ABCF->ab-angle a

Percentage Accurate: 19.1% → 62.0%

Time: 13.8s

Alternatives: 12

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: 62.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C\\ t_1 := -\sqrt{F}\\ t_2 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ t_3 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(t\_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_3}\\ \mathbf{if}\;t\_4 \leq -1 \cdot 10^{-194}:\\ \;\;\;\;\frac{\left(\sqrt{t\_0 \cdot 2} \cdot t\_1\right) \cdot \sqrt{t\_2}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\frac{\sqrt{\left(\left(C + -0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right) + C\right) \cdot t\_2} \cdot \left(-\sqrt{F \cdot 2}\right)}{t\_3}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{t\_0} \cdot \frac{\sqrt{\left(F \cdot 2\right) \cdot t\_2}}{\mathsf{fma}\left(-B\_m, B\_m, C \cdot \left(A \cdot 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (+ (+ (hypot (- A C) B_m) A) C))
        (t_1 (- (sqrt F)))
        (t_2 (fma (* -4.0 C) A (* B_m B_m)))
        (t_3 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* t_3 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_3))))
   (if (<= t_4 -1e-194)
     (/ (* (* (sqrt (* t_0 2.0)) t_1) (sqrt t_2)) t_3)
     (if (<= t_4 0.0)
       (/
        (*
         (sqrt (* (+ (+ C (* -0.5 (/ (* B_m B_m) A))) C) t_2))
         (- (sqrt (* F 2.0))))
        t_3)
       (if (<= t_4 INFINITY)
         (*
          (sqrt t_0)
          (/ (sqrt (* (* F 2.0) t_2)) (fma (- B_m) B_m (* C (* A 4.0)))))
         (* t_1 (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) {
	double t_0 = (hypot((A - C), B_m) + A) + C;
	double t_1 = -sqrt(F);
	double t_2 = fma((-4.0 * C), A, (B_m * B_m));
	double t_3 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_4 = sqrt(((2.0 * (t_3 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_3;
	double tmp;
	if (t_4 <= -1e-194) {
		tmp = ((sqrt((t_0 * 2.0)) * t_1) * sqrt(t_2)) / t_3;
	} else if (t_4 <= 0.0) {
		tmp = (sqrt((((C + (-0.5 * ((B_m * B_m) / A))) + C) * t_2)) * -sqrt((F * 2.0))) / t_3;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt(t_0) * (sqrt(((F * 2.0) * t_2)) / fma(-B_m, B_m, (C * (A * 4.0))));
	} else {
		tmp = t_1 * sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)
	t_1 = Float64(-sqrt(F))
	t_2 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	t_3 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_3 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_3))
	tmp = 0.0
	if (t_4 <= -1e-194)
		tmp = Float64(Float64(Float64(sqrt(Float64(t_0 * 2.0)) * t_1) * sqrt(t_2)) / t_3);
	elseif (t_4 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(C + Float64(-0.5 * Float64(Float64(B_m * B_m) / A))) + C) * t_2)) * Float64(-sqrt(Float64(F * 2.0)))) / t_3);
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(t_0) * Float64(sqrt(Float64(Float64(F * 2.0) * t_2)) / fma(Float64(-B_m), B_m, Float64(C * Float64(A * 4.0)))));
	else
		tmp = Float64(t_1 * sqrt(Float64(2.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[F], $MachinePrecision])}, Block[{t$95$2 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$3)), $MachinePrecision]}, If[LessEqual[t$95$4, -1e-194], N[(N[(N[(N[Sqrt[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(N[(N[Sqrt[N[(N[(N[(C + N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + C), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[t$95$0], $MachinePrecision] * N[(N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision] / N[((-B$95$m) * B$95$m + N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.00000000000000002e-194

   1. Initial program 43.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 60.6%

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

   6. Applied rewrites 76.1%

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

   if -1.00000000000000002e-194 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 3.7%

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

   6. Applied rewrites 17.8%

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

   7. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 32.9

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

   9. Applied rewrites 32.9%

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

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

   1. Initial program 33.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 83.9%

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

   6. Applied rewrites 83.9%

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

   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 19.4

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

   5. Applied rewrites 19.4%

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

   7. Applied rewrites 19.4%

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

   9. Applied rewrites 23.1%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-194}:\\ \;\;\;\;\frac{\left(\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot 2} \cdot \left(-\sqrt{F}\right)\right) \cdot \sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 0:\\ \;\;\;\;\frac{\sqrt{\left(\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right) + C\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \left(-\sqrt{F \cdot 2}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C} \cdot \frac{\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}}{\mathsf{fma}\left(-B, B, C \cdot \left(A \cdot 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 59.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := {B\_m}^{2} - t\_1\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_2}\\ t_4 := \left(C + -0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right) + C\\ t_5 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{t\_4 \cdot 2} \cdot \left(-\sqrt{F \cdot t\_0}\right)}{t\_2}\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-194}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(B\_m, A - C\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot t\_0\right)}}{\mathsf{fma}\left(-B\_m, B\_m, t\_1\right)}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\sqrt{t\_4 \cdot t\_5} \cdot \left(-\sqrt{F \cdot 2}\right)}{t\_2}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C} \cdot \frac{\sqrt{\left(F \cdot 2\right) \cdot t\_5}}{\mathsf{fma}\left(-B\_m, B\_m, C \cdot \left(A \cdot 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (* (* 4.0 A) C))
        (t_2 (- (pow B_m 2.0) t_1))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* t_2 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_2)))
        (t_4 (+ (+ C (* -0.5 (/ (* B_m B_m) A))) C))
        (t_5 (fma (* -4.0 C) A (* B_m B_m))))
   (if (<= t_3 (- INFINITY))
     (/ (* (sqrt (* t_4 2.0)) (- (sqrt (* F t_0)))) t_2)
     (if (<= t_3 -1e-194)
       (/
        (sqrt (* (+ (+ (hypot B_m (- A C)) A) C) (* (* 2.0 F) t_0)))
        (fma (- B_m) B_m t_1))
       (if (<= t_3 0.0)
         (/ (* (sqrt (* t_4 t_5)) (- (sqrt (* F 2.0)))) t_2)
         (if (<= t_3 INFINITY)
           (*
            (sqrt (+ (+ (hypot (- A C) B_m) A) C))
            (/ (sqrt (* (* F 2.0) t_5)) (fma (- B_m) B_m (* C (* A 4.0)))))
           (* (- (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) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = (4.0 * A) * C;
	double t_2 = pow(B_m, 2.0) - t_1;
	double t_3 = sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_2;
	double t_4 = (C + (-0.5 * ((B_m * B_m) / A))) + C;
	double t_5 = fma((-4.0 * C), A, (B_m * B_m));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (sqrt((t_4 * 2.0)) * -sqrt((F * t_0))) / t_2;
	} else if (t_3 <= -1e-194) {
		tmp = sqrt((((hypot(B_m, (A - C)) + A) + C) * ((2.0 * F) * t_0))) / fma(-B_m, B_m, t_1);
	} else if (t_3 <= 0.0) {
		tmp = (sqrt((t_4 * t_5)) * -sqrt((F * 2.0))) / t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((hypot((A - C), B_m) + A) + C)) * (sqrt(((F * 2.0) * t_5)) / fma(-B_m, B_m, (C * (A * 4.0))));
	} else {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64((B_m ^ 2.0) - t_1)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_2))
	t_4 = Float64(Float64(C + Float64(-0.5 * Float64(Float64(B_m * B_m) / A))) + C)
	t_5 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(t_4 * 2.0)) * Float64(-sqrt(Float64(F * t_0)))) / t_2);
	elseif (t_3 <= -1e-194)
		tmp = Float64(sqrt(Float64(Float64(Float64(hypot(B_m, Float64(A - C)) + A) + C) * Float64(Float64(2.0 * F) * t_0))) / fma(Float64(-B_m), B_m, t_1));
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(t_4 * t_5)) * Float64(-sqrt(Float64(F * 2.0)))) / t_2);
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)) * Float64(sqrt(Float64(Float64(F * 2.0) * t_5)) / fma(Float64(-B_m), B_m, Float64(C * Float64(A * 4.0)))));
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / 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[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$2)), $MachinePrecision]}, Block[{t$95$4 = N[(N[(C + N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + C), $MachinePrecision]}, Block[{t$95$5 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[Sqrt[N[(t$95$4 * 2.0), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(F * t$95$0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, -1e-194], N[(N[Sqrt[N[(N[(N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[((-B$95$m) * B$95$m + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(N[Sqrt[N[(t$95$4 * t$95$5), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$5), $MachinePrecision]], $MachinePrecision] / N[((-B$95$m) * B$95$m + N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / 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))) < -inf.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
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 27.9

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

   7. Applied rewrites 27.9%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.00000000000000002e-194

   1. Initial program 99.6%

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

   4. Applied rewrites 99.5%

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

   if -1.00000000000000002e-194 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 3.7%

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

   6. Applied rewrites 17.8%

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

   7. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 32.9

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

   9. Applied rewrites 32.9%

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

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

   1. Initial program 33.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 83.9%

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

   6. Applied rewrites 83.9%

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

   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 19.4

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

   5. Applied rewrites 19.4%

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

   7. Applied rewrites 19.4%

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

   9. Applied rewrites 23.1%

      \[\leadsto -\sqrt{F} \cdot \sqrt{\frac{2}{B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\left(\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right) + C\right) \cdot 2} \cdot \left(-\sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-194}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(B, A - C\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(-B, B, \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 0:\\ \;\;\;\;\frac{\sqrt{\left(\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right) + C\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \left(-\sqrt{F \cdot 2}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C} \cdot \frac{\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}}{\mathsf{fma}\left(-B, B, C \cdot \left(A \cdot 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ t_1 := \left(C + -0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right) + C\\ t_2 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_3 := \left(4 \cdot A\right) \cdot C\\ t_4 := \mathsf{fma}\left(-B\_m, B\_m, t\_3\right)\\ t_5 := {B\_m}^{2} - t\_3\\ t_6 := \frac{\sqrt{\left(2 \cdot \left(t\_5 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_5}\\ \mathbf{if}\;t\_6 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot 2} \cdot \left(-\sqrt{F \cdot t\_2}\right)}{t\_5}\\ \mathbf{elif}\;t\_6 \leq -1 \cdot 10^{-194}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(B\_m, A - C\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot t\_2\right)}}{t\_4}\\ \mathbf{elif}\;t\_6 \leq 10^{+177}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(\left(F \cdot t\_1\right) \cdot 2\right)}}{t\_4}\\ \mathbf{elif}\;t\_6 \leq \infty:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C} \cdot \frac{\sqrt{\left(F \cdot 2\right) \cdot t\_0}}{\mathsf{fma}\left(-B\_m, B\_m, C \cdot \left(A \cdot 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B_m B_m)))
        (t_1 (+ (+ C (* -0.5 (/ (* B_m B_m) A))) C))
        (t_2 (fma -4.0 (* C A) (* B_m B_m)))
        (t_3 (* (* 4.0 A) C))
        (t_4 (fma (- B_m) B_m t_3))
        (t_5 (- (pow B_m 2.0) t_3))
        (t_6
         (/
          (sqrt
           (*
            (* 2.0 (* t_5 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_5))))
   (if (<= t_6 (- INFINITY))
     (/ (* (sqrt (* t_1 2.0)) (- (sqrt (* F t_2)))) t_5)
     (if (<= t_6 -1e-194)
       (/ (sqrt (* (+ (+ (hypot B_m (- A C)) A) C) (* (* 2.0 F) t_2))) t_4)
       (if (<= t_6 1e+177)
         (/ (sqrt (* t_0 (* (* F t_1) 2.0))) t_4)
         (if (<= t_6 INFINITY)
           (*
            (sqrt (+ (+ (hypot (- A C) B_m) A) C))
            (/ (sqrt (* (* F 2.0) t_0)) (fma (- B_m) B_m (* C (* A 4.0)))))
           (* (- (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) {
	double t_0 = fma((-4.0 * C), A, (B_m * B_m));
	double t_1 = (C + (-0.5 * ((B_m * B_m) / A))) + C;
	double t_2 = fma(-4.0, (C * A), (B_m * B_m));
	double t_3 = (4.0 * A) * C;
	double t_4 = fma(-B_m, B_m, t_3);
	double t_5 = pow(B_m, 2.0) - t_3;
	double t_6 = sqrt(((2.0 * (t_5 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_5;
	double tmp;
	if (t_6 <= -((double) INFINITY)) {
		tmp = (sqrt((t_1 * 2.0)) * -sqrt((F * t_2))) / t_5;
	} else if (t_6 <= -1e-194) {
		tmp = sqrt((((hypot(B_m, (A - C)) + A) + C) * ((2.0 * F) * t_2))) / t_4;
	} else if (t_6 <= 1e+177) {
		tmp = sqrt((t_0 * ((F * t_1) * 2.0))) / t_4;
	} else if (t_6 <= ((double) INFINITY)) {
		tmp = sqrt(((hypot((A - C), B_m) + A) + C)) * (sqrt(((F * 2.0) * t_0)) / fma(-B_m, B_m, (C * (A * 4.0))));
	} else {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	t_1 = Float64(Float64(C + Float64(-0.5 * Float64(Float64(B_m * B_m) / A))) + C)
	t_2 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_3 = Float64(Float64(4.0 * A) * C)
	t_4 = fma(Float64(-B_m), B_m, t_3)
	t_5 = Float64((B_m ^ 2.0) - t_3)
	t_6 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_5 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_5))
	tmp = 0.0
	if (t_6 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(t_1 * 2.0)) * Float64(-sqrt(Float64(F * t_2)))) / t_5);
	elseif (t_6 <= -1e-194)
		tmp = Float64(sqrt(Float64(Float64(Float64(hypot(B_m, Float64(A - C)) + A) + C) * Float64(Float64(2.0 * F) * t_2))) / t_4);
	elseif (t_6 <= 1e+177)
		tmp = Float64(sqrt(Float64(t_0 * Float64(Float64(F * t_1) * 2.0))) / t_4);
	elseif (t_6 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)) * Float64(sqrt(Float64(Float64(F * 2.0) * t_0)) / fma(Float64(-B_m), B_m, Float64(C * Float64(A * 4.0)))));
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(C + N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + C), $MachinePrecision]}, Block[{t$95$2 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$4 = N[((-B$95$m) * B$95$m + t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$3), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$5 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$5)), $MachinePrecision]}, If[LessEqual[t$95$6, (-Infinity)], N[(N[(N[Sqrt[N[(t$95$1 * 2.0), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(F * t$95$2), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$5), $MachinePrecision], If[LessEqual[t$95$6, -1e-194], N[(N[Sqrt[N[(N[(N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$6, 1e+177], N[(N[Sqrt[N[(t$95$0 * N[(N[(F * t$95$1), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$6, Infinity], N[(N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / N[((-B$95$m) * B$95$m + N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / 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))) < -inf.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
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 27.9

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

   7. Applied rewrites 27.9%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.00000000000000002e-194

   1. Initial program 99.6%

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

   4. Applied rewrites 99.5%

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

   if -1.00000000000000002e-194 < (/.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))) < 1e177

   1. Initial program 19.5%

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

   4. Applied rewrites 22.2%

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

   6. Applied rewrites 22.2%

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

   7. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 29.0

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

   9. Applied rewrites 29.0%

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

   if 1e177 < (/.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 3.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 82.2%

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

   6. Applied rewrites 82.5%

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

   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 19.4

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

   5. Applied rewrites 19.4%

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

   7. Applied rewrites 19.4%

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

   9. Applied rewrites 23.1%

      \[\leadsto -\sqrt{F} \cdot \sqrt{\frac{2}{B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\left(\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right) + C\right) \cdot 2} \cdot \left(-\sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-194}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(B, A - C\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(-B, B, \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 10^{+177}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(\left(F \cdot \left(\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right) + C\right)\right) \cdot 2\right)}}{\mathsf{fma}\left(-B, B, \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C} \cdot \frac{\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}}{\mathsf{fma}\left(-B, B, C \cdot \left(A \cdot 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \mathsf{fma}\left(-B\_m, B\_m, t\_0\right)\\ t_2 := {B\_m}^{2} - t\_0\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_2}\\ t_4 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_5 := \frac{\sqrt{\left(2 \cdot C\right) \cdot 2} \cdot \left(-\sqrt{F \cdot t\_4}\right)}{t\_2}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-194}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(B\_m, A - C\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot t\_4\right)}}{t\_1}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right) \cdot \left(\left(F \cdot \left(\left(C + -0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right) + C\right)\right) \cdot 2\right)}}{t\_1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (fma (- B_m) B_m t_0))
        (t_2 (- (pow B_m 2.0) t_0))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* t_2 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_2)))
        (t_4 (fma -4.0 (* C A) (* B_m B_m)))
        (t_5 (/ (* (sqrt (* (* 2.0 C) 2.0)) (- (sqrt (* F t_4)))) t_2)))
   (if (<= t_3 (- INFINITY))
     t_5
     (if (<= t_3 -1e-194)
       (/ (sqrt (* (+ (+ (hypot B_m (- A C)) A) C) (* (* 2.0 F) t_4))) t_1)
       (if (<= t_3 5e-16)
         (/
          (sqrt
           (*
            (fma (* -4.0 C) A (* B_m B_m))
            (* (* F (+ (+ C (* -0.5 (/ (* B_m B_m) A))) C)) 2.0)))
          t_1)
         (if (<= t_3 INFINITY) t_5 (* (- (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) {
	double t_0 = (4.0 * A) * C;
	double t_1 = fma(-B_m, B_m, t_0);
	double t_2 = pow(B_m, 2.0) - t_0;
	double t_3 = sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_2;
	double t_4 = fma(-4.0, (C * A), (B_m * B_m));
	double t_5 = (sqrt(((2.0 * C) * 2.0)) * -sqrt((F * t_4))) / t_2;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_5;
	} else if (t_3 <= -1e-194) {
		tmp = sqrt((((hypot(B_m, (A - C)) + A) + C) * ((2.0 * F) * t_4))) / t_1;
	} else if (t_3 <= 5e-16) {
		tmp = sqrt((fma((-4.0 * C), A, (B_m * B_m)) * ((F * ((C + (-0.5 * ((B_m * B_m) / A))) + C)) * 2.0))) / t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_5;
	} else {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = fma(Float64(-B_m), B_m, t_0)
	t_2 = Float64((B_m ^ 2.0) - t_0)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_2))
	t_4 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_5 = Float64(Float64(sqrt(Float64(Float64(2.0 * C) * 2.0)) * Float64(-sqrt(Float64(F * t_4)))) / t_2)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_5;
	elseif (t_3 <= -1e-194)
		tmp = Float64(sqrt(Float64(Float64(Float64(hypot(B_m, Float64(A - C)) + A) + C) * Float64(Float64(2.0 * F) * t_4))) / t_1);
	elseif (t_3 <= 5e-16)
		tmp = Float64(sqrt(Float64(fma(Float64(-4.0 * C), A, Float64(B_m * B_m)) * Float64(Float64(F * Float64(Float64(C + Float64(-0.5 * Float64(Float64(B_m * B_m) / A))) + C)) * 2.0))) / t_1);
	elseif (t_3 <= Inf)
		tmp = t_5;
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[((-B$95$m) * B$95$m + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$2)), $MachinePrecision]}, Block[{t$95$4 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[Sqrt[N[(N[(2.0 * C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(F * t$95$4), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$5, If[LessEqual[t$95$3, -1e-194], N[(N[Sqrt[N[(N[(N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 5e-16], N[(N[Sqrt[N[(N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[(C + N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$5, N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0 or 5.0000000000000004e-16 < (/.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 9.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 48.2%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 27.7

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

   7. Applied rewrites 27.7%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.00000000000000002e-194

   1. Initial program 99.6%

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

   4. Applied rewrites 99.5%

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

   if -1.00000000000000002e-194 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 5.0000000000000004e-16

   1. Initial program 9.9%

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

   4. Applied rewrites 13.0%

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

   6. Applied rewrites 13.0%

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

   7. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 29.5

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

   9. Applied rewrites 29.5%

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

   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 19.4

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

   5. Applied rewrites 19.4%

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

   7. Applied rewrites 19.4%

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

   9. Applied rewrites 23.1%

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

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot 2} \cdot \left(-\sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-194}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(B, A - C\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(-B, B, \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(\left(F \cdot \left(\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right) + C\right)\right) \cdot 2\right)}}{\mathsf{fma}\left(-B, B, \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot 2} \cdot \left(-\sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := -t\_0\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_1}\\ t_3 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-194}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right) \cdot 2} \cdot \sqrt{F \cdot \left(\left(\mathsf{hypot}\left(B\_m, A - C\right) + A\right) + C\right)}}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\sqrt{\left(\left(C + -0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right) + C\right) \cdot t\_3} \cdot \left(-\sqrt{F \cdot 2}\right)}{t\_0}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C} \cdot \frac{\sqrt{\left(F \cdot 2\right) \cdot t\_3}}{\mathsf{fma}\left(-B\_m, B\_m, C \cdot \left(A \cdot 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_1 (- t_0))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* t_0 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          t_1))
        (t_3 (fma (* -4.0 C) A (* B_m B_m))))
   (if (<= t_2 -1e-194)
     (/
      (*
       (sqrt (* (fma -4.0 (* C A) (* B_m B_m)) 2.0))
       (sqrt (* F (+ (+ (hypot B_m (- A C)) A) C))))
      t_1)
     (if (<= t_2 0.0)
       (/
        (*
         (sqrt (* (+ (+ C (* -0.5 (/ (* B_m B_m) A))) C) t_3))
         (- (sqrt (* F 2.0))))
        t_0)
       (if (<= t_2 INFINITY)
         (*
          (sqrt (+ (+ (hypot (- A C) B_m) A) C))
          (/ (sqrt (* (* F 2.0) t_3)) (fma (- B_m) B_m (* C (* A 4.0)))))
         (* (- (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) {
	double t_0 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = -t_0;
	double t_2 = sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_1;
	double t_3 = fma((-4.0 * C), A, (B_m * B_m));
	double tmp;
	if (t_2 <= -1e-194) {
		tmp = (sqrt((fma(-4.0, (C * A), (B_m * B_m)) * 2.0)) * sqrt((F * ((hypot(B_m, (A - C)) + A) + C)))) / t_1;
	} else if (t_2 <= 0.0) {
		tmp = (sqrt((((C + (-0.5 * ((B_m * B_m) / A))) + C) * t_3)) * -sqrt((F * 2.0))) / t_0;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((hypot((A - C), B_m) + A) + C)) * (sqrt(((F * 2.0) * t_3)) / fma(-B_m, B_m, (C * (A * 4.0))));
	} else {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = Float64(-t_0)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / t_1)
	t_3 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	tmp = 0.0
	if (t_2 <= -1e-194)
		tmp = Float64(Float64(sqrt(Float64(fma(-4.0, Float64(C * A), Float64(B_m * B_m)) * 2.0)) * sqrt(Float64(F * Float64(Float64(hypot(B_m, Float64(A - C)) + A) + C)))) / t_1);
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(C + Float64(-0.5 * Float64(Float64(B_m * B_m) / A))) + C) * t_3)) * Float64(-sqrt(Float64(F * 2.0)))) / t_0);
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)) * Float64(sqrt(Float64(Float64(F * 2.0) * t_3)) / fma(Float64(-B_m), B_m, Float64(C * Float64(A * 4.0)))));
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = 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$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-194], N[(N[(N[Sqrt[N[(N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * N[(N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(N[Sqrt[N[(N[(N[(C + N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + C), $MachinePrecision] * t$95$3), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision]], $MachinePrecision] / N[((-B$95$m) * B$95$m + N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.00000000000000002e-194

   1. Initial program 43.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. sqrt-prod N/A

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

      8. pow1/2 N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 61.9%

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

   if -1.00000000000000002e-194 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 3.7%

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

   6. Applied rewrites 17.8%

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

   7. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 32.9

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

   9. Applied rewrites 32.9%

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

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

   1. Initial program 33.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 83.9%

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

   6. Applied rewrites 83.9%

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

   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 19.4

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

   5. Applied rewrites 19.4%

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

   7. Applied rewrites 19.4%

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

   9. Applied rewrites 23.1%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-194}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2} \cdot \sqrt{F \cdot \left(\left(\mathsf{hypot}\left(B, A - C\right) + A\right) + C\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 0:\\ \;\;\;\;\frac{\sqrt{\left(\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right) + C\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \left(-\sqrt{F \cdot 2}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C} \cdot \frac{\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}}{\mathsf{fma}\left(-B, B, C \cdot \left(A \cdot 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(\left(F \cdot \left(\left(C + -0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right) + C\right)\right) \cdot 2\right)}}{\mathsf{fma}\left(-B\_m, B\_m, \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+71}:\\ \;\;\;\;\left(-\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C\right) \cdot 2}\right) \cdot \frac{\sqrt{F \cdot t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B_m B_m))))
   (if (<= (pow B_m 2.0) 2e-95)
     (/
      (sqrt (* t_0 (* (* F (+ (+ C (* -0.5 (/ (* B_m B_m) A))) C)) 2.0)))
      (fma (- B_m) B_m (* (* 4.0 A) C)))
     (if (<= (pow B_m 2.0) 4e+71)
       (*
        (- (sqrt (* (+ (+ (hypot (- A C) B_m) A) C) 2.0)))
        (/ (sqrt (* F t_0)) t_0))
       (* (- (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) {
	double t_0 = fma((-4.0 * C), A, (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-95) {
		tmp = sqrt((t_0 * ((F * ((C + (-0.5 * ((B_m * B_m) / A))) + C)) * 2.0))) / fma(-B_m, B_m, ((4.0 * A) * C));
	} else if (pow(B_m, 2.0) <= 4e+71) {
		tmp = -sqrt((((hypot((A - C), B_m) + A) + C) * 2.0)) * (sqrt((F * t_0)) / t_0);
	} else {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-95)
		tmp = Float64(sqrt(Float64(t_0 * Float64(Float64(F * Float64(Float64(C + Float64(-0.5 * Float64(Float64(B_m * B_m) / A))) + C)) * 2.0))) / fma(Float64(-B_m), B_m, Float64(Float64(4.0 * A) * C)));
	elseif ((B_m ^ 2.0) <= 4e+71)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C) * 2.0))) * Float64(sqrt(Float64(F * t_0)) / t_0));
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-95], N[(N[Sqrt[N[(t$95$0 * N[(N[(F * N[(N[(C + N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[((-B$95$m) * B$95$m + N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+71], N[((-N[Sqrt[N[(N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]) * N[(N[Sqrt[N[(F * t$95$0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.99999999999999998e-95

   1. Initial program 18.2%

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

   4. Applied rewrites 26.7%

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

   6. Applied rewrites 26.1%

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

   7. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 24.7

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

   9. Applied rewrites 24.7%

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

   if 1.99999999999999998e-95 < (pow.f64 B #s(literal 2 binary64)) < 4.0000000000000002e71

   1. Initial program 46.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 59.7%

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

   6. Applied rewrites 59.7%

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

   if 4.0000000000000002e71 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 31.8

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

   5. Applied rewrites 31.8%

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

   7. Applied rewrites 31.9%

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

   9. Applied rewrites 36.0%

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

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(\left(F \cdot \left(\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right) + C\right)\right) \cdot 2\right)}}{\mathsf{fma}\left(-B, B, \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+71}:\\ \;\;\;\;\left(-\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot 2}\right) \cdot \frac{\sqrt{F \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right) \cdot \left(\left(F \cdot \left(\left(C + -0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right) + C\right)\right) \cdot 2\right)}}{\mathsf{fma}\left(-B\_m, B\_m, \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+71}:\\ \;\;\;\;\sqrt{\left(2 \cdot F\right) \cdot t\_0} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(B\_m, A - C\right) + A\right) + C}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m))))
   (if (<= (pow B_m 2.0) 2e-95)
     (/
      (sqrt
       (*
        (fma (* -4.0 C) A (* B_m B_m))
        (* (* F (+ (+ C (* -0.5 (/ (* B_m B_m) A))) C)) 2.0)))
      (fma (- B_m) B_m (* (* 4.0 A) C)))
     (if (<= (pow B_m 2.0) 4e+71)
       (*
        (sqrt (* (* 2.0 F) t_0))
        (/ (sqrt (+ (+ (hypot B_m (- A C)) A) C)) (- t_0)))
       (* (- (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) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-95) {
		tmp = sqrt((fma((-4.0 * C), A, (B_m * B_m)) * ((F * ((C + (-0.5 * ((B_m * B_m) / A))) + C)) * 2.0))) / fma(-B_m, B_m, ((4.0 * A) * C));
	} else if (pow(B_m, 2.0) <= 4e+71) {
		tmp = sqrt(((2.0 * F) * t_0)) * (sqrt(((hypot(B_m, (A - C)) + A) + C)) / -t_0);
	} else {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-95)
		tmp = Float64(sqrt(Float64(fma(Float64(-4.0 * C), A, Float64(B_m * B_m)) * Float64(Float64(F * Float64(Float64(C + Float64(-0.5 * Float64(Float64(B_m * B_m) / A))) + C)) * 2.0))) / fma(Float64(-B_m), B_m, Float64(Float64(4.0 * A) * C)));
	elseif ((B_m ^ 2.0) <= 4e+71)
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * t_0)) * Float64(sqrt(Float64(Float64(hypot(B_m, Float64(A - C)) + A) + C)) / Float64(-t_0)));
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / 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], 2e-95], N[(N[Sqrt[N[(N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[(C + N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[((-B$95$m) * B$95$m + N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+71], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.99999999999999998e-95

   1. Initial program 18.2%

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

   4. Applied rewrites 26.7%

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

   6. Applied rewrites 26.1%

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

   7. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 24.7

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

   9. Applied rewrites 24.7%

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

   if 1.99999999999999998e-95 < (pow.f64 B #s(literal 2 binary64)) < 4.0000000000000002e71

   1. Initial program 46.0%

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

   4. Applied rewrites 59.4%

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

   if 4.0000000000000002e71 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 31.8

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

   5. Applied rewrites 31.8%

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

   7. Applied rewrites 31.9%

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

   9. Applied rewrites 36.0%

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

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(\left(F \cdot \left(\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right) + C\right)\right) \cdot 2\right)}}{\mathsf{fma}\left(-B, B, \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+71}:\\ \;\;\;\;\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(B, A - C\right) + A\right) + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-B\_m, B\_m, \left(4 \cdot A\right) \cdot C\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right) \cdot \left(\left(F \cdot \left(\left(C + -0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right) + C\right)\right) \cdot 2\right)}}{t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(B\_m, A - C\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (- B_m) B_m (* (* 4.0 A) C))))
   (if (<= (pow B_m 2.0) 2e-95)
     (/
      (sqrt
       (*
        (fma (* -4.0 C) A (* B_m B_m))
        (* (* F (+ (+ C (* -0.5 (/ (* B_m B_m) A))) C)) 2.0)))
      t_0)
     (if (<= (pow B_m 2.0) 2e+51)
       (/
        (sqrt
         (*
          (+ (+ (hypot B_m (- A C)) A) C)
          (* (* 2.0 F) (fma -4.0 (* C A) (* B_m B_m)))))
        t_0)
       (* (- (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) {
	double t_0 = fma(-B_m, B_m, ((4.0 * A) * C));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-95) {
		tmp = sqrt((fma((-4.0 * C), A, (B_m * B_m)) * ((F * ((C + (-0.5 * ((B_m * B_m) / A))) + C)) * 2.0))) / t_0;
	} else if (pow(B_m, 2.0) <= 2e+51) {
		tmp = sqrt((((hypot(B_m, (A - C)) + A) + C) * ((2.0 * F) * fma(-4.0, (C * A), (B_m * B_m))))) / t_0;
	} else {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

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

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.99999999999999998e-95

   1. Initial program 18.2%

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

   4. Applied rewrites 26.7%

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

   6. Applied rewrites 26.1%

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

   7. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 24.7

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

   9. Applied rewrites 24.7%

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

   if 1.99999999999999998e-95 < (pow.f64 B #s(literal 2 binary64)) < 2e51

   1. Initial program 50.9%

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

   4. Applied rewrites 55.0%

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

   if 2e51 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 13.9%

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

   3. Taylor expanded in 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 31.0

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

   5. Applied rewrites 31.0%

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

   7. Applied rewrites 31.1%

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

   9. Applied rewrites 35.1%

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

4. Final simplification 32.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(\left(F \cdot \left(\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right) + C\right)\right) \cdot 2\right)}}{\mathsf{fma}\left(-B, B, \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(B, A - C\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{\mathsf{fma}\left(-B, B, \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{+21}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\right)}}{\mathsf{fma}\left(-B\_m, B\_m, \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 2e+21)
   (/
    (sqrt (* (* 2.0 C) (* (* 2.0 F) (fma -4.0 (* C A) (* B_m B_m)))))
    (fma (- B_m) B_m (* (* 4.0 A) C)))
   (* (- (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) {
	double tmp;
	if (pow(B_m, 2.0) <= 2e+21) {
		tmp = sqrt(((2.0 * C) * ((2.0 * F) * fma(-4.0, (C * A), (B_m * B_m))))) / fma(-B_m, B_m, ((4.0 * A) * C));
	} else {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e+21)
		tmp = Float64(sqrt(Float64(Float64(2.0 * C) * Float64(Float64(2.0 * F) * fma(-4.0, Float64(C * A), Float64(B_m * B_m))))) / fma(Float64(-B_m), B_m, Float64(Float64(4.0 * A) * C)));
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / 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[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+21], N[(N[Sqrt[N[(N[(2.0 * C), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[((-B$95$m) * B$95$m + N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2e21

   1. Initial program 22.8%

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

   4. Applied rewrites 30.9%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 24.0

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

   7. Applied rewrites 24.0%

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

   if 2e21 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 15.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 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 31.2%

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

   9. Applied rewrites 35.0%

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

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+21}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{\mathsf{fma}\left(-B, B, \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}{\mathsf{fma}\left(-B\_m, B\_m, \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 2e-90)
   (/ (sqrt (* -16.0 (* A (* (* C C) F)))) (fma (- B_m) B_m (* (* 4.0 A) C)))
   (* (- (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) {
	double tmp;
	if (pow(B_m, 2.0) <= 2e-90) {
		tmp = sqrt((-16.0 * (A * ((C * C) * F)))) / fma(-B_m, B_m, ((4.0 * A) * C));
	} else {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-90)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(A * Float64(Float64(C * C) * F)))) / fma(Float64(-B_m), B_m, Float64(Float64(4.0 * A) * C)));
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / 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[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-90], N[(N[Sqrt[N[(-16.0 * N[(A * N[(N[(C * C), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[((-B$95$m) * B$95$m + N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.99999999999999999e-90

   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 26.5%

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

   5. Taylor expanded in C around 0

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

      1. lower-+.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 16.5

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

   7. Applied rewrites 16.5%

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

   8. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 14.3

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

   10. Applied rewrites 14.3%

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

   if 1.99999999999999999e-90 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 21.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 28.1

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

   5. Applied rewrites 28.1%

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

   7. Applied rewrites 28.2%

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

   9. Applied rewrites 31.4%

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

4. Final simplification 23.2%

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

Alternative 11: 35.8% 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 19.7%

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

3. Taylor expanded in 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 16.3

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

5. Applied rewrites 16.3%

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

7. Applied rewrites 16.3%

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

9. Applied rewrites 18.4%

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

10. Final simplification 18.4%

    \[\leadsto \left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}} \]
11. Add Preprocessing

Alternative 12: 27.2% accurate, 16.9× speedup

Math

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

FPCore

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

C

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

Fortran

B_m = 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(F * Float64(2.0 / B_m))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.7%

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

3. Taylor expanded in 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 16.3

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

5. Applied rewrites 16.3%

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

7. Applied rewrites 16.3%

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

9. Applied rewrites 16.3%

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

Reproduce

herbie shell --seed 2024326 
(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))))