ABCF->ab-angle b

Percentage Accurate: 18.7% → 52.1%

Time: 15.2s

Alternatives: 18

Speedup: 7.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.7% 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: 52.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_2 = sqrt(((2.0 * (t_1 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = sqrt((F * (((C + A) - hypot((A - C), B_m)) / t_0))) * -sqrt(2.0);
	} else if (t_2 <= -1e-194) {
		tmp = t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((fma((A * C), -4.0, (B_m * B_m)) * ((F * 2.0) * (fma(((B_m * B_m) / C), -0.5, A) + A)))) / -t_0;
	} else {
		tmp = (sqrt(2.0) / -B_m) * sqrt(((A - hypot(B_m, A)) * F));
	}
	return tmp;
}

Julia

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

Wolfram

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

   1. Initial program 3.1%

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

   3. Taylor expanded in F around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 55.6%

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

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

   1. Initial program 99.1%

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

   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))) < +inf.0

   1. Initial program 16.7%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 22.6%

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

   5. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 30.9

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

   7. Applied rewrites 30.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

   9. Applied rewrites 36.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 17.2

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

   5. Applied rewrites 17.2%

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

4. Final simplification 39.6%

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

Alternative 2: 52.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_0;
	double t_2 = fma(-4.0, (C * A), (B_m * B_m));
	double t_3 = -t_2;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = sqrt((F * (((C + A) - hypot((A - C), B_m)) / t_2))) * -sqrt(2.0);
	} else if (t_1 <= -1e-194) {
		tmp = sqrt((((C + A) - hypot(B_m, (A - C))) * ((2.0 * F) * t_2))) / t_3;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((fma((A * C), -4.0, (B_m * B_m)) * ((F * 2.0) * (fma(((B_m * B_m) / C), -0.5, A) + A)))) / t_3;
	} else {
		tmp = (sqrt(2.0) / -B_m) * sqrt(((A - hypot(B_m, A)) * F));
	}
	return tmp;
}

Julia

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

Wolfram

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

   1. Initial program 3.1%

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

   3. Taylor expanded in F around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 55.6%

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

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

   1. Initial program 99.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.1%

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

   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))) < +inf.0

   1. Initial program 16.7%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 22.6%

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

   5. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 30.9

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

   7. Applied rewrites 30.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

   9. Applied rewrites 36.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 17.2

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

   5. Applied rewrites 17.2%

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

4. Final simplification 39.6%

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

Alternative 3: 33.5% accurate, 0.4× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* C A) -4.0 (* B_m B_m)))
        (t_1 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* t_1 F))
            (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_1)))
        (t_3 (/ (* B_m B_m) C))
        (t_4 (fma -4.0 (* C A) (* B_m B_m)))
        (t_5 (- t_4)))
   (if (<= t_2 (- INFINITY))
     (* (sqrt t_0) (/ (sqrt (* (+ (fma -0.5 t_3 A) A) (* 2.0 F))) (- t_0)))
     (if (<= t_2 -1e-194)
       (/
        (sqrt (* (* B_m (+ (/ A B_m) (- (/ C B_m) 1.0))) (* (* 2.0 F) t_4)))
        t_5)
       (/
        (sqrt
         (*
          (fma (* A C) -4.0 (* B_m B_m))
          (* (* F 2.0) (+ (fma t_3 -0.5 A) A))))
        t_5)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((C * A), -4.0, (B_m * B_m));
	double t_1 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_2 = sqrt(((2.0 * (t_1 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_1;
	double t_3 = (B_m * B_m) / C;
	double t_4 = fma(-4.0, (C * A), (B_m * B_m));
	double t_5 = -t_4;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = sqrt(t_0) * (sqrt(((fma(-0.5, t_3, A) + A) * (2.0 * F))) / -t_0);
	} else if (t_2 <= -1e-194) {
		tmp = sqrt(((B_m * ((A / B_m) + ((C / B_m) - 1.0))) * ((2.0 * F) * t_4))) / t_5;
	} else {
		tmp = sqrt((fma((A * C), -4.0, (B_m * B_m)) * ((F * 2.0) * (fma(t_3, -0.5, A) + A)))) / t_5;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_1 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_1))
	t_3 = Float64(Float64(B_m * B_m) / C)
	t_4 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_5 = Float64(-t_4)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(sqrt(t_0) * Float64(sqrt(Float64(Float64(fma(-0.5, t_3, A) + A) * Float64(2.0 * F))) / Float64(-t_0)));
	elseif (t_2 <= -1e-194)
		tmp = Float64(sqrt(Float64(Float64(B_m * Float64(Float64(A / B_m) + Float64(Float64(C / B_m) - 1.0))) * Float64(Float64(2.0 * F) * t_4))) / t_5);
	else
		tmp = Float64(sqrt(Float64(fma(Float64(A * C), -4.0, Float64(B_m * B_m)) * Float64(Float64(F * 2.0) * Float64(fma(t_3, -0.5, A) + A)))) / t_5);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$1 * 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[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision]}, Block[{t$95$4 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = (-t$95$4)}, If[LessEqual[t$95$2, (-Infinity)], N[(N[Sqrt[t$95$0], $MachinePrecision] * N[(N[Sqrt[N[(N[(N[(-0.5 * t$95$3 + A), $MachinePrecision] + A), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-194], N[(N[Sqrt[N[(N[(B$95$m * N[(N[(A / B$95$m), $MachinePrecision] + N[(N[(C / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(F * 2.0), $MachinePrecision] * N[(N[(t$95$3 * -0.5 + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 3.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 14.4%

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

   5. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 18.2

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

   7. Applied rewrites 18.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

   9. Applied rewrites 18.3%

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

   11. Applied rewrites 27.2%

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

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

   1. Initial program 99.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 40.2

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

   7. Applied rewrites 40.2%

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

   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)))

   1. Initial program 5.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 7.0%

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

   5. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 11.2

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

   7. Applied rewrites 11.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

   9. Applied rewrites 12.5%

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\right)\right)}}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 19.2%

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

Alternative 4: 31.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(\frac{B\_m \cdot B\_m}{C}, -0.5, A\right) + A\\ t_2 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_3 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(t\_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_3}\\ t_5 := -t\_2\\ \mathbf{if}\;t\_4 \leq -1 \cdot 10^{+277}:\\ \;\;\;\;\frac{-1}{t\_0} \cdot \sqrt{\left(\left(t\_0 \cdot F\right) \cdot 2\right) \cdot t\_1}\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-194}:\\ \;\;\;\;\frac{\sqrt{\left(B\_m \cdot \left(\frac{A}{B\_m} + \left(\frac{C}{B\_m} - 1\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot t\_2\right)}}{t\_5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(\left(F \cdot 2\right) \cdot t\_1\right)}}{t\_5}\\ \end{array} \end{array} \]

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 (* A C) -4.0 (* B_m B_m)))
        (t_1 (+ (fma (/ (* B_m B_m) C) -0.5 A) A))
        (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)))
        (t_5 (- t_2)))
   (if (<= t_4 -1e+277)
     (* (/ -1.0 t_0) (sqrt (* (* (* t_0 F) 2.0) t_1)))
     (if (<= t_4 -1e-194)
       (/
        (sqrt (* (* B_m (+ (/ A B_m) (- (/ C B_m) 1.0))) (* (* 2.0 F) t_2)))
        t_5)
       (/ (sqrt (* t_0 (* (* F 2.0) t_1))) t_5)))))

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((A * C), -4.0, (B_m * B_m));
	double t_1 = fma(((B_m * B_m) / C), -0.5, A) + A;
	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 t_5 = -t_2;
	double tmp;
	if (t_4 <= -1e+277) {
		tmp = (-1.0 / t_0) * sqrt((((t_0 * F) * 2.0) * t_1));
	} else if (t_4 <= -1e-194) {
		tmp = sqrt(((B_m * ((A / B_m) + ((C / B_m) - 1.0))) * ((2.0 * F) * t_2))) / t_5;
	} else {
		tmp = sqrt((t_0 * ((F * 2.0) * t_1))) / t_5;
	}
	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(A * C), -4.0, Float64(B_m * B_m))
	t_1 = Float64(fma(Float64(Float64(B_m * B_m) / C), -0.5, A) + A)
	t_2 = fma(-4.0, Float64(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))
	t_5 = Float64(-t_2)
	tmp = 0.0
	if (t_4 <= -1e+277)
		tmp = Float64(Float64(-1.0 / t_0) * sqrt(Float64(Float64(Float64(t_0 * F) * 2.0) * t_1)));
	elseif (t_4 <= -1e-194)
		tmp = Float64(sqrt(Float64(Float64(B_m * Float64(Float64(A / B_m) + Float64(Float64(C / B_m) - 1.0))) * Float64(Float64(2.0 * F) * t_2))) / t_5);
	else
		tmp = Float64(sqrt(Float64(t_0 * Float64(Float64(F * 2.0) * t_1))) / t_5);
	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[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] * -0.5 + A), $MachinePrecision] + A), $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[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]}, Block[{t$95$5 = (-t$95$2)}, If[LessEqual[t$95$4, -1e+277], N[(N[(-1.0 / t$95$0), $MachinePrecision] * N[Sqrt[N[(N[(N[(t$95$0 * F), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -1e-194], N[(N[Sqrt[N[(N[(B$95$m * N[(N[(A / B$95$m), $MachinePrecision] + N[(N[(C / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision], N[(N[Sqrt[N[(t$95$0 * N[(N[(F * 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 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))) < -1e277

   1. Initial program 4.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 16.0%

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

   5. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 17.9

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

   7. Applied rewrites 17.9%

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

   9. Applied rewrites 17.9%

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

   if -1e277 < (/.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 99.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 40.8

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

   7. Applied rewrites 40.8%

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

   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)))

   1. Initial program 5.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 7.0%

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

   5. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 11.2

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

   7. Applied rewrites 11.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

   9. Applied rewrites 12.5%

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\right)\right)}}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 17.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{+277}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \sqrt{\left(\left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot F\right) \cdot 2\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-194}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot \left(\frac{A}{B} + \left(\frac{C}{B} - 1\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 29.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = (2.0 * F) * t_0;
	double t_2 = -t_0;
	double t_3 = (4.0 * A) * C;
	double t_4 = pow(B_m, 2.0) - t_3;
	double t_5 = sqrt(((2.0 * (t_4 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_4;
	double tmp;
	if (t_5 <= -1e-16) {
		tmp = sqrt(((A + A) * t_1)) / t_2;
	} else if (t_5 <= -1e-194) {
		tmp = sqrt((-B_m * t_1)) / t_2;
	} else {
		tmp = sqrt((fma((A * C), -4.0, (B_m * B_m)) * ((F * 2.0) * (fma(((B_m * B_m) / C), -0.5, A) + A)))) / t_3;
	}
	return tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.9999999999999998e-17

   1. Initial program 27.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 35.4%

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

   5. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 17.5

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

   7. Applied rewrites 17.5%

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

   if -9.9999999999999998e-17 < (/.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 99.8%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 46.7

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

   7. Applied rewrites 46.7%

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

   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)))

   1. Initial program 5.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 7.0%

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

   5. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 11.2

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

   7. Applied rewrites 11.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

   9. Applied rewrites 12.5%

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

   10. Taylor expanded in A around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 11.7

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

   12. Applied rewrites 11.7%

       \[\leadsto \frac{\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\right)\right)}}{\color{blue}{\left(4 \cdot A\right) \cdot C}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 15.4%

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

Alternative 6: 29.2% accurate, 0.5× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (* (* 2.0 F) t_0))
        (t_2 (- t_0))
        (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-16)
     (/ (sqrt (* (+ A A) t_1)) t_2)
     (if (<= t_4 -1e-194)
       (/ (sqrt (* (- B_m) t_1)) t_2)
       (/
        (sqrt (* (fma (* A C) -4.0 (* B_m B_m)) (* (* F 2.0) (+ A A))))
        t_2)))))

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 = (2.0 * F) * t_0;
	double t_2 = -t_0;
	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-16) {
		tmp = sqrt(((A + A) * t_1)) / t_2;
	} else if (t_4 <= -1e-194) {
		tmp = sqrt((-B_m * t_1)) / t_2;
	} else {
		tmp = sqrt((fma((A * C), -4.0, (B_m * B_m)) * ((F * 2.0) * (A + A)))) / t_2;
	}
	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(2.0 * F) * t_0)
	t_2 = Float64(-t_0)
	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-16)
		tmp = Float64(sqrt(Float64(Float64(A + A) * t_1)) / t_2);
	elseif (t_4 <= -1e-194)
		tmp = Float64(sqrt(Float64(Float64(-B_m) * t_1)) / t_2);
	else
		tmp = Float64(sqrt(Float64(fma(Float64(A * C), -4.0, Float64(B_m * B_m)) * Float64(Float64(F * 2.0) * Float64(A + A)))) / t_2);
	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[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = (-t$95$0)}, 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-16], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, -1e-194], N[(N[Sqrt[N[((-B$95$m) * t$95$1), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(F * 2.0), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.9999999999999998e-17

   1. Initial program 27.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 35.4%

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

   5. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 17.5

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

   7. Applied rewrites 17.5%

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

   if -9.9999999999999998e-17 < (/.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 99.8%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 46.7

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

   7. Applied rewrites 46.7%

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

   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)))

   1. Initial program 5.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 7.0%

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

   5. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 11.2

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

   7. Applied rewrites 11.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

   9. Applied rewrites 12.5%

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

   10. Taylor expanded in C around inf

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

       1. lower--.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lower-neg.f64 11.4

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

   12. Applied rewrites 11.4%

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

4. Final simplification 15.2%

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

Alternative 7: 29.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := \left(2 \cdot F\right) \cdot t\_0\\ t_2 := -t\_0\\ t_3 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(t\_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_3}\\ \mathbf{if}\;t\_4 \leq -1 \cdot 10^{-16}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot t\_1}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-194}:\\ \;\;\;\;\frac{\sqrt{\left(-B\_m\right) \cdot t\_1}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}{t\_2}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (* (* 2.0 F) t_0))
        (t_2 (- t_0))
        (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-16)
     (/ (sqrt (* (+ A A) t_1)) t_2)
     (if (<= t_4 -1e-194)
       (/ (sqrt (* (- B_m) t_1)) t_2)
       (/ (sqrt (* -8.0 (* A (* C (* F (+ A A)))))) t_2)))))

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 = (2.0 * F) * t_0;
	double t_2 = -t_0;
	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-16) {
		tmp = sqrt(((A + A) * t_1)) / t_2;
	} else if (t_4 <= -1e-194) {
		tmp = sqrt((-B_m * t_1)) / t_2;
	} else {
		tmp = sqrt((-8.0 * (A * (C * (F * (A + A)))))) / t_2;
	}
	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(2.0 * F) * t_0)
	t_2 = Float64(-t_0)
	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-16)
		tmp = Float64(sqrt(Float64(Float64(A + A) * t_1)) / t_2);
	elseif (t_4 <= -1e-194)
		tmp = Float64(sqrt(Float64(Float64(-B_m) * t_1)) / t_2);
	else
		tmp = Float64(sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A + A)))))) / t_2);
	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[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = (-t$95$0)}, 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-16], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, -1e-194], N[(N[Sqrt[N[((-B$95$m) * t$95$1), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.9999999999999998e-17

   1. Initial program 27.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 35.4%

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

   5. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 17.5

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

   7. Applied rewrites 17.5%

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

   if -9.9999999999999998e-17 < (/.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 99.8%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 46.7

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

   7. Applied rewrites 46.7%

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

   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)))

   1. Initial program 5.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 7.0%

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

   5. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-+.f64 10.6

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

   7. Applied rewrites 10.6%

      \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 14.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-16}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-194}:\\ \;\;\;\;\frac{\sqrt{\left(-B\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 26.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := -t\_0\\ t_2 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_2}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{+277}:\\ \;\;\;\;\frac{\sqrt{\left(-8 \cdot \left(A \cdot \left(C \cdot \left(A + A\right)\right)\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)}\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-194}:\\ \;\;\;\;\frac{\sqrt{\left(-B\_m\right) \cdot \left(\left(2 \cdot F\right) \cdot t\_0\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}{t\_1}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (- t_0))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* t_2 F))
            (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_2))))
   (if (<= t_3 -1e+277)
     (/
      (sqrt (* (* -8.0 (* A (* C (+ A A)))) F))
      (- (fma (* C A) -4.0 (* B_m B_m))))
     (if (<= t_3 -1e-194)
       (/ (sqrt (* (- B_m) (* (* 2.0 F) t_0))) t_1)
       (/ (sqrt (* -8.0 (* A (* C (* F (+ A A)))))) t_1)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = -t_0;
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	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 tmp;
	if (t_3 <= -1e+277) {
		tmp = sqrt(((-8.0 * (A * (C * (A + A)))) * F)) / -fma((C * A), -4.0, (B_m * B_m));
	} else if (t_3 <= -1e-194) {
		tmp = sqrt((-B_m * ((2.0 * F) * t_0))) / t_1;
	} else {
		tmp = sqrt((-8.0 * (A * (C * (F * (A + A)))))) / t_1;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64(-t_0)
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	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))
	tmp = 0.0
	if (t_3 <= -1e+277)
		tmp = Float64(sqrt(Float64(Float64(-8.0 * Float64(A * Float64(C * Float64(A + A)))) * F)) / Float64(-fma(Float64(C * A), -4.0, Float64(B_m * B_m))));
	elseif (t_3 <= -1e-194)
		tmp = Float64(sqrt(Float64(Float64(-B_m) * Float64(Float64(2.0 * F) * t_0))) / t_1);
	else
		tmp = Float64(sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A + A)))))) / t_1);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $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]}, If[LessEqual[t$95$3, -1e+277], N[(N[Sqrt[N[(N[(-8.0 * N[(A * N[(C * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] / (-N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$3, -1e-194], N[(N[Sqrt[N[((-B$95$m) * N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 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))) < -1e277

   1. Initial program 4.9%

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

   3. Taylor expanded in A around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 16.1

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

   5. Applied rewrites 16.1%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. lift--.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

   7. Applied rewrites 12.7%

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

   8. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 14.6

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

   10. Applied rewrites 14.6%

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

   if -1e277 < (/.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 99.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 35.1

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

   7. Applied rewrites 35.1%

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

   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)))

   1. Initial program 5.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 7.0%

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

   5. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-+.f64 10.6

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

   7. Applied rewrites 10.6%

      \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 14.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{+277}:\\ \;\;\;\;\frac{\sqrt{\left(-8 \cdot \left(A \cdot \left(C \cdot \left(A + A\right)\right)\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-194}:\\ \;\;\;\;\frac{\sqrt{\left(-B\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-96}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(A + A\right)\right)}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+208}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\_m\right)}{t\_0}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{\left(A - \mathsf{hypot}\left(B\_m, A\right)\right) \cdot F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m))))
   (if (<= (pow B_m 2.0) 2e-96)
     (/
      (sqrt (* (fma (* A C) -4.0 (* B_m B_m)) (* (* F 2.0) (+ A A))))
      (- t_0))
     (if (<= (pow B_m 2.0) 5e+208)
       (* (sqrt (* F (/ (- (+ C A) (hypot (- A C) B_m)) t_0))) (- (sqrt 2.0)))
       (* (/ (sqrt 2.0) (- B_m)) (sqrt (* (- A (hypot B_m A)) F)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-96) {
		tmp = sqrt((fma((A * C), -4.0, (B_m * B_m)) * ((F * 2.0) * (A + A)))) / -t_0;
	} else if (pow(B_m, 2.0) <= 5e+208) {
		tmp = sqrt((F * (((C + A) - hypot((A - C), B_m)) / t_0))) * -sqrt(2.0);
	} else {
		tmp = (sqrt(2.0) / -B_m) * sqrt(((A - hypot(B_m, A)) * F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-96)
		tmp = Float64(sqrt(Float64(fma(Float64(A * C), -4.0, Float64(B_m * B_m)) * Float64(Float64(F * 2.0) * Float64(A + A)))) / Float64(-t_0));
	elseif ((B_m ^ 2.0) <= 5e+208)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(C + A) - hypot(Float64(A - C), B_m)) / t_0))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(Float64(A - hypot(B_m, A)) * F)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-96], N[(N[Sqrt[N[(N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(F * 2.0), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+208], N[(N[Sqrt[N[(F * N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999998e-96

   1. Initial program 19.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 23.8%

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

   5. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 22.9

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

   7. Applied rewrites 22.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

   9. Applied rewrites 24.8%

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

   10. Taylor expanded in C around inf

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

       1. lower--.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lower-neg.f64 25.0

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

   12. Applied rewrites 25.0%

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

   if 1.9999999999999998e-96 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000004e208

   1. Initial program 30.6%

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

   3. Taylor expanded in F around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 48.3%

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

   if 5.0000000000000004e208 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 3.9%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 23.0

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

   5. Applied rewrites 23.0%

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

4. Final simplification 29.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-96}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(A + A\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+208}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-118}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(A + A\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{\left(A - \mathsf{hypot}\left(B\_m, A\right)\right) \cdot F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 2e-118)
   (/
    (sqrt (* (fma (* A C) -4.0 (* B_m B_m)) (* (* F 2.0) (+ A A))))
    (- (fma -4.0 (* C A) (* B_m B_m))))
   (* (/ (sqrt 2.0) (- B_m)) (sqrt (* (- A (hypot B_m A)) F)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 2e-118) {
		tmp = sqrt((fma((A * C), -4.0, (B_m * B_m)) * ((F * 2.0) * (A + A)))) / -fma(-4.0, (C * A), (B_m * B_m));
	} else {
		tmp = (sqrt(2.0) / -B_m) * sqrt(((A - hypot(B_m, A)) * F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-118)
		tmp = Float64(sqrt(Float64(fma(Float64(A * C), -4.0, Float64(B_m * B_m)) * Float64(Float64(F * 2.0) * Float64(A + A)))) / Float64(-fma(-4.0, Float64(C * A), Float64(B_m * B_m))));
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(Float64(A - hypot(B_m, A)) * F)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-118], N[(N[Sqrt[N[(N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(F * 2.0), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.99999999999999997e-118

   1. Initial program 18.8%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 23.7%

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

   5. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 22.7

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

   7. Applied rewrites 22.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

   9. Applied rewrites 24.8%

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

   10. Taylor expanded in C around inf

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

       1. lower--.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lower-neg.f64 24.9

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

   12. Applied rewrites 24.9%

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

   if 1.99999999999999997e-118 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 15.1%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 21.9

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

   5. Applied rewrites 21.9%

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

4. Final simplification 23.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-118}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(A + A\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 28.5% accurate, 4.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := -t\_0\\ t_2 := \mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;C \leq -3.4 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(\left(F \cdot 2\right) \cdot \left(A + A\right)\right)}}{t\_1}\\ \mathbf{elif}\;C \leq 2.65 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{t\_2} \cdot \sqrt{\left(-B\_m\right) \cdot F}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{C}, A\right) + A\right) \cdot \left(\left(2 \cdot F\right) \cdot t\_0\right)}}{t\_1}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (- t_0))
        (t_2 (fma (* A C) -4.0 (* B_m B_m))))
   (if (<= C -3.4e-128)
     (/ (sqrt (* t_2 (* (* F 2.0) (+ A A)))) t_1)
     (if (<= C 2.65e-70)
       (* (/ (sqrt 2.0) -1.0) (/ (* (sqrt t_2) (sqrt (* (- B_m) F))) t_0))
       (/
        (sqrt (* (+ (fma -0.5 (/ (* B_m B_m) C) A) A) (* (* 2.0 F) t_0)))
        t_1)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = -t_0;
	double t_2 = fma((A * C), -4.0, (B_m * B_m));
	double tmp;
	if (C <= -3.4e-128) {
		tmp = sqrt((t_2 * ((F * 2.0) * (A + A)))) / t_1;
	} else if (C <= 2.65e-70) {
		tmp = (sqrt(2.0) / -1.0) * ((sqrt(t_2) * sqrt((-B_m * F))) / t_0);
	} else {
		tmp = sqrt(((fma(-0.5, ((B_m * B_m) / C), A) + A) * ((2.0 * F) * t_0))) / t_1;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64(-t_0)
	t_2 = fma(Float64(A * C), -4.0, Float64(B_m * B_m))
	tmp = 0.0
	if (C <= -3.4e-128)
		tmp = Float64(sqrt(Float64(t_2 * Float64(Float64(F * 2.0) * Float64(A + A)))) / t_1);
	elseif (C <= 2.65e-70)
		tmp = Float64(Float64(sqrt(2.0) / -1.0) * Float64(Float64(sqrt(t_2) * sqrt(Float64(Float64(-B_m) * F))) / t_0));
	else
		tmp = Float64(sqrt(Float64(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / C), A) + A) * Float64(Float64(2.0 * F) * t_0))) / t_1);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, -3.4e-128], N[(N[Sqrt[N[(t$95$2 * N[(N[(F * 2.0), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[C, 2.65e-70], N[(N[(N[Sqrt[2.0], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[(N[Sqrt[t$95$2], $MachinePrecision] * N[Sqrt[N[((-B$95$m) * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] + A), $MachinePrecision] + A), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if C < -3.39999999999999975e-128

   1. Initial program 23.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 30.8%

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

   5. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 4.5

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

   7. Applied rewrites 4.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

   9. Applied rewrites 5.7%

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

   10. Taylor expanded in C around inf

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

       1. lower--.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lower-neg.f64 6.1

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

   12. Applied rewrites 6.1%

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

   if -3.39999999999999975e-128 < C < 2.64999999999999992e-70

   1. Initial program 17.5%

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

   4. Applied rewrites 21.1%

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

   5. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 8.3

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

   7. Applied rewrites 8.3%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. unpow-prod-down N/A

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

      8. lower-*.f64 N/A

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

   9. Applied rewrites 8.8%

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

   if 2.64999999999999992e-70 < C

   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
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 10.5%

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

   5. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 26.1

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

   7. Applied rewrites 26.1%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right)} \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 14.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -3.4 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(A + A\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;C \leq 2.65 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \cdot \sqrt{\left(-B\right) \cdot F}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 29.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := -t\_0\\ \mathbf{if}\;C \leq 2.9 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(A + A\right)\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{C}, A\right) + A\right) \cdot \left(\left(2 \cdot F\right) \cdot t\_0\right)}}{t\_1}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m))) (t_1 (- t_0)))
   (if (<= C 2.9e-65)
     (/ (sqrt (* (fma (* A C) -4.0 (* B_m B_m)) (* (* F 2.0) (+ A A)))) t_1)
     (/
      (sqrt (* (+ (fma -0.5 (/ (* B_m B_m) C) A) A) (* (* 2.0 F) t_0)))
      t_1))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = -t_0;
	double tmp;
	if (C <= 2.9e-65) {
		tmp = sqrt((fma((A * C), -4.0, (B_m * B_m)) * ((F * 2.0) * (A + A)))) / t_1;
	} else {
		tmp = sqrt(((fma(-0.5, ((B_m * B_m) / C), A) + A) * ((2.0 * F) * t_0))) / t_1;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64(-t_0)
	tmp = 0.0
	if (C <= 2.9e-65)
		tmp = Float64(sqrt(Float64(fma(Float64(A * C), -4.0, Float64(B_m * B_m)) * Float64(Float64(F * 2.0) * Float64(A + A)))) / t_1);
	else
		tmp = Float64(sqrt(Float64(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / C), A) + A) * Float64(Float64(2.0 * F) * t_0))) / t_1);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, If[LessEqual[C, 2.9e-65], N[(N[Sqrt[N[(N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(F * 2.0), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] + A), $MachinePrecision] + A), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if C < 2.8999999999999998e-65

   1. Initial program 20.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 25.4%

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

   5. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 5.4

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

   7. Applied rewrites 5.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

   9. Applied rewrites 7.1%

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

   10. Taylor expanded in C around inf

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

       1. lower--.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lower-neg.f64 8.6

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

   12. Applied rewrites 8.6%

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

   if 2.8999999999999998e-65 < C

   1. Initial program 10.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 10.8%

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

   5. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 26.9

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

   7. Applied rewrites 26.9%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right)} \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 2.9 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(A + A\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 17.2% accurate, 7.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq 3.7 \cdot 10^{-56}:\\ \;\;\;\;\frac{\sqrt{\left(-16 \cdot \left(\left(A \cdot A\right) \cdot C\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= A 3.7e-56)
   (/ (sqrt (* (* -16.0 (* (* A A) C)) F)) (- (fma (* C A) -4.0 (* B_m B_m))))
   (/
    (sqrt (* (* -16.0 (* A A)) (* C F)))
    (- (fma -4.0 (* C A) (* B_m 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 (A <= 3.7e-56) {
		tmp = sqrt(((-16.0 * ((A * A) * C)) * F)) / -fma((C * A), -4.0, (B_m * B_m));
	} else {
		tmp = sqrt(((-16.0 * (A * A)) * (C * F))) / -fma(-4.0, (C * A), (B_m * 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 (A <= 3.7e-56)
		tmp = Float64(sqrt(Float64(Float64(-16.0 * Float64(Float64(A * A) * C)) * F)) / Float64(-fma(Float64(C * A), -4.0, Float64(B_m * B_m))));
	else
		tmp = Float64(sqrt(Float64(Float64(-16.0 * Float64(A * A)) * Float64(C * F))) / Float64(-fma(-4.0, Float64(C * A), Float64(B_m * 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[A, 3.7e-56], N[(N[Sqrt[N[(N[(-16.0 * N[(N[(A * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] / (-N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(N[(-16.0 * N[(A * A), $MachinePrecision]), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < 3.7000000000000002e-56

   1. Initial program 20.7%

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

   3. Taylor expanded in A around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 9.2

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

   5. Applied rewrites 9.2%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. lift--.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

   7. Applied rewrites 7.5%

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

   8. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 11.2

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

   10. Applied rewrites 11.2%

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

   if 3.7000000000000002e-56 < A

   1. Initial program 7.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 9.4%

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

   5. Taylor expanded in A around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 4.2

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

   7. Applied rewrites 4.2%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right)}}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 23.1% accurate, 7.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (/
  (sqrt (* -8.0 (* A (* C (* F (+ A A))))))
  (- (fma -4.0 (* C A) (* B_m 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((-8.0 * (A * (C * (F * (A + A)))))) / -fma(-4.0, (C * A), (B_m * 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(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A + A)))))) / Float64(-fma(-4.0, Float64(C * A), Float64(B_m * B_m))))
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 16.7%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-neg.f64 N/A

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

   4. remove-double-neg N/A

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

   5. lower-/.f64 N/A

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

4. Applied rewrites 20.5%

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

5. Taylor expanded in C around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. sub-neg N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-+.f64 11.2

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

7. Applied rewrites 11.2%

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

Alternative 15: 16.2% accurate, 8.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (/ (sqrt (* (* -16.0 (* A A)) (* C F))) (- (fma -4.0 (* C A) (* B_m 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(((-16.0 * (A * A)) * (C * F))) / -fma(-4.0, (C * A), (B_m * B_m));
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(sqrt(Float64(Float64(-16.0 * Float64(A * A)) * Float64(C * F))) / Float64(-fma(-4.0, Float64(C * A), Float64(B_m * B_m))))
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[N[(N[(-16.0 * N[(A * A), $MachinePrecision]), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 16.7%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-neg.f64 N/A

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

   4. remove-double-neg N/A

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

   5. lower-/.f64 N/A

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

4. Applied rewrites 20.5%

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

5. Taylor expanded in A around -inf

   \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot {A}^{2}\right) \cdot \left(C \cdot F\right)}}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot {A}^{2}\right) \cdot \left(C \cdot F\right)}}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot {A}^{2}\right)} \cdot \left(C \cdot F\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   4. unpow2 N/A

      \[\leadsto \frac{\sqrt{\left(-16 \cdot \color{blue}{\left(A \cdot A\right)}\right) \cdot \left(C \cdot F\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\sqrt{\left(-16 \cdot \color{blue}{\left(A \cdot A\right)}\right) \cdot \left(C \cdot F\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   6. lower-*.f64 8.3

      \[\leadsto \frac{\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \color{blue}{\left(C \cdot F\right)}}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

7. Applied rewrites 8.3%

   \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right)}}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
8. Add Preprocessing

Alternative 16: 2.1% accurate, 8.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{\sqrt{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (/ (sqrt (* -16.0 (* A (* (* C C) F)))) (- (fma -4.0 (* C A) (* B_m 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((-16.0 * (A * ((C * C) * F)))) / -fma(-4.0, (C * A), (B_m * 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(-16.0 * Float64(A * Float64(Float64(C * C) * F)))) / Float64(-fma(-4.0, Float64(C * A), Float64(B_m * B_m))))
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[N[(-16.0 * N[(A * N[(N[(C * C), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 16.7%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

   2. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)\right)}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   3. lift-neg.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)\right)}\right)}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

   4. remove-double-neg N/A

      \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

4. Applied rewrites 20.5%

   \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

5. Taylor expanded in B around 0

   \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(A \cdot \left({C}^{2} \cdot F\right)\right)}}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\sqrt{-16 \cdot \left(A \cdot \color{blue}{\left({C}^{2} \cdot F\right)}\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   4. unpow2 N/A

      \[\leadsto \frac{\sqrt{-16 \cdot \left(A \cdot \left(\color{blue}{\left(C \cdot C\right)} \cdot F\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   5. lower-*.f64 10.1

      \[\leadsto \frac{\sqrt{-16 \cdot \left(A \cdot \left(\color{blue}{\left(C \cdot C\right)} \cdot F\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

7. Applied rewrites 10.1%

   \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
8. Add Preprocessing

Alternative 17: 1.6% accurate, 14.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \sqrt{\frac{2}{\frac{B\_m}{F}}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (sqrt (/ 2.0 (/ B_m F))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return sqrt((2.0 / (B_m / F)));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt((2.0d0 / (b_m / f)))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt((2.0 / (B_m / F)));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return math.sqrt((2.0 / (B_m / F)))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return sqrt(Float64(2.0 / Float64(B_m / F)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = sqrt((2.0 / (B_m / F)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[Sqrt[N[(2.0 / N[(B$95$m / F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 16.7%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]

   2. lower-neg.f64 N/A

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)} \]

   3. *-commutative N/A

      \[\leadsto -\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]

   4. lower-*.f64 N/A

      \[\leadsto -\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]

   5. *-commutative N/A

      \[\leadsto -\color{blue}{\left(\sqrt{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

   6. unpow2 N/A

      \[\leadsto -\left(\sqrt{2} \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right) \cdot \sqrt{\frac{F}{B}} \]

   7. rem-square-sqrt N/A

      \[\leadsto -\left(\sqrt{2} \cdot \color{blue}{-1}\right) \cdot \sqrt{\frac{F}{B}} \]

   8. lower-*.f64 N/A

      \[\leadsto -\color{blue}{\left(\sqrt{2} \cdot -1\right)} \cdot \sqrt{\frac{F}{B}} \]

   9. lower-sqrt.f64 N/A

      \[\leadsto -\left(\color{blue}{\sqrt{2}} \cdot -1\right) \cdot \sqrt{\frac{F}{B}} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto -\left(\sqrt{2} \cdot -1\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

   11. lower-/.f64 2.1

       \[\leadsto -\left(\sqrt{2} \cdot -1\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 2.1%

   \[\leadsto \color{blue}{-\left(\sqrt{2} \cdot -1\right) \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

7. Applied rewrites 2.1%

   \[\leadsto -\sqrt{\frac{F}{B} \cdot 2} \cdot -1 \]
8. Step-by-step derivation

9. Applied rewrites 2.1%

   \[\leadsto \sqrt{\frac{F}{B} \cdot 2} \]
10. Step-by-step derivation

11. Applied rewrites 2.1%

    \[\leadsto \sqrt{\frac{2}{\frac{B}{F}}} \]
12. Add Preprocessing

Alternative 18: 1.5% accurate, 18.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \sqrt{F \cdot \frac{2}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (sqrt (* F (/ 2.0 B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return sqrt((F * (2.0 / B_m)));
}

Fortran

B_m = 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 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 16.7%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]

   2. lower-neg.f64 N/A

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)} \]

   3. *-commutative N/A

      \[\leadsto -\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]

   4. lower-*.f64 N/A

      \[\leadsto -\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]

   5. *-commutative N/A

      \[\leadsto -\color{blue}{\left(\sqrt{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

   6. unpow2 N/A

      \[\leadsto -\left(\sqrt{2} \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right) \cdot \sqrt{\frac{F}{B}} \]

   7. rem-square-sqrt N/A

      \[\leadsto -\left(\sqrt{2} \cdot \color{blue}{-1}\right) \cdot \sqrt{\frac{F}{B}} \]

   8. lower-*.f64 N/A

      \[\leadsto -\color{blue}{\left(\sqrt{2} \cdot -1\right)} \cdot \sqrt{\frac{F}{B}} \]

   9. lower-sqrt.f64 N/A

      \[\leadsto -\left(\color{blue}{\sqrt{2}} \cdot -1\right) \cdot \sqrt{\frac{F}{B}} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto -\left(\sqrt{2} \cdot -1\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

   11. lower-/.f64 2.1

       \[\leadsto -\left(\sqrt{2} \cdot -1\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 2.1%

   \[\leadsto \color{blue}{-\left(\sqrt{2} \cdot -1\right) \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

7. Applied rewrites 2.1%

   \[\leadsto -\sqrt{\frac{F}{B} \cdot 2} \cdot -1 \]
8. Step-by-step derivation

9. Applied rewrites 2.1%

   \[\leadsto \sqrt{\frac{F}{B} \cdot 2} \]
10. Step-by-step derivation

11. Applied rewrites 2.1%

    \[\leadsto \sqrt{F \cdot \frac{2}{B}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024322 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))