))) t_2) (- (sqrt (* C 2.0))))
           (* (- (pow B_m -0.5)) (sqrt (* F 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 = C * (A * 4.0);
	double t_1 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * ((F * (pow(B_m, 2.0) - t_0)) * 2.0))) / (t_0 - pow(B_m, 2.0));
	double t_2 = fma(-4.0, (C * A), (B_m * B_m));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((sqrt((fma((C * A), -4.0, (B_m * B_m)) * 2.0)) * sqrt(F)) / t_2) * (sqrt(C) * -sqrt(2.0));
	} else if (t_1 <= -2e-227) {
		tmp = sqrt((((hypot(C, B_m) + C) * F) * 2.0)) / -B_m;
	} else if (t_1 <= 5.0) {
		tmp = (sqrt(((t_2 * F) * (((((B_m * B_m) / A) * -0.5) + C) + C))) / -t_2) * sqrt(2.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (sqrt((t_2 * (F * 2.0))) / t_2) * -sqrt((C * 2.0));
	} else {
		tmp = -pow(B_m, -0.5) * sqrt((F * 2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(C * Float64(A * 4.0))
	t_1 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_0)) * 2.0))) / Float64(t_0 - (B_m ^ 2.0)))
	t_2 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(sqrt(Float64(fma(Float64(C * A), -4.0, Float64(B_m * B_m)) * 2.0)) * sqrt(F)) / t_2) * Float64(sqrt(C) * Float64(-sqrt(2.0))));
	elseif (t_1 <= -2e-227)
		tmp = Float64(sqrt(Float64(Float64(Float64(hypot(C, B_m) + C) * F) * 2.0)) / Float64(-B_m));
	elseif (t_1 <= 5.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(t_2 * F) * Float64(Float64(Float64(Float64(Float64(B_m * B_m) / A) * -0.5) + C) + C))) / Float64(-t_2)) * sqrt(2.0));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(t_2 * Float64(F * 2.0))) / t_2) * Float64(-sqrt(Float64(C * 2.0))));
	else
		tmp = Float64(Float64(-(B_m ^ -0.5)) * sqrt(Float64(F * 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[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[Sqrt[N[(N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(N[Sqrt[C], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e-227], N[(N[Sqrt[N[(N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], If[LessEqual[t$95$1, 5.0], N[(N[(N[Sqrt[N[(N[(t$95$2 * F), $MachinePrecision] * N[(N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$2)), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[Sqrt[N[(t$95$2 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision] * (-N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[((-N[Power[B$95$m, -0.5], $MachinePrecision]) * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.0%

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

   4. Applied rewrites 33.8%

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

   5. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 22.4

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

   7. Applied rewrites 22.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. sqrt-prod N/A

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

      6. lift-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. sqrt-prod N/A

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

      17. associate-*r* N/A

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

   9. Applied rewrites 32.4%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.99999999999999989e-227

   1. Initial program 97.2%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 36.1

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

   5. Applied rewrites 36.1%

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

   7. Applied rewrites 36.2%

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

   if -1.99999999999999989e-227 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 5

   1. Initial program 7.6%

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

   4. Applied rewrites 11.9%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(C + \color{blue}{\frac{-1}{2} \cdot \frac{{B}^{2}}{A}}\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(C + \frac{-1}{2} \cdot \color{blue}{\frac{{B}^{2}}{A}}\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(C + \frac{-1}{2} \cdot \frac{\color{blue}{B \cdot B}}{A}\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-*.f64 29.5

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(C + -0.5 \cdot \frac{\color{blue}{B \cdot B}}{A}\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 29.5%

      \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\color{blue}{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)} + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 5 < (/.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 27.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 85.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-1} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{C} \cdot \sqrt{2}\right)\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{C} \cdot \sqrt{2}\right)\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\sqrt{C} \cdot \sqrt{2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{C} \cdot \sqrt{2}}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{C}} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-sqrt.f64 42.5

         \[\leadsto \left(-\sqrt{C} \cdot \color{blue}{\sqrt{2}}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 42.5%

      \[\leadsto \color{blue}{\left(-\sqrt{C} \cdot \sqrt{2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 42.8%

      \[\leadsto \color{blue}{\left(-\sqrt{C \cdot 2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

      5. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{F}{B}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

      8. lower-/.f64 14.2

         \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 14.2%

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 14.3%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 22.1%

      \[\leadsto -{B}^{-0.5} \cdot \sqrt{F \cdot 2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 29.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2} \cdot \sqrt{F}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{C} \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F\right) \cdot 2}}{-B}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 5:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F\right) \cdot \left(\left(\frac{B \cdot B}{A} \cdot -0.5 + C\right) + C\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-{B}^{-0.5}\right) \cdot \sqrt{F \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := -t\_0\\ t_2 := C \cdot \left(A \cdot 4\right)\\ t_3 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_2\right)\right) \cdot 2\right)}}{t\_2 - {B\_m}^{2}}\\ t_4 := t\_0 \cdot F\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+202}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right) \cdot 2} \cdot \sqrt{F}}{t\_0} \cdot \left(\sqrt{C} \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\frac{A}{B\_m} - -1\right) \cdot B\_m + C\right) \cdot t\_4}}{t\_1} \cdot \sqrt{2}\\ \mathbf{elif}\;t\_3 \leq 5:\\ \;\;\;\;\frac{\sqrt{t\_4 \cdot \left(\left(\frac{B\_m \cdot B\_m}{A} \cdot -0.5 + C\right) + C\right)}}{t\_1} \cdot \sqrt{2}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(F \cdot 2\right)}}{t\_0} \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-{B\_m}^{-0.5}\right) \cdot \sqrt{F \cdot 2}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (- t_0))
        (t_2 (* C (* A 4.0)))
        (t_3
         (/
          (sqrt
           (*
            (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))
            (* (* F (- (pow B_m 2.0) t_2)) 2.0)))
          (- t_2 (pow B_m 2.0))))
        (t_4 (* t_0 F)))
   (if (<= t_3 -5e+202)
     (*
      (/ (* (sqrt (* (fma (* C A) -4.0 (* B_m B_m)) 2.0)) (sqrt F)) t_0)
      (* (sqrt C) (- (sqrt 2.0))))
     (if (<= t_3 -2e-227)
       (* (/ (sqrt (* (+ (* (- (/ A B_m) -1.0) B_m) C) t_4)) t_1) (sqrt 2.0))
       (if (<= t_3 5.0)
         (*
          (/ (sqrt (* t_4 (+ (+ (* (/ (* B_m B_m) A) -0.5) C) C))) t_1)
          (sqrt 2.0))
         (if (<= t_3 INFINITY)
           (* (/ (sqrt (* t_0 (* F 2.0))) t_0) (- (sqrt (* C 2.0))))
           (* (- (pow B_m -0.5)) (sqrt (* F 2.0)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = -t_0;
	double t_2 = C * (A * 4.0);
	double t_3 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * ((F * (pow(B_m, 2.0) - t_2)) * 2.0))) / (t_2 - pow(B_m, 2.0));
	double t_4 = t_0 * F;
	double tmp;
	if (t_3 <= -5e+202) {
		tmp = ((sqrt((fma((C * A), -4.0, (B_m * B_m)) * 2.0)) * sqrt(F)) / t_0) * (sqrt(C) * -sqrt(2.0));
	} else if (t_3 <= -2e-227) {
		tmp = (sqrt((((((A / B_m) - -1.0) * B_m) + C) * t_4)) / t_1) * sqrt(2.0);
	} else if (t_3 <= 5.0) {
		tmp = (sqrt((t_4 * (((((B_m * B_m) / A) * -0.5) + C) + C))) / t_1) * sqrt(2.0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt((t_0 * (F * 2.0))) / t_0) * -sqrt((C * 2.0));
	} else {
		tmp = -pow(B_m, -0.5) * sqrt((F * 2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64(-t_0)
	t_2 = Float64(C * Float64(A * 4.0))
	t_3 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_2)) * 2.0))) / Float64(t_2 - (B_m ^ 2.0)))
	t_4 = Float64(t_0 * F)
	tmp = 0.0
	if (t_3 <= -5e+202)
		tmp = Float64(Float64(Float64(sqrt(Float64(fma(Float64(C * A), -4.0, Float64(B_m * B_m)) * 2.0)) * sqrt(F)) / t_0) * Float64(sqrt(C) * Float64(-sqrt(2.0))));
	elseif (t_3 <= -2e-227)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(Float64(Float64(A / B_m) - -1.0) * B_m) + C) * t_4)) / t_1) * sqrt(2.0));
	elseif (t_3 <= 5.0)
		tmp = Float64(Float64(sqrt(Float64(t_4 * Float64(Float64(Float64(Float64(Float64(B_m * B_m) / A) * -0.5) + C) + C))) / t_1) * sqrt(2.0));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(t_0 * Float64(F * 2.0))) / t_0) * Float64(-sqrt(Float64(C * 2.0))));
	else
		tmp = Float64(Float64(-(B_m ^ -0.5)) * sqrt(Float64(F * 2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 * F), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+202], N[(N[(N[(N[Sqrt[N[(N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[Sqrt[C], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -2e-227], N[(N[(N[Sqrt[N[(N[(N[(N[(N[(A / B$95$m), $MachinePrecision] - -1.0), $MachinePrecision] * B$95$m), $MachinePrecision] + C), $MachinePrecision] * t$95$4), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5.0], N[(N[(N[Sqrt[N[(t$95$4 * N[(N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] * (-N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[((-N[Power[B$95$m, -0.5], $MachinePrecision]) * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.9999999999999999e202

   1. Initial program 4.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 34.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-1} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{C} \cdot \sqrt{2}\right)\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{C} \cdot \sqrt{2}\right)\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\sqrt{C} \cdot \sqrt{2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{C} \cdot \sqrt{2}}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{C}} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-sqrt.f64 22.1

         \[\leadsto \left(-\sqrt{C} \cdot \color{blue}{\sqrt{2}}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 22.1%

      \[\leadsto \color{blue}{\left(-\sqrt{C} \cdot \sqrt{2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\color{blue}{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\color{blue}{\left(2 \cdot F\right)} \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. associate-*l* N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\color{blue}{2 \cdot \left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. sqrt-prod N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\left(-4 \cdot \left(C \cdot A\right) + B \cdot B\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      7. +-commutative N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\left(B \cdot B + -4 \cdot \left(C \cdot A\right)\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      8. metadata-eval N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(B \cdot B + \color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \left(C \cdot A\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(\color{blue}{B \cdot B} - 4 \cdot \left(C \cdot A\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      11. pow2 N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(\color{blue}{{B}^{2}} - 4 \cdot \left(C \cdot A\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \left({B}^{2} - 4 \cdot \color{blue}{\left(C \cdot A\right)}\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      13. *-commutative N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \left({B}^{2} - 4 \cdot \color{blue}{\left(A \cdot C\right)}\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      14. associate-*l* N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \left({B}^{2} - \color{blue}{\left(4 \cdot A\right) \cdot C}\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      15. *-commutative N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{\color{blue}{\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      16. sqrt-prod N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      17. associate-*r* N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\color{blue}{\left(2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right) \cdot F}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   9. Applied rewrites 31.9%

      \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\color{blue}{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2} \cdot \sqrt{F}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if -4.9999999999999999e202 < (/.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))) < -1.99999999999999989e-227

   1. Initial program 97.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 97.2%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in B around inf

      \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\color{blue}{B \cdot \left(1 + \frac{A}{B}\right)} + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\color{blue}{B \cdot \left(1 + \frac{A}{B}\right)} + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(B \cdot \color{blue}{\left(1 + \frac{A}{B}\right)} + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-/.f64 33.0

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(B \cdot \left(1 + \color{blue}{\frac{A}{B}}\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 33.0%

      \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\color{blue}{B \cdot \left(1 + \frac{A}{B}\right)} + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if -1.99999999999999989e-227 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 5

   1. Initial program 7.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 11.9%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\color{blue}{\left(C + \frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)} + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\color{blue}{\left(C + \frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)} + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(C + \color{blue}{\frac{-1}{2} \cdot \frac{{B}^{2}}{A}}\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(C + \frac{-1}{2} \cdot \color{blue}{\frac{{B}^{2}}{A}}\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(C + \frac{-1}{2} \cdot \frac{\color{blue}{B \cdot B}}{A}\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-*.f64 29.5

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(C + -0.5 \cdot \frac{\color{blue}{B \cdot B}}{A}\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 29.5%

      \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\color{blue}{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)} + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 5 < (/.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 27.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 85.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-1} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{C} \cdot \sqrt{2}\right)\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{C} \cdot \sqrt{2}\right)\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\sqrt{C} \cdot \sqrt{2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{C} \cdot \sqrt{2}}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{C}} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-sqrt.f64 42.5

         \[\leadsto \left(-\sqrt{C} \cdot \color{blue}{\sqrt{2}}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 42.5%

      \[\leadsto \color{blue}{\left(-\sqrt{C} \cdot \sqrt{2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 42.8%

      \[\leadsto \color{blue}{\left(-\sqrt{C \cdot 2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

      5. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{F}{B}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

      8. lower-/.f64 14.2

         \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 14.2%

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 14.3%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 22.1%

      \[\leadsto -{B}^{-0.5} \cdot \sqrt{F \cdot 2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 28.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -5 \cdot 10^{+202}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2} \cdot \sqrt{F}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{C} \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\frac{A}{B} - -1\right) \cdot B + C\right) \cdot \left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 5:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F\right) \cdot \left(\left(\frac{B \cdot B}{A} \cdot -0.5 + C\right) + C\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-{B}^{-0.5}\right) \cdot \sqrt{F \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := -t\_0\\ t_2 := C \cdot \left(A \cdot 4\right)\\ t_3 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_2\right)\right) \cdot 2\right)}}{t\_2 - {B\_m}^{2}}\\ t_4 := t\_0 \cdot F\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+202}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right) \cdot 2} \cdot \sqrt{F}}{t\_0} \cdot \left(\sqrt{C} \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\frac{A}{B\_m} - -1\right) \cdot B\_m + C\right) \cdot t\_4}}{t\_1} \cdot \sqrt{2}\\ \mathbf{elif}\;t\_3 \leq 5:\\ \;\;\;\;\frac{\sqrt{t\_4 \cdot \left(\left(\frac{B\_m \cdot B\_m}{A} \cdot -0.5 + C\right) + C\right)}}{t\_1} \cdot \sqrt{2}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(F \cdot 2\right)}}{t\_0} \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (- t_0))
        (t_2 (* C (* A 4.0)))
        (t_3
         (/
          (sqrt
           (*
            (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))
            (* (* F (- (pow B_m 2.0) t_2)) 2.0)))
          (- t_2 (pow B_m 2.0))))
        (t_4 (* t_0 F)))
   (if (<= t_3 -5e+202)
     (*
      (/ (* (sqrt (* (fma (* C A) -4.0 (* B_m B_m)) 2.0)) (sqrt F)) t_0)
      (* (sqrt C) (- (sqrt 2.0))))
     (if (<= t_3 -2e-227)
       (* (/ (sqrt (* (+ (* (- (/ A B_m) -1.0) B_m) C) t_4)) t_1) (sqrt 2.0))
       (if (<= t_3 5.0)
         (*
          (/ (sqrt (* t_4 (+ (+ (* (/ (* B_m B_m) A) -0.5) C) C))) t_1)
          (sqrt 2.0))
         (if (<= t_3 INFINITY)
           (* (/ (sqrt (* t_0 (* F 2.0))) t_0) (- (sqrt (* C 2.0))))
           (/ (sqrt (* F 2.0)) (- (sqrt B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = -t_0;
	double t_2 = C * (A * 4.0);
	double t_3 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * ((F * (pow(B_m, 2.0) - t_2)) * 2.0))) / (t_2 - pow(B_m, 2.0));
	double t_4 = t_0 * F;
	double tmp;
	if (t_3 <= -5e+202) {
		tmp = ((sqrt((fma((C * A), -4.0, (B_m * B_m)) * 2.0)) * sqrt(F)) / t_0) * (sqrt(C) * -sqrt(2.0));
	} else if (t_3 <= -2e-227) {
		tmp = (sqrt((((((A / B_m) - -1.0) * B_m) + C) * t_4)) / t_1) * sqrt(2.0);
	} else if (t_3 <= 5.0) {
		tmp = (sqrt((t_4 * (((((B_m * B_m) / A) * -0.5) + C) + C))) / t_1) * sqrt(2.0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt((t_0 * (F * 2.0))) / t_0) * -sqrt((C * 2.0));
	} else {
		tmp = sqrt((F * 2.0)) / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64(-t_0)
	t_2 = Float64(C * Float64(A * 4.0))
	t_3 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_2)) * 2.0))) / Float64(t_2 - (B_m ^ 2.0)))
	t_4 = Float64(t_0 * F)
	tmp = 0.0
	if (t_3 <= -5e+202)
		tmp = Float64(Float64(Float64(sqrt(Float64(fma(Float64(C * A), -4.0, Float64(B_m * B_m)) * 2.0)) * sqrt(F)) / t_0) * Float64(sqrt(C) * Float64(-sqrt(2.0))));
	elseif (t_3 <= -2e-227)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(Float64(Float64(A / B_m) - -1.0) * B_m) + C) * t_4)) / t_1) * sqrt(2.0));
	elseif (t_3 <= 5.0)
		tmp = Float64(Float64(sqrt(Float64(t_4 * Float64(Float64(Float64(Float64(Float64(B_m * B_m) / A) * -0.5) + C) + C))) / t_1) * sqrt(2.0));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(t_0 * Float64(F * 2.0))) / t_0) * Float64(-sqrt(Float64(C * 2.0))));
	else
		tmp = Float64(sqrt(Float64(F * 2.0)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 * F), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+202], N[(N[(N[(N[Sqrt[N[(N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[Sqrt[C], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -2e-227], N[(N[(N[Sqrt[N[(N[(N[(N[(N[(A / B$95$m), $MachinePrecision] - -1.0), $MachinePrecision] * B$95$m), $MachinePrecision] + C), $MachinePrecision] * t$95$4), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5.0], N[(N[(N[Sqrt[N[(t$95$4 * N[(N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] * (-N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.9999999999999999e202

   1. Initial program 4.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 34.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-1} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{C} \cdot \sqrt{2}\right)\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{C} \cdot \sqrt{2}\right)\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\sqrt{C} \cdot \sqrt{2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{C} \cdot \sqrt{2}}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{C}} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-sqrt.f64 22.1

         \[\leadsto \left(-\sqrt{C} \cdot \color{blue}{\sqrt{2}}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 22.1%

      \[\leadsto \color{blue}{\left(-\sqrt{C} \cdot \sqrt{2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\color{blue}{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\color{blue}{\left(2 \cdot F\right)} \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. associate-*l* N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\color{blue}{2 \cdot \left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. sqrt-prod N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\left(-4 \cdot \left(C \cdot A\right) + B \cdot B\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      7. +-commutative N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\left(B \cdot B + -4 \cdot \left(C \cdot A\right)\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      8. metadata-eval N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(B \cdot B + \color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \left(C \cdot A\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(\color{blue}{B \cdot B} - 4 \cdot \left(C \cdot A\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      11. pow2 N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(\color{blue}{{B}^{2}} - 4 \cdot \left(C \cdot A\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \left({B}^{2} - 4 \cdot \color{blue}{\left(C \cdot A\right)}\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      13. *-commutative N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \left({B}^{2} - 4 \cdot \color{blue}{\left(A \cdot C\right)}\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      14. associate-*l* N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \left({B}^{2} - \color{blue}{\left(4 \cdot A\right) \cdot C}\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      15. *-commutative N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{\color{blue}{\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      16. sqrt-prod N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      17. associate-*r* N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\color{blue}{\left(2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right) \cdot F}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   9. Applied rewrites 31.9%

      \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\color{blue}{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2} \cdot \sqrt{F}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if -4.9999999999999999e202 < (/.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))) < -1.99999999999999989e-227

   1. Initial program 97.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 97.2%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in B around inf

      \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\color{blue}{B \cdot \left(1 + \frac{A}{B}\right)} + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\color{blue}{B \cdot \left(1 + \frac{A}{B}\right)} + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(B \cdot \color{blue}{\left(1 + \frac{A}{B}\right)} + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-/.f64 33.0

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(B \cdot \left(1 + \color{blue}{\frac{A}{B}}\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 33.0%

      \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\color{blue}{B \cdot \left(1 + \frac{A}{B}\right)} + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if -1.99999999999999989e-227 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 5

   1. Initial program 7.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 11.9%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\color{blue}{\left(C + \frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)} + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\color{blue}{\left(C + \frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)} + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(C + \color{blue}{\frac{-1}{2} \cdot \frac{{B}^{2}}{A}}\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(C + \frac{-1}{2} \cdot \color{blue}{\frac{{B}^{2}}{A}}\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(C + \frac{-1}{2} \cdot \frac{\color{blue}{B \cdot B}}{A}\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-*.f64 29.5

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(C + -0.5 \cdot \frac{\color{blue}{B \cdot B}}{A}\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 29.5%

      \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\color{blue}{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)} + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 5 < (/.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 27.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 85.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-1} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{C} \cdot \sqrt{2}\right)\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{C} \cdot \sqrt{2}\right)\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\sqrt{C} \cdot \sqrt{2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{C} \cdot \sqrt{2}}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{C}} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-sqrt.f64 42.5

         \[\leadsto \left(-\sqrt{C} \cdot \color{blue}{\sqrt{2}}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 42.5%

      \[\leadsto \color{blue}{\left(-\sqrt{C} \cdot \sqrt{2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 42.8%

      \[\leadsto \color{blue}{\left(-\sqrt{C \cdot 2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

      5. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{F}{B}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

      8. lower-/.f64 14.2

         \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 14.2%

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 14.3%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 22.1%

      \[\leadsto \frac{\sqrt{F \cdot 2}}{\color{blue}{-\sqrt{B}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 28.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -5 \cdot 10^{+202}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2} \cdot \sqrt{F}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{C} \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\frac{A}{B} - -1\right) \cdot B + C\right) \cdot \left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 5:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F\right) \cdot \left(\left(\frac{B \cdot B}{A} \cdot -0.5 + C\right) + C\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 56.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_2 := C \cdot \left(A \cdot 4\right)\\ t_3 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_2\right)\right) \cdot 2\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+202}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot 2} \cdot \sqrt{F}}{t\_1} \cdot \left(\sqrt{C} \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\frac{A}{B\_m} - -1\right) \cdot B\_m + C\right) \cdot \left(t\_1 \cdot F\right)}}{-t\_1} \cdot \sqrt{2}\\ \mathbf{elif}\;t\_3 \leq 5:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C \cdot 2\right) \cdot t\_0\right) \cdot \left(F \cdot 2\right)}}{-t\_0}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(F \cdot 2\right)}}{t\_1} \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (fma (* C A) -4.0 (* B_m B_m)))
        (t_1 (fma -4.0 (* C A) (* B_m B_m)))
        (t_2 (* C (* A 4.0)))
        (t_3
         (/
          (sqrt
           (*
            (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))
            (* (* F (- (pow B_m 2.0) t_2)) 2.0)))
          (- t_2 (pow B_m 2.0)))))
   (if (<= t_3 -5e+202)
     (* (/ (* (sqrt (* t_0 2.0)) (sqrt F)) t_1) (* (sqrt C) (- (sqrt 2.0))))
     (if (<= t_3 -2e-227)
       (*
        (/ (sqrt (* (+ (* (- (/ A B_m) -1.0) B_m) C) (* t_1 F))) (- t_1))
        (sqrt 2.0))
       (if (<= t_3 5.0)
         (/
          (sqrt (* (* (fma -0.5 (/ (* B_m B_m) A) (* C 2.0)) t_0) (* F 2.0)))
          (- t_0))
         (if (<= t_3 INFINITY)
           (* (/ (sqrt (* t_1 (* F 2.0))) t_1) (- (sqrt (* C 2.0))))
           (/ (sqrt (* F 2.0)) (- (sqrt B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((C * A), -4.0, (B_m * B_m));
	double t_1 = fma(-4.0, (C * A), (B_m * B_m));
	double t_2 = C * (A * 4.0);
	double t_3 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * ((F * (pow(B_m, 2.0) - t_2)) * 2.0))) / (t_2 - pow(B_m, 2.0));
	double tmp;
	if (t_3 <= -5e+202) {
		tmp = ((sqrt((t_0 * 2.0)) * sqrt(F)) / t_1) * (sqrt(C) * -sqrt(2.0));
	} else if (t_3 <= -2e-227) {
		tmp = (sqrt((((((A / B_m) - -1.0) * B_m) + C) * (t_1 * F))) / -t_1) * sqrt(2.0);
	} else if (t_3 <= 5.0) {
		tmp = sqrt(((fma(-0.5, ((B_m * B_m) / A), (C * 2.0)) * t_0) * (F * 2.0))) / -t_0;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt((t_1 * (F * 2.0))) / t_1) * -sqrt((C * 2.0));
	} else {
		tmp = sqrt((F * 2.0)) / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_1 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_2 = Float64(C * Float64(A * 4.0))
	t_3 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_2)) * 2.0))) / Float64(t_2 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_3 <= -5e+202)
		tmp = Float64(Float64(Float64(sqrt(Float64(t_0 * 2.0)) * sqrt(F)) / t_1) * Float64(sqrt(C) * Float64(-sqrt(2.0))));
	elseif (t_3 <= -2e-227)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(Float64(Float64(A / B_m) - -1.0) * B_m) + C) * Float64(t_1 * F))) / Float64(-t_1)) * sqrt(2.0));
	elseif (t_3 <= 5.0)
		tmp = Float64(sqrt(Float64(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(C * 2.0)) * t_0) * Float64(F * 2.0))) / Float64(-t_0));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(t_1 * Float64(F * 2.0))) / t_1) * Float64(-sqrt(Float64(C * 2.0))));
	else
		tmp = Float64(sqrt(Float64(F * 2.0)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+202], N[(N[(N[(N[Sqrt[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[Sqrt[C], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -2e-227], N[(N[(N[Sqrt[N[(N[(N[(N[(N[(A / B$95$m), $MachinePrecision] - -1.0), $MachinePrecision] * B$95$m), $MachinePrecision] + C), $MachinePrecision] * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5.0], N[(N[Sqrt[N[(N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(t$95$1 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision] * (-N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.9999999999999999e202

   1. Initial program 4.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 34.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-1} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{C} \cdot \sqrt{2}\right)\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{C} \cdot \sqrt{2}\right)\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\sqrt{C} \cdot \sqrt{2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{C} \cdot \sqrt{2}}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{C}} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-sqrt.f64 22.1

         \[\leadsto \left(-\sqrt{C} \cdot \color{blue}{\sqrt{2}}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 22.1%

      \[\leadsto \color{blue}{\left(-\sqrt{C} \cdot \sqrt{2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\color{blue}{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\color{blue}{\left(2 \cdot F\right)} \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. associate-*l* N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\color{blue}{2 \cdot \left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. sqrt-prod N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\left(-4 \cdot \left(C \cdot A\right) + B \cdot B\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      7. +-commutative N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\left(B \cdot B + -4 \cdot \left(C \cdot A\right)\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      8. metadata-eval N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(B \cdot B + \color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \left(C \cdot A\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(\color{blue}{B \cdot B} - 4 \cdot \left(C \cdot A\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      11. pow2 N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(\color{blue}{{B}^{2}} - 4 \cdot \left(C \cdot A\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \left({B}^{2} - 4 \cdot \color{blue}{\left(C \cdot A\right)}\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      13. *-commutative N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \left({B}^{2} - 4 \cdot \color{blue}{\left(A \cdot C\right)}\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      14. associate-*l* N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \left({B}^{2} - \color{blue}{\left(4 \cdot A\right) \cdot C}\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      15. *-commutative N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{\color{blue}{\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      16. sqrt-prod N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      17. associate-*r* N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\color{blue}{\left(2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right) \cdot F}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   9. Applied rewrites 31.9%

      \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\color{blue}{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2} \cdot \sqrt{F}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if -4.9999999999999999e202 < (/.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))) < -1.99999999999999989e-227

   1. Initial program 97.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 97.2%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in B around inf

      \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\color{blue}{B \cdot \left(1 + \frac{A}{B}\right)} + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\color{blue}{B \cdot \left(1 + \frac{A}{B}\right)} + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(B \cdot \color{blue}{\left(1 + \frac{A}{B}\right)} + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-/.f64 33.0

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(B \cdot \left(1 + \color{blue}{\frac{A}{B}}\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 33.0%

      \[\leadsto \frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\color{blue}{B \cdot \left(1 + \frac{A}{B}\right)} + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if -1.99999999999999989e-227 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 5

   1. Initial program 7.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 11.9%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   6. Applied rewrites 7.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + \left(C + A\right)\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \frac{\sqrt{\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)} \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{{B}^{2}}{A}, 2 \cdot C\right)} \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{{B}^{2}}{A}}, 2 \cdot C\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{B \cdot B}}{A}, 2 \cdot C\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{B \cdot B}}{A}, 2 \cdot C\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      5. lower-*.f64 22.7

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, \color{blue}{2 \cdot C}\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   9. Applied rewrites 22.7%

      \[\leadsto \frac{\sqrt{\left(\color{blue}{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, 2 \cdot C\right)} \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if 5 < (/.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 27.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 85.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-1} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{C} \cdot \sqrt{2}\right)\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{C} \cdot \sqrt{2}\right)\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\sqrt{C} \cdot \sqrt{2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{C} \cdot \sqrt{2}}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{C}} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-sqrt.f64 42.5

         \[\leadsto \left(-\sqrt{C} \cdot \color{blue}{\sqrt{2}}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 42.5%

      \[\leadsto \color{blue}{\left(-\sqrt{C} \cdot \sqrt{2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 42.8%

      \[\leadsto \color{blue}{\left(-\sqrt{C \cdot 2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

      5. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{F}{B}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

      8. lower-/.f64 14.2

         \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 14.2%

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 14.3%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 22.1%

      \[\leadsto \frac{\sqrt{F \cdot 2}}{\color{blue}{-\sqrt{B}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -5 \cdot 10^{+202}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2} \cdot \sqrt{F}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{C} \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\frac{A}{B} - -1\right) \cdot B + C\right) \cdot \left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 5:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 56.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_2 := C \cdot \left(A \cdot 4\right)\\ t_3 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_2\right)\right) \cdot 2\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+202}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot 2} \cdot \sqrt{F}}{t\_1} \cdot \left(\sqrt{C} \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{\left(C + B\_m\right) \cdot 2}}{B\_m} \cdot \left(-\sqrt{F}\right)\\ \mathbf{elif}\;t\_3 \leq 5:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C \cdot 2\right) \cdot t\_0\right) \cdot \left(F \cdot 2\right)}}{-t\_0}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(F \cdot 2\right)}}{t\_1} \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (fma (* C A) -4.0 (* B_m B_m)))
        (t_1 (fma -4.0 (* C A) (* B_m B_m)))
        (t_2 (* C (* A 4.0)))
        (t_3
         (/
          (sqrt
           (*
            (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))
            (* (* F (- (pow B_m 2.0) t_2)) 2.0)))
          (- t_2 (pow B_m 2.0)))))
   (if (<= t_3 -5e+202)
     (* (/ (* (sqrt (* t_0 2.0)) (sqrt F)) t_1) (* (sqrt C) (- (sqrt 2.0))))
     (if (<= t_3 -2e-227)
       (* (/ (sqrt (* (+ C B_m) 2.0)) B_m) (- (sqrt F)))
       (if (<= t_3 5.0)
         (/
          (sqrt (* (* (fma -0.5 (/ (* B_m B_m) A) (* C 2.0)) t_0) (* F 2.0)))
          (- t_0))
         (if (<= t_3 INFINITY)
           (* (/ (sqrt (* t_1 (* F 2.0))) t_1) (- (sqrt (* C 2.0))))
           (/ (sqrt (* F 2.0)) (- (sqrt B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((C * A), -4.0, (B_m * B_m));
	double t_1 = fma(-4.0, (C * A), (B_m * B_m));
	double t_2 = C * (A * 4.0);
	double t_3 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * ((F * (pow(B_m, 2.0) - t_2)) * 2.0))) / (t_2 - pow(B_m, 2.0));
	double tmp;
	if (t_3 <= -5e+202) {
		tmp = ((sqrt((t_0 * 2.0)) * sqrt(F)) / t_1) * (sqrt(C) * -sqrt(2.0));
	} else if (t_3 <= -2e-227) {
		tmp = (sqrt(((C + B_m) * 2.0)) / B_m) * -sqrt(F);
	} else if (t_3 <= 5.0) {
		tmp = sqrt(((fma(-0.5, ((B_m * B_m) / A), (C * 2.0)) * t_0) * (F * 2.0))) / -t_0;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt((t_1 * (F * 2.0))) / t_1) * -sqrt((C * 2.0));
	} else {
		tmp = sqrt((F * 2.0)) / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_1 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_2 = Float64(C * Float64(A * 4.0))
	t_3 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_2)) * 2.0))) / Float64(t_2 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_3 <= -5e+202)
		tmp = Float64(Float64(Float64(sqrt(Float64(t_0 * 2.0)) * sqrt(F)) / t_1) * Float64(sqrt(C) * Float64(-sqrt(2.0))));
	elseif (t_3 <= -2e-227)
		tmp = Float64(Float64(sqrt(Float64(Float64(C + B_m) * 2.0)) / B_m) * Float64(-sqrt(F)));
	elseif (t_3 <= 5.0)
		tmp = Float64(sqrt(Float64(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(C * 2.0)) * t_0) * Float64(F * 2.0))) / Float64(-t_0));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(t_1 * Float64(F * 2.0))) / t_1) * Float64(-sqrt(Float64(C * 2.0))));
	else
		tmp = Float64(sqrt(Float64(F * 2.0)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+202], N[(N[(N[(N[Sqrt[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[Sqrt[C], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -2e-227], N[(N[(N[Sqrt[N[(N[(C + B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$3, 5.0], N[(N[Sqrt[N[(N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(t$95$1 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision] * (-N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.9999999999999999e202

   1. Initial program 4.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 34.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-1} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{C} \cdot \sqrt{2}\right)\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{C} \cdot \sqrt{2}\right)\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\sqrt{C} \cdot \sqrt{2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{C} \cdot \sqrt{2}}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{C}} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-sqrt.f64 22.1

         \[\leadsto \left(-\sqrt{C} \cdot \color{blue}{\sqrt{2}}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 22.1%

      \[\leadsto \color{blue}{\left(-\sqrt{C} \cdot \sqrt{2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\color{blue}{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\color{blue}{\left(2 \cdot F\right)} \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. associate-*l* N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\color{blue}{2 \cdot \left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. sqrt-prod N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\left(-4 \cdot \left(C \cdot A\right) + B \cdot B\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      7. +-commutative N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\left(B \cdot B + -4 \cdot \left(C \cdot A\right)\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      8. metadata-eval N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(B \cdot B + \color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \left(C \cdot A\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(\color{blue}{B \cdot B} - 4 \cdot \left(C \cdot A\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      11. pow2 N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(\color{blue}{{B}^{2}} - 4 \cdot \left(C \cdot A\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \left({B}^{2} - 4 \cdot \color{blue}{\left(C \cdot A\right)}\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      13. *-commutative N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \left({B}^{2} - 4 \cdot \color{blue}{\left(A \cdot C\right)}\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      14. associate-*l* N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{F \cdot \left({B}^{2} - \color{blue}{\left(4 \cdot A\right) \cdot C}\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      15. *-commutative N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{2} \cdot \sqrt{\color{blue}{\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      16. sqrt-prod N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      17. associate-*r* N/A

          \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\color{blue}{\left(2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right) \cdot F}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   9. Applied rewrites 31.9%

      \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\color{blue}{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2} \cdot \sqrt{F}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if -4.9999999999999999e202 < (/.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))) < -1.99999999999999989e-227

   1. Initial program 97.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      12. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{{C}^{2} + {B}^{2}}} + C\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{C \cdot C} + {B}^{2}} + C\right) \cdot F} \]

      14. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{C \cdot C + \color{blue}{B \cdot B}} + C\right) \cdot F} \]

      15. lower-hypot.f64 37.4

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \]

   5. Applied rewrites 37.4%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 37.3%

      \[\leadsto \left(\frac{\sqrt{2}}{-B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \color{blue}{\sqrt{F}} \]
   8. Step-by-step derivation

   9. Applied rewrites 37.3%

      \[\leadsto \frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \color{blue}{\sqrt{F}} \]

   10. Taylor expanded in C around 0

       \[\leadsto \frac{\sqrt{\left(B + C\right) \cdot 2}}{-B} \cdot \sqrt{F} \]
   11. Step-by-step derivation

   12. Applied rewrites 30.6%

       \[\leadsto \frac{\sqrt{\left(C + B\right) \cdot 2}}{-B} \cdot \sqrt{F} \]

   if -1.99999999999999989e-227 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 5

   1. Initial program 7.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 11.9%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   6. Applied rewrites 7.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + \left(C + A\right)\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \frac{\sqrt{\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)} \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{{B}^{2}}{A}, 2 \cdot C\right)} \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{{B}^{2}}{A}}, 2 \cdot C\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{B \cdot B}}{A}, 2 \cdot C\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{B \cdot B}}{A}, 2 \cdot C\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      5. lower-*.f64 22.7

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, \color{blue}{2 \cdot C}\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   9. Applied rewrites 22.7%

      \[\leadsto \frac{\sqrt{\left(\color{blue}{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, 2 \cdot C\right)} \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if 5 < (/.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 27.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 85.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-1} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{C} \cdot \sqrt{2}\right)\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{C} \cdot \sqrt{2}\right)\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\sqrt{C} \cdot \sqrt{2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{C} \cdot \sqrt{2}}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{C}} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-sqrt.f64 42.5

         \[\leadsto \left(-\sqrt{C} \cdot \color{blue}{\sqrt{2}}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 42.5%

      \[\leadsto \color{blue}{\left(-\sqrt{C} \cdot \sqrt{2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 42.8%

      \[\leadsto \color{blue}{\left(-\sqrt{C \cdot 2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

      5. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{F}{B}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

      8. lower-/.f64 14.2

         \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 14.2%

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 14.3%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 22.1%

      \[\leadsto \frac{\sqrt{F \cdot 2}}{\color{blue}{-\sqrt{B}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 27.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -5 \cdot 10^{+202}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2} \cdot \sqrt{F}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{C} \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{\left(C + B\right) \cdot 2}}{B} \cdot \left(-\sqrt{F}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 5:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 50.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := \sqrt{C} \cdot \left(-\sqrt{2}\right)\\ t_2 := C \cdot \left(A \cdot 4\right)\\ t_3 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_2\right)\right) \cdot 2\right)}}{t\_2 - {B\_m}^{2}}\\ t_4 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\left(\sqrt{\frac{F}{t\_0}} \cdot \sqrt{2}\right) \cdot t\_1\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{\left(C + B\_m\right) \cdot 2}}{B\_m} \cdot \left(-\sqrt{F}\right)\\ \mathbf{elif}\;t\_3 \leq 10^{+220}:\\ \;\;\;\;\frac{\sqrt{\left(\left(C \cdot 2\right) \cdot t\_4\right) \cdot \left(F \cdot 2\right)}}{-t\_4}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot C\right) \cdot A\right) \cdot -8}}{t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (* (sqrt C) (- (sqrt 2.0))))
        (t_2 (* C (* A 4.0)))
        (t_3
         (/
          (sqrt
           (*
            (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))
            (* (* F (- (pow B_m 2.0) t_2)) 2.0)))
          (- t_2 (pow B_m 2.0))))
        (t_4 (fma (* C A) -4.0 (* B_m B_m))))
   (if (<= t_3 (- INFINITY))
     (* (* (sqrt (/ F t_0)) (sqrt 2.0)) t_1)
     (if (<= t_3 -2e-227)
       (* (/ (sqrt (* (+ C B_m) 2.0)) B_m) (- (sqrt F)))
       (if (<= t_3 1e+220)
         (/ (sqrt (* (* (* C 2.0) t_4) (* F 2.0))) (- t_4))
         (if (<= t_3 INFINITY)
           (* (/ (sqrt (* (* (* F C) A) -8.0)) t_0) t_1)
           (/ (sqrt (* F 2.0)) (- (sqrt B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = sqrt(C) * -sqrt(2.0);
	double t_2 = C * (A * 4.0);
	double t_3 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * ((F * (pow(B_m, 2.0) - t_2)) * 2.0))) / (t_2 - pow(B_m, 2.0));
	double t_4 = fma((C * A), -4.0, (B_m * B_m));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (sqrt((F / t_0)) * sqrt(2.0)) * t_1;
	} else if (t_3 <= -2e-227) {
		tmp = (sqrt(((C + B_m) * 2.0)) / B_m) * -sqrt(F);
	} else if (t_3 <= 1e+220) {
		tmp = sqrt((((C * 2.0) * t_4) * (F * 2.0))) / -t_4;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt((((F * C) * A) * -8.0)) / t_0) * t_1;
	} else {
		tmp = sqrt((F * 2.0)) / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64(sqrt(C) * Float64(-sqrt(2.0)))
	t_2 = Float64(C * Float64(A * 4.0))
	t_3 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_2)) * 2.0))) / Float64(t_2 - (B_m ^ 2.0)))
	t_4 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(F / t_0)) * sqrt(2.0)) * t_1);
	elseif (t_3 <= -2e-227)
		tmp = Float64(Float64(sqrt(Float64(Float64(C + B_m) * 2.0)) / B_m) * Float64(-sqrt(F)));
	elseif (t_3 <= 1e+220)
		tmp = Float64(sqrt(Float64(Float64(Float64(C * 2.0) * t_4) * Float64(F * 2.0))) / Float64(-t_4));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(F * C) * A) * -8.0)) / t_0) * t_1);
	else
		tmp = Float64(sqrt(Float64(F * 2.0)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[C], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$2 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[Sqrt[N[(F / t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$3, -2e-227], N[(N[(N[Sqrt[N[(N[(C + B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$3, 1e+220], N[(N[Sqrt[N[(N[(N[(C * 2.0), $MachinePrecision] * t$95$4), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$4)), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(N[(N[(F * C), $MachinePrecision] * A), $MachinePrecision] * -8.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0

   1. Initial program 3.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 33.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-1} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{C} \cdot \sqrt{2}\right)\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{C} \cdot \sqrt{2}\right)\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\sqrt{C} \cdot \sqrt{2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{C} \cdot \sqrt{2}}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{C}} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-sqrt.f64 22.4

         \[\leadsto \left(-\sqrt{C} \cdot \color{blue}{\sqrt{2}}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 22.4%

      \[\leadsto \color{blue}{\left(-\sqrt{C} \cdot \sqrt{2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   8. Taylor expanded in F around 0

      \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sqrt{\frac{F}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sqrt{\frac{F}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}\right)} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\sqrt{\frac{F}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}}} \cdot \sqrt{2}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \left(\sqrt{\color{blue}{\frac{F}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}}} \cdot \sqrt{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \left(\sqrt{\frac{F}{\color{blue}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}}} \cdot \sqrt{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \left(\sqrt{\frac{F}{\mathsf{fma}\left(-4, \color{blue}{A \cdot C}, {B}^{2}\right)}} \cdot \sqrt{2}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \left(\sqrt{\frac{F}{\mathsf{fma}\left(-4, A \cdot C, \color{blue}{B \cdot B}\right)}} \cdot \sqrt{2}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \left(\sqrt{\frac{F}{\mathsf{fma}\left(-4, A \cdot C, \color{blue}{B \cdot B}\right)}} \cdot \sqrt{2}\right) \]

      8. lower-sqrt.f64 26.1

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \left(\sqrt{\frac{F}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \cdot \color{blue}{\sqrt{2}}\right) \]

   10. Applied rewrites 26.1%

       \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sqrt{\frac{F}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \cdot \sqrt{2}\right)} \]

   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))) < -1.99999999999999989e-227

   1. Initial program 97.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      12. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{{C}^{2} + {B}^{2}}} + C\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{C \cdot C} + {B}^{2}} + C\right) \cdot F} \]

      14. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{C \cdot C + \color{blue}{B \cdot B}} + C\right) \cdot F} \]

      15. lower-hypot.f64 36.1

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \]

   5. Applied rewrites 36.1%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 36.1%

      \[\leadsto \left(\frac{\sqrt{2}}{-B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \color{blue}{\sqrt{F}} \]
   8. Step-by-step derivation

   9. Applied rewrites 36.0%

      \[\leadsto \frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \color{blue}{\sqrt{F}} \]

   10. Taylor expanded in C around 0

       \[\leadsto \frac{\sqrt{\left(B + C\right) \cdot 2}}{-B} \cdot \sqrt{F} \]
   11. Step-by-step derivation

   12. Applied rewrites 29.6%

       \[\leadsto \frac{\sqrt{\left(C + B\right) \cdot 2}}{-B} \cdot \sqrt{F} \]

   if -1.99999999999999989e-227 < (/.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))) < 1e220

   1. Initial program 16.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 20.2%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   6. Applied rewrites 16.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + \left(C + A\right)\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \frac{\sqrt{\left(\color{blue}{\left(2 \cdot C\right)} \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 24.7

         \[\leadsto \frac{\sqrt{\left(\color{blue}{\left(2 \cdot C\right)} \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   9. Applied rewrites 24.7%

      \[\leadsto \frac{\sqrt{\left(\color{blue}{\left(2 \cdot C\right)} \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if 1e220 < (/.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 3.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 80.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-1} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{C} \cdot \sqrt{2}\right)\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{C} \cdot \sqrt{2}\right)\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\sqrt{C} \cdot \sqrt{2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{C} \cdot \sqrt{2}}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{C}} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-sqrt.f64 35.9

         \[\leadsto \left(-\sqrt{C} \cdot \color{blue}{\sqrt{2}}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 35.9%

      \[\leadsto \color{blue}{\left(-\sqrt{C} \cdot \sqrt{2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   8. Taylor expanded in A around inf

      \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{-8 \cdot \color{blue}{\left(A \cdot \left(C \cdot F\right)\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-*.f64 36.0

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{-8 \cdot \left(A \cdot \color{blue}{\left(C \cdot F\right)}\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   10. Applied rewrites 36.0%

       \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

      5. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{F}{B}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

      8. lower-/.f64 14.2

         \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 14.2%

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 14.3%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 22.1%

      \[\leadsto \frac{\sqrt{F \cdot 2}}{\color{blue}{-\sqrt{B}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 25.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\left(\sqrt{\frac{F}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \cdot \sqrt{2}\right) \cdot \left(\sqrt{C} \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{\left(C + B\right) \cdot 2}}{B} \cdot \left(-\sqrt{F}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 10^{+220}:\\ \;\;\;\;\frac{\sqrt{\left(\left(C \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot C\right) \cdot A\right) \cdot -8}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{C} \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 48.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(A \cdot 4\right)\\ t_1 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_0\right)\right) \cdot 2\right)}}{t\_0 - {B\_m}^{2}}\\ t_2 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\sqrt{\frac{F}{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}} \cdot \sqrt{2}\right) \cdot \left(\sqrt{C} \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{\left(C + B\_m\right) \cdot 2}}{B\_m} \cdot \left(-\sqrt{F}\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\left(C \cdot 2\right) \cdot t\_2\right) \cdot \left(F \cdot 2\right)}}{-t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* C (* A 4.0)))
        (t_1
         (/
          (sqrt
           (*
            (+ (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))
            (* (* F (- (pow B_m 2.0) t_0)) 2.0)))
          (- t_0 (pow B_m 2.0))))
        (t_2 (fma (* C A) -4.0 (* B_m B_m))))
   (if (<= t_1 (- INFINITY))
     (*
      (* (sqrt (/ F (fma -4.0 (* C A) (* B_m B_m)))) (sqrt 2.0))
      (* (sqrt C) (- (sqrt 2.0))))
     (if (<= t_1 -2e-227)
       (* (/ (sqrt (* (+ C B_m) 2.0)) B_m) (- (sqrt F)))
       (if (<= t_1 INFINITY)
         (/ (sqrt (* (* (* C 2.0) t_2) (* F 2.0))) (- t_2))
         (/ (sqrt (* F 2.0)) (- (sqrt B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = C * (A * 4.0);
	double t_1 = sqrt(((sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) + (C + A)) * ((F * (pow(B_m, 2.0) - t_0)) * 2.0))) / (t_0 - pow(B_m, 2.0));
	double t_2 = fma((C * A), -4.0, (B_m * B_m));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (sqrt((F / fma(-4.0, (C * A), (B_m * B_m)))) * sqrt(2.0)) * (sqrt(C) * -sqrt(2.0));
	} else if (t_1 <= -2e-227) {
		tmp = (sqrt(((C + B_m) * 2.0)) / B_m) * -sqrt(F);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((((C * 2.0) * t_2) * (F * 2.0))) / -t_2;
	} else {
		tmp = sqrt((F * 2.0)) / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(C * Float64(A * 4.0))
	t_1 = Float64(sqrt(Float64(Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) + Float64(C + A)) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_0)) * 2.0))) / Float64(t_0 - (B_m ^ 2.0)))
	t_2 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(F / fma(-4.0, Float64(C * A), Float64(B_m * B_m)))) * sqrt(2.0)) * Float64(sqrt(C) * Float64(-sqrt(2.0))));
	elseif (t_1 <= -2e-227)
		tmp = Float64(Float64(sqrt(Float64(Float64(C + B_m) * 2.0)) / B_m) * Float64(-sqrt(F)));
	elseif (t_1 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(Float64(C * 2.0) * t_2) * Float64(F * 2.0))) / Float64(-t_2));
	else
		tmp = Float64(sqrt(Float64(F * 2.0)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[Sqrt[N[(F / N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[C], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e-227], N[(N[(N[Sqrt[N[(N[(C + B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[Sqrt[N[(N[(N[(C * 2.0), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$2)), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0

   1. Initial program 3.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 33.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-1} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{C} \cdot \sqrt{2}\right)\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{C} \cdot \sqrt{2}\right)\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\sqrt{C} \cdot \sqrt{2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{C} \cdot \sqrt{2}}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{C}} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-sqrt.f64 22.4

         \[\leadsto \left(-\sqrt{C} \cdot \color{blue}{\sqrt{2}}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 22.4%

      \[\leadsto \color{blue}{\left(-\sqrt{C} \cdot \sqrt{2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   8. Taylor expanded in F around 0

      \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sqrt{\frac{F}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sqrt{\frac{F}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \cdot \sqrt{2}\right)} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\sqrt{\frac{F}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}}} \cdot \sqrt{2}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \left(\sqrt{\color{blue}{\frac{F}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}}} \cdot \sqrt{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \left(\sqrt{\frac{F}{\color{blue}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}}} \cdot \sqrt{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \left(\sqrt{\frac{F}{\mathsf{fma}\left(-4, \color{blue}{A \cdot C}, {B}^{2}\right)}} \cdot \sqrt{2}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \left(\sqrt{\frac{F}{\mathsf{fma}\left(-4, A \cdot C, \color{blue}{B \cdot B}\right)}} \cdot \sqrt{2}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \left(\sqrt{\frac{F}{\mathsf{fma}\left(-4, A \cdot C, \color{blue}{B \cdot B}\right)}} \cdot \sqrt{2}\right) \]

      8. lower-sqrt.f64 26.1

         \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \left(\sqrt{\frac{F}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \cdot \color{blue}{\sqrt{2}}\right) \]

   10. Applied rewrites 26.1%

       \[\leadsto \left(-\sqrt{C} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sqrt{\frac{F}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \cdot \sqrt{2}\right)} \]

   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))) < -1.99999999999999989e-227

   1. Initial program 97.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      12. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{{C}^{2} + {B}^{2}}} + C\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{C \cdot C} + {B}^{2}} + C\right) \cdot F} \]

      14. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{C \cdot C + \color{blue}{B \cdot B}} + C\right) \cdot F} \]

      15. lower-hypot.f64 36.1

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \]

   5. Applied rewrites 36.1%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 36.1%

      \[\leadsto \left(\frac{\sqrt{2}}{-B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \color{blue}{\sqrt{F}} \]
   8. Step-by-step derivation

   9. Applied rewrites 36.0%

      \[\leadsto \frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \color{blue}{\sqrt{F}} \]

   10. Taylor expanded in C around 0

       \[\leadsto \frac{\sqrt{\left(B + C\right) \cdot 2}}{-B} \cdot \sqrt{F} \]
   11. Step-by-step derivation

   12. Applied rewrites 29.6%

       \[\leadsto \frac{\sqrt{\left(C + B\right) \cdot 2}}{-B} \cdot \sqrt{F} \]

   if -1.99999999999999989e-227 < (/.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 13.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 25.1%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   6. Applied rewrites 20.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + \left(C + A\right)\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \frac{\sqrt{\left(\color{blue}{\left(2 \cdot C\right)} \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 22.9

         \[\leadsto \frac{\sqrt{\left(\color{blue}{\left(2 \cdot C\right)} \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   9. Applied rewrites 22.9%

      \[\leadsto \frac{\sqrt{\left(\color{blue}{\left(2 \cdot C\right)} \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   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. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

      5. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{F}{B}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

      8. lower-/.f64 14.2

         \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 14.2%

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 14.3%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 22.1%

      \[\leadsto \frac{\sqrt{F \cdot 2}}{\color{blue}{-\sqrt{B}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 24.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\left(\sqrt{\frac{F}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \cdot \sqrt{2}\right) \cdot \left(\sqrt{C} \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{\left(C + B\right) \cdot 2}}{B} \cdot \left(-\sqrt{F}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\left(C \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 51.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(F \cdot 2\right)}}{t\_0} \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C + B\_m\right) \cdot 2}}{B\_m} \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m))))
   (if (<= (pow B_m 2.0) 2e+151)
     (* (/ (sqrt (* t_0 (* F 2.0))) t_0) (- (sqrt (* C 2.0))))
     (* (/ (sqrt (* (+ C B_m) 2.0)) B_m) (- (sqrt F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 2e+151) {
		tmp = (sqrt((t_0 * (F * 2.0))) / t_0) * -sqrt((C * 2.0));
	} else {
		tmp = (sqrt(((C + B_m) * 2.0)) / B_m) * -sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e+151)
		tmp = Float64(Float64(sqrt(Float64(t_0 * Float64(F * 2.0))) / t_0) * Float64(-sqrt(Float64(C * 2.0))));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(C + B_m) * 2.0)) / B_m) * Float64(-sqrt(F)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+151], N[(N[(N[Sqrt[N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] * (-N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(C + B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000003e151

   1. Initial program 19.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 35.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-1} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{C} \cdot \sqrt{2}\right)\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{C} \cdot \sqrt{2}\right)\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\sqrt{C} \cdot \sqrt{2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{C} \cdot \sqrt{2}}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{C}} \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-sqrt.f64 18.1

         \[\leadsto \left(-\sqrt{C} \cdot \color{blue}{\sqrt{2}}\right) \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 18.1%

      \[\leadsto \color{blue}{\left(-\sqrt{C} \cdot \sqrt{2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 18.2%

      \[\leadsto \color{blue}{\left(-\sqrt{C \cdot 2}\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 2.00000000000000003e151 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 7.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      12. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{{C}^{2} + {B}^{2}}} + C\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{C \cdot C} + {B}^{2}} + C\right) \cdot F} \]

      14. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{C \cdot C + \color{blue}{B \cdot B}} + C\right) \cdot F} \]

      15. lower-hypot.f64 24.0

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \]

   5. Applied rewrites 24.0%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 35.6%

      \[\leadsto \left(\frac{\sqrt{2}}{-B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \color{blue}{\sqrt{F}} \]
   8. Step-by-step derivation

   9. Applied rewrites 35.7%

      \[\leadsto \frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \color{blue}{\sqrt{F}} \]

   10. Taylor expanded in C around 0

       \[\leadsto \frac{\sqrt{\left(B + C\right) \cdot 2}}{-B} \cdot \sqrt{F} \]
   11. Step-by-step derivation

   12. Applied rewrites 32.0%

       \[\leadsto \frac{\sqrt{\left(C + B\right) \cdot 2}}{-B} \cdot \sqrt{F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(-\sqrt{C \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C + B\right) \cdot 2}}{B} \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 43.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{-216}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(C \cdot C\right) \cdot F\right) \cdot A\right) \cdot -16}}{-\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C + B\_m\right) \cdot 2}}{B\_m} \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 4e-216)
   (/ (sqrt (* (* (* (* C C) F) A) -16.0)) (- (fma (* C A) -4.0 (* B_m B_m))))
   (* (/ (sqrt (* (+ C B_m) 2.0)) B_m) (- (sqrt F)))))

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 (pow(B_m, 2.0) <= 4e-216) {
		tmp = sqrt(((((C * C) * F) * A) * -16.0)) / -fma((C * A), -4.0, (B_m * B_m));
	} else {
		tmp = (sqrt(((C + B_m) * 2.0)) / B_m) * -sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4e-216)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(C * C) * F) * A) * -16.0)) / Float64(-fma(Float64(C * A), -4.0, Float64(B_m * B_m))));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(C + B_m) * 2.0)) / B_m) * Float64(-sqrt(F)));
	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[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-216], N[(N[Sqrt[N[(N[(N[(N[(C * C), $MachinePrecision] * F), $MachinePrecision] * A), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision] / (-N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(C + B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.0000000000000002e-216

   1. Initial program 9.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 17.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   6. Applied rewrites 11.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + \left(C + A\right)\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(A \cdot \left({C}^{2} \cdot F\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \left(A \cdot \color{blue}{\left({C}^{2} \cdot F\right)}\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \left(A \cdot \left(\color{blue}{\left(C \cdot C\right)} \cdot F\right)\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      5. lower-*.f64 12.8

         \[\leadsto \frac{\sqrt{-16 \cdot \left(A \cdot \left(\color{blue}{\left(C \cdot C\right)} \cdot F\right)\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   9. Applied rewrites 12.8%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if 4.0000000000000002e-216 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 17.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      12. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{{C}^{2} + {B}^{2}}} + C\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{C \cdot C} + {B}^{2}} + C\right) \cdot F} \]

      14. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{C \cdot C + \color{blue}{B \cdot B}} + C\right) \cdot F} \]

      15. lower-hypot.f64 19.3

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \]

   5. Applied rewrites 19.3%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 25.6%

      \[\leadsto \left(\frac{\sqrt{2}}{-B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \color{blue}{\sqrt{F}} \]
   8. Step-by-step derivation

   9. Applied rewrites 25.7%

      \[\leadsto \frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \color{blue}{\sqrt{F}} \]

   10. Taylor expanded in C around 0

       \[\leadsto \frac{\sqrt{\left(B + C\right) \cdot 2}}{-B} \cdot \sqrt{F} \]
   11. Step-by-step derivation

   12. Applied rewrites 21.8%

       \[\leadsto \frac{\sqrt{\left(C + B\right) \cdot 2}}{-B} \cdot \sqrt{F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{-216}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(C \cdot C\right) \cdot F\right) \cdot A\right) \cdot -16}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C + B\right) \cdot 2}}{B} \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 48.7% accurate, 6.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 8.8 \cdot 10^{+22}:\\ \;\;\;\;\frac{\sqrt{\left(\left(C \cdot 2\right) \cdot t\_0\right) \cdot \left(F \cdot 2\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C + B\_m\right) \cdot 2}}{B\_m} \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

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 (* C A) -4.0 (* B_m B_m))))
   (if (<= B_m 8.8e+22)
     (/ (sqrt (* (* (* C 2.0) t_0) (* F 2.0))) (- t_0))
     (* (/ (sqrt (* (+ C B_m) 2.0)) B_m) (- (sqrt F))))))

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((C * A), -4.0, (B_m * B_m));
	double tmp;
	if (B_m <= 8.8e+22) {
		tmp = sqrt((((C * 2.0) * t_0) * (F * 2.0))) / -t_0;
	} else {
		tmp = (sqrt(((C + B_m) * 2.0)) / B_m) * -sqrt(F);
	}
	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(C * A), -4.0, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 8.8e+22)
		tmp = Float64(sqrt(Float64(Float64(Float64(C * 2.0) * t_0) * Float64(F * 2.0))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(C + B_m) * 2.0)) / B_m) * Float64(-sqrt(F)));
	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] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 8.8e+22], N[(N[Sqrt[N[(N[(N[(C * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(C + B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 8.8e22

   1. Initial program 17.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 24.8%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   6. Applied rewrites 19.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + \left(C + A\right)\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \frac{\sqrt{\left(\color{blue}{\left(2 \cdot C\right)} \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 14.1

         \[\leadsto \frac{\sqrt{\left(\color{blue}{\left(2 \cdot C\right)} \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   9. Applied rewrites 14.1%

      \[\leadsto \frac{\sqrt{\left(\color{blue}{\left(2 \cdot C\right)} \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if 8.8e22 < B

   1. Initial program 4.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      12. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{{C}^{2} + {B}^{2}}} + C\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{C \cdot C} + {B}^{2}} + C\right) \cdot F} \]

      14. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{C \cdot C + \color{blue}{B \cdot B}} + C\right) \cdot F} \]

      15. lower-hypot.f64 43.6

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \]

   5. Applied rewrites 43.6%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 64.0%

      \[\leadsto \left(\frac{\sqrt{2}}{-B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \color{blue}{\sqrt{F}} \]
   8. Step-by-step derivation

   9. Applied rewrites 64.3%

      \[\leadsto \frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \color{blue}{\sqrt{F}} \]

   10. Taylor expanded in C around 0

       \[\leadsto \frac{\sqrt{\left(B + C\right) \cdot 2}}{-B} \cdot \sqrt{F} \]
   11. Step-by-step derivation

   12. Applied rewrites 58.8%

       \[\leadsto \frac{\sqrt{\left(C + B\right) \cdot 2}}{-B} \cdot \sqrt{F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8.8 \cdot 10^{+22}:\\ \;\;\;\;\frac{\sqrt{\left(\left(C \cdot 2\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C + B\right) \cdot 2}}{B} \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 43.3% accurate, 6.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 5.6 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(C \cdot C\right) \cdot A\right) \cdot -8\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C + B\_m\right) \cdot 2}}{B\_m} \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 5.6e-107)
   (/
    (sqrt (* (* (* (* C C) A) -8.0) (* F 2.0)))
    (- (fma (* C A) -4.0 (* B_m B_m))))
   (* (/ (sqrt (* (+ C B_m) 2.0)) B_m) (- (sqrt F)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 5.6e-107) {
		tmp = sqrt(((((C * C) * A) * -8.0) * (F * 2.0))) / -fma((C * A), -4.0, (B_m * B_m));
	} else {
		tmp = (sqrt(((C + B_m) * 2.0)) / B_m) * -sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 5.6e-107)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(C * C) * A) * -8.0) * Float64(F * 2.0))) / Float64(-fma(Float64(C * A), -4.0, Float64(B_m * B_m))));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(C + B_m) * 2.0)) / B_m) * Float64(-sqrt(F)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 5.6e-107], N[(N[Sqrt[N[(N[(N[(N[(C * C), $MachinePrecision] * A), $MachinePrecision] * -8.0), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(C + B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 5.5999999999999998e-107

   1. Initial program 14.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 21.3%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   6. Applied rewrites 15.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + \left(C + A\right)\right) \cdot \mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot \left(A \cdot {C}^{2}\right)\right)} \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot \left(A \cdot {C}^{2}\right)\right)} \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot \color{blue}{\left(A \cdot {C}^{2}\right)}\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot \left(A \cdot \color{blue}{\left(C \cdot C\right)}\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      4. lower-*.f64 9.3

         \[\leadsto \frac{\sqrt{\left(-8 \cdot \left(A \cdot \color{blue}{\left(C \cdot C\right)}\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   9. Applied rewrites 9.3%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)} \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if 5.5999999999999998e-107 < B

   1. Initial program 16.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      12. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{{C}^{2} + {B}^{2}}} + C\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{C \cdot C} + {B}^{2}} + C\right) \cdot F} \]

      14. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{C \cdot C + \color{blue}{B \cdot B}} + C\right) \cdot F} \]

      15. lower-hypot.f64 41.5

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \]

   5. Applied rewrites 41.5%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 56.1%

      \[\leadsto \left(\frac{\sqrt{2}}{-B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \color{blue}{\sqrt{F}} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.2%

      \[\leadsto \frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \color{blue}{\sqrt{F}} \]

   10. Taylor expanded in C around 0

       \[\leadsto \frac{\sqrt{\left(B + C\right) \cdot 2}}{-B} \cdot \sqrt{F} \]
   11. Step-by-step derivation

   12. Applied rewrites 50.3%

       \[\leadsto \frac{\sqrt{\left(C + B\right) \cdot 2}}{-B} \cdot \sqrt{F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.6 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(C \cdot C\right) \cdot A\right) \cdot -8\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C + B\right) \cdot 2}}{B} \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 36.9% accurate, 7.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 2.35 \cdot 10^{+145}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C \cdot 2} \cdot \frac{-\sqrt{2}}{B\_m}\right) \cdot \sqrt{F}\\ \end{array} \end{array} \]

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 (<= C 2.35e+145)
   (/ (sqrt (* F 2.0)) (- (sqrt B_m)))
   (* (* (sqrt (* C 2.0)) (/ (- (sqrt 2.0)) B_m)) (sqrt F))))

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 (C <= 2.35e+145) {
		tmp = sqrt((F * 2.0)) / -sqrt(B_m);
	} else {
		tmp = (sqrt((C * 2.0)) * (-sqrt(2.0) / B_m)) * sqrt(F);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    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 (c <= 2.35d+145) then
        tmp = sqrt((f * 2.0d0)) / -sqrt(b_m)
    else
        tmp = (sqrt((c * 2.0d0)) * (-sqrt(2.0d0) / b_m)) * sqrt(f)
    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 (C <= 2.35e+145) {
		tmp = Math.sqrt((F * 2.0)) / -Math.sqrt(B_m);
	} else {
		tmp = (Math.sqrt((C * 2.0)) * (-Math.sqrt(2.0) / B_m)) * Math.sqrt(F);
	}
	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 C <= 2.35e+145:
		tmp = math.sqrt((F * 2.0)) / -math.sqrt(B_m)
	else:
		tmp = (math.sqrt((C * 2.0)) * (-math.sqrt(2.0) / B_m)) * math.sqrt(F)
	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 (C <= 2.35e+145)
		tmp = Float64(sqrt(Float64(F * 2.0)) / Float64(-sqrt(B_m)));
	else
		tmp = Float64(Float64(sqrt(Float64(C * 2.0)) * Float64(Float64(-sqrt(2.0)) / B_m)) * sqrt(F));
	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 (C <= 2.35e+145)
		tmp = sqrt((F * 2.0)) / -sqrt(B_m);
	else
		tmp = (sqrt((C * 2.0)) * (-sqrt(2.0) / B_m)) * sqrt(F);
	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[C, 2.35e+145], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 2.3500000000000001e145

   1. Initial program 17.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

      5. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{F}{B}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

      8. lower-/.f64 14.7

         \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 14.7%

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 14.8%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 18.3%

      \[\leadsto \frac{\sqrt{F \cdot 2}}{\color{blue}{-\sqrt{B}}} \]

   if 2.3500000000000001e145 < C

   1. Initial program 1.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      12. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{{C}^{2} + {B}^{2}}} + C\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{C \cdot C} + {B}^{2}} + C\right) \cdot F} \]

      14. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{C \cdot C + \color{blue}{B \cdot B}} + C\right) \cdot F} \]

      15. lower-hypot.f64 7.7

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \]

   5. Applied rewrites 7.7%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 10.7%

      \[\leadsto \left(\frac{\sqrt{2}}{-B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \color{blue}{\sqrt{F}} \]

   8. Taylor expanded in B around 0

      \[\leadsto \left(\frac{\sqrt{2}}{-B} \cdot \sqrt{2 \cdot C}\right) \cdot \sqrt{F} \]
   9. Step-by-step derivation

   10. Applied rewrites 10.7%

       \[\leadsto \left(\frac{\sqrt{2}}{-B} \cdot \sqrt{2 \cdot C}\right) \cdot \sqrt{F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 2.35 \cdot 10^{+145}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C \cdot 2} \cdot \frac{-\sqrt{2}}{B}\right) \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 35.8% accurate, 10.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{\sqrt{\left(C + B\_m\right) \cdot 2}}{B\_m} \cdot \left(-\sqrt{F}\right) \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (* (/ (sqrt (* (+ C B_m) 2.0)) B_m) (- (sqrt F))))

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(((C + B_m) * 2.0)) / B_m) * -sqrt(F);
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (sqrt(((c + b_m) * 2.0d0)) / b_m) * -sqrt(f)
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(((C + B_m) * 2.0)) / B_m) * -Math.sqrt(F);
}

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(((C + B_m) * 2.0)) / B_m) * -math.sqrt(F)

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(Float64(sqrt(Float64(Float64(C + B_m) * 2.0)) / B_m) * Float64(-sqrt(F)))
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(((C + B_m) * 2.0)) / B_m) * -sqrt(F);
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[(N[Sqrt[N[(N[(C + B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 15.3%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in A around 0

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

   3. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

   4. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

   5. lower-/.f64 N/A

      \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

   8. *-commutative N/A

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

   9. lower-*.f64 N/A

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

   10. +-commutative N/A

       \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

   11. lower-+.f64 N/A

       \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

   12. +-commutative N/A

       \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{{C}^{2} + {B}^{2}}} + C\right) \cdot F} \]

   13. unpow2 N/A

       \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{C \cdot C} + {B}^{2}} + C\right) \cdot F} \]

   14. unpow2 N/A

       \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{C \cdot C + \color{blue}{B \cdot B}} + C\right) \cdot F} \]

   15. lower-hypot.f64 15.2

       \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \]

5. Applied rewrites 15.2%

   \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F}} \]
6. Step-by-step derivation

7. Applied rewrites 19.6%

   \[\leadsto \left(\frac{\sqrt{2}}{-B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \color{blue}{\sqrt{F}} \]
8. Step-by-step derivation

9. Applied rewrites 19.7%

   \[\leadsto \frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \color{blue}{\sqrt{F}} \]

10. Taylor expanded in C around 0

    \[\leadsto \frac{\sqrt{\left(B + C\right) \cdot 2}}{-B} \cdot \sqrt{F} \]
11. Step-by-step derivation

12. Applied rewrites 15.9%

    \[\leadsto \frac{\sqrt{\left(C + B\right) \cdot 2}}{-B} \cdot \sqrt{F} \]

13. Final simplification 15.9%

    \[\leadsto \frac{\sqrt{\left(C + B\right) \cdot 2}}{B} \cdot \left(-\sqrt{F}\right) \]
14. Add Preprocessing

Alternative 26: 35.2% accurate, 12.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}} \end{array} \]

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)) (- (sqrt B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return sqrt((F * 2.0)) / -sqrt(B_m);
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    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)) / -sqrt(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)) / -Math.sqrt(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)) / -math.sqrt(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(F * 2.0)) / Float64(-sqrt(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)) / -sqrt(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[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 15.3%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   2. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   3. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   5. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(-\color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{F}{B}} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left(-\sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

   8. lower-/.f64 13.2

      \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 13.2%

   \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

7. Applied rewrites 13.3%

   \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
8. Step-by-step derivation

9. Applied rewrites 16.2%

   \[\leadsto \frac{\sqrt{F \cdot 2}}{\color{blue}{-\sqrt{B}}} \]
10. Add Preprocessing

Alternative 27: 35.2% accurate, 12.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}} \end{array} \]

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)) (sqrt (/ 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) * sqrt((2.0 / B_m));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -sqrt(f) * sqrt((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) * Math.sqrt((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) * math.sqrt((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(Float64(-sqrt(F)) * sqrt(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) * sqrt((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[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.3%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   2. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   3. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   5. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(-\color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{F}{B}} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left(-\sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

   8. lower-/.f64 13.2

      \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 13.2%

   \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

7. Applied rewrites 13.3%

   \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
8. Step-by-step derivation

9. Applied rewrites 16.2%

   \[\leadsto -\sqrt{\frac{2}{B}} \cdot \sqrt{F} \]

10. Final simplification 16.2%

    \[\leadsto \left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}} \]
11. Add Preprocessing

Alternative 28: 27.0% accurate, 16.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -\sqrt{\frac{F \cdot 2}{B\_m}} \end{array} \]

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 (/ (* 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 = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    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 Float64(-sqrt(Float64(Float64(F * 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[(N[(F * 2.0), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 15.3%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   2. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   3. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   5. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(-\color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{F}{B}} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left(-\sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

   8. lower-/.f64 13.2

      \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 13.2%

   \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

7. Applied rewrites 13.3%

   \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
8. Add Preprocessing

Alternative 29: 27.0% accurate, 16.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -\sqrt{\frac{2}{B\_m} \cdot F} \end{array} \]

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))))

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));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    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))
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));
}

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))

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(2.0 / B_m) * F)))
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));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := (-N[Sqrt[N[(N[(2.0 / B$95$m), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 15.3%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   2. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   3. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   5. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(-\color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{F}{B}} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left(-\sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

   8. lower-/.f64 13.2

      \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 13.2%

   \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

7. Applied rewrites 13.3%

   \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
8. Step-by-step derivation

9. Applied rewrites 13.3%

   \[\leadsto -\sqrt{\frac{2}{B} \cdot F} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024304 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))