ABCF->ab-angle a

Percentage Accurate: 19.1% → 62.6%

Time: 16.5s

Alternatives: 27

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: 19.1% 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: 62.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := \sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)}\\ t_2 := \sqrt{F \cdot 2}\\ t_3 := C \cdot \left(A \cdot 4\right)\\ t_4 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_3\right)\right) \cdot 2\right)}}{t\_3 - {B\_m}^{2}}\\ t_5 := -t\_0\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{-189}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C}}{t\_0} \cdot \left(\left(-t\_1\right) \cdot t\_2\right)\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C\right) + C}}{t\_5} \cdot \left(t\_1 \cdot t\_2\right)\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot t\_0} \cdot \frac{\sqrt{C \cdot 2}}{t\_5}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{\mathsf{hypot}\left(B\_m, A\right) + A}\right) \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

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)))
        (t_1 (sqrt (fma (* -4.0 C) A (* B_m B_m))))
        (t_2 (sqrt (* F 2.0)))
        (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 (- t_0)))
   (if (<= t_4 -5e-189)
     (* (/ (sqrt (+ (+ (hypot (- A C) B_m) A) C)) t_0) (* (- t_1) t_2))
     (if (<= t_4 0.0)
       (* (/ (sqrt (+ (fma -0.5 (/ (* B_m B_m) A) C) C)) t_5) (* t_1 t_2))
       (if (<= t_4 INFINITY)
         (* (sqrt (* (* F 2.0) t_0)) (/ (sqrt (* C 2.0)) t_5))
         (*
          (* (/ (sqrt 2.0) B_m) (sqrt (+ (hypot B_m A) A)))
          (- (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(fma((-4.0 * C), A, (B_m * B_m)));
	double t_2 = sqrt((F * 2.0));
	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 = -t_0;
	double tmp;
	if (t_4 <= -5e-189) {
		tmp = (sqrt(((hypot((A - C), B_m) + A) + C)) / t_0) * (-t_1 * t_2);
	} else if (t_4 <= 0.0) {
		tmp = (sqrt((fma(-0.5, ((B_m * B_m) / A), C) + C)) / t_5) * (t_1 * t_2);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt(((F * 2.0) * t_0)) * (sqrt((C * 2.0)) / t_5);
	} else {
		tmp = ((sqrt(2.0) / B_m) * sqrt((hypot(B_m, A) + A))) * -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 = sqrt(fma(Float64(-4.0 * C), A, Float64(B_m * B_m)))
	t_2 = sqrt(Float64(F * 2.0))
	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(-t_0)
	tmp = 0.0
	if (t_4 <= -5e-189)
		tmp = Float64(Float64(sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)) / t_0) * Float64(Float64(-t_1) * t_2));
	elseif (t_4 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / A), C) + C)) / t_5) * Float64(t_1 * t_2));
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(F * 2.0) * t_0)) * Float64(sqrt(Float64(C * 2.0)) / t_5));
	else
		tmp = Float64(Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(hypot(B_m, A) + A))) * 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]}, Block[{t$95$1 = N[Sqrt[N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]}, 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 = (-t$95$0)}, If[LessEqual[t$95$4, -5e-189], N[(N[(N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] * N[((-t$95$1) * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(N[(N[Sqrt[N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision] * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision] + A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[F], $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))) < -4.9999999999999997e-189

   1. Initial program 35.1%

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

   4. Applied rewrites 64.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. /-rgt-identity N/A

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

      5. lift-sqrt.f64 N/A

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

      6. pow1/2 N/A

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

      7. lift-*.f64 N/A

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

      8. unpow-prod-down N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. pow1/2 N/A

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

      18. lower-sqrt.f64 84.2

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*r* N/A

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

      22. lift-*.f64 N/A

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

      23. pow2 N/A

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

      24. lift-pow.f64 N/A

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

   6. Applied rewrites 84.2%

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

   if -4.9999999999999997e-189 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.4%

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

   4. Applied rewrites 7.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. /-rgt-identity N/A

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

      5. lift-sqrt.f64 N/A

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

      6. pow1/2 N/A

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

      7. lift-*.f64 N/A

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

      8. unpow-prod-down N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. pow1/2 N/A

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

      18. lower-sqrt.f64 23.6

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*r* N/A

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

      22. lift-*.f64 N/A

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

      23. pow2 N/A

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

      24. lift-pow.f64 N/A

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

   6. Applied rewrites 23.6%

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

   7. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 27.2

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

   9. Applied rewrites 27.2%

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

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

   1. Initial program 49.6%

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

   4. Applied rewrites 80.8%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 57.8

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

   7. Applied rewrites 57.8%

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

   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 C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 14.6

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

   5. Applied rewrites 14.6%

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

   7. Applied rewrites 22.8%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -5 \cdot 10^{-189}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\left(-\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\right) \cdot \sqrt{F \cdot 2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right) + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \sqrt{F \cdot 2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{C \cdot 2}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\mathsf{hypot}\left(B, A\right) + A}\right) \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 59.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(A \cdot 4\right)\\ t_1 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_0\right)\right) \cdot 2\right)}}{t\_0 - {B\_m}^{2}}\\ t_2 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_3 := -t\_2\\ t_4 := \sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)} \cdot \sqrt{F \cdot 2}\\ t_5 := \sqrt{\left(F \cdot 2\right) \cdot t\_2}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(0, \frac{A}{C}, 1\right) \cdot C + C}}{t\_3} \cdot t\_4\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-189}:\\ \;\;\;\;\left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C}\right) \cdot \frac{t\_5}{t\_2}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C\right) + C}}{t\_3} \cdot t\_4\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_5 \cdot \frac{\sqrt{C \cdot 2}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{\mathsf{hypot}\left(B\_m, A\right) + A}\right) \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

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 (* 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)))
        (t_3 (- t_2))
        (t_4 (* (sqrt (fma (* -4.0 C) A (* B_m B_m))) (sqrt (* F 2.0))))
        (t_5 (sqrt (* (* F 2.0) t_2))))
   (if (<= t_1 (- INFINITY))
     (* (/ (sqrt (+ (* (fma 0.0 (/ A C) 1.0) C) C)) t_3) t_4)
     (if (<= t_1 -5e-189)
       (* (- (sqrt (+ (+ (hypot (- A C) B_m) A) C))) (/ t_5 t_2))
       (if (<= t_1 0.0)
         (* (/ (sqrt (+ (fma -0.5 (/ (* B_m B_m) A) C) C)) t_3) t_4)
         (if (<= t_1 INFINITY)
           (* t_5 (/ (sqrt (* C 2.0)) t_3))
           (*
            (* (/ (sqrt 2.0) B_m) (sqrt (+ (hypot B_m A) A)))
            (- (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 = 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 t_3 = -t_2;
	double t_4 = sqrt(fma((-4.0 * C), A, (B_m * B_m))) * sqrt((F * 2.0));
	double t_5 = sqrt(((F * 2.0) * t_2));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (sqrt(((fma(0.0, (A / C), 1.0) * C) + C)) / t_3) * t_4;
	} else if (t_1 <= -5e-189) {
		tmp = -sqrt(((hypot((A - C), B_m) + A) + C)) * (t_5 / t_2);
	} else if (t_1 <= 0.0) {
		tmp = (sqrt((fma(-0.5, ((B_m * B_m) / A), C) + C)) / t_3) * t_4;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_5 * (sqrt((C * 2.0)) / t_3);
	} else {
		tmp = ((sqrt(2.0) / B_m) * sqrt((hypot(B_m, A) + A))) * -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(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))
	t_3 = Float64(-t_2)
	t_4 = Float64(sqrt(fma(Float64(-4.0 * C), A, Float64(B_m * B_m))) * sqrt(Float64(F * 2.0)))
	t_5 = sqrt(Float64(Float64(F * 2.0) * t_2))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(Float64(fma(0.0, Float64(A / C), 1.0) * C) + C)) / t_3) * t_4);
	elseif (t_1 <= -5e-189)
		tmp = Float64(Float64(-sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C))) * Float64(t_5 / t_2));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / A), C) + C)) / t_3) * t_4);
	elseif (t_1 <= Inf)
		tmp = Float64(t_5 * Float64(sqrt(Float64(C * 2.0)) / t_3));
	else
		tmp = Float64(Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(hypot(B_m, A) + A))) * 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[(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]}, Block[{t$95$3 = (-t$95$2)}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[Sqrt[N[(N[(N[(0.0 * N[(A / C), $MachinePrecision] + 1.0), $MachinePrecision] * C), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision] * t$95$4), $MachinePrecision], If[LessEqual[t$95$1, -5e-189], N[((-N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision]) * N[(t$95$5 / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[Sqrt[N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision] * t$95$4), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(t$95$5 * N[(N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision] + A), $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 3.5%

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

   4. Applied rewrites 47.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. /-rgt-identity N/A

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

      5. lift-sqrt.f64 N/A

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

      6. pow1/2 N/A

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

      7. lift-*.f64 N/A

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

      8. unpow-prod-down N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. pow1/2 N/A

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

      18. lower-sqrt.f64 76.7

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*r* N/A

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

      22. lift-*.f64 N/A

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

      23. pow2 N/A

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

      24. lift-pow.f64 N/A

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

   6. Applied rewrites 76.7%

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

   7. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \left(\color{blue}{0} \cdot \frac{A}{C} + 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \color{blue}{\mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-/.f64 27.3

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \mathsf{fma}\left(0, \color{blue}{\frac{A}{C}}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   9. Applied rewrites 27.3%

      \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\color{blue}{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   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))) < -4.9999999999999997e-189

   1. Initial program 98.4%

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

   4. Applied rewrites 99.3%

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

   if -4.9999999999999997e-189 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.4%

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

   4. Applied rewrites 7.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. /-rgt-identity N/A

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

      5. lift-sqrt.f64 N/A

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

      6. pow1/2 N/A

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

      7. lift-*.f64 N/A

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

      8. unpow-prod-down N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. pow1/2 N/A

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

      18. lower-sqrt.f64 23.6

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*r* N/A

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

      22. lift-*.f64 N/A

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

      23. pow2 N/A

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

      24. lift-pow.f64 N/A

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

   6. Applied rewrites 23.6%

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

   7. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 27.2

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

   9. Applied rewrites 27.2%

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

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

   1. Initial program 49.6%

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

   4. Applied rewrites 80.8%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 57.8

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

   7. Applied rewrites 57.8%

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

   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 C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 14.6

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

   5. Applied rewrites 14.6%

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

   7. Applied rewrites 22.8%

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

4. Final simplification 35.7%

   \[\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 -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(0, \frac{A}{C}, 1\right) \cdot C + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \sqrt{F \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 -5 \cdot 10^{-189}:\\ \;\;\;\;\left(-\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}\right) \cdot \frac{\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\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{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right) + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \sqrt{F \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 \infty:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{C \cdot 2}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\mathsf{hypot}\left(B, A\right) + A}\right) \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)} \cdot \sqrt{F \cdot 2}\\ t_1 := C \cdot \left(A \cdot 4\right)\\ t_2 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_1\right)\right) \cdot 2\right)}}{t\_1 - {B\_m}^{2}}\\ t_3 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_4 := -t\_3\\ t_5 := \sqrt{\left(F \cdot 2\right) \cdot t\_3}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(0, \frac{A}{C}, 1\right) \cdot C + C}}{t\_4} \cdot t\_0\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-189}:\\ \;\;\;\;t\_5 \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C}}{t\_4}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C\right) + C}}{t\_4} \cdot t\_0\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_5 \cdot \frac{\sqrt{C \cdot 2}}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{\mathsf{hypot}\left(B\_m, A\right) + A}\right) \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

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 (* (sqrt (fma (* -4.0 C) A (* B_m B_m))) (sqrt (* F 2.0))))
        (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)))
        (t_4 (- t_3))
        (t_5 (sqrt (* (* F 2.0) t_3))))
   (if (<= t_2 (- INFINITY))
     (* (/ (sqrt (+ (* (fma 0.0 (/ A C) 1.0) C) C)) t_4) t_0)
     (if (<= t_2 -5e-189)
       (* t_5 (/ (sqrt (+ (+ (hypot (- A C) B_m) A) C)) t_4))
       (if (<= t_2 0.0)
         (* (/ (sqrt (+ (fma -0.5 (/ (* B_m B_m) A) C) C)) t_4) t_0)
         (if (<= t_2 INFINITY)
           (* t_5 (/ (sqrt (* C 2.0)) t_4))
           (*
            (* (/ (sqrt 2.0) B_m) (sqrt (+ (hypot B_m A) A)))
            (- (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 = sqrt(fma((-4.0 * C), A, (B_m * B_m))) * sqrt((F * 2.0));
	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 t_4 = -t_3;
	double t_5 = sqrt(((F * 2.0) * t_3));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (sqrt(((fma(0.0, (A / C), 1.0) * C) + C)) / t_4) * t_0;
	} else if (t_2 <= -5e-189) {
		tmp = t_5 * (sqrt(((hypot((A - C), B_m) + A) + C)) / t_4);
	} else if (t_2 <= 0.0) {
		tmp = (sqrt((fma(-0.5, ((B_m * B_m) / A), C) + C)) / t_4) * t_0;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_5 * (sqrt((C * 2.0)) / t_4);
	} else {
		tmp = ((sqrt(2.0) / B_m) * sqrt((hypot(B_m, A) + A))) * -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(sqrt(fma(Float64(-4.0 * C), A, Float64(B_m * B_m))) * sqrt(Float64(F * 2.0)))
	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))
	t_4 = Float64(-t_3)
	t_5 = sqrt(Float64(Float64(F * 2.0) * t_3))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(Float64(fma(0.0, Float64(A / C), 1.0) * C) + C)) / t_4) * t_0);
	elseif (t_2 <= -5e-189)
		tmp = Float64(t_5 * Float64(sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)) / t_4));
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / A), C) + C)) / t_4) * t_0);
	elseif (t_2 <= Inf)
		tmp = Float64(t_5 * Float64(sqrt(Float64(C * 2.0)) / t_4));
	else
		tmp = Float64(Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(hypot(B_m, A) + A))) * 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[Sqrt[N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $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[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = (-t$95$3)}, Block[{t$95$5 = N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[Sqrt[N[(N[(N[(0.0 * N[(A / C), $MachinePrecision] + 1.0), $MachinePrecision] * C), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$2, -5e-189], N[(t$95$5 * N[(N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(N[Sqrt[N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(t$95$5 * N[(N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision] + A), $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 3.5%

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

   4. Applied rewrites 47.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. /-rgt-identity N/A

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

      5. lift-sqrt.f64 N/A

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

      6. pow1/2 N/A

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

      7. lift-*.f64 N/A

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

      8. unpow-prod-down N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. pow1/2 N/A

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

      18. lower-sqrt.f64 76.7

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*r* N/A

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

      22. lift-*.f64 N/A

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

      23. pow2 N/A

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

      24. lift-pow.f64 N/A

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

   6. Applied rewrites 76.7%

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

   7. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \left(\color{blue}{0} \cdot \frac{A}{C} + 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \color{blue}{\mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-/.f64 27.3

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \mathsf{fma}\left(0, \color{blue}{\frac{A}{C}}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   9. Applied rewrites 27.3%

      \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\color{blue}{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   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))) < -4.9999999999999997e-189

   1. Initial program 98.4%

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

   4. Applied rewrites 99.4%

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

   if -4.9999999999999997e-189 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.4%

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

   4. Applied rewrites 7.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. /-rgt-identity N/A

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

      5. lift-sqrt.f64 N/A

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

      6. pow1/2 N/A

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

      7. lift-*.f64 N/A

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

      8. unpow-prod-down N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. pow1/2 N/A

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

      18. lower-sqrt.f64 23.6

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*r* N/A

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

      22. lift-*.f64 N/A

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

      23. pow2 N/A

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

      24. lift-pow.f64 N/A

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

   6. Applied rewrites 23.6%

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

   7. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 27.2

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

   9. Applied rewrites 27.2%

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

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

   1. Initial program 49.6%

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

   4. Applied rewrites 80.8%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 57.8

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

   7. Applied rewrites 57.8%

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

   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 C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 14.6

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

   5. Applied rewrites 14.6%

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

   7. Applied rewrites 22.8%

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

4. Final simplification 35.8%

   \[\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(0, \frac{A}{C}, 1\right) \cdot C + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \sqrt{F \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 -5 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-\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{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right) + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \sqrt{F \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 \infty:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{C \cdot 2}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\mathsf{hypot}\left(B, A\right) + A}\right) \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ t_1 := \sqrt{t\_0} \cdot \sqrt{F \cdot 2}\\ t_2 := C \cdot \left(A \cdot 4\right)\\ t_3 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_2\right)\right) \cdot 2\right)}}{t\_2 - {B\_m}^{2}}\\ t_4 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_5 := -t\_4\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(0, \frac{A}{C}, 1\right) \cdot C + C}}{t\_5} \cdot t\_1\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-189}:\\ \;\;\;\;\frac{-1}{t\_4} \cdot \left(\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\_m\right) + C\right) + A\right) \cdot 2} \cdot \sqrt{t\_0 \cdot F}\right)\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C\right) + C}}{t\_5} \cdot t\_1\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot t\_4} \cdot \frac{\sqrt{C \cdot 2}}{t\_5}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{\mathsf{hypot}\left(B\_m, A\right) + A}\right) \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

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)))
        (t_1 (* (sqrt t_0) (sqrt (* F 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 -4.0 (* C A) (* B_m B_m)))
        (t_5 (- t_4)))
   (if (<= t_3 (- INFINITY))
     (* (/ (sqrt (+ (* (fma 0.0 (/ A C) 1.0) C) C)) t_5) t_1)
     (if (<= t_3 -5e-189)
       (*
        (/ -1.0 t_4)
        (* (sqrt (* (+ (+ (hypot (- A C) B_m) C) A) 2.0)) (sqrt (* t_0 F))))
       (if (<= t_3 0.0)
         (* (/ (sqrt (+ (fma -0.5 (/ (* B_m B_m) A) C) C)) t_5) t_1)
         (if (<= t_3 INFINITY)
           (* (sqrt (* (* F 2.0) t_4)) (/ (sqrt (* C 2.0)) t_5))
           (*
            (* (/ (sqrt 2.0) B_m) (sqrt (+ (hypot B_m A) A)))
            (- (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(t_0) * sqrt((F * 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(-4.0, (C * A), (B_m * B_m));
	double t_5 = -t_4;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (sqrt(((fma(0.0, (A / C), 1.0) * C) + C)) / t_5) * t_1;
	} else if (t_3 <= -5e-189) {
		tmp = (-1.0 / t_4) * (sqrt((((hypot((A - C), B_m) + C) + A) * 2.0)) * sqrt((t_0 * F)));
	} else if (t_3 <= 0.0) {
		tmp = (sqrt((fma(-0.5, ((B_m * B_m) / A), C) + C)) / t_5) * t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((F * 2.0) * t_4)) * (sqrt((C * 2.0)) / t_5);
	} else {
		tmp = ((sqrt(2.0) / B_m) * sqrt((hypot(B_m, A) + A))) * -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(-4.0 * C), A, Float64(B_m * B_m))
	t_1 = Float64(sqrt(t_0) * sqrt(Float64(F * 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(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_5 = Float64(-t_4)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(Float64(fma(0.0, Float64(A / C), 1.0) * C) + C)) / t_5) * t_1);
	elseif (t_3 <= -5e-189)
		tmp = Float64(Float64(-1.0 / t_4) * Float64(sqrt(Float64(Float64(Float64(hypot(Float64(A - C), B_m) + C) + A) * 2.0)) * sqrt(Float64(t_0 * F))));
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / A), C) + C)) / t_5) * t_1);
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(F * 2.0) * t_4)) * Float64(sqrt(Float64(C * 2.0)) / t_5));
	else
		tmp = Float64(Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(hypot(B_m, A) + A))) * 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[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $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[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = (-t$95$4)}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[Sqrt[N[(N[(N[(0.0 * N[(A / C), $MachinePrecision] + 1.0), $MachinePrecision] * C), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$3, -5e-189], N[(N[(-1.0 / t$95$4), $MachinePrecision] * N[(N[Sqrt[N[(N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] + A), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(N[Sqrt[N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$4), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision] + A), $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 3.5%

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

   4. Applied rewrites 47.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. /-rgt-identity N/A

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

      5. lift-sqrt.f64 N/A

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

      6. pow1/2 N/A

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

      7. lift-*.f64 N/A

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

      8. unpow-prod-down N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. pow1/2 N/A

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

      18. lower-sqrt.f64 76.7

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*r* N/A

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

      22. lift-*.f64 N/A

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

      23. pow2 N/A

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

      24. lift-pow.f64 N/A

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

   6. Applied rewrites 76.7%

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

   7. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \left(\color{blue}{0} \cdot \frac{A}{C} + 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \color{blue}{\mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-/.f64 27.3

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \mathsf{fma}\left(0, \color{blue}{\frac{A}{C}}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   9. Applied rewrites 27.3%

      \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\color{blue}{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   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))) < -4.9999999999999997e-189

   1. Initial program 98.4%

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

   4. Applied rewrites 98.6%

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

   6. Applied rewrites 99.2%

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

   if -4.9999999999999997e-189 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.4%

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

   4. Applied rewrites 7.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. /-rgt-identity N/A

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

      5. lift-sqrt.f64 N/A

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

      6. pow1/2 N/A

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

      7. lift-*.f64 N/A

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

      8. unpow-prod-down N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. pow1/2 N/A

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

      18. lower-sqrt.f64 23.6

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*r* N/A

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

      22. lift-*.f64 N/A

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

      23. pow2 N/A

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

      24. lift-pow.f64 N/A

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

   6. Applied rewrites 23.6%

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

   7. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 27.2

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

   9. Applied rewrites 27.2%

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

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

   1. Initial program 49.6%

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

   4. Applied rewrites 80.8%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 57.8

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

   7. Applied rewrites 57.8%

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

   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 C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 14.6

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

   5. Applied rewrites 14.6%

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

   7. Applied rewrites 22.8%

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

4. Final simplification 35.7%

   \[\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 -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(0, \frac{A}{C}, 1\right) \cdot C + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \sqrt{F \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 -5 \cdot 10^{-189}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + C\right) + A\right) \cdot 2} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot 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 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right) + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \sqrt{F \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 \infty:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{C \cdot 2}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\mathsf{hypot}\left(B, A\right) + A}\right) \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)} \cdot \sqrt{F \cdot 2}\\ t_1 := C \cdot \left(A \cdot 4\right)\\ t_2 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_1\right)\right) \cdot 2\right)}}{t\_1 - {B\_m}^{2}}\\ t_3 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_4 := -t\_3\\ t_5 := \left(F \cdot 2\right) \cdot t\_3\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(0, \frac{A}{C}, 1\right) \cdot C + C}}{t\_4} \cdot t\_0\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{t\_5 \cdot \left(\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C\right)} \cdot \frac{-1}{t\_3}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C\right) + C}}{t\_4} \cdot t\_0\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{t\_5} \cdot \frac{\sqrt{C \cdot 2}}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{\mathsf{hypot}\left(B\_m, A\right) + A}\right) \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

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 (* (sqrt (fma (* -4.0 C) A (* B_m B_m))) (sqrt (* F 2.0))))
        (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)))
        (t_4 (- t_3))
        (t_5 (* (* F 2.0) t_3)))
   (if (<= t_2 (- INFINITY))
     (* (/ (sqrt (+ (* (fma 0.0 (/ A C) 1.0) C) C)) t_4) t_0)
     (if (<= t_2 -5e-189)
       (* (sqrt (* t_5 (+ (+ (hypot (- A C) B_m) A) C))) (/ -1.0 t_3))
       (if (<= t_2 0.0)
         (* (/ (sqrt (+ (fma -0.5 (/ (* B_m B_m) A) C) C)) t_4) t_0)
         (if (<= t_2 INFINITY)
           (* (sqrt t_5) (/ (sqrt (* C 2.0)) t_4))
           (*
            (* (/ (sqrt 2.0) B_m) (sqrt (+ (hypot B_m A) A)))
            (- (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 = sqrt(fma((-4.0 * C), A, (B_m * B_m))) * sqrt((F * 2.0));
	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 t_4 = -t_3;
	double t_5 = (F * 2.0) * t_3;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (sqrt(((fma(0.0, (A / C), 1.0) * C) + C)) / t_4) * t_0;
	} else if (t_2 <= -5e-189) {
		tmp = sqrt((t_5 * ((hypot((A - C), B_m) + A) + C))) * (-1.0 / t_3);
	} else if (t_2 <= 0.0) {
		tmp = (sqrt((fma(-0.5, ((B_m * B_m) / A), C) + C)) / t_4) * t_0;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(t_5) * (sqrt((C * 2.0)) / t_4);
	} else {
		tmp = ((sqrt(2.0) / B_m) * sqrt((hypot(B_m, A) + A))) * -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(sqrt(fma(Float64(-4.0 * C), A, Float64(B_m * B_m))) * sqrt(Float64(F * 2.0)))
	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))
	t_4 = Float64(-t_3)
	t_5 = Float64(Float64(F * 2.0) * t_3)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(Float64(fma(0.0, Float64(A / C), 1.0) * C) + C)) / t_4) * t_0);
	elseif (t_2 <= -5e-189)
		tmp = Float64(sqrt(Float64(t_5 * Float64(Float64(hypot(Float64(A - C), B_m) + A) + C))) * Float64(-1.0 / t_3));
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / A), C) + C)) / t_4) * t_0);
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(t_5) * Float64(sqrt(Float64(C * 2.0)) / t_4));
	else
		tmp = Float64(Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(hypot(B_m, A) + A))) * 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[Sqrt[N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $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[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = (-t$95$3)}, Block[{t$95$5 = N[(N[(F * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[Sqrt[N[(N[(N[(0.0 * N[(A / C), $MachinePrecision] + 1.0), $MachinePrecision] * C), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$2, -5e-189], N[(N[Sqrt[N[(t$95$5 * N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(N[Sqrt[N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[t$95$5], $MachinePrecision] * N[(N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision] + A), $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 3.5%

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

   4. Applied rewrites 47.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. /-rgt-identity N/A

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

      5. lift-sqrt.f64 N/A

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

      6. pow1/2 N/A

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

      7. lift-*.f64 N/A

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

      8. unpow-prod-down N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. pow1/2 N/A

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

      18. lower-sqrt.f64 76.7

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*r* N/A

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

      22. lift-*.f64 N/A

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

      23. pow2 N/A

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

      24. lift-pow.f64 N/A

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

   6. Applied rewrites 76.7%

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

   7. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \left(\color{blue}{0} \cdot \frac{A}{C} + 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \color{blue}{\mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-/.f64 27.3

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \mathsf{fma}\left(0, \color{blue}{\frac{A}{C}}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   9. Applied rewrites 27.3%

      \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\color{blue}{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   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))) < -4.9999999999999997e-189

   1. Initial program 98.4%

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

   4. Applied rewrites 98.6%

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

   if -4.9999999999999997e-189 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.4%

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

   4. Applied rewrites 7.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. /-rgt-identity N/A

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

      5. lift-sqrt.f64 N/A

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

      6. pow1/2 N/A

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

      7. lift-*.f64 N/A

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

      8. unpow-prod-down N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. pow1/2 N/A

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

      18. lower-sqrt.f64 23.6

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*r* N/A

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

      22. lift-*.f64 N/A

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

      23. pow2 N/A

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

      24. lift-pow.f64 N/A

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

   6. Applied rewrites 23.6%

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

   7. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 27.2

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

   9. Applied rewrites 27.2%

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

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

   1. Initial program 49.6%

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

   4. Applied rewrites 80.8%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 57.8

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

   7. Applied rewrites 57.8%

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

   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 C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 14.6

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

   5. Applied rewrites 14.6%

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

   7. Applied rewrites 22.8%

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

4. Final simplification 35.7%

   \[\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 -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(0, \frac{A}{C}, 1\right) \cdot C + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \sqrt{F \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 -5 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{\left(\left(F \cdot 2\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)}\\ \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{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right) + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \sqrt{F \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 \infty:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{C \cdot 2}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\mathsf{hypot}\left(B, A\right) + A}\right) \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)} \cdot \sqrt{F \cdot 2}\\ t_1 := C \cdot \left(A \cdot 4\right)\\ t_2 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_1\right)\right) \cdot 2\right)}}{t\_1 - {B\_m}^{2}}\\ t_3 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_4 := -t\_3\\ t_5 := \left(F \cdot 2\right) \cdot t\_3\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(0, \frac{A}{C}, 1\right) \cdot C + C}}{t\_4} \cdot t\_0\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-189}:\\ \;\;\;\;\frac{\sqrt{t\_5 \cdot \left(\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C\right)}}{t\_4}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C\right) + C}}{t\_4} \cdot t\_0\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{t\_5} \cdot \frac{\sqrt{C \cdot 2}}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{\mathsf{hypot}\left(B\_m, A\right) + A}\right) \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

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 (* (sqrt (fma (* -4.0 C) A (* B_m B_m))) (sqrt (* F 2.0))))
        (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)))
        (t_4 (- t_3))
        (t_5 (* (* F 2.0) t_3)))
   (if (<= t_2 (- INFINITY))
     (* (/ (sqrt (+ (* (fma 0.0 (/ A C) 1.0) C) C)) t_4) t_0)
     (if (<= t_2 -5e-189)
       (/ (sqrt (* t_5 (+ (+ (hypot (- A C) B_m) A) C))) t_4)
       (if (<= t_2 0.0)
         (* (/ (sqrt (+ (fma -0.5 (/ (* B_m B_m) A) C) C)) t_4) t_0)
         (if (<= t_2 INFINITY)
           (* (sqrt t_5) (/ (sqrt (* C 2.0)) t_4))
           (*
            (* (/ (sqrt 2.0) B_m) (sqrt (+ (hypot B_m A) A)))
            (- (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 = sqrt(fma((-4.0 * C), A, (B_m * B_m))) * sqrt((F * 2.0));
	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 t_4 = -t_3;
	double t_5 = (F * 2.0) * t_3;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (sqrt(((fma(0.0, (A / C), 1.0) * C) + C)) / t_4) * t_0;
	} else if (t_2 <= -5e-189) {
		tmp = sqrt((t_5 * ((hypot((A - C), B_m) + A) + C))) / t_4;
	} else if (t_2 <= 0.0) {
		tmp = (sqrt((fma(-0.5, ((B_m * B_m) / A), C) + C)) / t_4) * t_0;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(t_5) * (sqrt((C * 2.0)) / t_4);
	} else {
		tmp = ((sqrt(2.0) / B_m) * sqrt((hypot(B_m, A) + A))) * -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(sqrt(fma(Float64(-4.0 * C), A, Float64(B_m * B_m))) * sqrt(Float64(F * 2.0)))
	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))
	t_4 = Float64(-t_3)
	t_5 = Float64(Float64(F * 2.0) * t_3)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(Float64(fma(0.0, Float64(A / C), 1.0) * C) + C)) / t_4) * t_0);
	elseif (t_2 <= -5e-189)
		tmp = Float64(sqrt(Float64(t_5 * Float64(Float64(hypot(Float64(A - C), B_m) + A) + C))) / t_4);
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / A), C) + C)) / t_4) * t_0);
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(t_5) * Float64(sqrt(Float64(C * 2.0)) / t_4));
	else
		tmp = Float64(Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(hypot(B_m, A) + A))) * 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[Sqrt[N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $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[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = (-t$95$3)}, Block[{t$95$5 = N[(N[(F * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[Sqrt[N[(N[(N[(0.0 * N[(A / C), $MachinePrecision] + 1.0), $MachinePrecision] * C), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$2, -5e-189], N[(N[Sqrt[N[(t$95$5 * N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(N[Sqrt[N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[t$95$5], $MachinePrecision] * N[(N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision] + A), $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 3.5%

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

   4. Applied rewrites 47.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. /-rgt-identity N/A

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

      5. lift-sqrt.f64 N/A

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

      6. pow1/2 N/A

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

      7. lift-*.f64 N/A

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

      8. unpow-prod-down N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. pow1/2 N/A

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

      18. lower-sqrt.f64 76.7

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*r* N/A

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

      22. lift-*.f64 N/A

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

      23. pow2 N/A

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

      24. lift-pow.f64 N/A

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

   6. Applied rewrites 76.7%

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

   7. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \left(\color{blue}{0} \cdot \frac{A}{C} + 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \color{blue}{\mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-/.f64 27.3

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \mathsf{fma}\left(0, \color{blue}{\frac{A}{C}}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   9. Applied rewrites 27.3%

      \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\color{blue}{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   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))) < -4.9999999999999997e-189

   1. Initial program 98.4%

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

   4. Applied rewrites 98.4%

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

   if -4.9999999999999997e-189 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.4%

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

   4. Applied rewrites 7.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. /-rgt-identity N/A

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

      5. lift-sqrt.f64 N/A

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

      6. pow1/2 N/A

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

      7. lift-*.f64 N/A

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

      8. unpow-prod-down N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. pow1/2 N/A

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

      18. lower-sqrt.f64 23.6

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*r* N/A

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

      22. lift-*.f64 N/A

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

      23. pow2 N/A

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

      24. lift-pow.f64 N/A

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

   6. Applied rewrites 23.6%

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

   7. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 27.2

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

   9. Applied rewrites 27.2%

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

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

   1. Initial program 49.6%

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

   4. Applied rewrites 80.8%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 57.8

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

   7. Applied rewrites 57.8%

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

   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 C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 14.6

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

   5. Applied rewrites 14.6%

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

   7. Applied rewrites 22.8%

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

4. Final simplification 35.7%

   \[\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 -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(0, \frac{A}{C}, 1\right) \cdot C + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \sqrt{F \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 -5 \cdot 10^{-189}:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\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)}\\ \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{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right) + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \sqrt{F \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 \infty:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{C \cdot 2}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\mathsf{hypot}\left(B, A\right) + A}\right) \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := C \cdot \left(A \cdot 4\right)\\ t_2 := \sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)} \cdot \sqrt{F \cdot 2}\\ t_3 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_1\right)\right) \cdot 2\right)}}{t\_1 - {B\_m}^{2}}\\ t_4 := -t\_0\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{+157}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(0, \frac{A}{C}, 1\right) \cdot C + C}}{t\_4} \cdot t\_2\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{\frac{\left(\left(\mathsf{hypot}\left(A - C, B\_m\right) + C\right) + A\right) \cdot F}{t\_0}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C\right) + C}}{t\_4} \cdot t\_2\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot t\_0} \cdot \frac{\sqrt{C \cdot 2}}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{\mathsf{hypot}\left(B\_m, A\right) + A}\right) \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

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)))
        (t_1 (* C (* A 4.0)))
        (t_2 (* (sqrt (fma (* -4.0 C) A (* B_m B_m))) (sqrt (* F 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_1 (pow B_m 2.0))))
        (t_4 (- t_0)))
   (if (<= t_3 -1e+157)
     (* (/ (sqrt (+ (* (fma 0.0 (/ A C) 1.0) C) C)) t_4) t_2)
     (if (<= t_3 -5e-189)
       (* (sqrt (/ (* (+ (+ (hypot (- A C) B_m) C) A) F) t_0)) (- (sqrt 2.0)))
       (if (<= t_3 0.0)
         (* (/ (sqrt (+ (fma -0.5 (/ (* B_m B_m) A) C) C)) t_4) t_2)
         (if (<= t_3 INFINITY)
           (* (sqrt (* (* F 2.0) t_0)) (/ (sqrt (* C 2.0)) t_4))
           (*
            (* (/ (sqrt 2.0) B_m) (sqrt (+ (hypot B_m A) A)))
            (- (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(fma((-4.0 * C), A, (B_m * B_m))) * sqrt((F * 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_1 - pow(B_m, 2.0));
	double t_4 = -t_0;
	double tmp;
	if (t_3 <= -1e+157) {
		tmp = (sqrt(((fma(0.0, (A / C), 1.0) * C) + C)) / t_4) * t_2;
	} else if (t_3 <= -5e-189) {
		tmp = sqrt(((((hypot((A - C), B_m) + C) + A) * F) / t_0)) * -sqrt(2.0);
	} else if (t_3 <= 0.0) {
		tmp = (sqrt((fma(-0.5, ((B_m * B_m) / A), C) + C)) / t_4) * t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((F * 2.0) * t_0)) * (sqrt((C * 2.0)) / t_4);
	} else {
		tmp = ((sqrt(2.0) / B_m) * sqrt((hypot(B_m, A) + A))) * -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(fma(Float64(-4.0 * C), A, Float64(B_m * B_m))) * sqrt(Float64(F * 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))) / Float64(t_1 - (B_m ^ 2.0)))
	t_4 = Float64(-t_0)
	tmp = 0.0
	if (t_3 <= -1e+157)
		tmp = Float64(Float64(sqrt(Float64(Float64(fma(0.0, Float64(A / C), 1.0) * C) + C)) / t_4) * t_2);
	elseif (t_3 <= -5e-189)
		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 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / A), C) + C)) / t_4) * t_2);
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(F * 2.0) * t_0)) * Float64(sqrt(Float64(C * 2.0)) / t_4));
	else
		tmp = Float64(Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(hypot(B_m, A) + A))) * 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]}, Block[{t$95$1 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $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] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = (-t$95$0)}, If[LessEqual[t$95$3, -1e+157], N[(N[(N[Sqrt[N[(N[(N[(0.0 * N[(A / C), $MachinePrecision] + 1.0), $MachinePrecision] * C), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$3, -5e-189], 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, 0.0], N[(N[(N[Sqrt[N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision] + A), $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))) < -9.99999999999999983e156

   1. Initial program 8.1%

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

   4. Applied rewrites 49.7%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. /-rgt-identity N/A

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

      5. lift-sqrt.f64 N/A

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

      6. pow1/2 N/A

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

      7. lift-*.f64 N/A

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

      8. unpow-prod-down N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. pow1/2 N/A

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

      18. lower-sqrt.f64 77.7

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*r* N/A

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

      22. lift-*.f64 N/A

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

      23. pow2 N/A

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

      24. lift-pow.f64 N/A

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

   6. Applied rewrites 77.7%

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

   7. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \left(\color{blue}{0} \cdot \frac{A}{C} + 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \color{blue}{\mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-/.f64 26.1

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \mathsf{fma}\left(0, \color{blue}{\frac{A}{C}}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   9. Applied rewrites 26.1%

      \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\color{blue}{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if -9.99999999999999983e156 < (/.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))) < -4.9999999999999997e-189

   1. Initial program 98.3%

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

   3. Taylor expanded in F around 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 95.9%

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

   if -4.9999999999999997e-189 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.4%

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

   4. Applied rewrites 7.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. /-rgt-identity N/A

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

      5. lift-sqrt.f64 N/A

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

      6. pow1/2 N/A

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

      7. lift-*.f64 N/A

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

      8. unpow-prod-down N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. pow1/2 N/A

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

      18. lower-sqrt.f64 23.6

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*r* N/A

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

      22. lift-*.f64 N/A

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

      23. pow2 N/A

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

      24. lift-pow.f64 N/A

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

   6. Applied rewrites 23.6%

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

   7. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 27.2

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

   9. Applied rewrites 27.2%

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

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

   1. Initial program 49.6%

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

   4. Applied rewrites 80.8%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 57.8

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

   7. Applied rewrites 57.8%

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

   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 C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 14.6

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

   5. Applied rewrites 14.6%

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

   7. Applied rewrites 22.8%

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

4. Final simplification 34.3%

   \[\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 -1 \cdot 10^{+157}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(0, \frac{A}{C}, 1\right) \cdot C + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \sqrt{F \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 -5 \cdot 10^{-189}:\\ \;\;\;\;\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 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right) + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \sqrt{F \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 \infty:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{C \cdot 2}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\mathsf{hypot}\left(B, A\right) + A}\right) \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := C \cdot \left(A \cdot 4\right)\\ t_2 := \sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)} \cdot \sqrt{F \cdot 2}\\ t_3 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_1\right)\right) \cdot 2\right)}}{t\_1 - {B\_m}^{2}}\\ t_4 := -t\_0\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+87}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(0, \frac{A}{C}, 1\right) \cdot C + C}}{t\_4} \cdot t\_2\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot F} \cdot \frac{-\sqrt{2}}{B\_m}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C\right) + C}}{t\_4} \cdot t\_2\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot t\_0} \cdot \frac{\sqrt{C \cdot 2}}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{\mathsf{hypot}\left(B\_m, A\right) + A}\right) \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

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)))
        (t_1 (* C (* A 4.0)))
        (t_2 (* (sqrt (fma (* -4.0 C) A (* B_m B_m))) (sqrt (* F 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_1 (pow B_m 2.0))))
        (t_4 (- t_0)))
   (if (<= t_3 -2e+87)
     (* (/ (sqrt (+ (* (fma 0.0 (/ A C) 1.0) C) C)) t_4) t_2)
     (if (<= t_3 -5e-189)
       (* (sqrt (* (+ (hypot C B_m) C) F)) (/ (- (sqrt 2.0)) B_m))
       (if (<= t_3 0.0)
         (* (/ (sqrt (+ (fma -0.5 (/ (* B_m B_m) A) C) C)) t_4) t_2)
         (if (<= t_3 INFINITY)
           (* (sqrt (* (* F 2.0) t_0)) (/ (sqrt (* C 2.0)) t_4))
           (*
            (* (/ (sqrt 2.0) B_m) (sqrt (+ (hypot B_m A) A)))
            (- (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(fma((-4.0 * C), A, (B_m * B_m))) * sqrt((F * 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_1 - pow(B_m, 2.0));
	double t_4 = -t_0;
	double tmp;
	if (t_3 <= -2e+87) {
		tmp = (sqrt(((fma(0.0, (A / C), 1.0) * C) + C)) / t_4) * t_2;
	} else if (t_3 <= -5e-189) {
		tmp = sqrt(((hypot(C, B_m) + C) * F)) * (-sqrt(2.0) / B_m);
	} else if (t_3 <= 0.0) {
		tmp = (sqrt((fma(-0.5, ((B_m * B_m) / A), C) + C)) / t_4) * t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((F * 2.0) * t_0)) * (sqrt((C * 2.0)) / t_4);
	} else {
		tmp = ((sqrt(2.0) / B_m) * sqrt((hypot(B_m, A) + A))) * -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(fma(Float64(-4.0 * C), A, Float64(B_m * B_m))) * sqrt(Float64(F * 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))) / Float64(t_1 - (B_m ^ 2.0)))
	t_4 = Float64(-t_0)
	tmp = 0.0
	if (t_3 <= -2e+87)
		tmp = Float64(Float64(sqrt(Float64(Float64(fma(0.0, Float64(A / C), 1.0) * C) + C)) / t_4) * t_2);
	elseif (t_3 <= -5e-189)
		tmp = Float64(sqrt(Float64(Float64(hypot(C, B_m) + C) * F)) * Float64(Float64(-sqrt(2.0)) / B_m));
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / A), C) + C)) / t_4) * t_2);
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(F * 2.0) * t_0)) * Float64(sqrt(Float64(C * 2.0)) / t_4));
	else
		tmp = Float64(Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(hypot(B_m, A) + A))) * 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]}, Block[{t$95$1 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $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] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = (-t$95$0)}, If[LessEqual[t$95$3, -2e+87], N[(N[(N[Sqrt[N[(N[(N[(0.0 * N[(A / C), $MachinePrecision] + 1.0), $MachinePrecision] * C), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$3, -5e-189], N[(N[Sqrt[N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(N[Sqrt[N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision] + A), $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))) < -1.9999999999999999e87

   1. Initial program 15.9%

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

   4. Applied rewrites 54.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. /-rgt-identity N/A

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

      5. lift-sqrt.f64 N/A

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

      6. pow1/2 N/A

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

      7. lift-*.f64 N/A

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

      8. unpow-prod-down N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. pow1/2 N/A

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

      18. lower-sqrt.f64 79.6

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*r* N/A

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

      22. lift-*.f64 N/A

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

      23. pow2 N/A

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

      24. lift-pow.f64 N/A

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

   6. Applied rewrites 79.6%

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

   7. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \left(\color{blue}{0} \cdot \frac{A}{C} + 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \color{blue}{\mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-/.f64 27.0

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \mathsf{fma}\left(0, \color{blue}{\frac{A}{C}}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   9. Applied rewrites 27.0%

      \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\color{blue}{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if -1.9999999999999999e87 < (/.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))) < -4.9999999999999997e-189

   1. Initial program 98.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 48.9

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

   5. Applied rewrites 48.9%

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

   if -4.9999999999999997e-189 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.4%

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

   4. Applied rewrites 7.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. /-rgt-identity N/A

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

      5. lift-sqrt.f64 N/A

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

      6. pow1/2 N/A

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

      7. lift-*.f64 N/A

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

      8. unpow-prod-down N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. pow1/2 N/A

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

      18. lower-sqrt.f64 23.6

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*r* N/A

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

      22. lift-*.f64 N/A

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

      23. pow2 N/A

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

      24. lift-pow.f64 N/A

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

   6. Applied rewrites 23.6%

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

   7. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 27.2

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

   9. Applied rewrites 27.2%

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

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

   1. Initial program 49.6%

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

   4. Applied rewrites 80.8%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 57.8

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

   7. Applied rewrites 57.8%

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

   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 C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 14.6

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

   5. Applied rewrites 14.6%

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

   7. Applied rewrites 22.8%

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

4. Final simplification 29.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{+87}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(0, \frac{A}{C}, 1\right) \cdot C + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \sqrt{F \cdot 2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -5 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right) + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \sqrt{F \cdot 2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{C \cdot 2}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\mathsf{hypot}\left(B, A\right) + A}\right) \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := C \cdot \left(A \cdot 4\right)\\ t_2 := \sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)} \cdot \sqrt{F \cdot 2}\\ t_3 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_1\right)\right) \cdot 2\right)}}{t\_1 - {B\_m}^{2}}\\ t_4 := -t\_0\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+87}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(0, \frac{A}{C}, 1\right) \cdot C + C}}{t\_4} \cdot t\_2\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot F} \cdot \frac{-\sqrt{2}}{B\_m}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C\right) + C}}{t\_4} \cdot t\_2\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot t\_0} \cdot \frac{\sqrt{C \cdot 2}}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(B\_m, A\right) + A\right) \cdot 2}}{-B\_m} \cdot \sqrt{F}\\ \end{array} \end{array} \]

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 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (* C (* A 4.0)))
        (t_2 (* (sqrt (fma (* -4.0 C) A (* B_m B_m))) (sqrt (* F 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_1 (pow B_m 2.0))))
        (t_4 (- t_0)))
   (if (<= t_3 -2e+87)
     (* (/ (sqrt (+ (* (fma 0.0 (/ A C) 1.0) C) C)) t_4) t_2)
     (if (<= t_3 -5e-189)
       (* (sqrt (* (+ (hypot C B_m) C) F)) (/ (- (sqrt 2.0)) B_m))
       (if (<= t_3 0.0)
         (* (/ (sqrt (+ (fma -0.5 (/ (* B_m B_m) A) C) C)) t_4) t_2)
         (if (<= t_3 INFINITY)
           (* (sqrt (* (* F 2.0) t_0)) (/ (sqrt (* C 2.0)) t_4))
           (* (/ (sqrt (* (+ (hypot B_m A) A) 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 t_1 = C * (A * 4.0);
	double t_2 = sqrt(fma((-4.0 * C), A, (B_m * B_m))) * sqrt((F * 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_1 - pow(B_m, 2.0));
	double t_4 = -t_0;
	double tmp;
	if (t_3 <= -2e+87) {
		tmp = (sqrt(((fma(0.0, (A / C), 1.0) * C) + C)) / t_4) * t_2;
	} else if (t_3 <= -5e-189) {
		tmp = sqrt(((hypot(C, B_m) + C) * F)) * (-sqrt(2.0) / B_m);
	} else if (t_3 <= 0.0) {
		tmp = (sqrt((fma(-0.5, ((B_m * B_m) / A), C) + C)) / t_4) * t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((F * 2.0) * t_0)) * (sqrt((C * 2.0)) / t_4);
	} else {
		tmp = (sqrt(((hypot(B_m, A) + A) * 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))
	t_1 = Float64(C * Float64(A * 4.0))
	t_2 = Float64(sqrt(fma(Float64(-4.0 * C), A, Float64(B_m * B_m))) * sqrt(Float64(F * 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))) / Float64(t_1 - (B_m ^ 2.0)))
	t_4 = Float64(-t_0)
	tmp = 0.0
	if (t_3 <= -2e+87)
		tmp = Float64(Float64(sqrt(Float64(Float64(fma(0.0, Float64(A / C), 1.0) * C) + C)) / t_4) * t_2);
	elseif (t_3 <= -5e-189)
		tmp = Float64(sqrt(Float64(Float64(hypot(C, B_m) + C) * F)) * Float64(Float64(-sqrt(2.0)) / B_m));
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / A), C) + C)) / t_4) * t_2);
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(F * 2.0) * t_0)) * Float64(sqrt(Float64(C * 2.0)) / t_4));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(hypot(B_m, A) + A) * 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[(-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[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $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] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = (-t$95$0)}, If[LessEqual[t$95$3, -2e+87], N[(N[(N[Sqrt[N[(N[(N[(0.0 * N[(A / C), $MachinePrecision] + 1.0), $MachinePrecision] * C), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$3, -5e-189], N[(N[Sqrt[N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(N[Sqrt[N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision] + A), $MachinePrecision] * 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))) < -1.9999999999999999e87

   1. Initial program 15.9%

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

   4. Applied rewrites 54.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. /-rgt-identity N/A

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

      5. lift-sqrt.f64 N/A

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

      6. pow1/2 N/A

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

      7. lift-*.f64 N/A

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

      8. unpow-prod-down N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. pow1/2 N/A

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

      18. lower-sqrt.f64 79.6

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*r* N/A

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

      22. lift-*.f64 N/A

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

      23. pow2 N/A

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

      24. lift-pow.f64 N/A

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

   6. Applied rewrites 79.6%

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

   7. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \left(\color{blue}{0} \cdot \frac{A}{C} + 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \color{blue}{\mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-/.f64 27.0

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \mathsf{fma}\left(0, \color{blue}{\frac{A}{C}}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   9. Applied rewrites 27.0%

      \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\color{blue}{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if -1.9999999999999999e87 < (/.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))) < -4.9999999999999997e-189

   1. Initial program 98.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 48.9

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

   5. Applied rewrites 48.9%

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

   if -4.9999999999999997e-189 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.4%

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

   4. Applied rewrites 7.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. /-rgt-identity N/A

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

      5. lift-sqrt.f64 N/A

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

      6. pow1/2 N/A

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

      7. lift-*.f64 N/A

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

      8. unpow-prod-down N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. pow1/2 N/A

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

      18. lower-sqrt.f64 23.6

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*r* N/A

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

      22. lift-*.f64 N/A

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

      23. pow2 N/A

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

      24. lift-pow.f64 N/A

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

   6. Applied rewrites 23.6%

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

   7. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 27.2

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

   9. Applied rewrites 27.2%

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

   if -0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 49.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 80.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{2 \cdot C}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 57.8

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{2 \cdot C}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 57.8%

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{2 \cdot C}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   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 C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(A + \sqrt{{A}^{2} + {B}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(A + \sqrt{{A}^{2} + {B}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{A}^{2} + {B}^{2}} + A\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{A}^{2} + {B}^{2}} + A\right)} \cdot F} \]

      12. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{A \cdot A} + {B}^{2}} + A\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{A \cdot A + \color{blue}{B \cdot B}} + A\right) \cdot F} \]

      14. lower-hypot.f64 14.6

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(A, B\right)} + A\right) \cdot F} \]

   5. Applied rewrites 14.6%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(A, B\right) + A\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 22.8%

      \[\leadsto \left(\frac{\sqrt{2}}{-B} \cdot \sqrt{\mathsf{hypot}\left(B, A\right) + A}\right) \cdot \color{blue}{\sqrt{F}} \]
   8. Step-by-step derivation

   9. Applied rewrites 22.8%

      \[\leadsto \frac{\sqrt{\left(\mathsf{hypot}\left(B, A\right) + A\right) \cdot 2}}{-B} \cdot \color{blue}{\sqrt{F}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 29.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{+87}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(0, \frac{A}{C}, 1\right) \cdot C + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \sqrt{F \cdot 2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -5 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right) + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \sqrt{F \cdot 2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{C \cdot 2}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(B, A\right) + A\right) \cdot 2}}{-B} \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := C \cdot \left(A \cdot 4\right)\\ t_2 := \sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)} \cdot \sqrt{F \cdot 2}\\ t_3 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_1\right)\right) \cdot 2\right)}}{t\_1 - {B\_m}^{2}}\\ t_4 := -t\_0\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+87}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(0, \frac{A}{C}, 1\right) \cdot C + C}}{t\_4} \cdot t\_2\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot F} \cdot \frac{-\sqrt{2}}{B\_m}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C\right) + C}}{t\_4} \cdot t\_2\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot t\_0} \cdot \frac{\sqrt{C \cdot 2}}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B\_m}}\\ \end{array} \end{array} \]

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 (* C (* A 4.0)))
        (t_2 (* (sqrt (fma (* -4.0 C) A (* B_m B_m))) (sqrt (* F 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_1 (pow B_m 2.0))))
        (t_4 (- t_0)))
   (if (<= t_3 -2e+87)
     (* (/ (sqrt (+ (* (fma 0.0 (/ A C) 1.0) C) C)) t_4) t_2)
     (if (<= t_3 -5e-189)
       (* (sqrt (* (+ (hypot C B_m) C) F)) (/ (- (sqrt 2.0)) B_m))
       (if (<= t_3 0.0)
         (* (/ (sqrt (+ (fma -0.5 (/ (* B_m B_m) A) C) C)) t_4) t_2)
         (if (<= t_3 INFINITY)
           (* (sqrt (* (* F 2.0) t_0)) (/ (sqrt (* C 2.0)) t_4))
           (/ (- (sqrt F)) (sqrt (* 0.5 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 = C * (A * 4.0);
	double t_2 = sqrt(fma((-4.0 * C), A, (B_m * B_m))) * sqrt((F * 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_1 - pow(B_m, 2.0));
	double t_4 = -t_0;
	double tmp;
	if (t_3 <= -2e+87) {
		tmp = (sqrt(((fma(0.0, (A / C), 1.0) * C) + C)) / t_4) * t_2;
	} else if (t_3 <= -5e-189) {
		tmp = sqrt(((hypot(C, B_m) + C) * F)) * (-sqrt(2.0) / B_m);
	} else if (t_3 <= 0.0) {
		tmp = (sqrt((fma(-0.5, ((B_m * B_m) / A), C) + C)) / t_4) * t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((F * 2.0) * t_0)) * (sqrt((C * 2.0)) / t_4);
	} else {
		tmp = -sqrt(F) / sqrt((0.5 * 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(C * Float64(A * 4.0))
	t_2 = Float64(sqrt(fma(Float64(-4.0 * C), A, Float64(B_m * B_m))) * sqrt(Float64(F * 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))) / Float64(t_1 - (B_m ^ 2.0)))
	t_4 = Float64(-t_0)
	tmp = 0.0
	if (t_3 <= -2e+87)
		tmp = Float64(Float64(sqrt(Float64(Float64(fma(0.0, Float64(A / C), 1.0) * C) + C)) / t_4) * t_2);
	elseif (t_3 <= -5e-189)
		tmp = Float64(sqrt(Float64(Float64(hypot(C, B_m) + C) * F)) * Float64(Float64(-sqrt(2.0)) / B_m));
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / A), C) + C)) / t_4) * t_2);
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(F * 2.0) * t_0)) * Float64(sqrt(Float64(C * 2.0)) / t_4));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(0.5 * 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[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $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] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = (-t$95$0)}, If[LessEqual[t$95$3, -2e+87], N[(N[(N[Sqrt[N[(N[(N[(0.0 * N[(A / C), $MachinePrecision] + 1.0), $MachinePrecision] * C), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$3, -5e-189], N[(N[Sqrt[N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(N[Sqrt[N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(0.5 * B$95$m), $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))) < -1.9999999999999999e87

   1. Initial program 15.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 54.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1}} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)}{\mathsf{neg}\left(-1\right)}} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)}{\color{blue}{1}} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. /-rgt-identity N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. pow1/2 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left({\color{blue}{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}^{\frac{1}{2}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      8. unpow-prod-down N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{\left(2 \cdot F\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left({\left(2 \cdot F\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left({\left(2 \cdot F\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      11. pow1/2 N/A

          \[\leadsto \left(\color{blue}{\sqrt{2 \cdot F}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{2 \cdot F}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{2 \cdot F}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      14. *-commutative N/A

          \[\leadsto \left(\sqrt{\color{blue}{F \cdot 2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{F \cdot 2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      16. lower-neg.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \color{blue}{\left(-{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)}\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      17. pow1/2 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\color{blue}{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      18. lower-sqrt.f64 79.6

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\color{blue}{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      19. lift-fma.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\color{blue}{-4 \cdot \left(C \cdot A\right) + B \cdot B}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      20. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{-4 \cdot \color{blue}{\left(C \cdot A\right)} + B \cdot B}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      21. associate-*r* N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\color{blue}{\left(-4 \cdot C\right) \cdot A} + B \cdot B}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      22. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\left(-4 \cdot C\right) \cdot A + \color{blue}{B \cdot B}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      23. pow2 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\left(-4 \cdot C\right) \cdot A + \color{blue}{{B}^{2}}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      24. lift-pow.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\left(-4 \cdot C\right) \cdot A + \color{blue}{{B}^{2}}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   6. Applied rewrites 79.6%

      \[\leadsto \color{blue}{\left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Taylor expanded in C around inf

      \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\color{blue}{C \cdot \left(1 + \left(-1 \cdot \frac{A}{C} + \frac{A}{C}\right)\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\color{blue}{C \cdot \left(1 + \left(-1 \cdot \frac{A}{C} + \frac{A}{C}\right)\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \color{blue}{\left(\left(-1 \cdot \frac{A}{C} + \frac{A}{C}\right) + 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. distribute-lft1-in N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \left(\color{blue}{\left(-1 + 1\right) \cdot \frac{A}{C}} + 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. metadata-eval N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \left(\color{blue}{0} \cdot \frac{A}{C} + 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \color{blue}{\mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-/.f64 27.0

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \mathsf{fma}\left(0, \color{blue}{\frac{A}{C}}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   9. Applied rewrites 27.0%

      \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\color{blue}{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if -1.9999999999999999e87 < (/.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))) < -4.9999999999999997e-189

   1. Initial program 98.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 48.9

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \]

   5. Applied rewrites 48.9%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F}} \]

   if -4.9999999999999997e-189 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 7.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1}} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)}{\mathsf{neg}\left(-1\right)}} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)}{\color{blue}{1}} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. /-rgt-identity N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. pow1/2 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left({\color{blue}{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}^{\frac{1}{2}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      8. unpow-prod-down N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{\left(2 \cdot F\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left({\left(2 \cdot F\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left({\left(2 \cdot F\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      11. pow1/2 N/A

          \[\leadsto \left(\color{blue}{\sqrt{2 \cdot F}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{2 \cdot F}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{2 \cdot F}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      14. *-commutative N/A

          \[\leadsto \left(\sqrt{\color{blue}{F \cdot 2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{F \cdot 2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      16. lower-neg.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \color{blue}{\left(-{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)}\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      17. pow1/2 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\color{blue}{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      18. lower-sqrt.f64 23.6

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\color{blue}{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      19. lift-fma.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\color{blue}{-4 \cdot \left(C \cdot A\right) + B \cdot B}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      20. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{-4 \cdot \color{blue}{\left(C \cdot A\right)} + B \cdot B}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      21. associate-*r* N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\color{blue}{\left(-4 \cdot C\right) \cdot A} + B \cdot B}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      22. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\left(-4 \cdot C\right) \cdot A + \color{blue}{B \cdot B}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      23. pow2 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\left(-4 \cdot C\right) \cdot A + \color{blue}{{B}^{2}}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      24. lift-pow.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\left(-4 \cdot C\right) \cdot A + \color{blue}{{B}^{2}}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   6. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\color{blue}{\left(C + \frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + C\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{{B}^{2}}{A}, C\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{{B}^{2}}{A}}, C\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. unpow2 N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{B \cdot B}}{A}, C\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-*.f64 27.2

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{\color{blue}{B \cdot B}}{A}, C\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   9. Applied rewrites 27.2%

      \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if -0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 49.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 80.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{2 \cdot C}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 57.8

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{2 \cdot C}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 57.8%

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{2 \cdot C}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   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 16.7

         \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 16.7%

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 16.8%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 16.8%

      \[\leadsto -\sqrt{\frac{2}{B} \cdot F} \]
   10. Step-by-step derivation

   11. Applied rewrites 21.2%

       \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{+87}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(0, \frac{A}{C}, 1\right) \cdot C + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \sqrt{F \cdot 2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -5 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right) + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \sqrt{F \cdot 2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{C \cdot 2}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := \sqrt{\left(F \cdot 2\right) \cdot t\_0}\\ t_2 := C \cdot \left(A \cdot 4\right)\\ t_3 := \sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)} \cdot \sqrt{F \cdot 2}\\ t_4 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_2\right)\right) \cdot 2\right)}}{t\_2 - {B\_m}^{2}}\\ t_5 := -t\_0\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+27}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(0, \frac{A}{C}, 1\right) \cdot C + C}}{t\_5} \cdot t\_3\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-189}:\\ \;\;\;\;t\_1 \cdot \frac{\sqrt{\left(1 + \frac{A}{B\_m}\right) \cdot B\_m + C}}{t\_5}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C\right) + C}}{t\_5} \cdot t\_3\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_1 \cdot \frac{\sqrt{C \cdot 2}}{t\_5}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B\_m}}\\ \end{array} \end{array} \]

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 (* (* F 2.0) t_0)))
        (t_2 (* C (* A 4.0)))
        (t_3 (* (sqrt (fma (* -4.0 C) A (* B_m B_m))) (sqrt (* F 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_2 (pow B_m 2.0))))
        (t_5 (- t_0)))
   (if (<= t_4 -5e+27)
     (* (/ (sqrt (+ (* (fma 0.0 (/ A C) 1.0) C) C)) t_5) t_3)
     (if (<= t_4 -5e-189)
       (* t_1 (/ (sqrt (+ (* (+ 1.0 (/ A B_m)) B_m) C)) t_5))
       (if (<= t_4 0.0)
         (* (/ (sqrt (+ (fma -0.5 (/ (* B_m B_m) A) C) C)) t_5) t_3)
         (if (<= t_4 INFINITY)
           (* t_1 (/ (sqrt (* C 2.0)) t_5))
           (/ (- (sqrt F)) (sqrt (* 0.5 B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = sqrt(((F * 2.0) * t_0));
	double t_2 = C * (A * 4.0);
	double t_3 = sqrt(fma((-4.0 * C), A, (B_m * B_m))) * sqrt((F * 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_2 - pow(B_m, 2.0));
	double t_5 = -t_0;
	double tmp;
	if (t_4 <= -5e+27) {
		tmp = (sqrt(((fma(0.0, (A / C), 1.0) * C) + C)) / t_5) * t_3;
	} else if (t_4 <= -5e-189) {
		tmp = t_1 * (sqrt((((1.0 + (A / B_m)) * B_m) + C)) / t_5);
	} else if (t_4 <= 0.0) {
		tmp = (sqrt((fma(-0.5, ((B_m * B_m) / A), C) + C)) / t_5) * t_3;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1 * (sqrt((C * 2.0)) / t_5);
	} else {
		tmp = -sqrt(F) / sqrt((0.5 * B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = sqrt(Float64(Float64(F * 2.0) * t_0))
	t_2 = Float64(C * Float64(A * 4.0))
	t_3 = Float64(sqrt(fma(Float64(-4.0 * C), A, Float64(B_m * B_m))) * sqrt(Float64(F * 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))) / Float64(t_2 - (B_m ^ 2.0)))
	t_5 = Float64(-t_0)
	tmp = 0.0
	if (t_4 <= -5e+27)
		tmp = Float64(Float64(sqrt(Float64(Float64(fma(0.0, Float64(A / C), 1.0) * C) + C)) / t_5) * t_3);
	elseif (t_4 <= -5e-189)
		tmp = Float64(t_1 * Float64(sqrt(Float64(Float64(Float64(1.0 + Float64(A / B_m)) * B_m) + C)) / t_5));
	elseif (t_4 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / A), C) + C)) / t_5) * t_3);
	elseif (t_4 <= Inf)
		tmp = Float64(t_1 * Float64(sqrt(Float64(C * 2.0)) / t_5));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(0.5 * B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $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] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = (-t$95$0)}, If[LessEqual[t$95$4, -5e+27], N[(N[(N[Sqrt[N[(N[(N[(0.0 * N[(A / C), $MachinePrecision] + 1.0), $MachinePrecision] * C), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[t$95$4, -5e-189], N[(t$95$1 * N[(N[Sqrt[N[(N[(N[(1.0 + N[(A / B$95$m), $MachinePrecision]), $MachinePrecision] * B$95$m), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(N[(N[Sqrt[N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(t$95$1 * N[(N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(0.5 * B$95$m), $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))) < -4.99999999999999979e27

   1. Initial program 21.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1}} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)}{\mathsf{neg}\left(-1\right)}} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)}{\color{blue}{1}} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. /-rgt-identity N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. pow1/2 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left({\color{blue}{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}^{\frac{1}{2}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      8. unpow-prod-down N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{\left(2 \cdot F\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left({\left(2 \cdot F\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left({\left(2 \cdot F\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      11. pow1/2 N/A

          \[\leadsto \left(\color{blue}{\sqrt{2 \cdot F}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{2 \cdot F}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{2 \cdot F}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      14. *-commutative N/A

          \[\leadsto \left(\sqrt{\color{blue}{F \cdot 2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{F \cdot 2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      16. lower-neg.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \color{blue}{\left(-{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)}\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      17. pow1/2 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\color{blue}{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      18. lower-sqrt.f64 80.9

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\color{blue}{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      19. lift-fma.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\color{blue}{-4 \cdot \left(C \cdot A\right) + B \cdot B}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      20. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{-4 \cdot \color{blue}{\left(C \cdot A\right)} + B \cdot B}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      21. associate-*r* N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\color{blue}{\left(-4 \cdot C\right) \cdot A} + B \cdot B}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      22. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\left(-4 \cdot C\right) \cdot A + \color{blue}{B \cdot B}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      23. pow2 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\left(-4 \cdot C\right) \cdot A + \color{blue}{{B}^{2}}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      24. lift-pow.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\left(-4 \cdot C\right) \cdot A + \color{blue}{{B}^{2}}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   6. Applied rewrites 80.9%

      \[\leadsto \color{blue}{\left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Taylor expanded in C around inf

      \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\color{blue}{C \cdot \left(1 + \left(-1 \cdot \frac{A}{C} + \frac{A}{C}\right)\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\color{blue}{C \cdot \left(1 + \left(-1 \cdot \frac{A}{C} + \frac{A}{C}\right)\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \color{blue}{\left(\left(-1 \cdot \frac{A}{C} + \frac{A}{C}\right) + 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. distribute-lft1-in N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \left(\color{blue}{\left(-1 + 1\right) \cdot \frac{A}{C}} + 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. metadata-eval N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \left(\color{blue}{0} \cdot \frac{A}{C} + 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \color{blue}{\mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-/.f64 25.3

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \mathsf{fma}\left(0, \color{blue}{\frac{A}{C}}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   9. Applied rewrites 25.3%

      \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\color{blue}{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if -4.99999999999999979e27 < (/.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))) < -4.9999999999999997e-189

   1. Initial program 97.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in B around inf

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{B \cdot \left(1 + \frac{A}{B}\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{B \cdot \left(1 + \frac{A}{B}\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{B \cdot \color{blue}{\left(1 + \frac{A}{B}\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-/.f64 48.4

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{B \cdot \left(1 + \color{blue}{\frac{A}{B}}\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 48.4%

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{B \cdot \left(1 + \frac{A}{B}\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if -4.9999999999999997e-189 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 7.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1}} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)}{\mathsf{neg}\left(-1\right)}} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)}{\color{blue}{1}} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. /-rgt-identity N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. pow1/2 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left({\color{blue}{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}^{\frac{1}{2}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      8. unpow-prod-down N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{\left(2 \cdot F\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left({\left(2 \cdot F\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left({\left(2 \cdot F\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      11. pow1/2 N/A

          \[\leadsto \left(\color{blue}{\sqrt{2 \cdot F}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{2 \cdot F}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{2 \cdot F}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      14. *-commutative N/A

          \[\leadsto \left(\sqrt{\color{blue}{F \cdot 2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{F \cdot 2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      16. lower-neg.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \color{blue}{\left(-{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)}\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      17. pow1/2 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\color{blue}{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      18. lower-sqrt.f64 23.6

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\color{blue}{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      19. lift-fma.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\color{blue}{-4 \cdot \left(C \cdot A\right) + B \cdot B}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      20. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{-4 \cdot \color{blue}{\left(C \cdot A\right)} + B \cdot B}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      21. associate-*r* N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\color{blue}{\left(-4 \cdot C\right) \cdot A} + B \cdot B}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      22. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\left(-4 \cdot C\right) \cdot A + \color{blue}{B \cdot B}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      23. pow2 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\left(-4 \cdot C\right) \cdot A + \color{blue}{{B}^{2}}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      24. lift-pow.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\left(-4 \cdot C\right) \cdot A + \color{blue}{{B}^{2}}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   6. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\color{blue}{\left(C + \frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + C\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{{B}^{2}}{A}, C\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{{B}^{2}}{A}}, C\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. unpow2 N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{B \cdot B}}{A}, C\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-*.f64 27.2

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{\color{blue}{B \cdot B}}{A}, C\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   9. Applied rewrites 27.2%

      \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if -0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 49.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 80.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{2 \cdot C}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 57.8

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{2 \cdot C}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 57.8%

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{2 \cdot C}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   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 16.7

         \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 16.7%

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 16.8%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 16.8%

      \[\leadsto -\sqrt{\frac{2}{B} \cdot F} \]
   10. Step-by-step derivation

   11. Applied rewrites 21.2%

       \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 27.4%

   \[\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^{+27}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(0, \frac{A}{C}, 1\right) \cdot C + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \sqrt{F \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 -5 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{\left(1 + \frac{A}{B}\right) \cdot B + C}}{-\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{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right) + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \sqrt{F \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 \infty:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{C \cdot 2}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := C \cdot \left(A \cdot 4\right)\\ t_2 := \frac{\sqrt{\left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} + \left(C + A\right)\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_1\right)\right) \cdot 2\right)}}{t\_1 - {B\_m}^{2}}\\ t_3 := -t\_0\\ t_4 := \left(F \cdot 2\right) \cdot t\_0\\ t_5 := \sqrt{t\_4}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+27}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(0, \frac{A}{C}, 1\right) \cdot C + C}}{t\_3} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)} \cdot \sqrt{F \cdot 2}\right)\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-189}:\\ \;\;\;\;t\_5 \cdot \frac{\sqrt{\left(1 + \frac{A}{B\_m}\right) \cdot B\_m + C}}{t\_3}\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;\sqrt{\left(\left(\frac{B\_m \cdot B\_m}{A} \cdot -0.5 + C\right) + C\right) \cdot t\_4} \cdot \frac{-1}{t\_0}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_5 \cdot \frac{\sqrt{C \cdot 2}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B\_m}}\\ \end{array} \end{array} \]

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 (* 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 (- t_0))
        (t_4 (* (* F 2.0) t_0))
        (t_5 (sqrt t_4)))
   (if (<= t_2 -5e+27)
     (*
      (/ (sqrt (+ (* (fma 0.0 (/ A C) 1.0) C) C)) t_3)
      (* (sqrt (fma (* -4.0 C) A (* B_m B_m))) (sqrt (* F 2.0))))
     (if (<= t_2 -5e-189)
       (* t_5 (/ (sqrt (+ (* (+ 1.0 (/ A B_m)) B_m) C)) t_3))
       (if (<= t_2 1.0)
         (* (sqrt (* (+ (+ (* (/ (* B_m B_m) A) -0.5) C) C) t_4)) (/ -1.0 t_0))
         (if (<= t_2 INFINITY)
           (* t_5 (/ (sqrt (* C 2.0)) t_3))
           (/ (- (sqrt F)) (sqrt (* 0.5 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 = 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 = -t_0;
	double t_4 = (F * 2.0) * t_0;
	double t_5 = sqrt(t_4);
	double tmp;
	if (t_2 <= -5e+27) {
		tmp = (sqrt(((fma(0.0, (A / C), 1.0) * C) + C)) / t_3) * (sqrt(fma((-4.0 * C), A, (B_m * B_m))) * sqrt((F * 2.0)));
	} else if (t_2 <= -5e-189) {
		tmp = t_5 * (sqrt((((1.0 + (A / B_m)) * B_m) + C)) / t_3);
	} else if (t_2 <= 1.0) {
		tmp = sqrt(((((((B_m * B_m) / A) * -0.5) + C) + C) * t_4)) * (-1.0 / t_0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_5 * (sqrt((C * 2.0)) / t_3);
	} else {
		tmp = -sqrt(F) / sqrt((0.5 * 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(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(-t_0)
	t_4 = Float64(Float64(F * 2.0) * t_0)
	t_5 = sqrt(t_4)
	tmp = 0.0
	if (t_2 <= -5e+27)
		tmp = Float64(Float64(sqrt(Float64(Float64(fma(0.0, Float64(A / C), 1.0) * C) + C)) / t_3) * Float64(sqrt(fma(Float64(-4.0 * C), A, Float64(B_m * B_m))) * sqrt(Float64(F * 2.0))));
	elseif (t_2 <= -5e-189)
		tmp = Float64(t_5 * Float64(sqrt(Float64(Float64(Float64(1.0 + Float64(A / B_m)) * B_m) + C)) / t_3));
	elseif (t_2 <= 1.0)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(Float64(B_m * B_m) / A) * -0.5) + C) + C) * t_4)) * Float64(-1.0 / t_0));
	elseif (t_2 <= Inf)
		tmp = Float64(t_5 * Float64(sqrt(Float64(C * 2.0)) / t_3));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(0.5 * 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[(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 = (-t$95$0)}, Block[{t$95$4 = N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[t$95$4], $MachinePrecision]}, If[LessEqual[t$95$2, -5e+27], N[(N[(N[Sqrt[N[(N[(N[(0.0 * N[(A / C), $MachinePrecision] + 1.0), $MachinePrecision] * C), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision] * N[(N[Sqrt[N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -5e-189], N[(t$95$5 * N[(N[Sqrt[N[(N[(N[(1.0 + N[(A / B$95$m), $MachinePrecision]), $MachinePrecision] * B$95$m), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1.0], N[(N[Sqrt[N[(N[(N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision] * t$95$4), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(t$95$5 * N[(N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(0.5 * B$95$m), $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))) < -4.99999999999999979e27

   1. Initial program 21.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1}} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)}{\mathsf{neg}\left(-1\right)}} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)}{\color{blue}{1}} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. /-rgt-identity N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. pow1/2 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left({\color{blue}{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}^{\frac{1}{2}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      8. unpow-prod-down N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{\left(2 \cdot F\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left({\left(2 \cdot F\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left({\left(2 \cdot F\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      11. pow1/2 N/A

          \[\leadsto \left(\color{blue}{\sqrt{2 \cdot F}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{2 \cdot F}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{2 \cdot F}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      14. *-commutative N/A

          \[\leadsto \left(\sqrt{\color{blue}{F \cdot 2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{F \cdot 2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      16. lower-neg.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \color{blue}{\left(-{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)}\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      17. pow1/2 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\color{blue}{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      18. lower-sqrt.f64 80.9

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\color{blue}{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      19. lift-fma.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\color{blue}{-4 \cdot \left(C \cdot A\right) + B \cdot B}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      20. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{-4 \cdot \color{blue}{\left(C \cdot A\right)} + B \cdot B}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      21. associate-*r* N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\color{blue}{\left(-4 \cdot C\right) \cdot A} + B \cdot B}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      22. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\left(-4 \cdot C\right) \cdot A + \color{blue}{B \cdot B}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      23. pow2 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\left(-4 \cdot C\right) \cdot A + \color{blue}{{B}^{2}}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      24. lift-pow.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\left(-4 \cdot C\right) \cdot A + \color{blue}{{B}^{2}}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   6. Applied rewrites 80.9%

      \[\leadsto \color{blue}{\left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Taylor expanded in C around inf

      \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\color{blue}{C \cdot \left(1 + \left(-1 \cdot \frac{A}{C} + \frac{A}{C}\right)\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\color{blue}{C \cdot \left(1 + \left(-1 \cdot \frac{A}{C} + \frac{A}{C}\right)\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \color{blue}{\left(\left(-1 \cdot \frac{A}{C} + \frac{A}{C}\right) + 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. distribute-lft1-in N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \left(\color{blue}{\left(-1 + 1\right) \cdot \frac{A}{C}} + 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. metadata-eval N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \left(\color{blue}{0} \cdot \frac{A}{C} + 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \color{blue}{\mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-/.f64 25.3

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{C \cdot \mathsf{fma}\left(0, \color{blue}{\frac{A}{C}}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   9. Applied rewrites 25.3%

      \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\color{blue}{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if -4.99999999999999979e27 < (/.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))) < -4.9999999999999997e-189

   1. Initial program 97.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in B around inf

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{B \cdot \left(1 + \frac{A}{B}\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{B \cdot \left(1 + \frac{A}{B}\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{B \cdot \color{blue}{\left(1 + \frac{A}{B}\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-/.f64 48.4

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{B \cdot \left(1 + \color{blue}{\frac{A}{B}}\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 48.4%

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{B \cdot \left(1 + \frac{A}{B}\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if -4.9999999999999997e-189 < (/.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

   1. Initial program 12.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 15.6%

      \[\leadsto \color{blue}{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \sqrt{\left(\color{blue}{\left(C + \frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(C + \frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(C + \color{blue}{\frac{-1}{2} \cdot \frac{{B}^{2}}{A}}\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(C + \frac{-1}{2} \cdot \color{blue}{\frac{{B}^{2}}{A}}\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(C + \frac{-1}{2} \cdot \frac{\color{blue}{B \cdot B}}{A}\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-*.f64 31.6

         \[\leadsto \sqrt{\left(\left(C + -0.5 \cdot \frac{\color{blue}{B \cdot B}}{A}\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 31.6%

      \[\leadsto \sqrt{\left(\color{blue}{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 1 < (/.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 30.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 81.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{2 \cdot C}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 64.2

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{2 \cdot C}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 64.2%

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{2 \cdot C}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   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 16.7

         \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 16.7%

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 16.8%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 16.8%

      \[\leadsto -\sqrt{\frac{2}{B} \cdot F} \]
   10. Step-by-step derivation

   11. Applied rewrites 21.2%

       \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 27.9%

   \[\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^{+27}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(0, \frac{A}{C}, 1\right) \cdot C + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \sqrt{F \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 -5 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{\left(1 + \frac{A}{B}\right) \cdot B + C}}{-\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 1:\\ \;\;\;\;\sqrt{\left(\left(\frac{B \cdot B}{A} \cdot -0.5 + C\right) + C\right) \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\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 \infty:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{C \cdot 2}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := \left(F \cdot 2\right) \cdot t\_0\\ \mathbf{if}\;{B\_m}^{2} \leq 1.5 \cdot 10^{-37}:\\ \;\;\;\;\sqrt{\left(\left(\frac{B\_m \cdot B\_m}{A} \cdot -0.5 + C\right) + C\right) \cdot t\_1} \cdot \frac{-1}{t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+171}:\\ \;\;\;\;\sqrt{t\_1} \cdot \frac{\sqrt{C \cdot 2}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 (* (* F 2.0) t_0)))
   (if (<= (pow B_m 2.0) 1.5e-37)
     (* (sqrt (* (+ (+ (* (/ (* B_m B_m) A) -0.5) C) C) t_1)) (/ -1.0 t_0))
     (if (<= (pow B_m 2.0) 5e+171)
       (* (sqrt t_1) (/ (sqrt (* C 2.0)) (- t_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 = (F * 2.0) * t_0;
	double tmp;
	if (pow(B_m, 2.0) <= 1.5e-37) {
		tmp = sqrt(((((((B_m * B_m) / A) * -0.5) + C) + C) * t_1)) * (-1.0 / t_0);
	} else if (pow(B_m, 2.0) <= 5e+171) {
		tmp = sqrt(t_1) * (sqrt((C * 2.0)) / -t_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(Float64(F * 2.0) * t_0)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1.5e-37)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(Float64(B_m * B_m) / A) * -0.5) + C) + C) * t_1)) * Float64(-1.0 / t_0));
	elseif ((B_m ^ 2.0) <= 5e+171)
		tmp = Float64(sqrt(t_1) * Float64(sqrt(Float64(C * 2.0)) / Float64(-t_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 = N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1.5e-37], N[(N[Sqrt[N[(N[(N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+171], N[(N[Sqrt[t$95$1], $MachinePrecision] * N[(N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.5e-37

   1. Initial program 19.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 25.3%

      \[\leadsto \color{blue}{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \sqrt{\left(\color{blue}{\left(C + \frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(C + \frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(C + \color{blue}{\frac{-1}{2} \cdot \frac{{B}^{2}}{A}}\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(C + \frac{-1}{2} \cdot \color{blue}{\frac{{B}^{2}}{A}}\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(C + \frac{-1}{2} \cdot \frac{\color{blue}{B \cdot B}}{A}\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-*.f64 20.5

         \[\leadsto \sqrt{\left(\left(C + -0.5 \cdot \frac{\color{blue}{B \cdot B}}{A}\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 20.5%

      \[\leadsto \sqrt{\left(\color{blue}{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 1.5e-37 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000004e171

   1. Initial program 32.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 57.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{2 \cdot C}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 21.3

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{2 \cdot C}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 21.3%

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{2 \cdot C}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 5.0000000000000004e171 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 4.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 31.1

         \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 31.1%

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 31.3%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 36.6%

      \[\leadsto \frac{\sqrt{2 \cdot F}}{\color{blue}{-\sqrt{B}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 26.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 1.5 \cdot 10^{-37}:\\ \;\;\;\;\sqrt{\left(\left(\frac{B \cdot B}{A} \cdot -0.5 + C\right) + C\right) \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+171}:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{C \cdot 2}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 51.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := \left(F \cdot 2\right) \cdot t\_0\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-139}:\\ \;\;\;\;\sqrt{\left(C \cdot 2\right) \cdot t\_1} \cdot \frac{-1}{t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+171}:\\ \;\;\;\;\sqrt{t\_1} \cdot \frac{\sqrt{C \cdot 2}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 (* (* F 2.0) t_0)))
   (if (<= (pow B_m 2.0) 1e-139)
     (* (sqrt (* (* C 2.0) t_1)) (/ -1.0 t_0))
     (if (<= (pow B_m 2.0) 5e+171)
       (* (sqrt t_1) (/ (sqrt (* C 2.0)) (- t_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 = (F * 2.0) * t_0;
	double tmp;
	if (pow(B_m, 2.0) <= 1e-139) {
		tmp = sqrt(((C * 2.0) * t_1)) * (-1.0 / t_0);
	} else if (pow(B_m, 2.0) <= 5e+171) {
		tmp = sqrt(t_1) * (sqrt((C * 2.0)) / -t_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(Float64(F * 2.0) * t_0)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-139)
		tmp = Float64(sqrt(Float64(Float64(C * 2.0) * t_1)) * Float64(-1.0 / t_0));
	elseif ((B_m ^ 2.0) <= 5e+171)
		tmp = Float64(sqrt(t_1) * Float64(sqrt(Float64(C * 2.0)) / Float64(-t_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 = N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-139], N[(N[Sqrt[N[(N[(C * 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+171], N[(N[Sqrt[t$95$1], $MachinePrecision] * N[(N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000003e-139

   1. Initial program 15.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 23.2%

      \[\leadsto \color{blue}{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot C\right)} \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 22.6

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot C\right)} \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 22.6%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot C\right)} \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 1.00000000000000003e-139 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000004e171

   1. Initial program 33.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 54.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{2 \cdot C}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 19.8

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{2 \cdot C}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 19.8%

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{2 \cdot C}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 5.0000000000000004e171 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 4.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 31.1

         \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 31.1%

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 31.3%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 36.6%

      \[\leadsto \frac{\sqrt{2 \cdot F}}{\color{blue}{-\sqrt{B}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 26.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-139}:\\ \;\;\;\;\sqrt{\left(C \cdot 2\right) \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+171}:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{C \cdot 2}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 51.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-159}:\\ \;\;\;\;\sqrt{\left(C \cdot 2\right) \cdot \left(\left(F \cdot 2\right) \cdot t\_0\right)} \cdot \frac{-1}{t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+173}:\\ \;\;\;\;\left(\frac{-\sqrt{2}}{B\_m} \cdot \sqrt{F}\right) \cdot \sqrt{\frac{B\_m \cdot B\_m}{A} \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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))))
   (if (<= (pow B_m 2.0) 1e-159)
     (* (sqrt (* (* C 2.0) (* (* F 2.0) t_0))) (/ -1.0 t_0))
     (if (<= (pow B_m 2.0) 2e+173)
       (*
        (* (/ (- (sqrt 2.0)) B_m) (sqrt F))
        (sqrt (* (/ (* B_m B_m) A) -0.5)))
       (/ (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 tmp;
	if (pow(B_m, 2.0) <= 1e-159) {
		tmp = sqrt(((C * 2.0) * ((F * 2.0) * t_0))) * (-1.0 / t_0);
	} else if (pow(B_m, 2.0) <= 2e+173) {
		tmp = ((-sqrt(2.0) / B_m) * sqrt(F)) * sqrt((((B_m * B_m) / A) * -0.5));
	} 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))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-159)
		tmp = Float64(sqrt(Float64(Float64(C * 2.0) * Float64(Float64(F * 2.0) * t_0))) * Float64(-1.0 / t_0));
	elseif ((B_m ^ 2.0) <= 2e+173)
		tmp = Float64(Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(F)) * sqrt(Float64(Float64(Float64(B_m * B_m) / A) * -0.5)));
	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]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-159], N[(N[Sqrt[N[(N[(C * 2.0), $MachinePrecision] * N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+173], N[(N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999989e-160

   1. Initial program 15.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 23.2%

      \[\leadsto \color{blue}{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot C\right)} \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 22.5

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot C\right)} \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 22.5%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot C\right)} \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 9.99999999999999989e-160 < (pow.f64 B #s(literal 2 binary64)) < 2e173

   1. Initial program 31.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(A + \sqrt{{A}^{2} + {B}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(A + \sqrt{{A}^{2} + {B}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{A}^{2} + {B}^{2}} + A\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{A}^{2} + {B}^{2}} + A\right)} \cdot F} \]

      12. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{A \cdot A} + {B}^{2}} + A\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{A \cdot A + \color{blue}{B \cdot B}} + A\right) \cdot F} \]

      14. lower-hypot.f64 15.0

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(A, B\right)} + A\right) \cdot F} \]

   5. Applied rewrites 15.0%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(A, B\right) + A\right) \cdot F}} \]

   6. Taylor expanded in A around -inf

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right) \cdot F} \]
   7. Step-by-step derivation

   8. Applied rewrites 12.6%

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(-0.5 \cdot \frac{B \cdot B}{A}\right) \cdot F} \]
   9. Step-by-step derivation

   10. Applied rewrites 13.2%

       \[\leadsto \sqrt{\frac{B \cdot B}{A} \cdot -0.5} \cdot \color{blue}{\left(\sqrt{F} \cdot \frac{\sqrt{2}}{-B}\right)} \]

   if 2e173 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 4.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 31.4

         \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 31.4%

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 31.6%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 37.0%

      \[\leadsto \frac{\sqrt{2 \cdot F}}{\color{blue}{-\sqrt{B}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 24.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-159}:\\ \;\;\;\;\sqrt{\left(C \cdot 2\right) \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+173}:\\ \;\;\;\;\left(\frac{-\sqrt{2}}{B} \cdot \sqrt{F}\right) \cdot \sqrt{\frac{B \cdot B}{A} \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 50.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := \left(F \cdot 2\right) \cdot t\_0\\ t_2 := \sqrt{F \cdot 2}\\ \mathbf{if}\;B\_m \leq 1.02 \cdot 10^{-256}:\\ \;\;\;\;\left(\sqrt{\frac{1}{C}} \cdot \left(\frac{\sqrt{2}}{A} \cdot -0.25\right)\right) \cdot \left(\left(-\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)}\right) \cdot t\_2\right)\\ \mathbf{elif}\;B\_m \leq 4.1 \cdot 10^{-19}:\\ \;\;\;\;\sqrt{\left(\left(\frac{B\_m \cdot B\_m}{A} \cdot -0.5 + C\right) + C\right) \cdot t\_1} \cdot \frac{-1}{t\_0}\\ \mathbf{elif}\;B\_m \leq 7 \cdot 10^{+85}:\\ \;\;\;\;\sqrt{t\_1} \cdot \frac{\sqrt{C \cdot 2}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 (* (* F 2.0) t_0))
        (t_2 (sqrt (* F 2.0))))
   (if (<= B_m 1.02e-256)
     (*
      (* (sqrt (/ 1.0 C)) (* (/ (sqrt 2.0) A) -0.25))
      (* (- (sqrt (fma (* -4.0 C) A (* B_m B_m)))) t_2))
     (if (<= B_m 4.1e-19)
       (* (sqrt (* (+ (+ (* (/ (* B_m B_m) A) -0.5) C) C) t_1)) (/ -1.0 t_0))
       (if (<= B_m 7e+85)
         (* (sqrt t_1) (/ (sqrt (* C 2.0)) (- t_0)))
         (/ t_2 (- (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 = (F * 2.0) * t_0;
	double t_2 = sqrt((F * 2.0));
	double tmp;
	if (B_m <= 1.02e-256) {
		tmp = (sqrt((1.0 / C)) * ((sqrt(2.0) / A) * -0.25)) * (-sqrt(fma((-4.0 * C), A, (B_m * B_m))) * t_2);
	} else if (B_m <= 4.1e-19) {
		tmp = sqrt(((((((B_m * B_m) / A) * -0.5) + C) + C) * t_1)) * (-1.0 / t_0);
	} else if (B_m <= 7e+85) {
		tmp = sqrt(t_1) * (sqrt((C * 2.0)) / -t_0);
	} else {
		tmp = t_2 / -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(Float64(F * 2.0) * t_0)
	t_2 = sqrt(Float64(F * 2.0))
	tmp = 0.0
	if (B_m <= 1.02e-256)
		tmp = Float64(Float64(sqrt(Float64(1.0 / C)) * Float64(Float64(sqrt(2.0) / A) * -0.25)) * Float64(Float64(-sqrt(fma(Float64(-4.0 * C), A, Float64(B_m * B_m)))) * t_2));
	elseif (B_m <= 4.1e-19)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(Float64(B_m * B_m) / A) * -0.5) + C) + C) * t_1)) * Float64(-1.0 / t_0));
	elseif (B_m <= 7e+85)
		tmp = Float64(sqrt(t_1) * Float64(sqrt(Float64(C * 2.0)) / Float64(-t_0)));
	else
		tmp = Float64(t_2 / 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[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B$95$m, 1.02e-256], N[(N[(N[Sqrt[N[(1.0 / C), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] / A), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * N[((-N[Sqrt[N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 4.1e-19], N[(N[Sqrt[N[(N[(N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 7e+85], N[(N[Sqrt[t$95$1], $MachinePrecision] * N[(N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 1.01999999999999993e-256

   1. Initial program 17.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 28.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1}} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)}{\mathsf{neg}\left(-1\right)}} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)}{\color{blue}{1}} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. /-rgt-identity N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. pow1/2 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left({\color{blue}{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}^{\frac{1}{2}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      8. unpow-prod-down N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{\left(2 \cdot F\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left({\left(2 \cdot F\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left({\left(2 \cdot F\right)}^{\frac{1}{2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      11. pow1/2 N/A

          \[\leadsto \left(\color{blue}{\sqrt{2 \cdot F}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{2 \cdot F}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{2 \cdot F}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      14. *-commutative N/A

          \[\leadsto \left(\sqrt{\color{blue}{F \cdot 2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{F \cdot 2}} \cdot \left(\mathsf{neg}\left({\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      16. lower-neg.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \color{blue}{\left(-{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{\frac{1}{2}}\right)}\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      17. pow1/2 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\color{blue}{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      18. lower-sqrt.f64 32.4

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\color{blue}{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      19. lift-fma.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\color{blue}{-4 \cdot \left(C \cdot A\right) + B \cdot B}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      20. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{-4 \cdot \color{blue}{\left(C \cdot A\right)} + B \cdot B}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      21. associate-*r* N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\color{blue}{\left(-4 \cdot C\right) \cdot A} + B \cdot B}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      22. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\left(-4 \cdot C\right) \cdot A + \color{blue}{B \cdot B}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      23. pow2 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\left(-4 \cdot C\right) \cdot A + \color{blue}{{B}^{2}}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      24. lift-pow.f64 N/A

          \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\left(-4 \cdot C\right) \cdot A + \color{blue}{{B}^{2}}}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   6. Applied rewrites 32.4%

      \[\leadsto \color{blue}{\left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \color{blue}{\left(\frac{-1}{4} \cdot \left(\frac{\sqrt{2}}{A} \cdot \sqrt{\frac{1}{C}}\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \color{blue}{\left(\left(\frac{-1}{4} \cdot \frac{\sqrt{2}}{A}\right) \cdot \sqrt{\frac{1}{C}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \color{blue}{\left(\left(\frac{-1}{4} \cdot \frac{\sqrt{2}}{A}\right) \cdot \sqrt{\frac{1}{C}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \left(\color{blue}{\left(\frac{-1}{4} \cdot \frac{\sqrt{2}}{A}\right)} \cdot \sqrt{\frac{1}{C}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \left(\left(\frac{-1}{4} \cdot \color{blue}{\frac{\sqrt{2}}{A}}\right) \cdot \sqrt{\frac{1}{C}}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \left(\left(\frac{-1}{4} \cdot \frac{\color{blue}{\sqrt{2}}}{A}\right) \cdot \sqrt{\frac{1}{C}}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \left(\left(\frac{-1}{4} \cdot \frac{\sqrt{2}}{A}\right) \cdot \color{blue}{\sqrt{\frac{1}{C}}}\right) \]

      7. lower-/.f64 11.3

         \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \left(\left(-0.25 \cdot \frac{\sqrt{2}}{A}\right) \cdot \sqrt{\color{blue}{\frac{1}{C}}}\right) \]

   9. Applied rewrites 11.3%

      \[\leadsto \left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\right)\right) \cdot \color{blue}{\left(\left(-0.25 \cdot \frac{\sqrt{2}}{A}\right) \cdot \sqrt{\frac{1}{C}}\right)} \]

   if 1.01999999999999993e-256 < B < 4.09999999999999985e-19

   1. Initial program 18.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 22.5%

      \[\leadsto \color{blue}{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \sqrt{\left(\color{blue}{\left(C + \frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(C + \frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(C + \color{blue}{\frac{-1}{2} \cdot \frac{{B}^{2}}{A}}\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(C + \frac{-1}{2} \cdot \color{blue}{\frac{{B}^{2}}{A}}\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(C + \frac{-1}{2} \cdot \frac{\color{blue}{B \cdot B}}{A}\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-*.f64 14.0

         \[\leadsto \sqrt{\left(\left(C + -0.5 \cdot \frac{\color{blue}{B \cdot B}}{A}\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 14.0%

      \[\leadsto \sqrt{\left(\color{blue}{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 4.09999999999999985e-19 < B < 7.0000000000000001e85

   1. Initial program 29.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 62.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{2 \cdot C}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 28.7

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{2 \cdot C}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 28.7%

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{2 \cdot C}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 7.0000000000000001e85 < 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 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 54.8

         \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 54.8%

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 55.1%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 65.7%

      \[\leadsto \frac{\sqrt{2 \cdot F}}{\color{blue}{-\sqrt{B}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 23.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.02 \cdot 10^{-256}:\\ \;\;\;\;\left(\sqrt{\frac{1}{C}} \cdot \left(\frac{\sqrt{2}}{A} \cdot -0.25\right)\right) \cdot \left(\left(-\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\right) \cdot \sqrt{F \cdot 2}\right)\\ \mathbf{elif}\;B \leq 4.1 \cdot 10^{-19}:\\ \;\;\;\;\sqrt{\left(\left(\frac{B \cdot B}{A} \cdot -0.5 + C\right) + C\right) \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{+85}:\\ \;\;\;\;\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{C \cdot 2}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 45.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.4 \cdot 10^{-166}:\\ \;\;\;\;\sqrt{\left(\left(\left(C \cdot C\right) \cdot A\right) \cdot F\right) \cdot -16} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}\\ \mathbf{elif}\;B\_m \leq 3.5 \cdot 10^{+88}:\\ \;\;\;\;\left(\frac{-\sqrt{2}}{B\_m} \cdot \sqrt{F}\right) \cdot \sqrt{\frac{B\_m \cdot B\_m}{A} \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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
 (if (<= B_m 2.4e-166)
   (*
    (sqrt (* (* (* (* C C) A) F) -16.0))
    (/ -1.0 (fma -4.0 (* C A) (* B_m B_m))))
   (if (<= B_m 3.5e+88)
     (* (* (/ (- (sqrt 2.0)) B_m) (sqrt F)) (sqrt (* (/ (* B_m B_m) A) -0.5)))
     (/ (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 tmp;
	if (B_m <= 2.4e-166) {
		tmp = sqrt(((((C * C) * A) * F) * -16.0)) * (-1.0 / fma(-4.0, (C * A), (B_m * B_m)));
	} else if (B_m <= 3.5e+88) {
		tmp = ((-sqrt(2.0) / B_m) * sqrt(F)) * sqrt((((B_m * B_m) / A) * -0.5));
	} 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)
	tmp = 0.0
	if (B_m <= 2.4e-166)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(C * C) * A) * F) * -16.0)) * Float64(-1.0 / fma(-4.0, Float64(C * A), Float64(B_m * B_m))));
	elseif (B_m <= 3.5e+88)
		tmp = Float64(Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(F)) * sqrt(Float64(Float64(Float64(B_m * B_m) / A) * -0.5)));
	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_] := If[LessEqual[B$95$m, 2.4e-166], N[(N[Sqrt[N[(N[(N[(N[(C * C), $MachinePrecision] * A), $MachinePrecision] * F), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 3.5e+88], N[(N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 2.3999999999999999e-166

   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

   4. Applied rewrites 22.8%

      \[\leadsto \color{blue}{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in B around inf

      \[\leadsto \sqrt{\left(\color{blue}{B \cdot \left(1 + \frac{A}{B}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{B \cdot \left(1 + \frac{A}{B}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \sqrt{\left(B \cdot \color{blue}{\left(1 + \frac{A}{B}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-/.f64 4.2

         \[\leadsto \sqrt{\left(B \cdot \left(1 + \color{blue}{\frac{A}{B}}\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 4.2%

      \[\leadsto \sqrt{\left(\color{blue}{B \cdot \left(1 + \frac{A}{B}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   8. Taylor expanded in A around -inf

      \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{-16 \cdot \color{blue}{\left(\left(A \cdot {C}^{2}\right) \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{-16 \cdot \color{blue}{\left(\left(A \cdot {C}^{2}\right) \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{-16 \cdot \left(\color{blue}{\left(A \cdot {C}^{2}\right)} \cdot F\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{-16 \cdot \left(\left(A \cdot \color{blue}{\left(C \cdot C\right)}\right) \cdot F\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-*.f64 11.1

         \[\leadsto \sqrt{-16 \cdot \left(\left(A \cdot \color{blue}{\left(C \cdot C\right)}\right) \cdot F\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   10. Applied rewrites 11.1%

       \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 2.3999999999999999e-166 < B < 3.4999999999999998e88

   1. Initial program 23.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(A + \sqrt{{A}^{2} + {B}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(A + \sqrt{{A}^{2} + {B}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{A}^{2} + {B}^{2}} + A\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{A}^{2} + {B}^{2}} + A\right)} \cdot F} \]

      12. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{A \cdot A} + {B}^{2}} + A\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{A \cdot A + \color{blue}{B \cdot B}} + A\right) \cdot F} \]

      14. lower-hypot.f64 22.0

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(A, B\right)} + A\right) \cdot F} \]

   5. Applied rewrites 22.0%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(A, B\right) + A\right) \cdot F}} \]

   6. Taylor expanded in A around -inf

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right) \cdot F} \]
   7. Step-by-step derivation

   8. Applied rewrites 15.7%

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(-0.5 \cdot \frac{B \cdot B}{A}\right) \cdot F} \]
   9. Step-by-step derivation

   10. Applied rewrites 18.6%

       \[\leadsto \sqrt{\frac{B \cdot B}{A} \cdot -0.5} \cdot \color{blue}{\left(\sqrt{F} \cdot \frac{\sqrt{2}}{-B}\right)} \]

   if 3.4999999999999998e88 < 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 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 54.8

         \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 54.8%

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 55.1%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 65.7%

      \[\leadsto \frac{\sqrt{2 \cdot F}}{\color{blue}{-\sqrt{B}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 23.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.4 \cdot 10^{-166}:\\ \;\;\;\;\sqrt{\left(\left(\left(C \cdot C\right) \cdot A\right) \cdot F\right) \cdot -16} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{+88}:\\ \;\;\;\;\left(\frac{-\sqrt{2}}{B} \cdot \sqrt{F}\right) \cdot \sqrt{\frac{B \cdot B}{A} \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 43.2% accurate, 6.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 5.4 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{\left(\left(\left(C \cdot C\right) \cdot A\right) \cdot F\right) \cdot -16} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}\\ \mathbf{elif}\;B\_m \leq 1.15 \cdot 10^{-25}:\\ \;\;\;\;\sqrt{\left(\left(\frac{B\_m}{A} \cdot B\_m\right) \cdot -0.5\right) \cdot F} \cdot \frac{-\sqrt{2}}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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
 (if (<= B_m 5.4e-167)
   (*
    (sqrt (* (* (* (* C C) A) F) -16.0))
    (/ -1.0 (fma -4.0 (* C A) (* B_m B_m))))
   (if (<= B_m 1.15e-25)
     (* (sqrt (* (* (* (/ B_m A) B_m) -0.5) F)) (/ (- (sqrt 2.0)) B_m))
     (/ (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 tmp;
	if (B_m <= 5.4e-167) {
		tmp = sqrt(((((C * C) * A) * F) * -16.0)) * (-1.0 / fma(-4.0, (C * A), (B_m * B_m)));
	} else if (B_m <= 1.15e-25) {
		tmp = sqrt(((((B_m / A) * B_m) * -0.5) * F)) * (-sqrt(2.0) / B_m);
	} 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)
	tmp = 0.0
	if (B_m <= 5.4e-167)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(C * C) * A) * F) * -16.0)) * Float64(-1.0 / fma(-4.0, Float64(C * A), Float64(B_m * B_m))));
	elseif (B_m <= 1.15e-25)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(B_m / A) * B_m) * -0.5) * F)) * Float64(Float64(-sqrt(2.0)) / B_m));
	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_] := If[LessEqual[B$95$m, 5.4e-167], N[(N[Sqrt[N[(N[(N[(N[(C * C), $MachinePrecision] * A), $MachinePrecision] * F), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.15e-25], N[(N[Sqrt[N[(N[(N[(N[(B$95$m / A), $MachinePrecision] * B$95$m), $MachinePrecision] * -0.5), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 5.4000000000000001e-167

   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

   4. Applied rewrites 22.8%

      \[\leadsto \color{blue}{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in B around inf

      \[\leadsto \sqrt{\left(\color{blue}{B \cdot \left(1 + \frac{A}{B}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{B \cdot \left(1 + \frac{A}{B}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \sqrt{\left(B \cdot \color{blue}{\left(1 + \frac{A}{B}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-/.f64 4.2

         \[\leadsto \sqrt{\left(B \cdot \left(1 + \color{blue}{\frac{A}{B}}\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 4.2%

      \[\leadsto \sqrt{\left(\color{blue}{B \cdot \left(1 + \frac{A}{B}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   8. Taylor expanded in A around -inf

      \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{-16 \cdot \color{blue}{\left(\left(A \cdot {C}^{2}\right) \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{-16 \cdot \color{blue}{\left(\left(A \cdot {C}^{2}\right) \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{-16 \cdot \left(\color{blue}{\left(A \cdot {C}^{2}\right)} \cdot F\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{-16 \cdot \left(\left(A \cdot \color{blue}{\left(C \cdot C\right)}\right) \cdot F\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-*.f64 11.1

         \[\leadsto \sqrt{-16 \cdot \left(\left(A \cdot \color{blue}{\left(C \cdot C\right)}\right) \cdot F\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   10. Applied rewrites 11.1%

       \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 5.4000000000000001e-167 < B < 1.15e-25

   1. Initial program 17.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(A + \sqrt{{A}^{2} + {B}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(A + \sqrt{{A}^{2} + {B}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{A}^{2} + {B}^{2}} + A\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{A}^{2} + {B}^{2}} + A\right)} \cdot F} \]

      12. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{A \cdot A} + {B}^{2}} + A\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{A \cdot A + \color{blue}{B \cdot B}} + A\right) \cdot F} \]

      14. lower-hypot.f64 13.4

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(A, B\right)} + A\right) \cdot F} \]

   5. Applied rewrites 13.4%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(A, B\right) + A\right) \cdot F}} \]

   6. Taylor expanded in A around -inf

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right) \cdot F} \]
   7. Step-by-step derivation

   8. Applied rewrites 17.0%

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(-0.5 \cdot \frac{B \cdot B}{A}\right) \cdot F} \]
   9. Step-by-step derivation

   10. Applied rewrites 18.2%

       \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(-0.5 \cdot \left(\frac{B}{A} \cdot B\right)\right) \cdot F} \]

   if 1.15e-25 < B

   1. Initial program 14.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 48.2

         \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 48.2%

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.5%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 55.9%

      \[\leadsto \frac{\sqrt{2 \cdot F}}{\color{blue}{-\sqrt{B}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.4 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{\left(\left(\left(C \cdot C\right) \cdot A\right) \cdot F\right) \cdot -16} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.15 \cdot 10^{-25}:\\ \;\;\;\;\sqrt{\left(\left(\frac{B}{A} \cdot B\right) \cdot -0.5\right) \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 43.6% accurate, 6.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.4 \cdot 10^{-166}:\\ \;\;\;\;\sqrt{\left(\left(\left(C \cdot C\right) \cdot A\right) \cdot F\right) \cdot -16} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}\\ \mathbf{elif}\;B\_m \leq 1.15 \cdot 10^{-25}:\\ \;\;\;\;\sqrt{\frac{\left(B\_m \cdot B\_m\right) \cdot F}{A} \cdot -0.5} \cdot \frac{-\sqrt{2}}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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
 (if (<= B_m 2.4e-166)
   (*
    (sqrt (* (* (* (* C C) A) F) -16.0))
    (/ -1.0 (fma -4.0 (* C A) (* B_m B_m))))
   (if (<= B_m 1.15e-25)
     (* (sqrt (* (/ (* (* B_m B_m) F) A) -0.5)) (/ (- (sqrt 2.0)) B_m))
     (/ (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 tmp;
	if (B_m <= 2.4e-166) {
		tmp = sqrt(((((C * C) * A) * F) * -16.0)) * (-1.0 / fma(-4.0, (C * A), (B_m * B_m)));
	} else if (B_m <= 1.15e-25) {
		tmp = sqrt(((((B_m * B_m) * F) / A) * -0.5)) * (-sqrt(2.0) / B_m);
	} 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)
	tmp = 0.0
	if (B_m <= 2.4e-166)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(C * C) * A) * F) * -16.0)) * Float64(-1.0 / fma(-4.0, Float64(C * A), Float64(B_m * B_m))));
	elseif (B_m <= 1.15e-25)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(B_m * B_m) * F) / A) * -0.5)) * Float64(Float64(-sqrt(2.0)) / B_m));
	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_] := If[LessEqual[B$95$m, 2.4e-166], N[(N[Sqrt[N[(N[(N[(N[(C * C), $MachinePrecision] * A), $MachinePrecision] * F), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.15e-25], N[(N[Sqrt[N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * F), $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 2.3999999999999999e-166

   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

   4. Applied rewrites 22.8%

      \[\leadsto \color{blue}{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in B around inf

      \[\leadsto \sqrt{\left(\color{blue}{B \cdot \left(1 + \frac{A}{B}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{B \cdot \left(1 + \frac{A}{B}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \sqrt{\left(B \cdot \color{blue}{\left(1 + \frac{A}{B}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-/.f64 4.2

         \[\leadsto \sqrt{\left(B \cdot \left(1 + \color{blue}{\frac{A}{B}}\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 4.2%

      \[\leadsto \sqrt{\left(\color{blue}{B \cdot \left(1 + \frac{A}{B}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   8. Taylor expanded in A around -inf

      \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{-16 \cdot \color{blue}{\left(\left(A \cdot {C}^{2}\right) \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{-16 \cdot \color{blue}{\left(\left(A \cdot {C}^{2}\right) \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{-16 \cdot \left(\color{blue}{\left(A \cdot {C}^{2}\right)} \cdot F\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{-16 \cdot \left(\left(A \cdot \color{blue}{\left(C \cdot C\right)}\right) \cdot F\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-*.f64 11.1

         \[\leadsto \sqrt{-16 \cdot \left(\left(A \cdot \color{blue}{\left(C \cdot C\right)}\right) \cdot F\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   10. Applied rewrites 11.1%

       \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 2.3999999999999999e-166 < B < 1.15e-25

   1. Initial program 17.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(A + \sqrt{{A}^{2} + {B}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(A + \sqrt{{A}^{2} + {B}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{A}^{2} + {B}^{2}} + A\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{A}^{2} + {B}^{2}} + A\right)} \cdot F} \]

      12. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{A \cdot A} + {B}^{2}} + A\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{A \cdot A + \color{blue}{B \cdot B}} + A\right) \cdot F} \]

      14. lower-hypot.f64 13.4

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(A, B\right)} + A\right) \cdot F} \]

   5. Applied rewrites 13.4%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(A, B\right) + A\right) \cdot F}} \]

   6. Taylor expanded in A around -inf

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\frac{-1}{2} \cdot \frac{{B}^{2} \cdot F}{A}} \]
   7. Step-by-step derivation

   8. Applied rewrites 19.4%

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{-0.5 \cdot \frac{\left(B \cdot B\right) \cdot F}{A}} \]

   if 1.15e-25 < B

   1. Initial program 14.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 48.2

         \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 48.2%

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.5%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 55.9%

      \[\leadsto \frac{\sqrt{2 \cdot F}}{\color{blue}{-\sqrt{B}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 24.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.4 \cdot 10^{-166}:\\ \;\;\;\;\sqrt{\left(\left(\left(C \cdot C\right) \cdot A\right) \cdot F\right) \cdot -16} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.15 \cdot 10^{-25}:\\ \;\;\;\;\sqrt{\frac{\left(B \cdot B\right) \cdot F}{A} \cdot -0.5} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 43.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.4 \cdot 10^{-166}:\\ \;\;\;\;\sqrt{\left(\left(\left(C \cdot C\right) \cdot A\right) \cdot F\right) \cdot -16} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}\\ \mathbf{elif}\;B\_m \leq 1.15 \cdot 10^{-25}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\frac{B\_m \cdot B\_m}{A} \cdot -0.5\right) \cdot F\right) \cdot 2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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
 (if (<= B_m 2.4e-166)
   (*
    (sqrt (* (* (* (* C C) A) F) -16.0))
    (/ -1.0 (fma -4.0 (* C A) (* B_m B_m))))
   (if (<= B_m 1.15e-25)
     (/ (sqrt (* (* (* (/ (* B_m B_m) A) -0.5) F) 2.0)) (- B_m))
     (/ (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 tmp;
	if (B_m <= 2.4e-166) {
		tmp = sqrt(((((C * C) * A) * F) * -16.0)) * (-1.0 / fma(-4.0, (C * A), (B_m * B_m)));
	} else if (B_m <= 1.15e-25) {
		tmp = sqrt((((((B_m * B_m) / A) * -0.5) * F) * 2.0)) / -B_m;
	} 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)
	tmp = 0.0
	if (B_m <= 2.4e-166)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(C * C) * A) * F) * -16.0)) * Float64(-1.0 / fma(-4.0, Float64(C * A), Float64(B_m * B_m))));
	elseif (B_m <= 1.15e-25)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(B_m * B_m) / A) * -0.5) * F) * 2.0)) / Float64(-B_m));
	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_] := If[LessEqual[B$95$m, 2.4e-166], N[(N[Sqrt[N[(N[(N[(N[(C * C), $MachinePrecision] * A), $MachinePrecision] * F), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.15e-25], N[(N[Sqrt[N[(N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 2.3999999999999999e-166

   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

   4. Applied rewrites 22.8%

      \[\leadsto \color{blue}{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in B around inf

      \[\leadsto \sqrt{\left(\color{blue}{B \cdot \left(1 + \frac{A}{B}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{B \cdot \left(1 + \frac{A}{B}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \sqrt{\left(B \cdot \color{blue}{\left(1 + \frac{A}{B}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-/.f64 4.2

         \[\leadsto \sqrt{\left(B \cdot \left(1 + \color{blue}{\frac{A}{B}}\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 4.2%

      \[\leadsto \sqrt{\left(\color{blue}{B \cdot \left(1 + \frac{A}{B}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   8. Taylor expanded in A around -inf

      \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{-16 \cdot \color{blue}{\left(\left(A \cdot {C}^{2}\right) \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{-16 \cdot \color{blue}{\left(\left(A \cdot {C}^{2}\right) \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{-16 \cdot \left(\color{blue}{\left(A \cdot {C}^{2}\right)} \cdot F\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{-16 \cdot \left(\left(A \cdot \color{blue}{\left(C \cdot C\right)}\right) \cdot F\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-*.f64 11.1

         \[\leadsto \sqrt{-16 \cdot \left(\left(A \cdot \color{blue}{\left(C \cdot C\right)}\right) \cdot F\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   10. Applied rewrites 11.1%

       \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 2.3999999999999999e-166 < B < 1.15e-25

   1. Initial program 17.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(A + \sqrt{{A}^{2} + {B}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(A + \sqrt{{A}^{2} + {B}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{A}^{2} + {B}^{2}} + A\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{A}^{2} + {B}^{2}} + A\right)} \cdot F} \]

      12. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{A \cdot A} + {B}^{2}} + A\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{A \cdot A + \color{blue}{B \cdot B}} + A\right) \cdot F} \]

      14. lower-hypot.f64 13.4

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(A, B\right)} + A\right) \cdot F} \]

   5. Applied rewrites 13.4%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(A, B\right) + A\right) \cdot F}} \]

   6. Taylor expanded in A around -inf

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right) \cdot F} \]
   7. Step-by-step derivation

   8. Applied rewrites 17.0%

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(-0.5 \cdot \frac{B \cdot B}{A}\right) \cdot F} \]
   9. Step-by-step derivation

   10. Applied rewrites 17.0%

       \[\leadsto \frac{\sqrt{\left(\left(\frac{B \cdot B}{A} \cdot -0.5\right) \cdot F\right) \cdot 2}}{\color{blue}{-B}} \]

   if 1.15e-25 < B

   1. Initial program 14.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 48.2

         \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 48.2%

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.5%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 55.9%

      \[\leadsto \frac{\sqrt{2 \cdot F}}{\color{blue}{-\sqrt{B}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 24.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.4 \cdot 10^{-166}:\\ \;\;\;\;\sqrt{\left(\left(\left(C \cdot C\right) \cdot A\right) \cdot F\right) \cdot -16} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.15 \cdot 10^{-25}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\frac{B \cdot B}{A} \cdot -0.5\right) \cdot F\right) \cdot 2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 42.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.4 \cdot 10^{-166}:\\ \;\;\;\;\sqrt{\left(\left(\left(C \cdot C\right) \cdot F\right) \cdot A\right) \cdot -16} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}\\ \mathbf{elif}\;B\_m \leq 1.15 \cdot 10^{-25}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\frac{B\_m \cdot B\_m}{A} \cdot -0.5\right) \cdot F\right) \cdot 2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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
 (if (<= B_m 2.4e-166)
   (*
    (sqrt (* (* (* (* C C) F) A) -16.0))
    (/ -1.0 (fma -4.0 (* C A) (* B_m B_m))))
   (if (<= B_m 1.15e-25)
     (/ (sqrt (* (* (* (/ (* B_m B_m) A) -0.5) F) 2.0)) (- B_m))
     (/ (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 tmp;
	if (B_m <= 2.4e-166) {
		tmp = sqrt(((((C * C) * F) * A) * -16.0)) * (-1.0 / fma(-4.0, (C * A), (B_m * B_m)));
	} else if (B_m <= 1.15e-25) {
		tmp = sqrt((((((B_m * B_m) / A) * -0.5) * F) * 2.0)) / -B_m;
	} 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)
	tmp = 0.0
	if (B_m <= 2.4e-166)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(C * C) * F) * A) * -16.0)) * Float64(-1.0 / fma(-4.0, Float64(C * A), Float64(B_m * B_m))));
	elseif (B_m <= 1.15e-25)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(B_m * B_m) / A) * -0.5) * F) * 2.0)) / Float64(-B_m));
	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_] := If[LessEqual[B$95$m, 2.4e-166], N[(N[Sqrt[N[(N[(N[(N[(C * C), $MachinePrecision] * F), $MachinePrecision] * A), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.15e-25], N[(N[Sqrt[N[(N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 2.3999999999999999e-166

   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

   4. Applied rewrites 22.8%

      \[\leadsto \color{blue}{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{-16 \cdot \color{blue}{\left(A \cdot \left({C}^{2} \cdot F\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{-16 \cdot \left(A \cdot \color{blue}{\left({C}^{2} \cdot F\right)}\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. unpow2 N/A

         \[\leadsto \sqrt{-16 \cdot \left(A \cdot \left(\color{blue}{\left(C \cdot C\right)} \cdot F\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-*.f64 11.6

         \[\leadsto \sqrt{-16 \cdot \left(A \cdot \left(\color{blue}{\left(C \cdot C\right)} \cdot F\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 11.6%

      \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 2.3999999999999999e-166 < B < 1.15e-25

   1. Initial program 17.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(A + \sqrt{{A}^{2} + {B}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(A + \sqrt{{A}^{2} + {B}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{A}^{2} + {B}^{2}} + A\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{A}^{2} + {B}^{2}} + A\right)} \cdot F} \]

      12. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{A \cdot A} + {B}^{2}} + A\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{A \cdot A + \color{blue}{B \cdot B}} + A\right) \cdot F} \]

      14. lower-hypot.f64 13.4

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(A, B\right)} + A\right) \cdot F} \]

   5. Applied rewrites 13.4%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(A, B\right) + A\right) \cdot F}} \]

   6. Taylor expanded in A around -inf

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right) \cdot F} \]
   7. Step-by-step derivation

   8. Applied rewrites 17.0%

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(-0.5 \cdot \frac{B \cdot B}{A}\right) \cdot F} \]
   9. Step-by-step derivation

   10. Applied rewrites 17.0%

       \[\leadsto \frac{\sqrt{\left(\left(\frac{B \cdot B}{A} \cdot -0.5\right) \cdot F\right) \cdot 2}}{\color{blue}{-B}} \]

   if 1.15e-25 < B

   1. Initial program 14.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 48.2

         \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 48.2%

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.5%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 55.9%

      \[\leadsto \frac{\sqrt{2 \cdot F}}{\color{blue}{-\sqrt{B}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 24.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.4 \cdot 10^{-166}:\\ \;\;\;\;\sqrt{\left(\left(\left(C \cdot C\right) \cdot F\right) \cdot A\right) \cdot -16} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.15 \cdot 10^{-25}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\frac{B \cdot B}{A} \cdot -0.5\right) \cdot F\right) \cdot 2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 36.5% accurate, 8.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.15 \cdot 10^{-25}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\frac{B\_m \cdot B\_m}{A} \cdot -0.5\right) \cdot F\right) \cdot 2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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
 (if (<= B_m 1.15e-25)
   (/ (sqrt (* (* (* (/ (* B_m B_m) A) -0.5) F) 2.0)) (- B_m))
   (/ (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 tmp;
	if (B_m <= 1.15e-25) {
		tmp = sqrt((((((B_m * B_m) / A) * -0.5) * F) * 2.0)) / -B_m;
	} else {
		tmp = sqrt((F * 2.0)) / -sqrt(B_m);
	}
	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 (b_m <= 1.15d-25) then
        tmp = sqrt((((((b_m * b_m) / a) * (-0.5d0)) * f) * 2.0d0)) / -b_m
    else
        tmp = sqrt((f * 2.0d0)) / -sqrt(b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.15e-25) {
		tmp = Math.sqrt((((((B_m * B_m) / A) * -0.5) * F) * 2.0)) / -B_m;
	} else {
		tmp = Math.sqrt((F * 2.0)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.15e-25:
		tmp = math.sqrt((((((B_m * B_m) / A) * -0.5) * F) * 2.0)) / -B_m
	else:
		tmp = math.sqrt((F * 2.0)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.15e-25)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(B_m * B_m) / A) * -0.5) * F) * 2.0)) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(F * 2.0)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1.15e-25)
		tmp = sqrt((((((B_m * B_m) / A) * -0.5) * F) * 2.0)) / -B_m;
	else
		tmp = sqrt((F * 2.0)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.15e-25], N[(N[Sqrt[N[(N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.15e-25

   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 C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(A + \sqrt{{A}^{2} + {B}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(A + \sqrt{{A}^{2} + {B}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{A}^{2} + {B}^{2}} + A\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{A}^{2} + {B}^{2}} + A\right)} \cdot F} \]

      12. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{A \cdot A} + {B}^{2}} + A\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{A \cdot A + \color{blue}{B \cdot B}} + A\right) \cdot F} \]

      14. lower-hypot.f64 4.9

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(A, B\right)} + A\right) \cdot F} \]

   5. Applied rewrites 4.9%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(A, B\right) + A\right) \cdot F}} \]

   6. Taylor expanded in A around -inf

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right) \cdot F} \]
   7. Step-by-step derivation

   8. Applied rewrites 7.5%

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(-0.5 \cdot \frac{B \cdot B}{A}\right) \cdot F} \]
   9. Step-by-step derivation

   10. Applied rewrites 7.5%

       \[\leadsto \frac{\sqrt{\left(\left(\frac{B \cdot B}{A} \cdot -0.5\right) \cdot F\right) \cdot 2}}{\color{blue}{-B}} \]

   if 1.15e-25 < B

   1. Initial program 14.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 48.2

         \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 48.2%

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.5%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 55.9%

      \[\leadsto \frac{\sqrt{2 \cdot F}}{\color{blue}{-\sqrt{B}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 20.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.15 \cdot 10^{-25}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\frac{B \cdot B}{A} \cdot -0.5\right) \cdot F\right) \cdot 2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 34.8% 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 16.0%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
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 16.5

      \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 16.5%

   \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

7. Applied rewrites 16.6%

   \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
8. Step-by-step derivation

9. Applied rewrites 18.7%

   \[\leadsto \frac{\sqrt{2 \cdot F}}{\color{blue}{-\sqrt{B}}} \]

10. Final simplification 18.7%

    \[\leadsto \frac{\sqrt{F \cdot 2}}{-\sqrt{B}} \]
11. Add Preprocessing

Alternative 24: 34.8% accurate, 12.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{-\sqrt{F}}{\sqrt{0.5 \cdot B\_m}} \end{array} \]

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 (* 0.5 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((0.5 * 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((0.5d0 * 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((0.5 * 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((0.5 * 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(0.5 * 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((0.5 * 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[(0.5 * B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.0%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
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 16.5

      \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 16.5%

   \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

7. Applied rewrites 16.6%

   \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
8. Step-by-step derivation

9. Applied rewrites 16.5%

   \[\leadsto -\sqrt{\frac{2}{B} \cdot F} \]
10. Step-by-step derivation

11. Applied rewrites 18.7%

    \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]

12. Final simplification 18.7%

    \[\leadsto \frac{-\sqrt{F}}{\sqrt{0.5 \cdot B}} \]
13. Add Preprocessing

Alternative 25: 34.8% 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 16.0%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
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 16.5

      \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 16.5%

   \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

7. Applied rewrites 16.6%

   \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
8. Step-by-step derivation

9. Applied rewrites 18.6%

   \[\leadsto -\sqrt{\frac{2}{B}} \cdot \sqrt{F} \]

10. Final simplification 18.6%

    \[\leadsto \left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}} \]
11. Add Preprocessing

Alternative 26: 26.5% 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 16.0%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
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 16.5

      \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 16.5%

   \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

7. Applied rewrites 16.6%

   \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
8. Add Preprocessing

Alternative 27: 26.4% 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 16.0%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
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 16.5

      \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 16.5%

   \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

7. Applied rewrites 16.6%

   \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
8. Step-by-step derivation

9. Applied rewrites 16.5%

   \[\leadsto -\sqrt{\frac{2}{B} \cdot F} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024296 
(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))))